Online note taking app - Notes.ionotes

Fast | Easy | Short

Online Note Services - notes.io

On The Problem Of Quantifying Quantum Correlations With Non-commutative Discord
In this work we analyze a non-commutativity measure of quantum correlations recently proposed by Y. Guo [Sci. Rep. 6, 25241 (2016)]. By recourse to a systematic survey of a two-qubit system, we detected an undesirable behavior of such a measure related to its representation-dependence. In the case of pure states, this dependence manifests as a non-satisfactory entanglement measure whenever a representation other than the Schmidt’s is used. In order to avoid this dependence on the basis, we argue that a minimization procedure over the set of all possible representations of the quantum state is required. In the case of pure states, this minimization can be analytically performed and the optimal basis turns out to be that of Schmidt’s. In addition, the resulting measure inherits the main properties of Guo’s measure and, unlike the latter, it reduces to a legitimate entanglement measure in the case of pure states. Some examples involving general mixed states are also analyzed considering such a optimization. The results show that, in most cases of interest, the use of Guo’s measure can result in a overestimation of quantum correlations. However, since Guo’s measure has the advantage of being easily computable, it might be used as a qualitative estimator of the presence of quantum correlations.

Quantum Information Theory (QIT) is concerned with the use of quantum resources to perform tasks of information processing which are either not feasible to be implemented classically or can be performed with classical devices in a way much less efficient. The fact that in most cases quantum protocols can outperform their classical counterparts (if such a thing is feasible to be done) is generally attributed to the existence of quantum correlations (QCs). For a long time, QCs were associated with the existence of entanglement in a composite quantum system, a feature that according to Schrödinger himself is “the characteristic trait of quantum mechanics” Schrod35a ; Schrod35b , and which has been extensively studied in connection with Bell’s inequalities Bell . On one hand, entangled states violating Bell’s inequalities Bell contain ‘nonlocal’ features which were initially considered as the necessary quantum resource to achieve a computational speedup over the best classical algorithm Nielsen&Chuang . On the other hand, since (mixed) separable states do not violate Bell’s inequalities and can be prepared by local operations and classical communication (LOCC), until very recently they were considered as purely classical and, in consequence, useless for tasks of quantum information processing. However, further research has provided a great amount of evidence supporting the idea that this is not the case Knill98 ; Braun99 ; Meyer00 ; Datta05 ; Datta07 ; Datta08 ; Lanyon08 . As a consequence, the study of entanglement measures was extended in order to include the quantification of more general quantum correlations. One of the widely used measure of quantum correlations in bipartite systems is the so-called (standard) quantum discord (QD) OZ02 ; HV01 . In few words, QD quantifies the discrepancy between the quantum versions of two classically equivalent expressions for mutual information. Even though, from a conceptual point of view, QD is of relevance in assessing possible non-classical resources for information processing, for a practical use it present some drawbacks. For example, at this moment, there is no straightforward criterion to verify the presence of discord in a given quantum state. As the evaluation of QD involves an optimization procedure, analytical results are known only in a few cases Luo08b ; Lang10 ; Cen11 ; Adesso10 ; Ali10 ; Shi11 ; Chen11 ; Lu11 ; Giro12 ; Li11 . Furthermore, in general, calculation of quantum discord is NP-complete since the optimization procedure needs to be done sweeping a complete set of measurements over one of the subsystems Huang14 . With the aim of finding a measure of QCs easier to calculate, several alternative measures to QD have been proposed DVB10 ; Brod10 ; Spehner14a ; Spehner14b ; Jakob14 ; Paula13 . For example, we can mention discord-like quantities Brod10 , geometric measures to quantify QD DVB10 ; LuoGD10 , and a measure based on Bures distance Spehner14a ; Spehner14b , among others. Unfortunately, alternative measures of QCs (if not ill-defined) become difficult to calculate since they also involve an optimization process either in a minimization or in a maximization scenario. Additionally, in some cases, undesired behaviors of the measures reduces the potential of their applicability. Furthermore, the great number of measures currently found in literature difficult the progress in the study of their properties in order to assure they provide trustful measures of QCs. Thus, at present, there is not a general agreement about which measure of QCs is the most suitable to be used in a practical way in an arbitrary composite quantum system. Hence, extreme caution should be exercised in devising new practical methods to quantify QCs in order to avoid undesired behaviors and subtleties in their properties. In summary, it seems that a careful examination of the properties of any new promising measure of QCs should be carefully addressed. Following this last direction, in this work we investigate various features of a non-commutativity measure of QCs, introduced by Guo in a recent work Guo16 . In that work, Guo introduced two QCs measures in terms of the non-commutativity of some operators quantified by the trace norm and the Hilbert-Schmidt norm. According to Guo16 , the non-commutative quantum discord (NCQD) measures can be computed directly for any arbitrary state without requiring any previous optimization procedure, as is the case with usual discord. In this work we show that, indeed, the NCQD measures have the drawback of depending upon the representation of the state, and suggest a new measure to overcome this undesirable feature. The paper is organized as follows. In section II, we review the definition and main properties of the NCQD measures. In section III, we discuss the representation-dependence resorting to computational and Schmidt representations of pure states, and propose a new measure that is representation-independent, and extends to the general (mixed, d𝑑ditalic_d-dimensional) case. In section IV, we calculate the new measure for some typical examples, comparing it with NCQD measure introduced by Guo. Finally, some conclusions are drawn in section V.

II Non-commutativity measure of quantum correlations

Let us consider a bipartite system (A+B𝐴𝐵A+Bitalic_A + italic_B) in an arbitrary quantum state ρ𝜌rhoitalic_ρ, defined on the Hilbert space ℋ=ℋA⊗ℋBℋtensor-productsubscriptℋ𝐴subscriptℋ𝐵mathcalH=mathcalH_AotimesmathcalH_Bcaligraphic_H = caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ caligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. If iA⟩ketsubscript𝑖𝐴 italic_i start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ stands for an orthonormal basis of ℋAsubscriptℋ𝐴mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, then ρ𝜌rhoitalic_ρ can be represented by

ρ=∑i,j|iA⟩⟨jA|⊗Bij,𝜌subscript𝑖𝑗tensor-productketsubscript𝑖𝐴brasubscript𝑗𝐴subscript𝐵𝑖𝑗rho=sum_i,j|i_Aranglelangle j_A|otimes B_ij,italic_ρ = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT | italic_i start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ ⟨ italic_j start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | ⊗ italic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , (1)

where Bij=TrA[(|jA⟩⟨iA|⊗𝕀B)ρ]subscript𝐵𝑖𝑗subscriptTr𝐴delimited-[]tensor-productketsubscript𝑗𝐴brasubscript𝑖𝐴subscript𝕀𝐵𝜌B_ij=mathrmTr_A[(|j_Aranglelangle i_A|otimesmathbbI_B)rho]italic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT [ ( | italic_j start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ ⟨ italic_i start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | ⊗ blackboard_I start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ ] or, equivalently, Bij=⟨iA|ρ|jA⟩.subscript𝐵𝑖𝑗quantum-operator-productsubscript𝑖𝐴𝜌subscript𝑗𝐴B_ij=langle i_A|rho|j_Arangle.italic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = ⟨ italic_i start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_ρ | italic_j start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ .

With the operators Bijsubscript𝐵𝑖𝑗B_ijitalic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT just defined, Guo Guo16 introduced two non-commutativity measures as follows:

DG(ρ):=∑Ω||[Bij,Bkl]||Tr,assignsubscript𝐷𝐺𝜌subscriptΩsubscriptnormsubscript𝐵𝑖𝑗subscript𝐵𝑘𝑙TrD_G(rho):=sum_Omega||[B_ij,B_kl]||_mathrmTr,italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) := ∑ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT | | [ italic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ] | | start_POSTSUBSCRIPT roman_Tr end_POSTSUBSCRIPT , (2)

DG′(ρ):=∑Ω||[Bij,Bkl]||2,assignsubscriptsuperscript𝐷′𝐺𝜌subscriptΩsubscriptnormsubscript𝐵𝑖𝑗subscript𝐵𝑘𝑙2D^prime_G(rho):=sum_Omega||[B_ij,B_kl]||_2,italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) := ∑ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT | | [ italic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT ] | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (3)

where ΩΩOmegaroman_Ω represents the set of all the possible pairs (regardless of the order), and ||⋅||Trfragments||⋅|subscript|Tr||cdot||_mathrmTr| | ⋅ | | start_POSTSUBSCRIPT roman_Tr end_POSTSUBSCRIPT and ||⋅||2fragments||⋅|subscript|2||cdot||_2| | ⋅ | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT denote, respectively, the trace and the Hilbert-Schmidt norm, i.e. ||A||Tr=TrA†Asubscriptnorm𝐴TrTrsuperscript𝐴†𝐴||A||_mathrmTr=mathrmTrsqrtA^daggerA| | italic_A | | start_POSTSUBSCRIPT roman_Tr end_POSTSUBSCRIPT = roman_Tr square-root start_ARG italic_A start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_A end_ARG, and ||A||2=Tr(A†A)subscriptnorm𝐴2Trsuperscript𝐴†𝐴||A||_2=sqrtmathrmTr(A^daggerA)| | italic_A | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = square-root start_ARG roman_Tr ( italic_A start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_A ) end_ARG.

Resorting to the fact that (usual) quantum discord D(ρ)𝐷𝜌D(rho)italic_D ( italic_ρ ) vanishes if and only if all the (B𝐵Bitalic_B)-local operators Bijsubscript𝐵𝑖𝑗B_ijitalic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT are mutually commuting normal operators DVB10 ; GH12 , Guo proposes the non-commuting measures (2) as measures of quantum discord. As explained in Guo16 , DG(ρ)subscript𝐷𝐺𝜌D_G(rho)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) and DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) satisfy the following properties: (i) DG(ρ)=DG′(ρ)=0subscript𝐷𝐺𝜌subscriptsuperscript𝐷′𝐺𝜌0D_G(rho)=D^prime_G(rho)=0italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) = italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) = 0 iff D(ρ)=0𝐷𝜌0D(rho)=0italic_D ( italic_ρ ) = 0; (ii) D(ρ)𝐷𝜌D(rho)italic_D ( italic_ρ ), DG(ρ)subscript𝐷𝐺𝜌D_G(rho)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) and DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) are invariant under local unitary operations, i.e., DG(ρ)=DG(UA⊗UBρUA†⊗UB†)subscript𝐷𝐺𝜌subscript𝐷𝐺tensor-producttensor-productsubscript𝑈𝐴subscript𝑈𝐵𝜌superscriptsubscript𝑈𝐴†superscriptsubscript𝑈𝐵†D_G(rho)=D_G(U_Aotimes U_Brho U_A^daggerotimes U_B^dagger)italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) = italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) and DG′(ρ)=DG′(UA⊗UBρUA†⊗UB†)subscriptsuperscript𝐷′𝐺𝜌subscriptsuperscript𝐷′𝐺tensor-producttensor-productsubscript𝑈𝐴subscript𝑈𝐵𝜌superscriptsubscript𝑈𝐴†superscriptsubscript𝑈𝐵†D^prime_G(rho)=D^prime_G(U_Aotimes U_Brho U_A^dagger% otimes U_B^dagger)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) = italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ). Unlike the usual quantum discord and other measures of non-classicality, according to Guo Guo16 , the proposed measures are not based on measurements performed on one of the subsystems, and can be computed directly for any arbitrary state without requiring any previous optimization procedure as usual discord does. Note however that the Hilbert-Schmidt norm is easier to calculate (compared with the trace norm), hence from now on we will focus on the Hilbert-Schmidt norm measure, DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ).

III Pure states. Representation-dependence of the noncommutativity measure

From Eq. (1) it follows that the operators Bijsubscript𝐵𝑖𝑗B_ijitalic_B start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT can be identified with blocks of the matrix ρ𝜌rhoitalic_ρ. In a two-qubit system, for example,

ρ=(B00B01B10B11),𝜌matrixsubscript𝐵00subscript𝐵01subscript𝐵10subscript𝐵11rho=beginpmatrixB_00&B_01\ B_10&B_11endpmatrix,italic_ρ = ( start_ARG start_ROW start_CELL italic_B start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_CELL start_CELL italic_B start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_B start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL start_CELL italic_B start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , (4)

where 0 and 1 denote each of the two (orthonormal) basis vectors iA⟩ketsubscript𝑖𝐴 , put in correspondence with the canonical basis |ϵ0⟩=(1,0)𝑇ketsubscriptitalic-ϵ0superscript10𝑇left|epsilon_0rightrangle=(1,0)^textitT| italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟩ = ( 1 , 0 ) start_POSTSUPERSCRIPT T end_POSTSUPERSCRIPT, and |ϵ1⟩=(0,1)𝑇ketsubscriptitalic-ϵ1superscript01𝑇left|epsilon_1rightrangle=(0,1)^textitT| italic_ϵ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ = ( 0 , 1 ) start_POSTSUPERSCRIPT T end_POSTSUPERSCRIPT. Now, let us consider a two-qubit system in an arbitrary pure state

|ψ⟩=a|00⟩+b|01⟩+c|10⟩+d|11⟩,ket𝜓𝑎ket00𝑏ket01𝑐ket10𝑑ket11|psirangle=a|00rangle+b|01rangle+c|10rangle+d|11rangle,| italic_ψ ⟩ = italic_a | 00 ⟩ + italic_b | 01 ⟩ + italic_c | 10 ⟩ + italic_d | 11 ⟩ , (5)

where a𝑎aitalic_a, b𝑏bitalic_b, c𝑐citalic_c, and d𝑑ditalic_d are complex numbers that satisfy |a|2+|b|2+|c|2+|d|2=1superscript𝑎2superscript𝑏2superscript𝑐2superscript𝑑21|a|^2+|b|^2+|c|^2+|d|^2=1| italic_a | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_b | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_c | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_d | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1. The density matrix ρ𝜌rhoitalic_ρ is thus given by Eq. (4), with

B00=(|a|2ab*a*b|b|2),subscript𝐵00matrixsuperscript𝑎2𝑎superscript𝑏superscript𝑎𝑏superscript𝑏2B_00=beginpmatrix|a|^2&ab^*\ a^*b&|b|^2endpmatrix,italic_B start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL | italic_a | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL italic_a italic_b start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_b end_CELL start_CELL | italic_b | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ) , (6)

B01=(ac*ad*bc*bd*),subscript𝐵01matrix𝑎superscript𝑐𝑎superscript𝑑𝑏superscript𝑐𝑏superscript𝑑B_01=beginpmatrixac^*&ad^*\ bc^*&bd^*endpmatrix,italic_B start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_a italic_c start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL italic_a italic_d start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_b italic_c start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL italic_b italic_d start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ) , (7)

B10=(a*cb*ca*db*d),subscript𝐵10matrixsuperscript𝑎𝑐superscript𝑏𝑐superscript𝑎𝑑superscript𝑏𝑑B_10=beginpmatrixa^*c&b^*c\ a^*d&b^*dendpmatrix,italic_B start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_c end_CELL start_CELL italic_b start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_c end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_d end_CELL start_CELL italic_b start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_d end_CELL end_ROW end_ARG ) , (8)

B11=(|c|2cd*c*d|d|2).subscript𝐵11matrixsuperscript𝑐2𝑐superscript𝑑superscript𝑐𝑑superscript𝑑2B_11=beginpmatrix|c|^2&cd^*\ c^*d&|d|^2endpmatrix.italic_B start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL | italic_c | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL italic_c italic_d start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_d end_CELL start_CELL | italic_d | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ) . (9)

After a direct calculation of the six commutators [B00,B01]subscript𝐵00subscript𝐵01[B_00,B_01][ italic_B start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ], [B00,B10]subscript𝐵00subscript𝐵10[B_00,B_10][ italic_B start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ], [B00,B11]subscript𝐵00subscript𝐵11[B_00,B_11][ italic_B start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ], [B01,B10]subscript𝐵01subscript𝐵10[B_01,B_10][ italic_B start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT ], [B01,B11]subscript𝐵01subscript𝐵11[B_01,B_11][ italic_B start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ], and [B10,B11]subscript𝐵10subscript𝐵11[B_10,B_11][ italic_B start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ], the measure DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) can be written as

DG′(ρ)=C[1+122(C2+4|(ρA)01|2+2|(ρA)01|)],subscriptsuperscript𝐷′𝐺𝜌𝐶delimited-[]1122superscript𝐶24superscriptsubscriptsubscript𝜌𝐴0122subscriptsubscript𝜌𝐴01D^prime_G(rho)=CBig[1+frac12sqrt2Big(sqrtC^2+4+2|(rho_A)_01|Big)Big],italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) = italic_C [ 1 + divide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 2 end_ARG end_ARG ( square-root start_ARG italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 | ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + 2 | ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT | ) ] , (10)

where ρA=TrBρsubscript𝜌𝐴subscriptTr𝐵𝜌rho_A=textrmTr_Brhoitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ is the (reduced) density matrix corresponding to subsystem A𝐴Aitalic_A, so that |(ρA)01|=|a*c+b*d|subscriptsubscript𝜌𝐴01superscript𝑎𝑐superscript𝑏𝑑|(rho_A)_01|=|a^*c+b^*d|| ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT | = | italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_c + italic_b start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_d | represents a measure of the coherence of ρAsubscript𝜌𝐴rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, and C=|ad-bc|𝐶𝑎𝑑𝑏𝑐C=|ad-bc|italic_C = | italic_a italic_d - italic_b italic_c | stands for Wooter’s concurrence W98 , which is a measure of the entanglement between A𝐴Aitalic_A and B𝐵Bitalic_B.

The fact that DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) depends upon a parameter related to ρAsubscript𝜌𝐴rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ensues from the fact that ρ𝜌rhoitalic_ρ has been decomposed in the form (1), associated with the bipartition A|Bconditional𝐴𝐵A|Bitalic_A | italic_B. The bipartition B|Aconditional𝐵𝐴B|Aitalic_B | italic_A corresponds to the decomposition [cf. Eq. (1)]

ρ=∑i,jAij⊗|iB⟩⟨jB|,𝜌subscript𝑖𝑗tensor-productsubscript𝐴𝑖𝑗ketsubscript𝑖𝐵brasubscript𝑗𝐵rho=sum_i,jA_ijotimes|i_Branglelangle j_B|,italic_ρ = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ⊗ | italic_i start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⟩ ⟨ italic_j start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT | , (11)

where Aij=TrB[(𝕀A⊗|jB⟩⟨iB|)ρ]subscript𝐴𝑖𝑗subscriptTr𝐵delimited-[]tensor-productsubscript𝕀𝐴ketsubscript𝑗𝐵brasubscript𝑖𝐵𝜌A_ij=mathrmTr_B[(mathbbI_Aotimes|j_Branglelangle i_B|)rho]italic_A start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ ( blackboard_I start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_j start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⟩ ⟨ italic_i start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT | ) italic_ρ ] or, equivalently, Aij=⟨iB|ρ|jB⟩subscript𝐴𝑖𝑗quantum-operator-productsubscript𝑖𝐵𝜌subscript𝑗𝐵A_ij=langle i_B|rho|j_Brangleitalic_A start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = ⟨ italic_i start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT | italic_ρ | italic_j start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⟩. If instead of considering the bipartition A|Bconditional𝐴𝐵A|Bitalic_A | italic_B we consider the bipartition B|Aconditional𝐵𝐴B|Aitalic_B | italic_A, the term (ρA)01subscriptsubscript𝜌𝐴01(rho_A)_01( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT in Eq. (10) must be replaced by (ρB)01=|a*b+c*d|subscriptsubscript𝜌𝐵01superscript𝑎𝑏superscript𝑐𝑑(rho_B)_01=|a^*b+c^*d|( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = | italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_b + italic_c start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_d |, i.e., the coherence of the reduced density matrix ρBsubscript𝜌𝐵rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. This means that the correlation between A𝐴Aitalic_A and B𝐵Bitalic_B and the correlation between B𝐵Bitalic_B and A𝐴Aitalic_A, as measured by DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ), do not coincide in general. This is an undesirable feature since, even though discord is known to be a non-symmetric measure of quantum correlations, for pure states -as the one considered here- it should reduce to an entanglement measure, which should be symmetric under the exchange A↔B↔𝐴𝐵Aleftrightarrow Bitalic_A ↔ italic_B (e.g., Wootters’ concurrence C𝐶Citalic_C). On the other hand, the coherence term present in Eq. (10), unlike the concurrence C𝐶Citalic_C, depends upon the specific representation of ρ𝜌rhoitalic_ρ. Thus, DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) becomes representation-dependent, which is another undesirable property for an entanglement measure. Therefore, we are led to conclude that for pure states DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) does not reduce to a good entanglement measure (i.e., symmetric and representation-independent). This fact puts at stake its adequacy when dealing with general (mixed) states. In what follows we discuss how these disadvantages can be surmounted.

According to the aforementioned observations, it is precisely the coherence term appearing in Eq. (10) what introduces the inconvenient properties in DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ). Thus, as a first step, we require that (ρA)01subscriptsubscript𝜌𝐴01(rho_A)_01( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT and (ρB)01subscriptsubscript𝜌𝐵01(rho_B)_01( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT shall reduce to zero for all ρ=ρ2𝜌superscript𝜌2rho=rho^2italic_ρ = italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. This condition is met only when both ρAsubscript𝜌𝐴rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and ρBsubscript𝜌𝐵rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT are both diagonal, which holds irrespective of the state whenever |ψ⟩ket𝜓left|psirightrangle| italic_ψ ⟩ is decomposed into its Schmidt form

|ψ⟩=∑n=0,1λn|vnA⟩⊗|unB⟩,ket𝜓subscript𝑛01tensor-productsubscript𝜆𝑛ketsubscriptsuperscript𝑣𝐴𝑛ketsubscriptsuperscript𝑢𝐵𝑛|psirangle=sum_n=0,1sqrtlambda_n|v^A_nrangleotimes|u^B_nrangle,| italic_ψ ⟩ = ∑ start_POSTSUBSCRIPT italic_n = 0 , 1 end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | italic_v start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ ⊗ | italic_u start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ , (12)

where λnsubscript𝜆𝑛lambda_nitalic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT stands for the eigenvalues of ρAsubscript𝜌𝐴rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and ρBsubscript𝜌𝐵rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, so that λ0+λ1=1subscript𝜆0subscript𝜆11lambda_0+lambda_1=1italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1, and vnA⟩ketsubscriptsuperscript𝑣𝐴𝑛 italic_v start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ , ketsubscriptsuperscript𝑢𝐵𝑛u^B_nrangle italic_u start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ are the corresponding (orthonormal) eigenvectors. Thus, the measure DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT reduces (in the Schmidt representation) to

DG′(ρSch)=2λ0λ1+2λ0λ1,subscriptsuperscript𝐷′𝐺subscript𝜌Sch2subscript𝜆0subscript𝜆12subscript𝜆0subscript𝜆1D^prime_G(rho_textrmSch)=2sqrtlambda_0lambda_1+sqrt2% lambda_0lambda_1,italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT Sch end_POSTSUBSCRIPT ) = 2 square-root start_ARG italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + square-root start_ARG 2 end_ARG italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (13)

in agreement with Guo’s result. Using the fact that C2=4λ0λ1superscript𝐶24subscript𝜆0subscript𝜆1C^2=4lambda_0lambda_1italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 4 italic_λ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we get

DG′(ρSch)=DG′(ρ)|(ρA)01=0=C(1+122C),subscriptsuperscript𝐷′𝐺subscript𝜌Schevaluated-atsubscriptsuperscript𝐷′𝐺𝜌subscriptsubscript𝜌𝐴010𝐶1122𝐶D^prime_G(rho_textrmSch)=D^prime_G(rho)|_(rho_A)_01=0=% Cleft(1+frac12sqrt2Cright),italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT Sch end_POSTSUBSCRIPT ) = italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) | start_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT = italic_C ( 1 + divide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 2 end_ARG end_ARG italic_C ) , (14)

as follows from Eq. (10) with (ρA)01=0subscriptsubscript𝜌𝐴010(rho_A)_01=0( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = 0. Thus, resorting to the Schmidt representation (the only one considered in Guo16 ), we see that DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) reduces to a monotonic function of concurrence -hence to a measure of entanglement-, whose maximum value is attained for maximally entangled states (i.e., C=1𝐶1C=1italic_C = 1). However, in any other representation this ceases to be the case (of course, this is due to the coherences aforementioned). In addition, it is straightforward to see that Eq. (10) (or the analogous equation corresponding to the bipartition B|Aconditional𝐵𝐴B|Aitalic_B | italic_A) is minimized, with C𝐶Citalic_C fixed, for (ρA)01=0subscriptsubscript𝜌𝐴010(rho_A)_01=0( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = 0 (or (ρB)01=0subscriptsubscript𝜌𝐵010(rho_B)_01=0( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = 0). This means that Eq. (10) attains its minimum (and symmetrical) value whenever ρ𝜌rhoitalic_ρ is expressed in its Schmidt representation, whence

DG′(ρSch)=minDG′(ρ),subscriptsuperscript𝐷′𝐺subscript𝜌Schsubscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho_textrmSch)=min D^prime_G(rho),italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT Sch end_POSTSUBSCRIPT ) = roman_min italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) , (15)

where the minimum is taken over all decompositions of |ψ⟩ket𝜓left|psirightrangle| italic_ψ ⟩. Consequently, the quantity

d′(ρ):=minℛDG′(ρ),assignsuperscript𝑑′𝜌subscriptℛsubscriptsuperscript𝐷′𝐺𝜌d^prime(rho):=min_calRD^prime_G(rho),italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ ) := roman_min start_POSTSUBSCRIPT caligraphic_R end_POSTSUBSCRIPT italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) , (16)

being ℛℛcalRcaligraphic_R the set of all possible representations of ρ𝜌rhoitalic_ρ, constitutes a non-commutativity measure that inherits the main properties of Guo’s measure DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT but, unlike the latter, reduces to a legitimate entanglement measure for pure states. Notice that d′(ρ)superscript𝑑′𝜌d^prime(rho)italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ ) applies for general (mixed and pure) states of bipartite systems of arbitrary dimensions. However, this measure requires a minimization procedure -as is the case for usual discord - that makes it difficult to calculate for general mixed states. In the next section we analyze some examples involving the evaluation of d′(ρ)superscript𝑑′𝜌d^prime(rho)italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ ) in the case of mixed states.

In Fig. 1 we show the effect of the representation-dependence of DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT. We generated 106superscript10610^610 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT two-qubit random pure states distributed according to the Haar measure ZHS98 ; PZK98 and computed the DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) using both, the computational and the Schmidt representations. We plotted the DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT as a function of concurrence C𝐶Citalic_C. Notice that, as expected, the values obtained in the Schmidt representation (purple squares) are in all cases lower than those corresponding (for the same state) to the computational representation (orange dots). The maximum value of DG′(ρSch)subscriptsuperscript𝐷′𝐺subscript𝜌SchD^prime_G(rho_textrmSch)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT Sch end_POSTSUBSCRIPT ) is 1.3535 and corresponds to states with C=1𝐶1C=1italic_C = 1, though DG′(ρComp)subscriptsuperscript𝐷′𝐺subscript𝜌CompD^prime_G(rho_textrmComp)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT Comp end_POSTSUBSCRIPT ) attains its maximum value (1.3964) not for a maximally entangled state, but for a state with concurrence C=0.9725𝐶0.9725C=0.9725italic_C = 0.9725.

Finally, notice that for maximally entangled states |ψ⟩ket𝜓left|psirightrangle| italic_ψ ⟩, the reduced density matrices ρAsubscript𝜌𝐴rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and ρBsubscript𝜌𝐵rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT do coincide, and the coherence terms reduce to zero. Hence, for maximally entangled pure states any decomposition will display the same value of DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ).

IV Representation-independent measure. Some examples for mixed states.

In Ref. Guo16 , DGsubscript𝐷𝐺D_Gitalic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT and DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT were computed and compared with the usual discord for several families of mixed states, namely Werner, isotropic, and Bell-diagonal states. Here we will focus our discussion on general (mixed) states of two qubits. In what follows, we briefly discuss how DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) depends upon the representation, and compare it with the measure d′(ρ)superscript𝑑′𝜌d^prime(rho)italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ ) introduced earlier in Sec. III [cf. Eq. (16)].

Let |iA′⟩=UA|iA⟩ketsubscriptsuperscript𝑖′𝐴subscript𝑈𝐴ketsubscript𝑖𝐴|i^prime_Arangle=U_A|i_Arangle| italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ = italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_i start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ be an arbitrary orthonormal basis of ℋ𝒜subscriptℋ𝒜calH_Acaligraphic_H start_POSTSUBSCRIPT caligraphic_A end_POSTSUBSCRIPT. Then, the state ρ𝜌rhoitalic_ρ can also be represented as [cf. Eq. (1)]

ρ=∑ij|iA′⟩⟨jA′|⊗Bij′.𝜌subscript𝑖𝑗tensor-productketsubscriptsuperscript𝑖′𝐴brasubscriptsuperscript𝑗′𝐴subscriptsuperscript𝐵′𝑖𝑗rho=sum_ij|i^prime_Aranglelangle j^prime_A|otimes B^prime_% ij.italic_ρ = ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ ⟨ italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | ⊗ italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT . (17)

In this new representation, the operators Bij′subscriptsuperscript𝐵′𝑖𝑗B^prime_ijitalic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT take the form

Bij′subscriptsuperscript𝐵′𝑖𝑗displaystyle B^prime_ijitalic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT =displaystyle== TrAjA′⟩⟨iA′subscriptTr𝐴tensor-productketsubscriptsuperscript𝑗′𝐴brasubscriptsuperscript𝑖′𝐴subscript𝕀𝐵𝜌displaystyletextrmTr_Aj^prime_Aranglelangle i^prime_ATr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT (18)

=displaystyle== ⟨iA′|ρ|jA′⟩=⟨iA|UA†ρUA|jA⟩,quantum-operator-productsubscriptsuperscript𝑖′𝐴𝜌subscriptsuperscript𝑗′𝐴quantum-operator-productsubscript𝑖𝐴superscriptsubscript𝑈𝐴†𝜌subscript𝑈𝐴subscript𝑗𝐴displaystylelangle i^prime_A|rho|j^prime_Arangle=langle i_A|U% _A^daggerrho U_A|j_Arangle,⟨ italic_i start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_ρ | italic_j start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ = ⟨ italic_i start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_ρ italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_j start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ ,

hence, the Bij′subscriptsuperscript𝐵′𝑖𝑗B^prime_ijitalic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT can be identified with the block components of the matrix [cf. Eq. (4)]

UA†ρUA=(B00′B01′B10′B11′).superscriptsubscript𝑈𝐴†𝜌subscript𝑈𝐴matrixsubscriptsuperscript𝐵′00subscriptsuperscript𝐵′01subscriptsuperscript𝐵′10subscriptsuperscript𝐵′11displaystyle U_A^daggerrho U_A=beginpmatrixB^prime_00&B^% prime_01\ B^prime_10&B^prime_11endpmatrix.italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_ρ italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_CELL start_CELL italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL start_CELL italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) . (21)

Now, with these primed operators we can compute DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT in the new representation. The optimization procedure in (16) is then reduced to search the minimum of DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT over the set of all possible basis of ℋ𝒜subscriptℋ𝒜calH_Acaligraphic_H start_POSTSUBSCRIPT caligraphic_A end_POSTSUBSCRIPT, that is, over all transformations belonging to SU(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 ) BZ06 .

In order to be more general than in Guo16 we will consider a mixed state ρ𝜌rhoitalic_ρ of the form

ρ=(1-p)𝕀4+p|ψ⟩⟨ψ|,𝜌1𝑝𝕀4𝑝ket𝜓bra𝜓rho=(1-p)fracmathbbI4+p|psiranglelanglepsi|,italic_ρ = ( 1 - italic_p ) divide start_ARG blackboard_I end_ARG start_ARG 4 end_ARG + italic_p | italic_ψ ⟩ ⟨ italic_ψ | , (22)

where |ψ⟩ket𝜓|psirangle| italic_ψ ⟩ is any arbitrary pure state. If |ψ⟩ket𝜓|psirangle| italic_ψ ⟩ corresponds to a Bell state, the state ρ𝜌rhoitalic_ρ becomes symmetric under the interchange of the subsystems, the measure DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT becomes symmetric in both bipartitions (A|Bconditional𝐴𝐵A|Bitalic_A | italic_B and B|Aconditional𝐵𝐴B|Aitalic_B | italic_A), and also representation-independent (see inset of Fig. 2). However, if we take for instance |ψ⟩=1/3(|00⟩+|01⟩+|10⟩)ket𝜓13ket00ket01ket10|psirangle=1/sqrt3(|00rangle+|01rangle+|10rangle)| italic_ψ ⟩ = 1 / square-root start_ARG 3 end_ARG ( | 00 ⟩ + | 01 ⟩ + | 10 ⟩ ), the measure becomes dependent upon the representation. In Fig. 2 we show this fact explicitly by plotting DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT as a function of p𝑝pitalic_p considering the computational representation (solid curve), and the measure d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT introduced in Eq. (16) (dashed curve). As p𝑝pitalic_p goes from p=0𝑝0p=0italic_p = 0 to p=1𝑝1p=1italic_p = 1 (i.e., as the state (22) goes from being a maximally mixed state to a pure state) the difference between DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT and d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT increases continuously. At p=1𝑝1p=1italic_p = 1, d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT reduces to DG′(ρSch)subscriptsuperscript𝐷′𝐺subscript𝜌SchD^prime_G(rho_textrmSch)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT Sch end_POSTSUBSCRIPT ) (orange dot), i.e., the measure considered in Ref. Guo16 for pure states. Nevertheless, DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT does not coincide with such a value, which means that unless the minimization in Eq. (16) is performed, the measure DG′(ρ)subscriptsuperscript𝐷′𝐺𝜌D^prime_G(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) will in general exhibit discontinuities when a mixed state transforms into a pure one.

(b) Colour online. DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT (solid orange line) and d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT (dotted purple line) as a function of p𝑝pitalic_p for ρ2subscript𝜌2rho_2italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. All plotted quantities are dimensionless.

Figure 3: The graphs pertain to the states given by Eqs.(23) and (24).

As a second example we will compute d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for some of the Bell-diagonal states analized in Ref. Guo16 . Specifically we will consider the states

ρ1=p|β11⟩⟨β11|+1-p2(|β01⟩⟨β01|+|β00⟩⟨β00|),subscript𝜌1𝑝ketsubscript𝛽11brasubscript𝛽111𝑝2ketsubscript𝛽01brasubscript𝛽01ketsubscript𝛽00brasubscript𝛽00rho_1=p|beta_11ranglelanglebeta_11|+frac1-p2(|beta_01% ranglelanglebeta_01|+|beta_00ranglelanglebeta_00|),italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_p | italic_β start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT | + divide start_ARG 1 - italic_p end_ARG start_ARG 2 end_ARG ( | italic_β start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT | + | italic_β start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT | ) , (23)

ρ2=p|β11⟩⟨β11|+(1-p)|β01⟩⟨β01|,subscript𝜌2𝑝ketsubscript𝛽11quantum-operator-productsubscript𝛽111𝑝subscript𝛽01brasubscript𝛽01rho_2=p|beta_11ranglelanglebeta_11|+(1-p)|beta_01ranglelangle% beta_01|,italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_p | italic_β start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT | + ( 1 - italic_p ) | italic_β start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT | , (24)

where βab⟩ketsubscript𝛽𝑎𝑏 are four Bell states |βab⟩≡12[|0,b⟩+(-1)a|1,1⊕b⟩]ketsubscript𝛽𝑎𝑏12delimited-[]ket0𝑏superscript1𝑎ket1direct-sum1𝑏|beta_abrangleequivfrac1sqrt2[|0,brangle+(-1)^a|1,1oplus brangle]| italic_β start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟩ ≡ divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG [ | 0 , italic_b ⟩ + ( - 1 ) start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT | 1 , 1 ⊕ italic_b ⟩ ].

In Figs. 3(a) and 3(b) we plotted DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT in the computational representation as in Ref. Guo16 , and d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT as a function of the parameter p𝑝pitalic_p for the states ρ1subscript𝜌1rho_1italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ρ2subscript𝜌2rho_2italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively. For both states the optimization was numerically performed on SU(2)𝑆𝑈2SU(2)italic_S italic_U ( 2 ). Here, unlike the previous example, in both cases DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT and d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT do coincide for p=1𝑝1p=1italic_p = 1, i.e., when the states become pure states. Of course, this is so because for p=1𝑝1p=1italic_p = 1 both ρ1subscript𝜌1rho_1italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ρ2subscript𝜌2rho_2italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT reduce to a maximally entangled pure states, for which all representations give the same measure (see last paragraph in Section III). However, for intermediate values of p𝑝pitalic_p, DG′(ρ1)subscriptsuperscript𝐷′𝐺subscript𝜌1D^prime_G(rho_1)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and DG′(ρ2)subscriptsuperscript𝐷′𝐺subscript𝜌2D^prime_G(rho_2)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ), overestimate the amount of QCs. For ρ1subscript𝜌1rho_1italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT the difference is quite significant and the larger discrepancy is reached at p=0.5𝑝0.5p=0.5italic_p = 0.5. Thus, DG′(ρ2)subscriptsuperscript𝐷′𝐺subscript𝜌2D^prime_G(rho_2)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) works as a tight upper bound for d′(ρ2)superscript𝑑′subscript𝜌2d^prime(rho_2)italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) and at p=0.5𝑝0.5p=0.5italic_p = 0.5 both measures do coincide. This is because for p=0.5𝑝0.5p=0.5italic_p = 0.5 ρ2subscript𝜌2rho_2italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is classically correlated (or zero discordant). Note that ρ2subscript𝜌2rho_2italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT becomes a pure state also when p=0𝑝0p=0italic_p = 0. Although DISCORD SERVER 𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT nearly coincides with d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for ρ2subscript𝜌2rho_2italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, our examples show that the measure is still representation-dependent when arbitrary mixed states are considered. At this point, it is important to realize that, given a state ρ𝜌rhoitalic_ρ represented in a given fixed basis, the measure D′(ρ)superscript𝐷′𝜌D^prime(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ ) will be invariant under local unitary operations. However, given two different representations of the state ρ𝜌rhoitalic_ρ, the two values of D′(ρ)superscript𝐷′𝜌D^prime(rho)italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_ρ ) will be in general different from each other.

In this work, we analyzed the measure DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT of quantum correlations recently proposed by Y. Guo in Ref. Guo16 . The measure DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT is based on the amount of non-commutativity quantified by the Hilbert-Schmidt norm. Our results show that, in general, the measure DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT depends upon the representation of the state. First, we focused our study on pure states and, by resorting to the computational representation, we showed that DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT is a function of both, Wootters’ concurrence C𝐶Citalic_C of the pure state and the coherence of the reduced density matrix. In addition, due to this latter dependence, DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT becomes a representation-dependent quantity which, in most cases of interest, yields different results when the bipartition A|Bconditional𝐴𝐵A|Bitalic_A | italic_B or B|Aconditional𝐵𝐴B|Aitalic_B | italic_A is considered. These are undesirable features for any measure of QCs for pure states, since the measure does not reduce to a good measure of entanglement. Based on this findings, in order to overcome this undesirable behavior, we suggested an alternative measure d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, which inherits the main properties of Guo’s measure DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT. The proposed measure d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT involves a minimization procedure over the set of all local basis that, in the case of pure states, can be analytically performed. In that case, the optimal representation turns out to be that of Schmidt. In addition, unlike DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT, d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT reduces to a legitimate entanglement measure in the case of pure states. Next, we numerically computed the new measure d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT for some typical arbitrary (mixed) states and explicitly showed that also for mixed states DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT is representation-dependent. As a consequence, in most cases of interest, our results indicate that the use of DG′subscriptsuperscript𝐷′𝐺D^prime_Gitalic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT can result in a overestimation of quantum correlations. Nevertheless, regarding arbitrary mixed states, it is worth to mention that the optimization procedure involved in the calculation of d′superscript𝑑′d^primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT can be difficult to perform. As a final comment, since the NCQC measure introduced by Guo has the advantage of being easily computable, it might be used as a qualitative estimator of the presence of quantum correlations.

Acknowledgements.
A.P.M., D.B., T.M.O, and P.W.L. acknowledge the Argentinian agency SeCyT-UNC and CONICET for financial support. D. B. has a fellowship from CONICET. A.V.H. gratefully acknowledges financial support from DGAPA, UNAM through project PAPIIT IA101816.

Read More: https://discord-servers.biz/
     
 
what is notes.io
 

Notes.io is a web-based application for taking notes. You can take your notes and share with others people. If you like taking long notes, notes.io is designed for you. To date, over 8,000,000,000 notes created and continuing...

With notes.io;

  • * You can take a note from anywhere and any device with internet connection.
  • * You can share the notes in social platforms (YouTube, Facebook, Twitter, instagram etc.).
  • * You can quickly share your contents without website, blog and e-mail.
  • * You don't need to create any Account to share a note. As you wish you can use quick, easy and best shortened notes with sms, websites, e-mail, or messaging services (WhatsApp, iMessage, Telegram, Signal).
  • * Notes.io has fabulous infrastructure design for a short link and allows you to share the note as an easy and understandable link.

Fast: Notes.io is built for speed and performance. You can take a notes quickly and browse your archive.

Easy: Notes.io doesn’t require installation. Just write and share note!

Short: Notes.io’s url just 8 character. You’ll get shorten link of your note when you want to share. (Ex: notes.io/q )

Free: Notes.io works for 12 years and has been free since the day it was started.


You immediately create your first note and start sharing with the ones you wish. If you want to contact us, you can use the following communication channels;


Email: [email protected]

Twitter: http://twitter.com/notesio

Instagram: http://instagram.com/notes.io

Facebook: http://facebook.com/notesio



Regards;
Notes.io Team

     
 
Shortened Note Link
 
 
Looding Image
 
     
 
Long File
 
 

For written notes was greater than 18KB Unable to shorten.

To be smaller than 18KB, please organize your notes, or sign in.