Notes

Monogamy Of Quantum Discord
We explore all the trade-off relations of multipartite quantum discord proposed very recently in [Phys. Rev. Lett. 124, 110401 (2020)] and show that the multipartite quantum discord is completely monogamous provided that it does not increase under discard of subsystems. Here, a quantity of multipartite quantity, with the same spirit as established in [Phys. Rev. A. 101, 032301 (2020)], is said to be completely monogamous (i) if it does not increase under loss of subsystems and (ii) if some given combination of subsystems reach the total amout of correlation, then all other combination of subsystems that excluding the given subsystems do not contain such a correlation any more. In addition, we explore all the trade-off relations for the global quantum discord proposed in [Phys. Rev. A 84, 042109 (2011)] and show that the global quantum discord is not completely monogamous.

pacs:
03.67.Mn, 03.65.Db, 03.65.Ud.
††preprint: APS/123-QED

Quantum discord, as the foremost one of the quantum correlations beyond entanglement, has been extensively explored in the last two decades due to its remarkable applications in quantum information protocols [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. It is originally defined for bipartite system as the minimized difference between the quantum mutual information with and without a von Neumann projective measurement applied on one of the subsystems [1, 2]. Consequently, several multipartite generalizations have been proposed based on different scenarios [8, 9, 10, 11, 12, 13, 14, 15, 16].

An important issue closely related to a correlation measure of composite quantum system is to explore the distribution of the correlation throughout a multipartite state [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. In this context, there are two ways to describe these distribution: one is the monogamy or polygamy relation by the bipartite correlation measure [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] and the other one is the relation based on the multipartite correlation measure [45, 43, 44]. For the bipartite measure of quantum discord, although the square of quantum discord obeys the monogamy relation for three qubit states [19], other systems always display polygamous behaviour [17, 18, 19, 20, 21]. In Ref. [45], it has been shown that the global quantum discord proposed in Ref. [8] acts as a monogamy bound for pairwise quantum discord in which one of the subsystem is fixed while the other part runs over all subystems provided that bipartite discord does not increase under discard of subsystems. The aim of this article is (i) to explore the distribution of discord contained in a multipartite state whenever it is measured by the multipartite quantum discord defined in Ref. [16], and (ii) to discuss the distribution of the global quantum discord in the framework of the complete monogamy relation.

The rest of this paper is organized as follows. We recall the definition of the original quantum discord and two multipartite generalizations we discussed in this article and then present the monogamy laws in literatures in Section II. In Section III, We discuss and establish the framework of complete monogamy relation for the two generalizations of quantum discord. Section IV explores the trade-off relation and complete monogamy relation of the multipartite quantum discord. It is divided into three subsections. The first subsection discusses the tripartite case, the second subsection deals with the four-partite case and the main conclusioin is given in the last subsection for the general multipartite case. In Section V, we deal with the global quantum discord in the framework of complete monogamy relation. Finally, we conclude.

In this section, we review the definition of quantum discord and its various generalization firstly, and then recall the monogamy relation and complete monogamy relation in literatures. Throughout this paper, we let ℋA1A2⋯Ansuperscriptℋsubscript𝐴1subscript𝐴2⋯subscript𝐴𝑛mathcalH^A_1A_2cdots A_ncaligraphic_H start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT be the Hilbert space corresponding to the n𝑛nitalic_n-parite quantum system with finite dimension. and let 𝒮Xsuperscript𝒮𝑋mathcalS^Xcaligraphic_S start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT be the set of density operators acting on ℋXsuperscriptℋ𝑋mathcalH^Xcaligraphic_H start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT. The tripartite and the four-partite systems are always denoted by ℋABCsuperscriptℋ𝐴𝐵𝐶mathcalH^ABCcaligraphic_H start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT and ℋABCDsuperscriptℋ𝐴𝐵𝐶𝐷mathcalH^ABCDcaligraphic_H start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT, respectively.

II.1 Quantum discord and its generalization

For any bipartite state ρ∈𝒮AB𝜌superscript𝒮𝐴𝐵rhoinmathcalS^ABitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, the original discord is defined as [1, 2]

DA;B(ρ)=minΠA[SB|ΠA(ρ)-SB|A(ρ)]subscript𝐷𝐴𝐵𝜌subscriptsuperscriptΠ𝐴subscript𝑆conditional𝐵superscriptΠ𝐴𝜌subscript𝑆conditional𝐵𝐴𝜌displaystyle D_A;B(rho)=min_Pi^Aleft[S_Pi^A(rho)-S_A(% rho)right]italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT ( italic_ρ ) = roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) - italic_S start_POSTSUBSCRIPT italic_B | italic_A end_POSTSUBSCRIPT ( italic_ρ ) ] (1)
where SB|A(ρ)=SAB(ρ)-SA(ρ)subscript𝑆conditional𝐵𝐴𝜌subscript𝑆𝐴𝐵𝜌subscript𝑆𝐴𝜌S_A(rho)=S_AB(rho)-S_A(rho)italic_S start_POSTSUBSCRIPT italic_B | italic_A end_POSTSUBSCRIPT ( italic_ρ ) = italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_ρ ) - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ ), S(ρ)=-Trρlogρ𝑆𝜌Tr𝜌𝜌S(rho)=-textTrrhologrhoitalic_S ( italic_ρ ) = - Tr italic_ρ roman_log italic_ρ is the von Neumann entropy, SB|ΠA(ρ)=∑jpjASAB(ΠjAρΠjA/pjA)=SAB(ρΠA)-SA(ρΠA)subscript𝑆conditional𝐵superscriptΠ𝐴𝜌subscript𝑗superscriptsubscript𝑝𝑗𝐴subscript𝑆𝐴𝐵superscriptsubscriptΠ𝑗𝐴𝜌superscriptsubscriptΠ𝑗𝐴superscriptsubscript𝑝𝑗𝐴subscript𝑆𝐴𝐵subscript𝜌superscriptΠ𝐴subscript𝑆𝐴subscript𝜌superscriptΠ𝐴S_Pi^A(rho)=sum_jp_j^AS_AB(Pi_j^ArhoPi_j^A/p_j^% A)=S_AB(rho_Pi^A)-S_A(rho_Pi^A)italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_ρ roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT / italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) , ΠjAsuperscriptsubscriptΠ𝑗𝐴Pi_j^Aroman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT is a one-dimensional von Neumann projection operator on subsystem A𝐴Aitalic_A and pjA=Tr(ΠjAρΠjA)superscriptsubscript𝑝𝑗𝐴TrsuperscriptsubscriptΠ𝑗𝐴𝜌superscriptsubscriptΠ𝑗𝐴p_j^A=textTr(Pi_j^ArhoPi_j^A)italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = Tr ( roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_ρ roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ), ρΠA=∑jΠjAρΠjAsubscript𝜌superscriptΠ𝐴subscript𝑗superscriptsubscriptΠ𝑗𝐴𝜌superscriptsubscriptΠ𝑗𝐴rho_Pi^A=sum_jPi_j^ArhoPi_j^Aitalic_ρ start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_ρ roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT. Hereafter, we denote S(ρX)𝑆superscript𝜌𝑋S(rho^X)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) by SX(ρ)subscript𝑆𝑋𝜌S_X(rho)italic_S start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_ρ ) sometimes for simplicity.

For multipartite systems, the (n-1)𝑛1(n-1)( italic_n - 1 )-partite measurement is written [16]

Πj1…jn-1A1…An-1=Πj1A1⊗Πj2|j1A2⋯⊗Πjn-1|j1…jn-2An-1,subscriptsuperscriptΠsubscript𝐴1…subscript𝐴𝑛1subscript𝑗1…subscript𝑗𝑛1tensor-producttensor-productsubscriptsuperscriptΠsubscript𝐴1subscript𝑗1subscriptsuperscriptΠsubscript𝐴2conditionalsubscript𝑗2subscript𝑗1⋯subscriptsuperscriptΠsubscript𝐴𝑛1conditionalsubscript𝑗𝑛1subscript𝑗1…subscript𝑗𝑛2displaystylePi^A_1dots A_n-1_j_1dots j_n-1=Pi^A_1_j_1% otimesPi^A_2_j_1dotsotimesPi^A_n-1_j_1dots j% _n-2,roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_j start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ ⊗ roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT | italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_j start_POSTSUBSCRIPT italic_n - 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , (2)
where the n𝑛nitalic_n subsystems are labeled as Aisubscript𝐴𝑖A_iitalic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, the measurements take place in the order A1→A2→…An-1→subscript𝐴1subscript𝐴2→…subscript𝐴𝑛1A_1rightarrow A_2rightarrowdots A_n-1italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → … italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT. For ρ∈𝒮A1A2⋯An𝜌superscript𝒮subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛rhoinmathcalS^A_1A_2cdots A_nitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, the n𝑛nitalic_n-partite discord is defined by [16]

DA1;A2;…;An(ρ)=minΠA1…An-1[-SA2…An|A1(ρ)fragmentssubscript𝐷subscript𝐴1subscript𝐴2…subscript𝐴𝑛fragments(ρ)subscriptsuperscriptΠsubscript𝐴1…subscript𝐴𝑛1fragments[subscript𝑆conditionalsubscript𝐴2…subscript𝐴𝑛subscript𝐴1fragments(ρ)displaystyle D_A_1;A_2;dots;A_n(rho)=min_Pi^A_1dots A_n-1% Big[-S_A_2dots A_n(rho)italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) = roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ - italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ )

+SA2|ΠA1(ρ)⋯+SAn|ΠA1…An-1(ρ)]fragmentssubscript𝑆conditionalsubscript𝐴2superscriptΠsubscript𝐴1fragments(ρ)⋯subscript𝑆conditionalsubscript𝐴𝑛superscriptΠsubscript𝐴1…subscript𝐴𝑛1fragments(ρ)]displaystyle+S_Pi^A_1(rho)dots+S_Pi^A_1dots A_n-1% (rho)Big]+ italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) ⋯ + italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) ] (3)
up to the measurement ordering A1→A2→…→An-1→subscript𝐴1subscript𝐴2→…→subscript𝐴𝑛1A_1rightarrow A_2rightarrowdotsrightarrow A_n-1italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → … → italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT, where SAk|ΠA1…Ak-1(ρ)=∑j1…jk-1p𝒋(k-1)SA1…Ak(Π𝒋(k-1)ρΠ𝒋(k-1)/p𝒋(k-1))subscript𝑆conditionalsubscript𝐴𝑘superscriptΠsubscript𝐴1…subscript𝐴𝑘1𝜌subscriptsubscript𝑗1…subscript𝑗𝑘1superscriptsubscript𝑝𝒋𝑘1subscript𝑆subscript𝐴1…subscript𝐴𝑘subscriptsuperscriptΠ𝑘1𝒋𝜌subscriptsuperscriptΠ𝑘1𝒋superscriptsubscript𝑝𝒋𝑘1S_Pi^A_1dots A_k-1(rho)=sum_j_1dots j_k-1p_bmj^% (k-1)S_A_1dots A_k(Pi^(k-1)_bmjrhoPi^(k-1)_bmj/p_bm% j^(k-1))italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) = ∑ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_j start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k - 1 ) end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_Π start_POSTSUPERSCRIPT ( italic_k - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT italic_ρ roman_Π start_POSTSUPERSCRIPT ( italic_k - 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT / italic_p start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k - 1 ) end_POSTSUPERSCRIPT ) with Π𝒋(k)≡Πj1…jkA1…AksubscriptsuperscriptΠ𝑘𝒋subscriptsuperscriptΠsubscript𝐴1…subscript𝐴𝑘subscript𝑗1…subscript𝑗𝑘Pi^(k)_bmjequivPi^A_1dots A_k_j_1dots j_kroman_Π start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT ≡ roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT, p𝒋(k)=Tr(Π𝒋(k)ρΠ𝒋(k))superscriptsubscript𝑝𝒋𝑘TrsubscriptsuperscriptΠ𝑘𝒋𝜌subscriptsuperscriptΠ𝑘𝒋p_bmj^(k)=textTr(Pi^(k)_bmjrhoPi^(k)_bmj)italic_p start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = Tr ( roman_Π start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT italic_ρ roman_Π start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_italic_j end_POSTSUBSCRIPT ).

The global quantum discord DA1:⋯:Ansubscript𝐷:subscript𝐴1⋯:subscript𝐴𝑛D_A_1:cdots:A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT for an arbitrary state ρ∈𝒮A1⋯An𝜌superscript𝒮subscript𝐴1⋯subscript𝐴𝑛rhoinmathcalS^A_1cdots A_nitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT under a set of local measurements Πj1A1⊗⋯⊗ΠjnAntensor-productsubscriptsuperscriptΠsubscript𝐴1subscript𝑗1⋯subscriptsuperscriptΠsubscript𝐴𝑛subscript𝑗𝑛\Pi^A_1_j_1otimescdotsotimesPi^A_n_j_n roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ ⋯ ⊗ roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is defined as [8, 45]

DA1:⋯:An(ρ)=minΦ[I(ρ)-I(Φ(ρ))],subscript𝐷:subscript𝐴1⋯:subscript𝐴𝑛𝜌subscriptΦ𝐼𝜌𝐼Φ𝜌D_A_1:cdots:A_n(rho)=min_Phileft[I(rho)-I(Phi(rho))right],italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) = roman_min start_POSTSUBSCRIPT roman_Φ end_POSTSUBSCRIPT [ italic_I ( italic_ρ ) - italic_I ( roman_Φ ( italic_ρ ) ) ] , (4)
where

Φ(ρ)=∑kΠkρΠk,Φ𝜌subscript𝑘subscriptΠ𝑘𝜌subscriptΠ𝑘Phi(rho)=sum_k,Pi_krho,Pi_k,roman_Φ ( italic_ρ ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ρ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , (5)
with Πk=Πj1A1⊗⋯⊗ΠjnAnsubscriptΠ𝑘tensor-productsubscriptsuperscriptΠsubscript𝐴1subscript𝑗1⋯subscriptsuperscriptΠsubscript𝐴𝑛subscript𝑗𝑛Pi_k=Pi^A_1_j_1otimescdotsotimesPi^A_n_j_nroman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ ⋯ ⊗ roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT and k𝑘kitalic_k denoting the index string (j1⋯jnfragments(subscript𝑗1⋯subscript𝑗𝑛(j_1cdots j_n( italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_j start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT), the mutual information I(ρ)𝐼𝜌I(rho)italic_I ( italic_ρ ) is defined by [46]

I(ρ)𝐼𝜌displaystyle I(rho)italic_I ( italic_ρ ) =displaystyle== ∑k=1nSAk(ρ)-SA1⋯An(ρ),superscriptsubscript𝑘1𝑛subscript𝑆subscript𝐴𝑘𝜌subscript𝑆subscript𝐴1⋯subscript𝐴𝑛𝜌displaystylesum_k=1^nS_A_kleft(rhoright)-S_A_1cdots A_n% left(rhoright),∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) - italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ ) , (6)
where

Φ(ρAk)=∑k′Πk′AkρAkΠk′Ak.Φsuperscript𝜌subscript𝐴𝑘subscriptsuperscript𝑘′subscriptsuperscriptΠsubscript𝐴𝑘superscript𝑘′superscript𝜌subscript𝐴𝑘subscriptsuperscriptΠsubscript𝐴𝑘superscript𝑘′Phileft(rho^A_kright)=sum_k^primePi^A_k_k^prime,% rho^A_k,Pi^A_k_k^prime.roman_Φ ( italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT . (7)

II.2 Monogamy relation

For a given bipartite measure Q𝑄Qitalic_Q, Q𝑄Qitalic_Q is said to be monogamous (we take the tripartite case for example) if [22, 25]

Q(A|BC)≥Q(AB)+Q(AC),𝑄conditional𝐴𝐵𝐶𝑄𝐴𝐵𝑄𝐴𝐶displaystyle Q(A|BC)geq Q(AB)+Q(AC),italic_Q ( italic_A | italic_B italic_C ) ≥ italic_Q ( italic_A italic_B ) + italic_Q ( italic_A italic_C ) , (8)
where the vertical bar indicates the bipartite split across which the (bipartite) entanglement is measured. However, Eq. (8) is not valid for many entanglement measures [22, 33, 31, 38, 30, 34, 38, 32, 29] and quantum discord [19] but some power function of Q𝑄Qitalic_Q admits the monogamy relation [i.e., Qα(A|BC)≥Qα(AB)+Qα(AC)superscript𝑄𝛼conditional𝐴𝐵𝐶superscript𝑄𝛼𝐴𝐵superscript𝑄𝛼𝐴𝐶Q^alpha(A|BC)geq Q^alpha(AB)+Q^alpha(AC)italic_Q start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_A | italic_B italic_C ) ≥ italic_Q start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_A italic_B ) + italic_Q start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_A italic_C ) for some α>0𝛼0alpha>0italic_α >0]. In Ref. [40], we address this issue by proposing an improved definition of monogamy (without inequalities): A measure of entanglement E𝐸Eitalic_E is monogamous if for any ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT that satisfies the disentangling condition, i.e.,

E(ρA|BC)=E(ρAB),𝐸superscript𝜌conditional𝐴𝐵𝐶𝐸superscript𝜌𝐴𝐵E(rho^BC)=E(rho^AB),italic_E ( italic_ρ start_POSTSUPERSCRIPT italic_A | italic_B italic_C end_POSTSUPERSCRIPT ) = italic_E ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) , (9)
we have that E(ρAC)=0𝐸superscript𝜌𝐴𝐶0E(rho^AC)=0italic_E ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = 0. With respect to this definition, a continuous measure E𝐸Eitalic_E is monogamous according to this definition if and only if there exists 0<α<∞0𝛼00 <∞ such that

Eα(ρA|BC)≥Eα(ρAB)+Eα(ρAC),superscript𝐸𝛼superscript𝜌conditional𝐴𝐵𝐶superscript𝐸𝛼superscript𝜌𝐴𝐵superscript𝐸𝛼superscript𝜌𝐴𝐶E^alpha(rho^BC)geq E^alpha(rho^AB)+E^alpha(rho^AC),italic_E start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A | italic_B italic_C end_POSTSUPERSCRIPT ) ≥ italic_E start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + italic_E start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) , (10)
for all ρABCsuperscript𝜌𝐴𝐵𝐶rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT acting on the state space ℋABCsuperscriptℋ𝐴𝐵𝐶mathcalH^ABCcaligraphic_H start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT with fixed dimℋABC=d<∞dimensionsuperscriptℋ𝐴𝐵𝐶𝑑dimmathcalH^ABC=droman_dim caligraphic_H start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = italic_d <∞ (see Theorem 1 in Ref. [40]). Notice that, for these bipartite measures,only the relation between A|BCconditional𝐴𝐵𝐶A|BCitalic_A | italic_B italic_C, AB𝐴𝐵ABitalic_A italic_B and AC𝐴𝐶ACitalic_A italic_C are revealed, the global correlation in ABC and the correlation contained in part BC𝐵𝐶BCitalic_B italic_C is missed [43]. That is, the monogamy relation in such a sense is not “complete”. Recently, we established a complete monogamy relation for entanglement in Ref. [43]. For a unified tripartite entanglement measure E(3)superscript𝐸3E^(3)italic_E start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT, it is said to be completely monogamous if for any ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT that satisfies [43]

E(3)(ρABC)=E(2)(ρAB)superscript𝐸3superscript𝜌𝐴𝐵𝐶superscript𝐸2superscript𝜌𝐴𝐵E^(3)(rho^ABC)=E^(2)(rho^AB)italic_E start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) = italic_E start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) (11)
we have that E(2)(ρAC)=E(2)(ρBC)=0superscript𝐸2superscript𝜌𝐴𝐶superscript𝐸2superscript𝜌𝐵𝐶0E^(2)(rho^AC)=E^(2)(rho^BC)=0italic_E start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_E start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0. If E(3)superscript𝐸3E^(3)italic_E start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT is a continuous unified tripartite entanglement measure. Then, E(3)superscript𝐸3E^(3)italic_E start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT is completely monogamous if and only if there exists 0<α<∞0𝛼00 <∞ such that [43]

Eα(ρABC)≥Eα(ρAB)+Eα(ρAC)+Eα(ρBC),superscript𝐸𝛼superscript𝜌𝐴𝐵𝐶superscript𝐸𝛼superscript𝜌𝐴𝐵superscript𝐸𝛼superscript𝜌𝐴𝐶superscript𝐸𝛼superscript𝜌𝐵𝐶displaystyle E^alpha(rho^ABC)geq E^alpha(rho^AB)+E^alpha(rho% ^AC)+E^alpha(rho^BC),italic_E start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ≥ italic_E start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + italic_E start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) + italic_E start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) , (12)
for all ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT with fixed dimℋABC=d<∞dimensionsuperscriptℋ𝐴𝐵𝐶𝑑dimmathcalH^ABC=droman_dim caligraphic_H start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = italic_d <∞, here we omitted the superscript (2,3)23^(2,3)start_FLOATSUPERSCRIPT ( 2 , 3 ) end_FLOATSUPERSCRIPT of E(2,3)superscript𝐸23E^(2,3)italic_E start_POSTSUPERSCRIPT ( 2 , 3 ) end_POSTSUPERSCRIPT for brevity. This complete monogamy relation displays the distribution of entanglement throughly. We thus adopt this scenario to describe the monogamy of quantum discord.

III Framework of complete monogamy relation for quantum discord

When we cope with the complete monogamy relation, we need to take into account the unification of the measure at first [43, 44], i.e., whether Q(k+1)superscript𝑄𝑘1Q^(k+1)italic_Q start_POSTSUPERSCRIPT ( italic_k + 1 ) end_POSTSUPERSCRIPT is consistent with Q(k)superscript𝑄𝑘Q^(k)italic_Q start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT, k≥2𝑘2kgeq 2italic_k ≥ 2. It is proved in Ref. [16] that DA;B;C=DX;Ysubscript𝐷𝐴𝐵𝐶subscript𝐷𝑋𝑌D_A;B;C=D_X;Yitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_X ; italic_Y end_POSTSUBSCRIPT for any ρ=ρXY⊗ρZ∈𝒮ABC𝜌tensor-productsuperscript𝜌𝑋𝑌superscript𝜌𝑍superscript𝒮𝐴𝐵𝐶rho=rho^XYotimesrho^ZinmathcalS^ABCitalic_ρ = italic_ρ start_POSTSUPERSCRIPT italic_X italic_Y end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_Z end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT, X,Y,Z=A,B,C𝑋𝑌𝑍𝐴𝐵𝐶X,Y,Z=A,B,C italic_X , italic_Y , italic_Z = italic_A , italic_B , italic_C . Going further, for any bipartite state ρ∈𝒮AB𝜌superscript𝒮𝐴𝐵rhoinmathcalS^ABitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, for any given decomposition ρ=∑ipiρiAB𝜌subscript𝑖subscript𝑝𝑖superscriptsubscript𝜌𝑖𝐴𝐵rho=sum_ip_irho_i^ABitalic_ρ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, it can be extended by adding an auxiliary system C𝐶Citalic_C that does not correlate with AB𝐴𝐵ABitalic_A italic_B as

ρ=∑ipi|i⟩⟨i|C⊗ρiAB.𝜌subscript𝑖tensor-productsubscript𝑝𝑖ket𝑖superscriptbra𝑖𝐶superscriptsubscript𝜌𝑖𝐴𝐵displaystylerho=sum_ip_i|iranglelangle i|^Cotimesrho_i^AB.italic_ρ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_i ⟩ ⟨ italic_i | start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT . (13)
That is, DC;AB=0subscript𝐷𝐶𝐴𝐵0D_C;AB=0italic_D start_POSTSUBSCRIPT italic_C ; italic_A italic_B end_POSTSUBSCRIPT = 0. One can easily show that

DC;A;B=DA;Bsubscript𝐷𝐶𝐴𝐵subscript𝐷𝐴𝐵displaystyle D_C;A;B=D_A;Bitalic_D start_POSTSUBSCRIPT italic_C ; italic_A ; italic_B end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT (14)
for such a state. We can also show that (see in Proposition 1 and Proposition 2 in the next Section)

DA;C;B≥DA;Bsubscript𝐷𝐴𝐶𝐵subscript𝐷𝐴𝐵displaystyle D_A;C;Bgeq D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_C ; italic_B end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT (15)
with the equality holds if and only if DA;C=0subscript𝐷𝐴𝐶0D_A;C=0italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT = 0 for such a state. Eqs. (14,15) imply that: (i) when an auxiliary particle classically correlated with the state is added and it does not disturb the measurement ordering A→B→𝐴𝐵Arightarrow Bitalic_A → italic_B, then the multipartite discord equals to the pre-state; (ii) when the auxiliary particle disturb the previous measurement ordering A→B→𝐴𝐵Arightarrow Bitalic_A → italic_B, then the multipartite discord does not decrease. In other words, the multipartite quantum discord is a unified measure of quantum correlation. The global quantum discord is defined in the same way as that of the bipartite symmetric discord and thus it can be regarded as a unified measure as well. (However, it is easy to see that, when an auxiliary particle is added, the global discord always increases except for the state after extension is a fully separable pure state. From this point of view, the unification property of DA1;A2;⋯;Ansubscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛D_A_1;A_2;cdots;A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is better than DA1:A2:⋯:Ansubscript𝐷:subscript𝐴1subscript𝐴2:⋯:subscript𝐴𝑛D_A_1:A_2:cdots:A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT.)

We are now ready to discuss the complete monogamy of these two generalizations of quantum discord. With the same spirit as the monogamy of entanglement discussed in [40, 41, 43], we can now give the definition of complete monogamy for the multipartite quantum discord and the global quantum discord.

Let DA1|A2|⋯|Ansubscript𝐷conditionalsubscript𝐴1subscript𝐴2normal-⋯subscript𝐴𝑛D_A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT be DA1;A2;⋯;Ansubscript𝐷subscript𝐴1subscript𝐴2normal-⋯subscript𝐴𝑛D_A_1;A_2;cdots;A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT or DA1:A2:⋯:Ansubscript𝐷normal-:subscript𝐴1subscript𝐴2normal-:normal-⋯normal-:subscript𝐴𝑛D_A_1:A_2:cdots:A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT. DA1|A2|⋯|Ansubscript𝐷conditionalsubscript𝐴1subscript𝐴2normal-⋯subscript𝐴𝑛D_cdotsitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is said to be completely monogamous if

•
Monotonic under discard of subsystems: For any state ρ∈𝒮A1A2⋯An𝜌superscript𝒮subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛rhoinmathcalS^A_1A_2cdots A_nitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, DA1|A2|⋯|An≥DAi1|Ai2|⋯|Aik≥DAi1′|Ai2′|⋯|Aip′subscript𝐷conditionalsubscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷conditionalsubscript𝐴subscript𝑖1subscript𝐴subscript𝑖2⋯subscript𝐴subscript𝑖𝑘subscript𝐷conditionalsubscript𝐴subscriptsuperscript𝑖′1subscript𝐴subscriptsuperscript𝑖′2⋯subscript𝐴subscriptsuperscript𝑖′𝑝D_cdotsgeqD_cdots% geq D_A_i^prime_pitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT for any k

•
Dis-correlated condition: For any state ρ∈𝒮A1A2⋯An𝜌superscript𝒮subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛rhoinmathcalS^A_1A_2cdots A_nitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT that satisfies DA1|A2|⋯|An=DAi1|Ai2|⋯|Aiksubscript𝐷conditionalsubscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷conditionalsubscript𝐴subscript𝑖1subscript𝐴subscript𝑖2⋯subscript𝐴subscript𝑖𝑘D_A_2=D_A_i_1italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT, k

(i) DAj1|Aj2|⋯|Ajl=0subscript𝐷conditionalsubscript𝐴subscript𝑗1subscript𝐴subscript𝑗2⋯subscript𝐴subscript𝑗𝑙0D_A_j_l=0italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 for any 1≤j1 <⋯

(ii) DAi1′|Ai2′|⋯|Aip′|Aj1|Aj2|⋯|Ajq=0subscript𝐷conditionalsubscript𝐴subscriptsuperscript𝑖′1subscript𝐴subscriptsuperscript𝑖′2⋯subscript𝐴subscriptsuperscript𝑖′𝑝subscript𝐴subscript𝑗1subscript𝐴subscript𝑗2⋯subscript𝐴subscript𝑗𝑞0D_A_i^prime_1=0italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | ⋯ | italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 for any i1′,i2′,…,ip′⊆i1,i2,…,iksubscriptsuperscript𝑖′1subscriptsuperscript𝑖′2…subscriptsuperscript𝑖′𝑝subscript𝑖1subscript𝑖2…subscript𝑖𝑘i^prime_1,i^prime_2,dots,i^prime_p\subseteqi_1,i_2,% dots,i_k italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ⊆ italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and 1≤j1 <⋯

We illustrate Definition 1 with the four partite case. DA;B;C;Dsubscript𝐷𝐴𝐵𝐶𝐷D_A;B;C;Ditalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT is completely monogamous if

•
Monotonicity: For any state ρ∈𝒮ABCD𝜌superscript𝒮𝐴𝐵𝐶𝐷rhoinmathcalS^ABCDitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT, DA;B;C;D≥DX;Y;Z≥DM;Nsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝑋𝑌𝑍subscript𝐷𝑀𝑁D_A;B;C;DgeqD_X;Y;ZgeqD_M;Nitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_X ; italic_Y ; italic_Z end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_M ; italic_N end_POSTSUBSCRIPT for any M,N⊆X,Y,Z⊆A,B,C,D𝑀𝑁𝑋𝑌𝑍𝐴𝐵𝐶𝐷M,N\subseteqX,Y,Z\subseteqA,B,C,D italic_M , italic_N ⊆ italic_X , italic_Y , italic_Z ⊆ italic_A , italic_B , italic_C , italic_D .

•
Dis-correlated condition: For any state ρ∈𝒮ABCD𝜌superscript𝒮𝐴𝐵𝐶𝐷rhoinmathcalS^ABCDitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT that satisfies DA;B;C;D=DA;B;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐶D_A;B;C;D=D_A;B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT (here we take X,Y,Z=A,B,C𝑋𝑌𝑍𝐴𝐵𝐶X,Y,Z=A,B,C italic_X , italic_Y , italic_Z = italic_A , italic_B , italic_C for example, the other cases can be argued similarly), we have that DA;B;D=DA;C;D=DB;C;D=DA;D=DB;D=DC;D=0subscript𝐷𝐴𝐵𝐷subscript𝐷𝐴𝐶𝐷subscript𝐷𝐵𝐶𝐷subscript𝐷𝐴𝐷subscript𝐷𝐵𝐷subscript𝐷𝐶𝐷0D_A;B;D=D_A;C;D=D_B;C;D=D_A;D=D_B;D=D_C;D=0italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_C ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_C ; italic_D end_POSTSUBSCRIPT = 0 . For any state ρ∈𝒮ABCD𝜌superscript𝒮𝐴𝐵𝐶𝐷rhoinmathcalS^ABCDitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT that satisfies DA;B;C;D=DABsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵D_A;B;C;D=D_ABitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT (here we take M,N=A,B𝑀𝑁𝐴𝐵M,N=A,B italic_M , italic_N = italic_A , italic_B for example, the other cases can be argued similarly), we have that DC;D=DA;C;D=DB;C;D=DA;C=DA;D=DB;C=DB;D=0subscript𝐷𝐶𝐷subscript𝐷𝐴𝐶𝐷subscript𝐷𝐵𝐶𝐷subscript𝐷𝐴𝐶subscript𝐷𝐴𝐷subscript𝐷𝐵𝐶subscript𝐷𝐵𝐷0D_C;D=D_A;C;D=D_B;C;D=D_A;C=D_A;D=D_B;C=D_B;D=0italic_D start_POSTSUBSCRIPT italic_C ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_C ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_D end_POSTSUBSCRIPT = 0.

For any entanglement measure E𝐸Eitalic_E, EA1A2⋯An(n)≥EAi1Ai2⋯Aik(k)subscriptsuperscript𝐸𝑛subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscriptsuperscript𝐸𝑘subscript𝐴subscript𝑖1subscript𝐴subscript𝑖2⋯subscript𝐴subscript𝑖𝑘E^(n)_A_1A_2cdots A_ngeqE^(k)_A_i_1A_i_2cdots A_i% _kitalic_E start_POSTSUPERSCRIPT ( italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_E start_POSTSUPERSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT is valid naturally since tracing out on any subsystem is a LOCC (local operation with classical communications) and entanglement can not increase under any LOCC. But this fact may be not true for other quantum correlation. For example, there exists three-qubit state ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT such that DA:BC DM:Nsubscript𝐷:𝐴𝐵:𝐶subscript𝐷:𝑀𝑁D_A:B:C>D_M:Nitalic_D start_POSTSUBSCRIPT italic_A : italic_B : italic_C end_POSTSUBSCRIPT >italic_D start_POSTSUBSCRIPT italic_M : italic_N end_POSTSUBSCRIPT for any tripartite state, M,N⊆A,B,C𝑀𝑁𝐴𝐵𝐶M,N\subseteqA,B,C italic_M , italic_N ⊆ italic_A , italic_B , italic_C . That is, the monotonicity should holds whenever the measurement on both sides are the same one. We illustrate it with a counterexample: in DA:BCsubscript𝐷:𝐴𝐵𝐶D_A:BCitalic_D start_POSTSUBSCRIPT italic_A : italic_B italic_C end_POSTSUBSCRIPT as above, the local measurement is acting on the composite system AB𝐴𝐵ABitalic_A italic_B as a single system on the left side while in DA:Bsubscript𝐷:𝐴𝐵D_A:Bitalic_D start_POSTSUBSCRIPT italic_A : italic_B end_POSTSUBSCRIPT the local measurement is acting on system A𝐴Aitalic_A and B𝐵Bitalic_B separately. So the measurement on these two sides are different.

IV Monogamy of the multipartite quantum discord

Hereafter, we call the generalization of quantum discord in Ref. [16] multipartite quantum discord. For convenience, we fix some notations. For any local measurement ΠAi1⋯AiksuperscriptΠsubscript𝐴subscript𝑖1⋯subscript𝐴subscript𝑖𝑘Pi^A_i_1cdots A_i_kroman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT acting on the reduced state ρAi1⋯AikAik+1superscript𝜌subscript𝐴subscript𝑖1⋯subscript𝐴subscript𝑖𝑘subscript𝐴subscript𝑖𝑘1rho^A_i_1cdots A_i_kA_i_k+1italic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT of ρA1A2⋯An∈𝒮A1A2⋯Ansuperscript𝜌subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛superscript𝒮subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛rho^A_1A_2cdots A_ninmathcalS^A_1A_2cdots A_nitalic_ρ start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, the conditional mutual information changes by an amount

dAi1…Aik;Aik+1=SAik+1|ΠAi1…Aik-SAik+1|Ai1…Aik,subscript𝑑subscript𝐴subscript𝑖1…subscript𝐴subscript𝑖𝑘subscript𝐴subscript𝑖𝑘1subscript𝑆conditionalsubscript𝐴subscript𝑖𝑘1superscriptΠsubscript𝐴subscript𝑖1…subscript𝐴subscript𝑖𝑘subscript𝑆conditionalsubscript𝐴subscript𝑖𝑘1subscript𝐴subscript𝑖1…subscript𝐴subscript𝑖𝑘missing-subexpressionmissing-subexpressionbeginarray[]rcld_A_i_1dots A_i_k;A_i_k+1=S_% Pi^A_i_1dots A_i_k-S_A_i_k+1,endarraystart_ARRAY start_ROW start_CELL italic_d start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | roman_Π start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT , end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW end_ARRAY (16)
where 1≤i1 <⋯ik

IV.1 The tripartite case

We start by the trade-off relation. Different from multipartite entanglement, it is unknown whether the correlation decreases under tracing out subsystems.

For any state ρABCsuperscript𝜌𝐴𝐵𝐶rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT in 𝒮ABCsuperscript𝒮𝐴𝐵𝐶mathcalS^ABCcaligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT, the following trade-off relations hold:

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DA;B;C≥DA;Bsubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐵D_A;B;Cgeq D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT, DA;B;C≥DA;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐶D_A;B;Cgeq D_A;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT.

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DA;B;C≥DB;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐵𝐶D_A;B;Cgeq D_B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT provided that dAB;C≥dB;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐵𝐶d_AB;Cgeq d_B;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT.

We assume that DA;B(ρAB)=S(ρ′AB)-S(ρ′A)-S(ρAB)+S(ρA)subscript𝐷𝐴𝐵superscript𝜌𝐴𝐵𝑆superscript𝜌′𝐴𝐵𝑆superscript𝜌′𝐴𝑆superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴D_A;B(rho^AB)=S(rho^prime AB)-S(rho^prime A)-S(rho^AB)+S(rho^% A)italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) for some ΠA(ρAB)=ρ′ABsuperscriptΠ𝐴superscript𝜌𝐴𝐵superscript𝜌′𝐴𝐵Pi^A(rho^AB)=rho^prime ABroman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_ρ start_POSTSUPERSCRIPT ′ italic_A italic_B end_POSTSUPERSCRIPT, and that DA;B;C(ρABC)=S(ρ′′AB)-S(ρ′′A)+S(ρ′′′ABC)-S(ρ′′′AB)-S(ρABC)+S(ρA)subscript𝐷𝐴𝐵𝐶superscript𝜌𝐴𝐵𝐶𝑆superscript𝜌′′𝐴𝐵𝑆superscript𝜌′′𝐴𝑆superscript𝜌′′′𝐴𝐵𝐶𝑆superscript𝜌′′′𝐴𝐵𝑆superscript𝜌𝐴𝐵𝐶𝑆superscript𝜌𝐴D_A;B;C(rho^ABC)=S(rho^primeprime AB)-S(rho^primeprime A)+S(% rho^primeprimeprime ABC)-S(rho^primeprimeprime AB)-S(rho^ABC)+S% (rho^A)italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ italic_A end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_B italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) for some ΠABsuperscriptΠ𝐴𝐵Pi^ABroman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT. For simplicity, we denote S(ρ*X)𝑆superscript𝜌absent𝑋S(rho^*X)italic_S ( italic_ρ start_POSTSUPERSCRIPT * italic_X end_POSTSUPERSCRIPT ) by SX*subscriptsuperscript𝑆𝑋S^*_Xitalic_S start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT hereafter (e.g., we denote S(ρ′′AB)𝑆superscript𝜌′′𝐴𝐵S(rho^primeprime AB)italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ italic_A italic_B end_POSTSUPERSCRIPT ) by SAB′′subscriptsuperscript𝑆′′𝐴𝐵S^primeprime_ABitalic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT for brevity). It is straightforward that, for any given ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT, we have

DA;B;C-DA;Bsubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐵displaystyle D_A;B;C-D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT

=displaystyle== minΠAB[-SBC|A+SB|ΠA+SC|ΠAB]subscriptsuperscriptΠ𝐴𝐵subscript𝑆conditional𝐵𝐶𝐴subscript𝑆conditional𝐵superscriptΠ𝐴subscript𝑆conditional𝐶superscriptΠ𝐴𝐵displaystylemin_Pi^AB[-S_A+S_Pi^A+S_C]roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ - italic_S start_POSTSUBSCRIPT italic_B italic_C | italic_A end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_C | roman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ]

-minΠA[SB|ΠA-SB|A]subscriptsuperscriptΠ𝐴subscript𝑆conditional𝐵superscriptΠ𝐴subscript𝑆conditional𝐵𝐴displaystyle-min_Pi^A[S_B-S_B]- roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT [ italic_S start_POSTSUBSCRIPT italic_B | roman_Π start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_B | italic_A end_POSTSUBSCRIPT ]

-[SAB′-SA′-SAB+SA]delimited-[]subscriptsuperscript𝑆′𝐴𝐵subscriptsuperscript𝑆′𝐴subscript𝑆𝐴𝐵subscript𝑆𝐴displaystyle-[S^prime_AB-S^prime_A-S_AB+S_A]- [ italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

-[SAB′′-SA′′-SAB+SA]delimited-[]subscriptsuperscript𝑆′′𝐴𝐵subscriptsuperscript𝑆′′𝐴subscript𝑆𝐴𝐵subscript𝑆𝐴displaystyle-[S^primeprime_AB-S^primeprime_A-S_AB+S_A]- [ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

=displaystyle== SABC′′′-SAB′′′-SABC+SABsubscriptsuperscript𝑆′′′𝐴𝐵𝐶subscriptsuperscript𝑆′′′𝐴𝐵subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵displaystyle S^primeprimeprime_ABC-S^primeprimeprime_AB-S_ABC% +S_ABitalic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT

DA;B;C-DA;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐶displaystyle D_A;B;C-D_A;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT

-[SAC′′-SA′′-SAC+SA]delimited-[]subscriptsuperscript𝑆′′𝐴𝐶subscriptsuperscript𝑆′′𝐴subscript𝑆𝐴𝐶subscript𝑆𝐴displaystyle-[S^primeprime_AC-S^primeprime_A-S_AC+S_A]- [ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

+[SABC′′-SAC′′-SABC+SAC]delimited-[]subscriptsuperscript𝑆′′𝐴𝐵𝐶subscriptsuperscript𝑆′′𝐴𝐶subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐶displaystyle+[S^primeprime_ABC-S^primeprime_AC-S_ABC+S_AC]+ [ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT ]

-[SABC′′-SAB′′-SABC+SAB]delimited-[]subscriptsuperscript𝑆′′𝐴𝐵𝐶subscriptsuperscript𝑆′′𝐴𝐵subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵displaystyle-[S^primeprime_ABC-S^primeprime_AB-S_ABC+S_AB]- [ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ]

=displaystyle== SABC′′′-SAB′′′-SABC′′+SAB′′subscriptsuperscript𝑆′′′𝐴𝐵𝐶subscriptsuperscript𝑆′′′𝐴𝐵subscriptsuperscript𝑆′′𝐴𝐵𝐶subscriptsuperscript𝑆′′𝐴𝐵displaystyle S^primeprimeprime_ABC-S^primeprimeprime_AB-S^% primeprime_ABC+S^primeprime_ABitalic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT

≥displaystylegeq≥ 0.0displaystyle 0.0 .
Here, the second inequality holds since the mutual information always decreases under local operation [55] (which is equivalent to SXY′′-SX′′-SXY+SX≥0subscriptsuperscript𝑆′′𝑋𝑌subscriptsuperscript𝑆′′𝑋subscript𝑆𝑋𝑌subscript𝑆𝑋0S^primeprime_XY-S^primeprime_X-S_XY+S_Xgeq 0italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ≥ 0). Observe that

DA;B;C-DB;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐵𝐶displaystyle D_A;B;C-D_B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT (17)

≥displaystylegeq≥ [SAB′′-SA′′+SABC′′′-SAB′′′-SABC+SA]delimited-[]subscriptsuperscript𝑆′′𝐴𝐵subscriptsuperscript𝑆′′𝐴subscriptsuperscript𝑆′′′𝐴𝐵𝐶subscriptsuperscript𝑆′′′𝐴𝐵subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴displaystyle[S^primeprime_AB-S^primeprime_A+S^primeprimeprime% _ABC-S^primeprimeprime_AB-S_ABC+S_A][ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

-[SBC′′′-SB′′′-SBC′′+SB′′]delimited-[]subscriptsuperscript𝑆′′′𝐵𝐶subscriptsuperscript𝑆′′′𝐵subscriptsuperscript𝑆′′𝐵𝐶subscriptsuperscript𝑆′′𝐵displaystyle-[S^primeprimeprime_BC-S^primeprimeprime_B-S^% primeprime_BC+S^primeprime_B]- [ italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ]

=displaystyle== [SAB′′-SA′′+SABC′′′-SAB′′′-SABC+SA]delimited-[]subscriptsuperscript𝑆′′𝐴𝐵subscriptsuperscript𝑆′′𝐴subscriptsuperscript𝑆′′′𝐴𝐵𝐶subscriptsuperscript𝑆′′′𝐴𝐵subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴displaystyle[S^primeprime_AB-S^primeprime_A+S^primeprimeprime% _ABC-S^primeprimeprime_AB-S_ABC+S_A][ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

-[SBC′′′-SB′′′-SBC+SB]delimited-[]subscriptsuperscript𝑆′′′𝐵𝐶subscriptsuperscript𝑆′′′𝐵subscript𝑆𝐵𝐶subscript𝑆𝐵displaystyle-[S^primeprimeprime_BC-S^primeprimeprime_B-S_BC+S% _B]- [ italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ]

=displaystyle== [SABC′′′-SAB′′′-SABC+SAB]delimited-[]subscriptsuperscript𝑆′′′𝐴𝐵𝐶subscriptsuperscript𝑆′′′𝐴𝐵subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵displaystyle[S^primeprimeprime_ABC-S^primeprimeprime_AB-S_ABC% +S_AB][ italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ]

+[SAB′′-SA′′-SAB+SA]delimited-[]subscriptsuperscript𝑆′′𝐴𝐵subscriptsuperscript𝑆′′𝐴subscript𝑆𝐴𝐵subscript𝑆𝐴displaystyle+[S^primeprime_AB-S^primeprime_A-S_AB+S_A]+ [ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

-[SBC′′′-SB′′′-SBC+SB],delimited-[]subscriptsuperscript𝑆′′′𝐵𝐶subscriptsuperscript𝑆′′′𝐵subscript𝑆𝐵𝐶subscript𝑆𝐵displaystyle-[S^primeprimeprime_BC-S^primeprimeprime_B-S_BC+S% _B],- [ italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ] ,
thus DA;B;C≥DB;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐵𝐶D_A;B;Cgeq D_B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT since dAB;C≥dB;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐵𝐶d_AB;Cgeq d_B;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT by assumption. ∎

DA;B;Csubscript𝐷𝐴𝐵𝐶D_A;B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT is completely monogamous provided that DA;B;C≥DB;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐵𝐶D_A;B;Cgeq D_B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT.

We only need to show that then DX;Z=DY;Z=0subscript𝐷𝑋𝑍subscript𝐷𝑌𝑍0D_X;Z=D_Y;Z=0italic_D start_POSTSUBSCRIPT italic_X ; italic_Z end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_Y ; italic_Z end_POSTSUBSCRIPT = 0 whenever DA;B;C=DX;Ysubscript𝐷𝐴𝐵𝐶subscript𝐷𝑋𝑌D_A;B;C=D_X;Yitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_X ; italic_Y end_POSTSUBSCRIPT for any X𝑋Xitalic_X, Y∈A,B,C𝑌𝐴𝐵𝐶YinA,B,Citalic_Y ∈ italic_A , italic_B , italic_C . For implicity, we use the notations in the proof of Proposition 1. Case 1: DA;B;C=DA;Bsubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐵D_A;B;C=D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT. If DA;B;C(ρABC)-DA;B(ρAB)subscript𝐷𝐴𝐵𝐶superscript𝜌𝐴𝐵𝐶subscript𝐷𝐴𝐵superscript𝜌𝐴𝐵D_A;B;C(rho^ABC)-D_A;B(rho^AB)italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) - italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), we get S(ρ′′′ABC)-S(ρ′′′AB)-S(ρABC)+S(ρAB)=0𝑆superscript𝜌′′′𝐴𝐵𝐶𝑆superscript𝜌′′′𝐴𝐵𝑆superscript𝜌𝐴𝐵𝐶𝑆superscript𝜌𝐴𝐵0S(rho^primeprimeprime ABC)-S(rho^primeprimeprime AB)-S(rho^ABC)% +S(rho^AB)=0italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_B italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 0 for some von Neumann measurement ΠABsuperscriptΠ𝐴𝐵Pi^ABroman_Π start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, which implies that ρABC=∑k,jpk,j|k⟩⟨k|A⊗|j⟩⟨j|B⊗ρk,jCsuperscript𝜌𝐴𝐵𝐶subscript𝑘𝑗tensor-producttensor-productsubscript𝑝𝑘𝑗ket𝑘superscriptbra𝑘𝐴ket𝑗superscriptbra𝑗𝐵superscriptsubscript𝜌𝑘𝑗𝐶rho^ABC=sum_k,jp_k,j|kranglelangle k|^Aotimes|jranglelangle j|^% Botimesrho_k,j^Citalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_k , italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k , italic_j end_POSTSUBSCRIPT | italic_k ⟩ ⟨ italic_k | start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ | italic_j ⟩ ⟨ italic_j | start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_k , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT. It is clear that DA;C=DB;C=0subscript𝐷𝐴𝐶subscript𝐷𝐵𝐶0D_A;C=D_B;C=0italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT = 0 for such a state. Case 2: DA;B;C=DB;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐵𝐶D_A;B;C=D_B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT. According to Eq. (17), for any state ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT, if DA;B;C=DB;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐵𝐶D_A;B;C=D_B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT, then SABC′′-SA′′-SABC+SA=IBC|A-IBC|A′′=0subscriptsuperscript𝑆′′𝐴𝐵𝐶subscriptsuperscript𝑆′′𝐴subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴subscript𝐼conditional𝐵𝐶𝐴subscriptsuperscript𝐼′′conditional𝐵𝐶𝐴0S^primeprime_ABC-S^primeprime_A-S_ABC+S_A=I_A-I^prime% prime_A=0italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_I start_POSTSUBSCRIPT italic_B italic_C | italic_A end_POSTSUBSCRIPT - italic_I start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_C | italic_A end_POSTSUBSCRIPT = 0. Therefore ρABC=∑kpk|k⟩⟨k|A⊗ρkBCsuperscript𝜌𝐴𝐵𝐶subscript𝑘tensor-productsubscript𝑝𝑘ket𝑘superscriptbra𝑘𝐴superscriptsubscript𝜌𝑘𝐵𝐶rho^ABC=sum_kp_k|kranglelangle k|^Aotimesrho_k^BCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_k ⟩ ⟨ italic_k | start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT for some basis k⟩Asuperscriptket𝑘𝐴krangle^A of ℋAsuperscriptℋ𝐴mathcalH^Acaligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT, and thus DA;B=DA;C=0subscript𝐷𝐴𝐵subscript𝐷𝐴𝐶0D_A;B=D_A;C=0italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT = 0. Case 3: If DA;B;C=DA;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐶D_A;B;C=D_A;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT, one can easily check that ρABC=∑k,jpk,j|k⟩⟨k|A⊗|j⟩⟨j|B⊗ρk,jCsuperscript𝜌𝐴𝐵𝐶subscript𝑘𝑗tensor-producttensor-productsubscript𝑝𝑘𝑗ket𝑘superscriptbra𝑘𝐴ket𝑗superscriptbra𝑗𝐵superscriptsubscript𝜌𝑘𝑗𝐶rho^ABC=sum_k,jp_k,j|kranglelangle k|^Aotimes|jranglelangle j|^% Botimesrho_k,j^Citalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_k , italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k , italic_j end_POSTSUBSCRIPT | italic_k ⟩ ⟨ italic_k | start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ | italic_j ⟩ ⟨ italic_j | start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_k , italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT for some basis j⟩Bsuperscriptket𝑘𝐴superscriptket𝑗𝐵jrangle^B italic_k ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT in ℋABsuperscriptℋ𝐴𝐵mathcalH^ABcaligraphic_H start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT. It follows that DA;B=DB;C=0subscript𝐷𝐴𝐵subscript𝐷𝐵𝐶0D_A;B=D_B;C=0italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT = 0. ∎

Using the same notations as in the proof of Proposition 1, it is clear that, for any tripartite state ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT,

DA;B;C-DA;B-DA;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐵subscript𝐷𝐴𝐶displaystyle D_A;B;C-D_A;B-D_A;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT

≥displaystylegeq≥ [SABC′′′-SAB′′′-SABC+SAB]delimited-[]subscriptsuperscript𝑆′′′𝐴𝐵𝐶subscriptsuperscript𝑆′′′𝐴𝐵subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵displaystyle[S^primeprimeprime_ABC-S^primeprimeprime_AB-S_ABC% +S_AB][ italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ]

-[S′′′AC-S′′′A-SAC+SA]delimited-[]superscript𝑆′′′𝐴𝐶superscript𝑆′′′𝐴subscript𝑆𝐴𝐶subscript𝑆𝐴displaystyle-[S^primeprimeprime AC-S^primeprimeprime A-S_AC+S_A]- [ italic_S start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_C end_POSTSUPERSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ italic_A end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

≥displaystylegeq≥ 00displaystyle 0
whenever dAB;C≥dA;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐴𝐶d_AB;Cgeq d_A;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT. That is

Proposition 3.

For any state ρABC∈𝒮ABCsuperscript𝜌𝐴𝐵𝐶superscript𝒮𝐴𝐵𝐶rho^ABCinmathcalS^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT that satisfies dAB;C≥dA;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐴𝐶d_AB;Cgeq d_A;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT, we have

DA;B;C≥DA;B+DA;C.subscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐵subscript𝐷𝐴𝐶displaystyle D_A;B;Cgeq D_A;B+D_A;C.italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT . (18)

IV.2 The four-partite case

We propose the trade-off relation at first. With the increasing of particles involved, the hierarchy of trade-off relations become more complicated.

Proposition 4.

For any state ρABCDsuperscript𝜌𝐴𝐵𝐶𝐷rho^ABCDitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT in 𝒮ABCDsuperscript𝒮𝐴𝐵𝐶𝐷mathcalS^ABCDcaligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT, the following trade-off relations hold:

•
DA;B;C;D≥DA;B;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐶D_A;B;C;Dgeq D_A;B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT.

•
DA;B;C;D≥DA;B;D+DAB;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐷subscript𝐷𝐴𝐵𝐶D_A;B;C;Dgeq D_A;B;D+D_AB;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_D end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT provided that dABC;D≥dAB;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐴𝐵𝐷d_ABC;Dgeq d_AB;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_D end_POSTSUBSCRIPT.

•
DA;B;C;D≥DA;C;D+DA;Bsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐶𝐷subscript𝐷𝐴𝐵D_A;B;C;Dgeq D_A;C;D+D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_C ; italic_D end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT provided that dABC;D≥dAC;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐴𝐶𝐷d_ABC;Dgeq d_AC;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A italic_C ; italic_D end_POSTSUBSCRIPT and dAB;C≥dA;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐴𝐶d_AB;Cgeq d_A;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT.

•
DA;B;C;D≥DB;C;D+DA;Bsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐵𝐶𝐷subscript𝐷𝐴𝐵D_A;B;C;Dgeq D_B;C;D+D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT provided that dABC;D≥dBC;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐵𝐶𝐷d_ABC;Dgeq d_BC;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B italic_C ; italic_D end_POSTSUBSCRIPT and dAB;C≥dB;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐵𝐶d_AB;Cgeq d_B;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT.

•
DA;B;C;D≥DA;Bsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵D_A;B;C;Dgeq D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT, DA;B;C;D≥DA;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐶D_A;B;C;Dgeq D_A;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT.

•
DA;B;C;D≥DA;Dsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐷D_A;B;C;Dgeq D_A;Ditalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_D end_POSTSUBSCRIPT provided that dABC;D≥dA;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐴𝐷d_ABC;Dgeq d_A;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A ; italic_D end_POSTSUBSCRIPT.

•
DA;B;C;D≥DB;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐵𝐶D_A;B;C;Dgeq D_B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT provided that dAB;C≥dB;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐵𝐶d_AB;Cgeq d_B;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT.

•
DA;B;C;D≥DB;Dsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐵𝐷D_A;B;C;Dgeq D_B;Ditalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_B ; italic_D end_POSTSUBSCRIPT provided that dABC;D≥dB;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐵𝐷d_ABC;Dgeq d_B;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B ; italic_D end_POSTSUBSCRIPT.

•
DA;B;C;D≥DC;Dsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐶𝐷D_A;B;C;Dgeq D_C;Ditalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_C ; italic_D end_POSTSUBSCRIPT provided that dABC;D≥dA;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐴𝐷d_ABC;Dgeq d_A;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A ; italic_D end_POSTSUBSCRIPT.

For any ρ∈𝒮ABCD𝜌superscript𝒮𝐴𝐵𝐶𝐷rhoinmathcalS^ABCDitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT, we assume that DX;Y(ρXY)=S(ρ′XY)-S(ρ′X)-S(ρXY)+S(ρX)subscript𝐷𝑋𝑌superscript𝜌𝑋𝑌𝑆superscript𝜌′𝑋𝑌𝑆superscript𝜌′𝑋𝑆superscript𝜌𝑋𝑌𝑆superscript𝜌𝑋D_X;Y(rho^XY)=S(rho^prime XY)-S(rho^prime X)-S(rho^XY)+S(rho^% X)italic_D start_POSTSUBSCRIPT italic_X ; italic_Y end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_X italic_Y end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ italic_X italic_Y end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ italic_X end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_X italic_Y end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ) for some ΠX(ρXY)=ρ′XYsuperscriptΠ𝑋superscript𝜌𝑋𝑌superscript𝜌′𝑋𝑌Pi^X(rho^XY)=rho^prime XYroman_Π start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_X italic_Y end_POSTSUPERSCRIPT ) = italic_ρ start_POSTSUPERSCRIPT ′ italic_X italic_Y end_POSTSUPERSCRIPT, and that DA;B;C;D(ρABCD)=S(ρ′′AB)-S(ρ′′A)+S(ρ′′′ABC)-S(ρ′′′AB)+S(ρ′′′′ABCD)-S(ρ′′′ABC)-S(ρABCD)+S(ρA)subscript𝐷𝐴𝐵𝐶𝐷superscript𝜌𝐴𝐵𝐶𝐷𝑆superscript𝜌′′𝐴𝐵𝑆superscript𝜌′′𝐴𝑆superscript𝜌′′′𝐴𝐵𝐶𝑆superscript𝜌′′′𝐴𝐵𝑆superscript𝜌′′′′𝐴𝐵𝐶𝐷𝑆superscript𝜌′′′𝐴𝐵𝐶𝑆superscript𝜌𝐴𝐵𝐶𝐷𝑆superscript𝜌𝐴D_A;B;C;D(rho^ABCD)=S(rho^primeprime AB)-S(rho^primeprime A)+S(% rho^primeprimeprime ABC)-S(rho^primeprimeprime AB)+S(rho^prime% primeprimeprime ABCD)-S(rho^primeprimeprime ABC)-S(rho^ABCD)+S(% rho^A)italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ italic_A end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_B italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_B end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ ′ italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ ′ ′ italic_A italic_B italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) for some ΠABCsuperscriptΠ𝐴𝐵𝐶Pi^ABCroman_Π start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT. It is straightforward that

DA;B;C:D-DA;B;Csubscript𝐷:𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐶displaystyle D_A;B;C:D-D_A;B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C : italic_D end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT

≥displaystylegeq≥ [-SABCD+SA+SAB′′-SA′′fragments[subscript𝑆𝐴𝐵𝐶𝐷subscript𝑆𝐴superscriptsubscript𝑆𝐴𝐵′′superscriptsubscript𝑆𝐴′′displaystyle[-S_ABCD+S_A+S_AB^primeprime-S_A^primeprime[ - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT

+SABC′′′-SAB′′′+SABCD′′′′-SABC′′′′]fragmentssuperscriptsubscript𝑆𝐴𝐵𝐶′′′superscriptsubscript𝑆𝐴𝐵′′′superscriptsubscript𝑆𝐴𝐵𝐶𝐷′′′′superscriptsubscript𝑆𝐴𝐵𝐶′′′′]displaystyle+S_ABC^primeprimeprime-S_AB^primeprimeprime+S_ABCD% ^primeprimeprimeprime-S_ABC^primeprimeprimeprime]+ italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT ]

-[SAB′′-SA′′+SABC′′′-SAB′′′-SBC+SA]delimited-[]subscriptsuperscript𝑆′′𝐴𝐵subscriptsuperscript𝑆′′𝐴subscriptsuperscript𝑆′′′𝐴𝐵𝐶subscriptsuperscript𝑆′′′𝐴𝐵subscript𝑆𝐵𝐶superscript𝑆𝐴displaystyle-[S^primeprime_AB-S^primeprime_A+S^primeprime% prime_ABC-S^primeprimeprime_AB-S_BC+S^A]- [ italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT - italic_S start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ]

-[SABC′′′-SAB′′′-SABC+SAB]delimited-[]superscriptsubscript𝑆𝐴𝐵𝐶′′′superscriptsubscript𝑆𝐴𝐵′′′subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵displaystyle-[S_ABC^primeprimeprime-S_AB^primeprimeprime-S_ABC% +S_AB]- [ italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ]

=displaystyle== SABCD′′′′-SABC′′′′-SABCD+SABCsuperscriptsubscript𝑆𝐴𝐵𝐶𝐷′′′′superscriptsubscript𝑆𝐴𝐵𝐶′′′′subscript𝑆𝐴𝐵𝐶𝐷subscript𝑆𝐴𝐵𝐶displaystyle S_ABCD^primeprimeprimeprime-S_ABC^primeprimeprime% prime-S_ABCD+S_ABCitalic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT

≥displaystylegeq≥ 0.0displaystyle 0.0 .
Similarly,

DA;B;C:D-DA;B;Dsubscript𝐷:𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐷displaystyle D_A;B;C:D-D_A;B;Ditalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C : italic_D end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_D end_POSTSUBSCRIPT

-[SAB′′-SA′′-SAB+SA]delimited-[]superscriptsubscript𝑆𝐴𝐵′′superscriptsubscript𝑆𝐴′′subscript𝑆𝐴𝐵subscript𝑆𝐴displaystyle-[S_AB^primeprime-S_A^primeprime-S_AB+S_A]- [ italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

-[SABD′′′-SAB′′′-SABD+SAB]delimited-[]superscriptsubscript𝑆𝐴𝐵𝐷′′′superscriptsubscript𝑆𝐴𝐵′′′subscript𝑆𝐴𝐵𝐷subscript𝑆𝐴𝐵displaystyle-[S_ABD^primeprimeprime-S_AB^primeprimeprime-S_ABD% +S_AB]- [ italic_S start_POSTSUBSCRIPT italic_A italic_B italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ]

≥displaystylegeq≥ SABC′′′-SAB′′′-SABC+SAB≥DAB;Csuperscriptsubscript𝑆𝐴𝐵𝐶′′′superscriptsubscript𝑆𝐴𝐵′′′subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵subscript𝐷𝐴𝐵𝐶displaystyle S_ABC^primeprimeprime-S_AB^primeprimeprime-S_ABC% +S_ABgeq D_AB;Citalic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT
provided that dABC;D≥dAB;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐴𝐵𝐷d_ABC;Dgeq d_AB;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_D end_POSTSUBSCRIPT, and

DA;B;C:D-DA;C;Dsubscript𝐷:𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐶𝐷displaystyle D_A;B;C:D-D_A;C;Ditalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C : italic_D end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_A ; italic_C ; italic_D end_POSTSUBSCRIPT

≥displaystylegeq≥ (SABCD′′′′-SABC′′′′-SABCD+SABC)superscriptsubscript𝑆𝐴𝐵𝐶𝐷′′′′superscriptsubscript𝑆𝐴𝐵𝐶′′′′subscript𝑆𝐴𝐵𝐶𝐷subscript𝑆𝐴𝐵𝐶displaystyle(S_ABCD^primeprimeprimeprime-S_ABC^primeprimeprime% prime-S_ABCD+S_ABC)( italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT )

-[SACD′′′-SAC′′′-SACD+SAC]delimited-[]superscriptsubscript𝑆𝐴𝐶𝐷′′′superscriptsubscript𝑆𝐴𝐶′′′subscript𝑆𝐴𝐶𝐷subscript𝑆𝐴𝐶displaystyle-[S_ACD^primeprimeprime-S_AC^primeprimeprime-S_ACD% +S_AC]- [ italic_S start_POSTSUBSCRIPT italic_A italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT ]

=displaystyle== (SABCD′′′′-SABC′′′′-SABCD+SABC)superscriptsubscript𝑆𝐴𝐵𝐶𝐷′′′′superscriptsubscript𝑆𝐴𝐵𝐶′′′′subscript𝑆𝐴𝐵𝐶𝐷subscript𝑆𝐴𝐵𝐶displaystyle(S_ABCD^primeprimeprimeprime-S_ABC^primeprimeprime% prime-S_ABCD+S_ABC)( italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT )

-[SACD′′′-SAC′′′-SACD+SAB]delimited-[]superscriptsubscript𝑆𝐴𝐶𝐷′′′superscriptsubscript𝑆𝐴𝐶′′′subscript𝑆𝐴𝐶𝐷subscript𝑆𝐴𝐵displaystyle-[S_ACD^primeprimeprime-S_AC^primeprimeprime-S_ACD% +S_AB]- [ italic_S start_POSTSUBSCRIPT italic_A italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ]

+(SABC′′′-SAB′′′-SABC+SAB)superscriptsubscript𝑆𝐴𝐵𝐶′′′superscriptsubscript𝑆𝐴𝐵′′′subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵displaystyle+(S_ABC^primeprimeprime-S_AB^primeprimeprime-S_ABC% +S_AB)+ ( italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT )

-[SAC′′-SA′′-SAC+SA]delimited-[]superscriptsubscript𝑆𝐴𝐶′′superscriptsubscript𝑆𝐴′′subscript𝑆𝐴𝐶subscript𝑆𝐴displaystyle-[S_AC^primeprime-S_A^primeprime-S_AC+S_A]- [ italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ]

+(SAB′′-SA′′-SAB+SA)superscriptsubscript𝑆𝐴𝐵′′superscriptsubscript𝑆𝐴′′subscript𝑆𝐴𝐵subscript𝑆𝐴displaystyle+(S_AB^primeprime-S_A^primeprime-S_AB+S_A)+ ( italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT )

≥displaystylegeq≥ SAB′′-SA′′-SAB+SA≥DA;Bsuperscriptsubscript𝑆𝐴𝐵′′superscriptsubscript𝑆𝐴′′subscript𝑆𝐴𝐵subscript𝑆𝐴subscript𝐷𝐴𝐵displaystyle S_AB^primeprime-S_A^primeprime-S_AB+S_Ageq D_A% ;Bitalic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT
provided that dABC;D≥dAC;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐴𝐶𝐷d_ABC;Dgeq d_AC;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A italic_C ; italic_D end_POSTSUBSCRIPT and dAB;C≥dA;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐴𝐶d_AB;Cgeq d_A;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT. With the same argument, one can check that DA;B;C;D≥DB;C;D+DA;Bsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐵𝐶𝐷subscript𝐷𝐴𝐵D_A;B;C;Dgeq D_B;C;D+D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT provided that dABC;D≥dBC;Dsubscript𝑑𝐴𝐵𝐶𝐷subscript𝑑𝐵𝐶𝐷d_ABC;Dgeq d_BC;Ditalic_d start_POSTSUBSCRIPT italic_A italic_B italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B italic_C ; italic_D end_POSTSUBSCRIPT and dAB;C≥dB;Csubscript𝑑𝐴𝐵𝐶subscript𝑑𝐵𝐶d_AB;Cgeq d_B;Citalic_d start_POSTSUBSCRIPT italic_A italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT. Since DA;B;C;D≥DA;B;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐶D_A;B;C;Dgeq D_A;B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT and DA;B;C≥DA;Bsubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐵D_A;B;Cgeq D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT, DA;B;C≥DA;Csubscript𝐷𝐴𝐵𝐶subscript𝐷𝐴𝐶D_A;B;Cgeq D_A;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT, we get DA;B;C;D≥DA;Bsubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵D_A;B;C;Dgeq D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT and DA;B;C;D≥DA;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐶D_A;B;C;Dgeq D_A;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT directly. The last 4 items can also be easily checked, and thus the proof is completed. ∎

Proposition 5.

If DA;B;C;Dsubscript𝐷𝐴𝐵𝐶𝐷D_A;B;C;Ditalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT is monotonic under discard of subsystems, then it is completely monogamous.

For convenience, we use the same notations as that of the proof of Proposition 4. If DA;B;C;D(ρABCD)=DA;B(ρAB)subscript𝐷𝐴𝐵𝐶𝐷superscript𝜌𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵superscript𝜌𝐴𝐵D_A;B;C;D(rho^ABCD)=D_A;B(rho^AB)italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT ) = italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), then SABCD′′′′-SABC′′′′-SABCD+SABC=0superscriptsubscript𝑆𝐴𝐵𝐶𝐷′′′′superscriptsubscript𝑆𝐴𝐵𝐶′′′′subscript𝑆𝐴𝐵𝐶𝐷subscript𝑆𝐴𝐵𝐶0S_ABCD^primeprimeprimeprime-S_ABC^primeprimeprimeprime-S_ABCD% +S_ABC=0italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT = 0 and SABC′′-SAB′′-SABC+SAB=0superscriptsubscript𝑆𝐴𝐵𝐶′′superscriptsubscript𝑆𝐴𝐵′′subscript𝑆𝐴𝐵𝐶subscript𝑆𝐴𝐵0S_ABC^primeprime-S_AB^primeprime-S_ABC+S_AB=0italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = 0, which implies that ρABCD=∑k,j,lpk,j,l|k⟩⟨k|A⊗|j⟩⟨j|B⊗|l⟩⟨l|C⊗ρk,j,lDsuperscript𝜌𝐴𝐵𝐶𝐷subscript𝑘𝑗𝑙tensor-producttensor-producttensor-productsubscript𝑝𝑘𝑗𝑙ket𝑘superscriptbra𝑘𝐴ket𝑗superscriptbra𝑗𝐵ket𝑙superscriptbra𝑙𝐶superscriptsubscript𝜌𝑘𝑗𝑙𝐷rho^ABCD=sum_k,j,lp_k,j,l|kranglelangle k|^Aotimes|jrangle% langle j|^Botimes|lranglelangle l|^Cotimesrho_k,j,l^Ditalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_k , italic_j , italic_l end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k , italic_j , italic_l end_POSTSUBSCRIPT | italic_k ⟩ ⟨ italic_k | start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ | italic_j ⟩ ⟨ italic_j | start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⊗ | italic_l ⟩ ⟨ italic_l | start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_k , italic_j , italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT. Thus DA;C=DA;D=DB;C=DB;D=DC;D=0subscript𝐷𝐴𝐶subscript𝐷𝐴𝐷subscript𝐷𝐵𝐶subscript𝐷𝐵𝐷subscript𝐷𝐶𝐷0D_A;C=D_A;D=D_B;C=D_B;D=D_C;D=0italic_D start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_C ; italic_D end_POSTSUBSCRIPT = 0. If DA;B;C;D(ρABCD)=DA;B;C(ρABC)subscript𝐷𝐴𝐵𝐶𝐷superscript𝜌𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐶superscript𝜌𝐴𝐵𝐶D_A;B;C;D(rho^ABCD)=D_A;B;C(rho^ABC)italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT ) = italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ), then SABCD′′′′-SABC′′′′-SABCD+SABC=0superscriptsubscript𝑆𝐴𝐵𝐶𝐷′′′′superscriptsubscript𝑆𝐴𝐵𝐶′′′′subscript𝑆𝐴𝐵𝐶𝐷subscript𝑆𝐴𝐵𝐶0S_ABCD^primeprimeprimeprime-S_ABC^primeprimeprimeprime-S_ABCD% +S_ABC=0italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ ′ ′ ′ end_POSTSUPERSCRIPT - italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C italic_D end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT = 0, which implies that ρABCD=∑k,j,lpk,j,l|k⟩⟨k|A⊗|j⟩⟨j|B⊗|l⟩⟨l|C⊗ρk,j,lDsuperscript𝜌𝐴𝐵𝐶𝐷subscript𝑘𝑗𝑙tensor-producttensor-producttensor-productsubscript𝑝𝑘𝑗𝑙ket𝑘superscriptbra𝑘𝐴ket𝑗superscriptbra𝑗𝐵ket𝑙superscriptbra𝑙𝐶superscriptsubscript𝜌𝑘𝑗𝑙𝐷rho^ABCD=sum_k,j,lp_k,j,l|kranglelangle k|^Aotimes|jrangle% langle j|^Botimes|lranglelangle l|^Cotimesrho_k,j,l^Ditalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C italic_D end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_k , italic_j , italic_l end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k , italic_j , italic_l end_POSTSUBSCRIPT | italic_k ⟩ ⟨ italic_k | start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ | italic_j ⟩ ⟨ italic_j | start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⊗ | italic_l ⟩ ⟨ italic_l | start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_k , italic_j , italic_l end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT. Thus DA;D=DB;D=DC;D=0subscript𝐷𝐴𝐷subscript𝐷𝐵𝐷subscript𝐷𝐶𝐷0D_A;D=D_B;D=D_C;D=0italic_D start_POSTSUBSCRIPT italic_A ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_B ; italic_D end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_C ; italic_D end_POSTSUBSCRIPT = 0. Similarly, one can easily verify that all the other dis-correlate conditions are valid. ∎

IV.3 The n𝑛nitalic_n-partite case

It is easy to see that DA;B;C≥DAB|C+DA;Bsubscript𝐷𝐴𝐵𝐶subscript𝐷conditional𝐴𝐵𝐶subscript𝐷𝐴𝐵D_A;B;Cgeq D_AB+D_A;Bitalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A italic_B | italic_C end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT and DA;B;C;D≥DABC|D+DA;B;Csubscript𝐷𝐴𝐵𝐶𝐷subscript𝐷conditional𝐴𝐵𝐶𝐷subscript𝐷𝐴𝐵𝐶D_A;B;C;Dgeq D_D+D_A;B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C ; italic_D end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A italic_B italic_C | italic_D end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT. Moreover, we can conclude the following, which are the main results of this paper.

Let ℋA1A2⋯Ansuperscriptℋsubscript𝐴1subscript𝐴2normal-⋯subscript𝐴𝑛mathcalH^A_1A_2cdots A_ncaligraphic_H start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT be the state space of n𝑛nitalic_n-partite quanutm system with finite dimension. Then the following holds true for any n𝑛nitalic_n-partite state:

•
DA1;A2;⋯;An≥DA1A2⋯An-1|An+DA1;A2;⋯;An-1subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷conditionalsubscript𝐴1subscript𝐴2⋯subscript𝐴𝑛1subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛1D_A_1;A_2;cdots;A_ngeq D_A_1A_2cdots A_n-1+D_A_1;A% _2;cdots;A_n-1italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT | italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT.

•
DA1;A2;⋯;An≥DA1;A2subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2D_A_1;A_2;cdots;A_ngeq D_A_1;A_2italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT.

•
DA1;A2;⋯;An≥DA1;A3subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴3D_A_1;A_2;cdots;A_ngeq D_A_1;A_3italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT.

Moreover, for any tripartite system ℋXYZsuperscriptℋ𝑋𝑌𝑍mathcalH^XYZcaligraphic_H start_POSTSUPERSCRIPT italic_X italic_Y italic_Z end_POSTSUPERSCRIPT with X=Ai1⋯Aip𝑋subscript𝐴subscript𝑖1normal-⋯subscript𝐴subscript𝑖𝑝X=A_i_1cdots A_i_pitalic_X = italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT, Y=Aj1⋯Ajq𝑌subscript𝐴subscript𝑗1normal-⋯subscript𝐴subscript𝑗𝑞Y=A_j_1cdots A_j_qitalic_Y = italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT and Z=Ak𝑍subscript𝐴𝑘Z=A_kitalic_Z = italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, 1≤i1<⋯ <⋯

dXY;Z≥dX;Z,dXY;Z≥dY;Zformulae-sequencesubscript𝑑𝑋𝑌𝑍subscript𝑑𝑋𝑍subscript𝑑𝑋𝑌𝑍subscript𝑑𝑌𝑍displaystyle d_XY;Zgeq d_X;Z,quad d_XY;Zgeq d_Y;Zitalic_d start_POSTSUBSCRIPT italic_X italic_Y ; italic_Z end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_X ; italic_Z end_POSTSUBSCRIPT , italic_d start_POSTSUBSCRIPT italic_X italic_Y ; italic_Z end_POSTSUBSCRIPT ≥ italic_d start_POSTSUBSCRIPT italic_Y ; italic_Z end_POSTSUBSCRIPT (19)
holds for any multipartite von Neumann measurement ΠXYsuperscriptnormal-Π𝑋𝑌Pi^XYroman_Π start_POSTSUPERSCRIPT italic_X italic_Y end_POSTSUPERSCRIPT (i.e., ΠijXY=Πi1⋯ipj1⋯jqAi1⋯AipAj1⋯Ajqsubscriptsuperscriptnormal-Π𝑋𝑌𝑖𝑗superscriptsubscriptnormal-Πsubscript𝑖1normal-⋯subscript𝑖𝑝subscript𝑗1normal-⋯subscript𝑗𝑞subscript𝐴subscript𝑖1normal-⋯subscript𝐴subscript𝑖𝑝subscript𝐴subscript𝑗1normal-⋯subscript𝐴subscript𝑗𝑞Pi^XY_ij=Pi_i_1cdots i_pj_1cdots j_q^A_i_1cdots A_i_% pA_j_1cdots A_j_qroman_Π start_POSTSUPERSCRIPT italic_X italic_Y end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = roman_Π start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_j start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUPERSCRIPT), then

•
DA1;A2;⋯;An≥DA1;A2;⋯Ak;Ak+2;⋯;An-1;An+DA1A2⋯Ak;Ak+1subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑘subscript𝐴𝑘2⋯subscript𝐴𝑛1subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑘subscript𝐴𝑘1D_A_1;A_2;cdots;A_ngeq D_A_1;A_2;cdots A_k;A_k+2;cdots;A_% n-1;A_n+D_A_1A_2cdots A_k;A_k+1italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT italic_k + 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT.

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DA1;A2;⋯;An≥DAi1;Ai2;⋯Ain-2;An+DA1;A2subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴subscript𝑖1subscript𝐴subscript𝑖2⋯subscript𝐴subscript𝑖𝑛2subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2D_A_1;A_2;cdots;A_ngeq D_A_i_1;A_i_2;cdots A_i_n-2;A_% n+D_A_1;A_2italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_n - 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, where 1≤i1 <⋯

The multipartite discord DA1;A2;⋯;Ansubscript𝐷subscript𝐴1subscript𝐴2normal-⋯subscript𝐴𝑛D_A_1;A_2;cdots;A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is completely monogamous if it is monotonic under discard of subsystems.

Note here that, the monotonicity condition DA1;A2;⋯;An≥DA1;A2;⋯;An-1subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛1D_A_1;A_2;cdots;A_ngeq D_A_1;A_2;cdots;A_n-1italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, DA1;A2;⋯;An≥DA1;A2subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2D_A_1;A_2;cdots;A_ngeq D_A_1;A_2italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, and DA1;A2;⋯;An≥DA1;A3subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴3D_A_1;A_2;cdots;A_ngeq D_A_1;A_3italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT are true automatically. In Theorem 2, we only need the assumption of other monotonicity condition DA1;A2;⋯;An≥DAi;Ajsubscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴𝑖subscript𝐴𝑗D_A_1;A_2;cdots;A_ngeq D_A_i;A_jitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT for any 1≤ieq(1,3)( italic_i , italic_j ) ≠ ( 1 , 3 ) since DA1;A2;⋯;An≥DA1;A2subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴2D_A_1;A_2;cdots;A_ngeq D_A_1;A_2italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and DA1;A2;⋯;An≥DA1;A3subscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛subscript𝐷subscript𝐴1subscript𝐴3D_A_1;A_2;cdots;A_ngeq D_A_1;A_3italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT automatically. In addition, it is obvious that if the condition (19) is valid, then all the monotonicity conditions are valid. Therefore

Proposition 6.

The multipartite discord DA1;A2;⋯;Ansubscript𝐷subscript𝐴1subscript𝐴2normal-⋯subscript𝐴𝑛D_A_1;A_2;cdots;A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is completely monogamous provided that the condition (19) is valid.

Although DA1;A2;⋯;Ansubscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛D_A_1;A_2;cdots;A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is not continuous (since DA1;A2subscript𝐷subscript𝐴1subscript𝐴2D_A_1;A_2italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is not continuous [56, 57]), n>2𝑛2n>2italic_n >2, we still can, taking the tripartite case for example, get the following monogamy relation: If DA;B;Csubscript𝐷𝐴𝐵𝐶D_A;B;Citalic_D start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT is monotonic under discord of subsystems, then for any given ρ∈𝒮ABC𝜌superscript𝒮𝐴𝐵𝐶rhoinmathcalS^ABCitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT there exists 0<α<∞0𝛼00 <∞ such that

DA;B;Cα≥DA;Bα+DA;Cα+DB;Cα.subscriptsuperscript𝐷𝛼𝐴𝐵𝐶subscriptsuperscript𝐷𝛼𝐴𝐵subscriptsuperscript𝐷𝛼𝐴𝐶subscriptsuperscript𝐷𝛼𝐵𝐶displaystyle D^alpha_A;B;Cgeq D^alpha_A;B+D^alpha_A;C+D^% alpha_B;C.italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A ; italic_B ; italic_C end_POSTSUBSCRIPT ≥ italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A ; italic_B end_POSTSUBSCRIPT + italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A ; italic_C end_POSTSUBSCRIPT + italic_D start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B ; italic_C end_POSTSUBSCRIPT . (20)
That is, α𝛼alphaitalic_α is dependent not only on dimℋABCdimensionsuperscriptℋ𝐴𝐵𝐶dimmathcalH^ABCroman_dim caligraphic_H start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT but also on the given state. Note that, for the continuous measure, there exists a common minimal exponent α𝛼alphaitalic_α for all states ρ∈𝒮ABC𝜌superscript𝒮𝐴𝐵𝐶rhoinmathcalS^ABCitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT with fixed dimℋABC=ddimensionsuperscriptℋ𝐴𝐵𝐶𝑑dimmathcalH^ABC=droman_dim caligraphic_H start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = italic_d such that Eq. (20) holds according to the arguments in Refs. [40, 43].

V The trade-off relation of global quantum discord

In Ref. [45], the following monogamy bound is proved provided that the bipartite discord does not increase under loss of subsystems, i.e., DA1⋯Ak:Ak+1≥DA1:Ak+1subscript𝐷:subscript𝐴1⋯subscript𝐴𝑘subscript𝐴𝑘1subscript𝐷:subscript𝐴1subscript𝐴𝑘1D_A_1cdots A_k:A_k+1geq D_A_1:A_k+1italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, 2≤k

DA1:⋯:An≥∑k=1n-1DA1:Ak+1.subscript𝐷:subscript𝐴1⋯:subscript𝐴𝑛superscriptsubscript𝑘1𝑛1subscript𝐷:subscript𝐴1subscript𝐴𝑘1displaystyleD_A_1:cdots:A_ngeqsum_k=1^n-1D_A_1:A_k+1.italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT . (21)
However, the assumption DA1⋯Ak:Ak+1≥DA1:Ak+1subscript𝐷:subscript𝐴1⋯subscript𝐴𝑘subscript𝐴𝑘1subscript𝐷:subscript𝐴1subscript𝐴𝑘1D_A_1cdots A_k:A_k+1geq D_A_1:A_k+1italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is not valid in general. For example, consider a three qubit state ρABC=(1/2)(|000⟩⟨000|+|+11⟩⟨+11|)subscript𝜌𝐴𝐵𝐶12ket000bra000ket11bra11rho_ABC=(1/2)(|000ranglelangle 000|+|+11ranglelangle+11|)italic_ρ start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT = ( 1 / 2 ) ( | 000 ⟩ ⟨ 000 | + | + 11 ⟩ ⟨ + 11 | ), where |+⟩=(|0⟩+|1⟩)/2ketket0ket12|+rangle=(|0rangle+|1rangle)/sqrt2| + ⟩ = ( | 0 ⟩ + | 1 ⟩ ) / square-root start_ARG 2 end_ARG, it is evident that DAB:C=0 DISCORD SERVER ρ∈𝒮A1A2⋯An𝜌superscript𝒮subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛rhoinmathcalS^A_1A_2cdots A_nitalic_ρ ∈ caligraphic_S start_POSTSUPERSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT, we write state after measurement ΦΦPhiroman_Φ as in Eq. (5) as ρ′superscript𝜌′rho^primeitalic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, write S(ρ′XY∥ρ′X⊗ρ′Y)𝑆conditionalsuperscript𝜌′𝑋𝑌tensor-productsuperscript𝜌′𝑋superscript𝜌′𝑌S(rho^prime XY)italic_S ( italic_ρ start_POSTSUPERSCRIPT ′ italic_X italic_Y end_POSTSUPERSCRIPT ∥ italic_ρ start_POSTSUPERSCRIPT ′ italic_X end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT ′ italic_Y end_POSTSUPERSCRIPT ) as SXY∥X⊗Y′subscriptsuperscript𝑆′conditional𝑋𝑌tensor-product𝑋𝑌S^prime_Xotimes Yitalic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X italic_Y ∥ italic_X ⊗ italic_Y end_POSTSUBSCRIPT, and write I(ρ)-I(ρ′)𝐼𝜌𝐼superscript𝜌′I(rho)-I(rho^prime)italic_I ( italic_ρ ) - italic_I ( italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) as dA1:A2:⋯:AnΦsubscriptsuperscript𝑑Φ:subscript𝐴1subscript𝐴2:⋯:subscript𝐴𝑛d^Phi_A_1:A_2:cdots:A_nitalic_d start_POSTSUPERSCRIPT roman_Φ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Then, it is easy to argue that, for any k

dA1:A2:⋯:AnΦ-dAi1:Ai2:⋯:AikΦsubscriptsuperscript𝑑Φ:subscript𝐴1subscript𝐴2:⋯:subscript𝐴𝑛subscriptsuperscript𝑑Φ:subscript𝐴subscript𝑖1subscript𝐴subscript𝑖2:⋯:subscript𝐴subscript𝑖𝑘displaystyle d^Phi_A_1:A_2:cdots:A_n-d^Phi_A_i_1:A_i_2% :cdots:A_i_kitalic_d start_POSTSUPERSCRIPT roman_Φ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_d start_POSTSUPERSCRIPT roman_Φ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT (22)

=displaystyle== SA1A2⋯An∥Ai1Ai2⋯Aik⊗Aj1⊗Aj2⊗⋯⊗Ajn-ksubscript𝑆conditionalsubscript𝐴1subscript𝐴2⋯subscript𝐴𝑛tensor-productsubscript𝐴subscript𝑖1subscript𝐴subscript𝑖2⋯subscript𝐴subscript𝑖𝑘subscript𝐴subscript𝑗1subscript𝐴subscript𝑗2⋯subscript𝐴subscript𝑗𝑛𝑘displaystyle S_A_i_1A_i_2cdots A_i_k% otimesA_j_1otimesA_j_2otimescdotsotimesA_j_n-kitalic_S start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ ⋯ ⊗ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_n - italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT

-SA1A2⋯An∥Ai1Ai2⋯Aik⊗Aj1⊗Aj2⊗⋯⊗Ajn-k′.subscriptsuperscript𝑆′conditionalsubscript𝐴1subscript𝐴2⋯subscript𝐴𝑛tensor-productsubscript𝐴subscript𝑖1subscript𝐴subscript𝑖2⋯subscript𝐴subscript𝑖𝑘subscript𝐴subscript𝑗1subscript𝐴subscript𝑗2⋯subscript𝐴subscript𝑗𝑛𝑘displaystyle-S^prime_A_1A_2cdots A_n.- italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋯ italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ ⋯ ⊗ italic_A start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_n - italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT .
where js≠itsubscript𝑗𝑠subscript𝑖𝑡j_s
eq i_titalic_j start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ≠ italic_i start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT, 1≤t≤k1𝑡𝑘1leq tleq k1 ≤ italic_t ≤ italic_k, 1≤j≤n-k1𝑗𝑛𝑘1leq jleq n-k1 ≤ italic_j ≤ italic_n - italic_k. With these notations in mind, we can easily conclude the following trade-off relations.

Proposition 7.

Using the notations as Eq. (22), if dA1:A2:⋯:AnΦ≥dAi1:Ai2:⋯:AikΦsubscriptsuperscript𝑑normal-Φnormal-:subscript𝐴1subscript𝐴2normal-:normal-⋯normal-:subscript𝐴𝑛subscriptsuperscript𝑑normal-Φnormal-:subscript𝐴subscript𝑖1subscript𝐴subscript𝑖2normal-:normal-⋯normal-:subscript𝐴subscript𝑖𝑘d^Phi_A_1:A_2:cdots:A_ngeq d^Phi_A_i_1:A_i_2:cdots:A% _i_kitalic_d start_POSTSUPERSCRIPT roman_Φ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≥ italic_d start_POSTSUPERSCRIPT roman_Φ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT for any k

Next, we consider the monogamy of DA1:A2:⋯:Ansubscript𝐷:subscript𝐴1subscript𝐴2:⋯:subscript𝐴𝑛D_A_1:A_2:cdots:A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Take a three qubit state ρABC=(1/2)(|000⟩⟨000|+|1+1⟩⟨1+1|)subscript𝜌𝐴𝐵𝐶12ket000bra000ket11bra11rho_ABC=(1/2)(|000ranglelangle 000|+|1+1ranglelangle 1+1|)italic_ρ start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT = ( 1 / 2 ) ( | 000 ⟩ ⟨ 000 | + | 1 + 1 ⟩ ⟨ 1 + 1 | ) as in Ref. [45], where |+⟩=(|0⟩+|1⟩)/2ketket0ket12|+rangle=(|0rangle+|1rangle)/sqrt2| + ⟩ = ( | 0 ⟩ + | 1 ⟩ ) / square-root start_ARG 2 end_ARG. For this state, it is shown in Ref. [45] that DA:B:C=DA:B≈0.204subscript𝐷:𝐴𝐵:𝐶subscript𝐷:𝐴𝐵0.204D_A:B:C=D_A:Bapprox 0.204italic_D start_POSTSUBSCRIPT italic_A : italic_B : italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A : italic_B end_POSTSUBSCRIPT ≈ 0.204 and DA:C=0subscript𝐷:𝐴𝐶0D_A:C=0italic_D start_POSTSUBSCRIPT italic_A : italic_C end_POSTSUBSCRIPT = 0. It is easy to see that DB:C=DA:Bsubscript𝐷:𝐵𝐶subscript𝐷:𝐴𝐵D_B:C=D_A:Bitalic_D start_POSTSUBSCRIPT italic_B : italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A : italic_B end_POSTSUBSCRIPT. That is, DA:B:C=DA:Bsubscript𝐷:𝐴𝐵:𝐶subscript𝐷:𝐴𝐵D_A:B:C=D_A:Bitalic_D start_POSTSUBSCRIPT italic_A : italic_B : italic_C end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_A : italic_B end_POSTSUBSCRIPT but DB:C≠0subscript𝐷:𝐵𝐶0D_B:C
eq 0italic_D start_POSTSUBSCRIPT italic_B : italic_C end_POSTSUBSCRIPT ≠ 0. We thus obtain the following theorem.

DA1:⋯:Ansubscript𝐷:subscript𝐴1⋯:subscript𝐴𝑛D_A_1:cdots:A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is not completely monogamous.

Comparing with the multipartite quantum discord, we find that the multipartite quantum discord DA1;A2;⋯;Ansubscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛D_A_1;A_2;cdots;A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is better than the global quantum disocrd DA1:A2:⋯:Ansubscript𝐷:subscript𝐴1subscript𝐴2:⋯:subscript𝐴𝑛D_A_1:A_2:cdots:A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT in the sense of the complete monogamy framework. Of course that, DA1:A2:⋯:Ansubscript𝐷:subscript𝐴1subscript𝐴2:⋯:subscript𝐴𝑛D_A_1:A_2:cdots:A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : ⋯ : italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT has some merit, e.g., it is symmetric but DA1;A2;⋯;Ansubscript𝐷subscript𝐴1subscript𝐴2⋯subscript𝐴𝑛D_A_1;A_2;cdots;A_nitalic_D start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; ⋯ ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT is not symmetric.

VI Conclusions and discussions

In this paper we discussed the monogamy relation of the multipartite quantum discord and the global quantum discord in detail. Different from the monogamy scenario discussed in the previous literatures, we developed a complete monogamy framework for multipartite quantity associated with local measurement. In such a frame, to characterize the distribution of quantum discord, the first issue is to check whether the quantity is monotonic under loss of subsystems and the other issue it to investigate the dis-correlate condition. The dis-correlate condition of quantum discord is the counterpart to the disentanglement condition in entanglement measure. With the assumption of monotonicity under loss of subsystems, we proved that the multipartite quantum discord is completely monogamous while the global quantum discord is not. This fact also supports that the multipartite quantum discord in Ref. [16] is an excellent generalization of quantum discord.

Also note that our assumptions are reasonable at least intuitively since the monotonicity is expected to be valid under the same local measurement on the state and its reduced state (as we have discussed in Sec. IV, the monotonicity is not valid in general whenever the local measurements on the state and its reduced state are different). We conjecture that theses assumptions are true which seems difficult to prove and remains further investigation in the future.

Acknowledgements.
This work is supported by the National Natural Science Foundation of China under Grant No. 11971277, the Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi, the Scientific Innovation Foundation of the Higher Education Institutions of Shanxi Province under Grant Nos. 2019KJ034, 2019L0742, and 2020L0471, and the Science Technology Plan Project of Datong City, China under Grant Nos. 2018151 and 2020153.

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