Notes

Quantum Correlations Beyond Entanglement And Discord
Dissimilar notions of quantum correlations have been established, each of them being motivated through particular applications in quantum information science and each competing for being recognized as the most relevant measure of quantumness. In this contribution, we experimentally realize a form of quantum correlation that exists even in the absence of entanglement and discord. We certify the presence of such quantum correlations via negativities in the regularized two-mode Glauber-Sudarshan function. Our data show compatibility with an incoherent mixture of orthonormal photon-number states, ruling out quantum coherence and other kinds of quantum resources. By construction, the quantumness of our state is robust against dephasing, thus requiring less experimental resources to ensure stability. In addition, we theoretically show how multimode entanglement can be activated based on the generated, nonentangled state. Therefore, we implement a robust kind of nonclassical photon-photon correlated state with useful applications in quantum information processing.

The certification of quantum correlations is essential for the ever accelerating development of quantum technologies. Beyond this practical demand, a fundamental understanding of quantum correlations, including their characterization and quantification, plays a key role when exploring the boundary between classical theories and the unique features of quantum physics. Still, the question remains which kinds of correlations are genuinely quantum. That is, which of the many contenders-be it an established or recently proposed concept (e.g., quantum coherence and resource theory SAP17 ; CG19 , entanglement HHHH09 , discord MBCPV12 , etc.)-does describe the concept of a nonclassical correlation best? In this work, we show that the notion of nonclassicality in quantum optics TG65 ; M86 can supersede contemporary forms of quantumness in its ability to unveil quantum correlations.

In the context of quantum optics, any observations which cannot be fully described in terms of Maxwell’s wave theory of light are called nonclassical TG65 ; M86 . This well-established concept of nonclassicality is based on the impossibility of describing field correlations as done in classical electrodynamics, and it is commonly defined in terms of the Glauber-Sudarshan P𝑃Pitalic_P representation G63 ; S63 . The latter describes nonclassical light through phase-space distributions that are, in the case of quantum light, incompatible with a classical concept of a nonnegative probability distribution.

The more recently developed concept of quantum coherence adapts some ideas of the notion of nonclassicality to quantify resources required for quantum information processing SAP17 ; CG19 . In this framework, quantum superpositions equally serve as the origin of quantumness in a system. However, in most cases, the classical reference is defined through incoherent mixtures of orthonormal basis states, contrasting the notion of quantum-optical nonclassicality in terms of nonorthogonal eigenstates of the non-Hermitian annihilation operator.

Entanglement, which can be embedded into the concept of coherence SSDBA15 ; KSP16 , is by far the most frequently studied form of quantum correlation among the many contenders HHHH09 . This is due to its fundamental role as well as its many applications, e.g., in quantum metrology, cryptography, computing, and teleportation. The phenomenon of entanglement was discovered in early seminal discussions about the implications of quantum physics S35 ; EPR35 , long before the conception of the relatively young field of quantum information processing.

Many other notions and measures of quantum correlations have been proposed too MBCPV12 . For instance, discord is a feature which includes correlations caused by entangled but also by nonentangled states HV01 ; OZ01 , and it can be connected to quantum coherence as well MYGVG16 . In this context, it is worth mentioning that the label quantum for this sort of correlation is a topic of ongoing debates GOS15 . Nonetheless, it has been demonstrated that discord is maximally inequivalent to the notion of quantum-optical nonclassicality FP12 . To date, it remains an open problem to decide-not only in theory, but also experimentally-which of the candidates is best suited for characterizing quantum correlations.

In this Letter, we address this issue experimentally by realizing and analyzing a fully phase-randomized two-mode squeezed vacuum (TMSV) state, as theoretically proposed in Ref. ASV13 . This state of quantum light has the following properties: it is nonentangled; it has zero discord; it does not exhibit quantum coherence in the photon-number basis; its reduced single-mode states are classical; and it has a non-negative, two-mode Wigner function. Despite these strong signatures of classicality, we demonstrate the presence of quantum correlations as defined through the notion of nonclassicality in quantum optics with a statistical significance next to certainty. In addition, because of the phase independence of the generated state, its nonclassical feature is robust under dephasing. Furthermore, the activation of entanglement using this kind of state is developed to demonstrate the state’s usefulness for quantum information processing applications. Consequently, we realize a form of quantum correlation of light which is inaccessible with other notions of quantumness but can nevertheless serve as a versatile resource for quantum information science.

II Quantum correlations

For analyzing quantum correlations, we consider a class of two-mode states that are phase insensitive FP12 ; ASV13 . Still, intensity-intensity (likewise, photon-photon) correlations are present in such states. For instance, this can be achieved by a full phase randomization of a TMSV state, resulting in

ρ^=∑n∈ℕ(1-p)pn|n⟩⟨n|⊗|n⟩⟨n|,^𝜌subscript𝑛ℕtensor-product1𝑝superscript𝑝𝑛ket𝑛bra𝑛ket𝑛bra𝑛displaystylehatrho=sum_ninmathbbN(1-p)p^n|nranglelangle n|% otimes|nranglelangle n|,over^ start_ARG italic_ρ end_ARG = ∑ start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ( 1 - italic_p ) italic_p start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT | italic_n ⟩ ⟨ italic_n | ⊗ | italic_n ⟩ ⟨ italic_n | , (1)
which is a statistical mixture of photon-number product states, where p=tanh|ξ|𝑝𝜉p=tanh|xi|italic_p = roman_tanh | italic_ξ | is a value between zero and one and ξ𝜉xiitalic_ξ is the complex squeezing parameter. Such states are, for example, a relevant resource for boson sampling tasks SLR17 .

In terms of quantum correlations, it can be directly observed that the state in Eq. (1) is an incoherent mixture of photon-number states, thus exhibiting no quantum coherence in the form of quantum superpositions of photon-number states; it is a classical mixture of tensor-product states, thus exhibiting no entanglement; and it has zero discord because ρ^=∑n∈ℕρ^A|n⊗|n⟩⟨n|^𝜌subscript𝑛ℕtensor-productsubscript^𝜌conditional𝐴𝑛ket𝑛bra𝑛hatrho=sum_ninmathbbNhatrho_Aotimes|nranglelangle n|over^ start_ARG italic_ρ end_ARG = ∑ start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_A | italic_n end_POSTSUBSCRIPT ⊗ | italic_n ⟩ ⟨ italic_n | holds true, where the photon-number states form the eigenbasis to ρ^A|n=(1^⊗⟨n|)ρ^(1^⊗|n⟩)subscript^𝜌conditional𝐴𝑛tensor-product^1bra𝑛^𝜌tensor-product^1ket𝑛hatrho_n=(hat1otimeslangle n|)hatrho(hat1otimes|nrangle)over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_A | italic_n end_POSTSUBSCRIPT = ( over^ start_ARG 1 end_ARG ⊗ ⟨ italic_n | ) over^ start_ARG italic_ρ end_ARG ( over^ start_ARG 1 end_ARG ⊗ | italic_n ⟩ ) and trAρ^subscripttr𝐴^𝜌mathrmtr_Ahatrhoroman_tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT over^ start_ARG italic_ρ end_ARG D11 . For those criteria of quantum correlations, it suffices in our scenario to consider the contribution of off-diagonal elements,

𝒞(ρ^)=def.∑m,n,k,l∈ℕ:m≠n,k≠l|(⟨m|⊗⟨k|)ρ^(|n⟩⊗|l⟩)|,superscriptdef.𝒞^𝜌subscript:𝑚𝑛𝑘𝑙ℕabsentformulae-sequence𝑚𝑛𝑘𝑙tensor-productbra𝑚bra𝑘^𝜌tensor-productket𝑛ket𝑙displaystylemathcalC(hatrho)stackrelscriptstyletextdef.=% sum_beginsubarraycm,n,k,linmathbbN:\ m
eq n,k
eq lendsubarraybig(langle m|otimeslangle k|)hatrho(|% nrangleotimes|lrangle)big,caligraphic_C ( over^ start_ARG italic_ρ end_ARG ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def. end_ARG end_RELOP ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_m , italic_n , italic_k , italic_l ∈ blackboard_N : end_CELL end_ROW start_ROW start_CELL italic_m ≠ italic_n , italic_k ≠ italic_l end_CELL end_ROW end_ARG end_POSTSUBSCRIPT | ( ⟨ italic_m | ⊗ ⟨ italic_k | ) over^ start_ARG italic_ρ end_ARG ( | italic_n ⟩ ⊗ | italic_l ⟩ ) | , (4)
which quantifies the coherent contributions BCP14 and becomes 𝒞(ρ^)=0𝒞^𝜌0mathcalC(hatrho)=0caligraphic_C ( over^ start_ARG italic_ρ end_ARG ) = 0. It is worth emphasizing that the nullity of coherence in the two-mode photon-number basis implies the nullity of discord which further implies no entanglement. In addition, the incoherent mixture of photon-number states under study further implies a classical interpretation in a particle picture SDNTBBS19 . Furthermore, the marginal states trAρ^subscripttr𝐴^𝜌mathrmtr_Ahatrhoroman_tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT over^ start_ARG italic_ρ end_ARG and trBρ^subscripttr𝐵^𝜌mathrmtr_Bhatrhoroman_tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT over^ start_ARG italic_ρ end_ARG are thermal states, thus classical too. Also, the two-mode state under study is a mixture of Gaussian TMSV states, implying a nonnegative Wigner function.

To this end, one sees no indication of quantum correlations. Yet, we have not considered the notion of nonclassicality in quantum optics so far. This concept is defined through the Glauber-Sudarshan P𝑃Pitalic_P representation G63 ; S63 ,

ρ^=∫d2α∫d2βP(α,β)|α⟩⟨α|⊗|β⟩⟨β|,^𝜌superscript𝑑2𝛼tensor-productsuperscript𝑑2𝛽𝑃𝛼𝛽ket𝛼bra𝛼ket𝛽bra𝛽displaystylehatrho=int d^2alphaint d^2beta,P(alpha,beta)|% alpharanglelanglealpha|otimes|betaranglelanglebeta|,over^ start_ARG italic_ρ end_ARG = ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_α ∫ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β italic_P ( italic_α , italic_β ) | italic_α ⟩ ⟨ italic_α | ⊗ | italic_β ⟩ ⟨ italic_β | , (5)
where |α⟩ket𝛼|alpharangle| italic_α ⟩ and |β⟩ket𝛽|betarangle| italic_β ⟩ denote classically coherent states of the harmonic oscillator S26 . Whenever P𝑃Pitalic_P cannot be interpreted as a classical probability density, the state of light ρ^^𝜌hatrhoover^ start_ARG italic_ρ end_ARG refers to as a nonclassical one TG65 ; M86 . Since the P𝑃Pitalic_P distribution is in many cases highly singular S16 , thus experimentally inaccessible, regularization and direct sampling procedures have been proposed and implemented to reconstruct a function PΩsubscript𝑃ΩP_Omegaitalic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT which is always regular, and nonnegative for any classical states of light KV10 ; KVHS11 . This is achieved by a convolution of the Glauber-Sudarshan P𝑃Pitalic_P function with a suitable, non-Gaussian kernel ΩΩOmegaroman_Ω, resulting in PΩsubscript𝑃ΩP_Omegaitalic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT. Our previous theoretical studies suggest that the state in Eq. (1) indeed demonstrates nonclassical correlations ASV13 , i.e.,

PΩ(α,β)

III Experimental implementation

On the basis of our theoretical study, we now consider the experimental preparation and detection of the quantum state given in Eq. (1). Figure 1 shows our experimental setup.

Two amplitude-squeezed fields at 1064nm1064nm1064,mathrmnm1064 roman_nm are produced by optical parametric amplifiers (OPAs). One OPA consists of a type-I hemilithic, standing wave, nonlinear cavity with a 7%percent77%7 % MgO:LiNbO3 crystal; the other OPA uses a periodically poled potassium titanyl phosphate crystal instead. The seed and pump powers are adjusted such that the squeezed output fields of the two crystals are of equal intensity and squeezing to achieve the challenging goal of using different sources of quantum light in one setup. For this purpose, the two pump powers of the second harmonic are chosen as 242mW242mW242,mathrmmW242 roman_mW and 50mW50mW50,mathrmmW50 roman_mW, respectively. The two squeezed fields are superposed with a visibility of 96.5%percent96.596.5%96.5 %-demonstrating a high compatibility of the output fields of the two distinct sources-and a relative phase of π/2𝜋2pi/2italic_π / 2 on a 50:50:505050:5050 : 50 beam splitter, resulting in a two-mode squeezed vacuum state,

|TMSV⟩=1cosh|ξ|∑n=0∞(eiargξtanh|ξ|)n|n⟩⊗|n⟩.ketTMSV1𝜉superscriptsubscript𝑛0tensor-productsuperscriptsuperscript𝑒𝑖𝜉𝜉𝑛ket𝑛ket𝑛displaystyle|mathrmTMSVrangle=frac1xisum_n=0^infty% left(e^iargxitanh|xi|right)^n|nrangleotimes|nrangle.| roman_TMSV ⟩ = divide start_ARG 1 end_ARG start_ARG roman_cosh | italic_ξ | end_ARG ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_e start_POSTSUPERSCRIPT italic_i roman_arg italic_ξ end_POSTSUPERSCRIPT roman_tanh | italic_ξ | ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT | italic_n ⟩ ⊗ | italic_n ⟩ . (7)

Both output modes A𝐴Aitalic_A and B𝐵Bitalic_B of the state are probed by balanced homodyne detectors. We observed (96.7±0.7)%percentplus-or-minus96.70.7(96.7pm 0.7)%( 96.7 ± 0.7 ) % visibility between the fields and their corresponding local oscillators. For one OPA, we measured a single-mode squeezing variance of -1.3dB1.3dB-1.3,mathrmdB- 1.3 roman_dB and +3.7dB3.7dB+3.7,mathrmdB+ 3.7 roman_dB antisqueezing with respect to the vacuum state. This yields an initial squeezing of -7.3dB7.3dB-7.3,mathrmdB- 7.3 roman_dB and an overall efficiency of (63±2)%percentplus-or-minus632(63pm 2)%( 63 ± 2 ) %. The latter figure was multiplied by two to compensate for the vacuum input because blocking the second OPA effectively introduces additional 50%percent5050%50 % loss at the first (i.e., leftmost) beam splitter in Fig. 1.

We use piezoelectric transducers to control the optical phases and realize a random phase shift δφ𝛿𝜑deltavarphiitalic_δ italic_φ in one of the arms. To achieve a uniform dephasing over the full 2π2𝜋2pi2 italic_π interval, white noise is applied with sufficiently high amplitude. Due to the bandwidth limitations of the transducers, a uniformly distributed phase-as required to exactly obtain the state in Eq. (1)-can only be approximated via long measurement times.

Figure 2 depicts the reconstructed two-mode density matrix of the TMSV state in photon-number basis, without (top panel) and with (bottom panel) phase randomization. Both plots visualize the first 625625625625 matrix elements, organized in blocks for mode B𝐵Bitalic_B, and inner elements represent mode A𝐴Aitalic_A. The initially generated state shows strong contributions of off-diagonal elements, relating to the presence of the resource quantum coherence, Eq. (4), resulting in the measured quantum coherence 𝒞(|TMSV⟩⟨TMSV|)=1.789±0.021𝒞ketTMSVbraTMSVplus-or-minus1.7890.021mathcalC(|mathrmTMSVranglelanglemathrmTMSV|)=1.789pm 0.021caligraphic_C ( | roman_TMSV ⟩ ⟨ roman_TMSV | ) = 1.789 ± 0.021. The implemented phase randomization then leads to a 45-fold suppression of the initial coherence, resulting in almost no subsisting overall coherence, 𝒞(ρ^)=0.041±0.005𝒞^𝜌plus-or-minus0.0410.005mathcalC(hatrho)=0.041pm 0.005caligraphic_C ( over^ start_ARG italic_ρ end_ARG ) = 0.041 ± 0.005, when compared to the initial TMSV state and as expected from the theory. (Errors have been obtained through a Monte Carlo approach; see the Supplemental Material SuppMat for details on the data processing and a discussion of the small amount of remaining quantum coherence, caused by experimental imperfections.) The loss of coherence implies that other forms of quantum correlations, such as entanglement and discord, do not contribute to the correlations of the phase-randomized state that is almost completely characterized by its diagonal elements in the photon-number basis; see Eq. (1) and bottom plot in Fig. 2.

To this end, we showed that quantum correlations in terms of quantum coherence are negligible for the produced state. That is, the phase-averaged state is mostly consistent with a classical statistical mixture of orthonormal two-mode, tensor-product, photon-number states, thus also ruling out entanglement and discord as a source of quantum correlations as discussed earlier. However, from our balanced homodyne detection data, we can further directly sample the regularized phase-space function PΩsubscript𝑃ΩP_Omegaitalic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT KVHS11 . For real-valued phase-space parameters α𝛼alphaitalic_α and β𝛽betaitalic_β, the resulting distribution is shown in Fig. 3. We found a maximal statistical significance of more than 150 standard deviations for the negativity of the reconstructed quasiprobability distribution, PΩ(α=0,β=1.5)=(-1.570±0.010)×10-3subscript𝑃Ωformulae-sequence𝛼0𝛽1.5plus-or-minus1.5700.010superscript103P_Omega(alpha=0,beta=1.5)=(-1.570pm 0.010)times 10^-3italic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( italic_α = 0 , italic_β = 1.5 ) = ( - 1.570 ± 0.010 ) × 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT, in Fig. 3. Because of these highly significant negativities, nonclassical quantum correlations in the generated state are confirmed beyond the previously considered notions. Thus, we experimentally generated quantum correlations which are inaccessible via other means, such as coherence, entanglement, and discord. Moreover, phase stability is not required for the kind of quantum effect we have just verified. In fact, we artificially introduced phase noise-which is often omnipresent in realistic quantum channels-to produce the sought-after state.

V Entanglement activation

Because of the ever growing importance for quantum information processing HHHH09 ; NC00 , the question arises if entanglement can be activated from the nonclassically correlated state under study-although it does not exhibit other forms of quantum correlations-as it was done for coherence and discord SSDBA15 ; KSP16 ; MYGVG16 . Achieving such an activation would render the fully phase-randomized TMSV state a useful resource for many quantum protocols.

For this purpose, let us recall that single-mode nonclassicality can be converted into entanglement via simple beam splitters X02 ; KSBK02 ; VS14 . Similarly, we consider combining each of our two modes separately on a 50:50 beam splitter with vacuum, where annihilation operators for the non-vacuum input map as a^↦(a^+a^′)/2maps-to^𝑎^𝑎superscript^𝑎′2hatamapsto(hata+hata^prime)/sqrt2over^ start_ARG italic_a end_ARG ↦ ( over^ start_ARG italic_a end_ARG + over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / square-root start_ARG 2 end_ARG and b^↦(b^+b^′)/2maps-to^𝑏^𝑏superscript^𝑏′2hatbmapsto(hatb+hatb^prime)/sqrt2over^ start_ARG italic_b end_ARG ↦ ( over^ start_ARG italic_b end_ARG + over^ start_ARG italic_b end_ARG start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / square-root start_ARG 2 end_ARG; additional modes obtained from the splitting are indicated by prime superscripts. Such operations are free (i.e., classical) ones with respect to the reference |α⟩⊗|α′⟩tensor-productket𝛼ketsuperscript𝛼′|alpharangleotimes|alpha^primerangle| italic_α ⟩ ⊗ | italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ of the Glauber-Sudarshan representation [Eq. (5)] since the beam-splitter output for mode A𝐴Aitalic_A remains in this family of states, |(α+α′)/2⟩⊗|(α-α′)/2⟩tensor-productket𝛼superscript𝛼′2ket𝛼superscript𝛼′2|(alpha+alpha^prime)/sqrt2rangleotimes|(alpha-alpha^prime)/% sqrt2rangle| ( italic_α + italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / square-root start_ARG 2 end_ARG ⟩ ⊗ | ( italic_α - italic_α start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / square-root start_ARG 2 end_ARG ⟩, likewise for B𝐵Bitalic_B. Furthermore, applied to a photon-number input state |n⟩⊗|0⟩tensor-productket𝑛ket0|nrangleotimes|0rangle| italic_n ⟩ ⊗ | 0 ⟩, the map yields the output state |Ψn⟩=2-n/2∑j=0n(nj)1/2(-1)n-j|j⟩⊗|n-j⟩′ketsubscriptΨ𝑛superscript2𝑛2superscriptsubscript𝑗0𝑛tensor-productsuperscriptbinomial𝑛𝑗12superscript1𝑛𝑗ket𝑗superscriptket𝑛𝑗′|Psi_nrangle=2^-n/2sum_j=0^nbinomnj^1/2(-1)^n-j|jrangle% otimes|n-jrangle^prime| roman_Ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ = 2 start_POSTSUPERSCRIPT - italic_n / 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( FRACOP start_ARG italic_n end_ARG start_ARG italic_j end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( - 1 ) start_POSTSUPERSCRIPT italic_n - italic_j end_POSTSUPERSCRIPT | italic_j ⟩ ⊗ | italic_n - italic_j ⟩ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Therefore, the state in Eq. (1) results in the final four-mode state

ρ^AA′BB′=∑n∈ℕ(1-p)pn|Ψn⟩⟨Ψn|⊗|Ψn⟩⟨Ψn|subscript^𝜌𝐴superscript𝐴′𝐵superscript𝐵′subscript𝑛ℕtensor-product1𝑝superscript𝑝𝑛ketsubscriptΨ𝑛brasubscriptΨ𝑛ketsubscriptΨ𝑛brasubscriptΨ𝑛displaystylehatrho_AA^primeBB^prime=sum_ninmathbbN(1-p)p^% n|Psi_nranglelanglePsi_n|otimes|Psi_nranglelanglePsi_n|over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_A italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ( 1 - italic_p ) italic_p start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT | roman_Ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ ⟨ roman_Ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⊗ | roman_Ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ ⟨ roman_Ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | (8)
for the free operations under consideration.

Clearly, this state is still nonentangled when separating the joint subsystems AA′𝐴superscript𝐴′AA^primeitalic_A italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and BB′𝐵superscript𝐵′BB^primeitalic_B italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT from each other. However, multimode entanglement is much richer since entanglement in various mode decompositions can be considered; see Ref. GSVCRTF15 for a full experimental characterization. Here, let us restrict to the question whether there is entanglement between the primed and unprimed modes-i.e., is the state in Eq. (8) entangled with respect to the separation of AB𝐴𝐵ABitalic_A italic_B and A′B′superscript𝐴′superscript𝐵′A^primeB^primeitalic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT?

To answer this question, we consider the partial transposition criterion P96 ; HHH96 . In one form HHH96 , this criterion states that a state is entangled if the expectation value of a so-called entanglement witness W^=(|Φ⟩⟨Φ|)PT′^𝑊superscriptketΦbraΦsuperscriptPT′hatW=(|PhiranglelanglePhi|)^mathrmPT^primeover^ start_ARG italic_W end_ARG = ( | roman_Φ ⟩ ⟨ roman_Φ | ) start_POSTSUPERSCRIPT roman_PT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT is negative, where PT′superscriptPT′mathrmPT^primeroman_PT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT denotes the partial transposition of the primed modes. For example, we can choose |Φ⟩=|0⟩⊗|0⟩′⊗|1⟩⊗|1⟩′-|1⟩⊗|1⟩′⊗|0⟩⊗|0⟩′ketΦtensor-productket0superscriptket0′ket1superscriptket1′tensor-productket1superscriptket1′ket0superscriptket0′|Phirangle=|0rangleotimes|0rangle^primeotimes|1rangleotimes|1% rangle^prime-|1rangleotimes|1rangle^primeotimes|0rangleotimes|0% rangle^prime| roman_Φ ⟩ = | 0 ⟩ ⊗ | 0 ⟩ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊗ | 1 ⟩ ⊗ | 1 ⟩ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - | 1 ⟩ ⊗ | 1 ⟩ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊗ | 0 ⟩ ⊗ | 0 ⟩ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, which results in

tr(W^ρ^AA′BB′)=-(1-p)p2<0,tr^𝑊subscript^𝜌𝐴superscript𝐴′𝐵superscript𝐵′1𝑝𝑝20displaystylemathrmtr(hatWhatrho_AA^primeBB^prime)=-frac(1% -p)p2<0,roman_tr ( over^ start_ARG italic_W end_ARG over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_A italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_B italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = - divide start_ARG ( 1 - italic_p ) italic_p end_ARG start_ARG 2 end_ARG <0 , (9)
for all nontrivial parameters 0
<10𝑝10<0 <1 that define our states in Eq. (1). Therefore, the two-mode nonclassical correlations of the generated state can be successfully activated to produce four-mode entanglement, being a useful quantum resource.

VI Discussion and conclusion

We experimentally realized a quantum state with quantum correlations that are inaccessible by means of two-mode coherence, thus entanglement and discord, but can be intuitively visualized by negative quasiprobabilities. Quantum coherence-a recently explored resource for quantum information processing-exists in terms of superpositions of orthogonal photon-number states of the inital TMSV state. But a phase averaging destroys this and related kinds of quantum correlations. Thus, such contemporary quantum-information-based concepts of correlation fail to certify the quantumness of the state under the challenging, but common scenario of dephasing. However, the Glauber-Sudarshan-based concept of nonclassiality of light-frequently considered to be a dated notion, or not being considered at all- is still capable of uncovering the quantum nature of the generated state. Since the Glauber-Sudarshan distribution often displays a highly singular behavior, we employ a technique which enables us to directly sample a regularized version of such a two-mode phase-space function, allowing us to certify nonclassical negativities with a statistical significance of more than 150 standard deviations. Thus, the phase-independent quantum-correlated state, generated by combining two squeezed states from distinct sources with a high overlap on a beam splitter and adding phase noise in one output, exhibits quantum correlations. Furthermore, we theoretically devised a method to activate multimode entanglement, only utilizing the produced state and simple beam splitter operations. This approach renders the produced state a valuable resource for quantum communications.

In conclusion, we realized a type of quantum correlation that can be accessed via phase-space approaches but not through more recent notions of quantumness. By construction, this kind of correlation is intrinsically robust against dephasing and can be easily converted into multimode entanglement. This finding offers a useful form of quantum correlation which can be realized without experimentally costly phase stabilization and proves its usefulness as a resource for realizing modern quantum information protocols.

E. A. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie IF InDiQE (EU project 845486). M. S. acknowledges financial support by the Deutsche Forschungsgemeinschaft through STA-543/9-1. J. S. thanks Torsten Meier and Tim Bartley for discussions. This work was supported by the Deutsche Forschungsgemeinschaft through SFB 652, project No. B12 and B13.

Supplemental Material

In this supplementary document, we provide additional details on the experiment and data processing. We address the experimental implementation and data handling for phase readout and randomization in Appendix A. Aspects of the reconstruction of the density matrix in the photon-number basis are provided in Appendix B. The reconstruction and optimization of the phase-space representation is briefly discussed in Appendix C.

Appendix A Phase readout and randomization

The density matrices were calculated from quadrature histograms using pattern functions for the photon-number basis expansion ALP95 ; LMKRR96 ; R96 . For this reconstruction, the quadrature data needed to be allocated to specific optical phases φLO,Asubscript𝜑LO𝐴varphi_mathrmLO,Aitalic_φ start_POSTSUBSCRIPT roman_LO , italic_A end_POSTSUBSCRIPT and φLO,Bsubscript𝜑LO𝐵varphi_mathrmLO,Bitalic_φ start_POSTSUBSCRIPT roman_LO , italic_B end_POSTSUBSCRIPT for modes A𝐴Aitalic_A and B𝐵Bitalic_B, respectively; see also Fig. 1 in the main text. The phase range from 00 to 2π2𝜋2pi2 italic_π was divided into 30303030 equidistant parts, resulting in 302=900superscript30290030^2=90030 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 900 histograms or phase combinations between A𝐴Aitalic_A and B𝐵Bitalic_B. To obtain the current optical phase within the balanced homodyne detectors, an auxiliary laser was coupled to the output field of an OPA creating a phase-locked loop. Accordingly, optical phases could be resolved by feeding the AC signals of the homodyne detectors into phase frequency detectors with same beat frequency.

The optical phase in arm A𝐴Aitalic_A was randomized within our experiment. For this purpose, a piezoelectric transducer was supplied with a low pass filtered white noise signal to match the constant part of the piezo’s transfer function. For small amplitudes, the phase fluctuates around the preset reference phase. With a sufficiently high amplitude, however, the phase fluctuates over several periods and becomes uniformly distributed. From Discord servers , we found that the standard deviation of the piezo movement needed to be larger than 3.7rad3.7rad3.7,mathrmrad3.7 roman_rad to approach uniformly distributed phases. Due to the bandwidth limitations of the transducers, a uniformly distributed phase is only achievable by measuring infinitely long. Therefore, even the smallest density matrix elements are expected to have a value differing from zero.

In contrast to the continuous phase recording in mode B𝐵Bitalic_B, the optical phase at A𝐴Aitalic_A was fixed for time intervals of 120ms120ms120,mathrmms120 roman_ms, and the phase value determined when the noise signal was zero. All data recorded during this time were allocated to that reference phase. Long-term drifts contribute to a change in the reference phase at A𝐴Aitalic_A. However, in addition, we applied a changing offset signal every time a new reference phase was set that allowed us to collect similar numbers of data points for each phase.

Appendix B Impurities in the suppression of off-diagonal elements and statistical analysis

As one can see in Fig. 2 of the main text, all off-diagonal elements in the photon-number basis of subsystem A𝐴Aitalic_A approach the expected value of zero. In the basis of subsystem B𝐵Bitalic_B, small nonzero off-diagonal elements remain. This may be unexpected as phase randomization in one subsystem should be sufficient for extinction of off-diagonal elements in both subsystems in theory. In the experimental data, these nonvanishing entries can be traced back to having not exactly the same amount of squeezing coming from the two distinct OPAs and potentially not having exactly π/2𝜋2pi/2italic_π / 2 phase shift between the two single-mode squeezed states at the first beam splitter (cf. Fig. 1 in the main text). This also results in slightly different Wigner functions and quadrature variances, as depicted in Fig. 4, where mode B𝐵Bitalic_B displays a residual amount of phase dependence.

Additional imperfections are caused by deviations from a perfectly uniform distribution of phases because of bandwidth limitations for implementing white noise with our transducers.

Further deviations from zero are due to inherent random errors. To show that the deviations from zero are mostly of statistical nature, we performed a Monte Carlo simulation using the experimental parameters. This allowed us to determine the standard deviation of each element in the density matrix. Furthermore, the simulation also shows that, with increasing amount of the quadrature data (factor N𝑁Nitalic_N) used to calculate the density matrix, the absolute values of the off-diagonal density matrix elements decrease accordingly, i.e., by the factor N𝑁sqrtNsquare-root start_ARG italic_N end_ARG. In Fig. 5, we have shown in a histogram the frequency of off-diagonal density matrix elements, normalized to their standard deviation. The distribution shows a similar shape for our experimental data and the Monte Carlo simulation in which the complex, off-diagonal elements are normally distributed around zero (solid curve in Fig. 5). This leads to a coherence level 𝒞MC=0.039subscript𝒞MC0.039mathcalC_mathrmMC=0.039caligraphic_C start_POSTSUBSCRIPT roman_MC end_POSTSUBSCRIPT = 0.039 [with an uncertainty σ(𝒞MC)=0.005𝜎subscript𝒞MC0.005sigma(mathcalC_mathrmMC)=0.005italic_σ ( caligraphic_C start_POSTSUBSCRIPT roman_MC end_POSTSUBSCRIPT ) = 0.005] for the state under study which is in agreement with the estimate 𝒞exp=0.041subscript𝒞exp0.041mathcalC_mathrmexp=0.041caligraphic_C start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT = 0.041 from our photon-number basis reconstruction.

Appendix C Phase-space distributions and statistical analysis

In order to make a statement about the significance of the negativities in the regularized P𝑃Pitalic_P function, we divided our measured data into N=141𝑁141N=141italic_N = 141 ensembles, 4×1074superscript1074times 10^74 × 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT quadrature data points each, and calculated PΩsubscript𝑃ΩP_Omegaitalic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT for each ensemble separately. The standard error of the mean can be expressed as σN=σ(PΩ(α,β))/Nsubscript𝜎𝑁𝜎subscript𝑃Ω𝛼𝛽𝑁sigma_N=sigma(P_Omega(alpha,beta))/sqrtNitalic_σ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = italic_σ ( italic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( italic_α , italic_β ) ) / square-root start_ARG italic_N end_ARG, where σ(PΩ(α,β))𝜎subscript𝑃Ω𝛼𝛽sigma(P_Omega(alpha,beta))italic_σ ( italic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( italic_α , italic_β ) ) is the standard deviation for the ensembles. The maximal significance of the negativities ΣΣSigmaroman_Σ for any points in phase space is Σ=maxα,β[-PΩ(α,β)/σN]Σsubscript𝛼𝛽subscript𝑃Ω𝛼𝛽subscript𝜎𝑁Sigma=max_alpha,beta[-P_Omega(alpha,beta)/sigma_N]roman_Σ = roman_max start_POSTSUBSCRIPT italic_α , italic_β end_POSTSUBSCRIPT [ - italic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( italic_α , italic_β ) / italic_σ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ].

The kernel ΩΩOmegaroman_Ω for regularizing the singular P𝑃Pitalic_P function is characterized by a width parameter w𝑤witalic_w KVHS11 . Note that this approach, like the photon-number based reconstruction, is based on pattern functions KVHS11 , also including the parameter w𝑤witalic_w. For characterizing the nonclassical features of the reconstructed quasiprobability distribution PΩsubscript𝑃ΩP_Omegaitalic_P start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT, one may choose a suitable w𝑤witalic_w such that ΣΣSigmaroman_Σ becomes maximal. For such an optimization, we found the maximum Σ=153Σ153Sigma=153roman_Σ = 153 for w=1.3𝑤1.3w=1.3italic_w = 1.3, cf. Fig. 6.

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