Gaussian Multipartite Quantum Discord From Classical Mutual Information

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Quantum discord is a measure of non-classical correlations, which are excess correlations inherent in quantum states that cannot be accessed by classical measurements. For multipartite states, the classically accessible correlations can be defined by the mutual information of the multipartite measurement outcomes. In general the quantum discord of an arbitrary quantum state involves an optimisation of over the classical measurements which is hard to compute. In this paper, we examine the quantum discord in the experimentally relevant case when the quantum states are Gaussian and the measurements are restricted to Gaussian measurements. We perform the optimisation over the measurements to find the Gaussian discord of the bipartite EPR state and tripartite GHZ state in the presence of different types of noise: uncorrelated noise, multiplicative noise and correlated noise. We find that by adding uncorrelated noise and multiplicative noise, the quantum discord always decreases. However, correlated noise can either increase or decrease the quantum discord. We also find that for low noise, the optimal classical measurements are single quadrature measurements. As the noise increases, a dual quadrature measurement becomes optimal.



A pair of quantum systems can be entangled Horodecki et al. (2009). Entangled quantum states posses a form of correlation not possible with classical systems. If two quantum states are not entangled, they are said to be separable. Separable quantum states can be created through local operations and classical communication. However, separable quantum states can still possess correlations that are not accessible through local measurements Bennett et al. (1999). Quantum discord (QD) was proposed by Ollivier and Zurek Ollivier and Zurek (2001) and Henderson and Vedral Henderson and Vedral (2001) as a means of quantifying the quantum correlations present in bipartite states that are not necessarily entangled. To quantify the locally accessible (classical) correlations, this quantification involves a measurement on one of the subsystem. This measurement is chosen to maximize the classical correlations. In general, this quantum discord will be different depending on which subsystem is measured. As such, we will refer to this as the asymmetric QD.



A desirable property of correlations might be for them to be symmetric and one way to impose this property is to require that both parties measure their subsystems. Such symmetric versions of the quantum discord have been proposed; the symmetric QD is defined by requiring a projective measurements of both subsystems Maziero et al. (2010). Alternatively, another version of the QD can be defined involving arbitrary measurements on each subsystem Piani et al. (2008); Wu et al. (2009); Terhal et al. (2002), we call this the extended symmetric QD.



QD can also be extended to more than two parties. The multipartite symmetric QD quantifies the correlations present when there are three or more parties, and when each party performs projective measurements on their subsystem Rulli and Sarandy (2011). It can also be defined for the situation in which each party performs arbitrary measurements Piani et al. (2008), which we call the multipartite extended symmetric QD.



Calculating the asymmetric QD is an NP-hard problem Huang (2014). The symmetric QD, and extended symmetric QD, and their multipartite extensions, are likely just as difficult. For continuous variable states, one can consider Gaussian versions of QD. If the state is Gaussian, restricting the measurement to Gaussian measurements give rise to the Gaussian QD Giorda and Paris (2010); Adesso and Datta (2010). This restriction significantly reduces the number of variables involved in the optimisation for finding the optimal measurement. The Gaussian discord is asymmetric as it involves a measurement on only one of the subsystems. In this paper, we define and investigate the symmetric and multipartite versions of the Gaussian QD.



There are many other ways of defining quantum discord-like measures. The quantum discord can be defined as the distance to the closest classical state in terms of relative entropy Modi et al. (2010), or trace distance which gives the geometric quantum discord Dakić et al. (2010). The quantum work deficit Oppenheim et al. (2002) describes the difference in work that can be extracted from a heat bath if one party is in possession of bath subsystems compared to when they are not. Measurement-induced nonlocality Luo and Fu (2011) quantifies the distance between the pre and post measurement state, when a local projective measurement is performed on one subsystem without disturbing the subsystem. The interferometric power Girolami et al. (2014) quantifies how helpful a quantum state is for estimating a parameter of a Hamiltonian that acts on one of the subsystems. See Bera et al. (2018) for a review of quantum discord measures.



This paper is organised as follows: In section II, we describe the asymmetric QD, the symmetric QD, extended symmetric QD, and multipartite extended symmetric QD. In section III, we introduce the Gaussian multipartite QD, describe its properties, and calculate it for a two-mode EPR state and a three-mode tripartite GHZ state subjected to different types of noise. Finally, we summarize our results in section IV.



II Background



II.1 Asymmetric quantum discord



The total correlations present in a bipartite quantum state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT is given by the quantum mutual information (MI),



IQ(A;B)=S(A)+S(B)-S(AB),subscript𝐼𝑄𝐴𝐵𝑆𝐴𝑆𝐵𝑆𝐴𝐵I_Q(A;B)=S(A)+S(B)-S(AB),italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_A ; italic_B ) = italic_S ( italic_A ) + italic_S ( italic_B ) - italic_S ( italic_A italic_B ) , (1) where S𝑆Sitalic_S is the von Neumann entropy given by



S(ρ)=∑ih(λi)𝑆𝜌subscript𝑖ℎsubscript𝜆𝑖S(\rho)=\sum_ih(\lambda_i)italic_S ( italic_ρ ) = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_h ( italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) (2) where λisubscript𝜆𝑖\lambda_iitalic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are the eigenvalues of the state ρ𝜌\rhoitalic_ρ and h(x)=-xlog2xℎ𝑥𝑥subscript2𝑥h(x)=-x\log_2xitalic_h ( italic_x ) = - italic_x roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x. But how can we divide the total correlations into a classical and a quantum part? This question was first answered by Henderson and Vedral Henderson and Vedral (2001) and Ollivier and Zurek Ollivier and Zurek (2001). They defined the classical correlations (CC) by



J(B|A)=S(B)-minΠa∑apaS(ρB|a).𝐽conditional𝐵𝐴𝑆𝐵subscriptsubscriptΠ𝑎subscript𝑎subscript𝑝𝑎𝑆subscript𝜌conditional𝐵𝑎J(B|A)=S(B)-\min_\\Pi_a\\sum_ap_aS(\rho_B).italic_J ( italic_B | italic_A ) = italic_S ( italic_B ) - roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B | italic_a end_POSTSUBSCRIPT ) . (3) where the ΠasubscriptΠ𝑎\\Pi_a\ roman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT is a positive operator valued measure (POVM) performed on subsystem A𝐴Aitalic_A. We refer to this measure of classical correlations as the asymmetric CC. A POVM describes a quantum measurement. It is set of nonnegative self-adjoint operators that satisfy ∑aΠa=Isubscript𝑎subscriptΠ𝑎𝐼\sum_a\Pi_a=I∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_I. The probability of measuring outcome a𝑎aitalic_a is pa=Tr(ρAΠa)subscript𝑝𝑎tracesubscript𝜌𝐴subscriptΠ𝑎p_a=\Tr\rho_A\Pi_aitalic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = roman_Tr ( start_ARG italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG ). The state of B𝐵Bitalic_B after a𝑎aitalic_a is measured on A𝐴Aitalic_A is given by



ρB|a=TrA((Ma⊗I)ρAB(Ma†⊗I)pa),subscript𝜌conditional𝐵𝑎subscripttrace𝐴tensor-productsubscript𝑀𝑎𝐼subscript𝜌𝐴𝐵tensor-productsuperscriptsubscript𝑀𝑎†𝐼subscript𝑝𝑎\rho_a=\Tr_A\left(\frac(M_a\otimes I)\rho_AB(M_a^\dagger\otimes I% )p_a\right),italic_ρ start_POSTSUBSCRIPT italic_B | italic_a end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( divide start_ARG ( italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_I ) italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ⊗ italic_I ) end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_ARG ) , (4) where Πa=MaMa†subscriptΠ𝑎subscript𝑀𝑎superscriptsubscript𝑀𝑎†\Pi_a=M_aM_a^\daggerroman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT. But what about the quantum correlations? They defined the quantum correlations or quantum discord (QD) as the total correlations minus the classical correlations,



δ(B|A)=IQ(A;B)-J(B|A).𝛿conditional𝐵𝐴subscript𝐼𝑄𝐴𝐵𝐽conditional𝐵𝐴\delta(B|A)=I_Q(A;B)-J(B|A).italic_δ ( italic_B | italic_A ) = italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_A ; italic_B ) - italic_J ( italic_B | italic_A ) . (5) Henderson and Vedral defined classical correlations in this way because it satisfied certain desirable properties, and Ollivier and Zurek came up with this definition by generalizing classical conditional entropy to a quantum version. Gaming news This is not the only way classical correlations can be defined.



One of the properties of the assymetric QD defined in Eq. (5) is that it is not symmetric. That is, δ(A|B)≠δ(B|A)𝛿conditional𝐴𝐵𝛿conditional𝐵𝐴\delta(A|B) eq\delta(B|A)italic_δ ( italic_A | italic_B ) ≠ italic_δ ( italic_B | italic_A ) in general. This is because the asymmetric CC defined in Eq. (3) are not symmetric. However, one desirable property of a measure of classical correlations C𝐶Citalic_C would be that the measure is symmetric. This view was expressed by Henderson and Vedral in their original paper Henderson and Vedral (2001): “It is also natural that the measure C𝐶Citalic_C should be symmetric under interchange of the subsystems A𝐴Aitalic_A and B𝐵Bitalic_B. This is because it should quantify the correlation between subsystems rather than a property of either subsystem.” It was not clear back then if the measure defined by Eq. (3) was symmetric or not.



II.2 Symmetric versions of quantum discord



There are several different ways of defining a symmetric version of the quantum discord, which turn out to be equivalent. We use the term symmetric QD when Alice and Bob are restricted to performing projective measurements, and extended symmetric QD when they can perform arbitrary POVM measurements.



II.2.1 Symmetric quantum discord



We first consider the approach of Ref. Maziero et al. (2010), which was also used by Ref. Rulli and Sarandy (2011) to define the multipartite global quantum discord.



Equation (3) can be written in an alternative form. Suppose Alice performs a projective measurement on her subsystem of ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. A projective measurement is a POVM in which the elements are Πa=|a⟩⟨a|subscriptΠ𝑎ket𝑎bra𝑎\Pi_a=\keta\braaroman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = | start_ARG italic_a end_ARG ⟩ ⟨ start_ARG italic_a end_ARG | where a⟩ket𝑎\\keta\ start_ARG italic_a end_ARG ⟩ are a set of states that form an orthonormal basis. The state after the measurement will be a classical-quantum state given by



ϕA(ρAB)=∑apa|a⟩⟨a|⊗ρB|a.subscriptitalic-ϕ𝐴subscript𝜌𝐴𝐵subscript𝑎tensor-productsubscript𝑝𝑎ket𝑎bra𝑎subscript𝜌conditional𝐵𝑎\phi_A(\rho_AB)=\sum_ap_a\keta\braa\otimes\rho_B.italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_a end_ARG ⟩ ⟨ start_ARG italic_a end_ARG | ⊗ italic_ρ start_POSTSUBSCRIPT italic_B | italic_a end_POSTSUBSCRIPT . (6) Here we have defined ϕA(ρ)subscriptitalic-ϕ𝐴𝜌\phi_A(\rho)italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ ) to be the state ρ𝜌\rhoitalic_ρ becomes after a measurement on A𝐴Aitalic_A. After the measurement, A𝐴Aitalic_A will be diagonal in the measurement basis:



ϕA(ρA)=∑apa|a⟩⟨a|,subscriptitalic-ϕ𝐴subscript𝜌𝐴subscript𝑎subscript𝑝𝑎ket𝑎bra𝑎\phi_A(\rho_A)=\sum_ap_a\keta\braa,italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_a end_ARG ⟩ ⟨ start_ARG italic_a end_ARG | , (7) with entropy S(ϕA(ρA))=∑ah(pa)𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐴subscript𝑎ℎsubscript𝑝𝑎S(\phi_A(\rho_A))=\sum_ah(p_a)italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) ) = ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ). B𝐵Bitalic_B will be unchanged by a measurement on A𝐴Aitalic_A: ϕA(ρB)=ρBsubscriptitalic-ϕ𝐴subscript𝜌𝐵subscript𝜌𝐵\phi_A(\rho_B)=\rho_Bitalic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) = italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Since ϕA(ρAB)subscriptitalic-ϕ𝐴subscript𝜌𝐴𝐵\phi_A(\rho_AB)italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) is a classical-quantum state, we can write its entropy as



S(ϕA(ρAB))=S(ϕA(ρA))+∑apaS(ρB|a).𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐴𝐵𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐴subscript𝑎subscript𝑝𝑎𝑆subscript𝜌conditional𝐵𝑎S(\phi_A(\rho_AB))=S(\phi_A(\rho_A))+\sum_ap_aS(\rho_B).italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) = italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) ) + ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B | italic_a end_POSTSUBSCRIPT ) . (8) The asymmetric CC Eq. (3), with the optimisation over projective measurements instead of POVMs, is equivalent to the quantum MI of state ϕA(ρAB)subscriptitalic-ϕ𝐴subscript𝜌𝐴𝐵\phi_A(\rho_AB)italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) maximized over all projective measurements on A𝐴Aitalic_A.



maxa⟩IQ(ϕA(ρAB))subscriptket𝑎subscript𝐼𝑄subscriptitalic-ϕ𝐴subscript𝜌𝐴𝐵\displaystyle\max_\\keta\I_Q(\phi_A(\rho_AB))roman_max start_POSTSUBSCRIPT start_ARG italic_a end_ARG ⟩ end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) (9)



=maxa⟩S(ϕA(ρA))+S(ϕA(ρB))-S(ϕA(ρAB))absentsubscriptket𝑎𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐴𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐵𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐴𝐵\displaystyle=\max_\\keta\S(\phi_A(\rho_A))+S(\phi_A(\rho_B))-S(% \phi_A(\rho_AB))= roman_max start_POSTSUBSCRIPT start_ARG italic_a end_ARG ⟩ end_POSTSUBSCRIPT italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) ) + italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) ) - italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) (10)



=maxa⟩S(ϕA(ρA))+S(ρB)-S(ϕA(ρA))-∑apaS(ρB|a)absentsubscriptket𝑎𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐴𝑆subscript𝜌𝐵𝑆subscriptitalic-ϕ𝐴subscript𝜌𝐴subscript𝑎subscript𝑝𝑎𝑆subscript𝜌conditional𝐵𝑎\displaystyle=\max_\\keta\S(\phi_A(\rho_A))+S(\rho_B)-S(\phi_A(% \rho_A))-\sum_ap_aS(\rho_B)= roman_max start_POSTSUBSCRIPT start_ARG italic_a end_ARG ⟩ end_POSTSUBSCRIPT italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) ) + italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) - italic_S ( italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) ) - ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B | italic_a end_POSTSUBSCRIPT ) (11)



=S(B)-min∑apaS(ρB|a)absent𝑆𝐵subscriptket𝑎subscript𝑎subscript𝑝𝑎𝑆subscript𝜌conditional𝐵𝑎\displaystyle=S(B)-\min_\\keta\\sum_ap_aS(\rho_a)= italic_S ( italic_B ) - roman_min start_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B | italic_a end_POSTSUBSCRIPT ) (12) The interpretation of the measurement that maximizes the above is that it is the projective measurement that least disturbs the state, that is, the projective measurement that results in the least loss in quantum MI.



The extension to a symmetric version is simple. In this case, a projective measurement is performed on both A𝐴Aitalic_A and B𝐵Bitalic_B. After the measurement, ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT will became be a classical-classical state given by



ϕAB(ρAB)=∑a,bpab|a⟩⟨a|⊗|b⟩⟨b|.subscriptitalic-ϕ𝐴𝐵subscript𝜌𝐴𝐵subscript𝑎𝑏tensor-productsubscript𝑝𝑎𝑏ket𝑎bra𝑎ket𝑏bra𝑏\phi_AB(\rho_AB)=\sum_a,bp_ab\keta\braa\otimes\ketb\brab.italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT | start_ARG italic_a end_ARG ⟩ ⟨ start_ARG italic_a end_ARG | ⊗ | start_ARG italic_b end_ARG ⟩ ⟨ start_ARG italic_b end_ARG | . (13) where pab=Tr(ρAB|a⟩⟨a|⊗|b⟩⟨b|)subscript𝑝𝑎𝑏tracetensor-productsubscript𝜌𝐴𝐵ket𝑎bra𝑎ket𝑏bra𝑏p_ab=\Tr\rho_AB\keta\braa\otimes\ketb\brabitalic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = roman_Tr ( start_ARG italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | start_ARG italic_a end_ARG ⟩ ⟨ start_ARG italic_a end_ARG | ⊗ | start_ARG italic_b end_ARG ⟩ ⟨ start_ARG italic_b end_ARG | end_ARG ). The symmetric CC is given by



JS(A;B)subscript𝐽𝑆𝐴𝐵\displaystyle J_S(A;B)italic_J start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) =max,IQ(ϕAB(ρAB))absentsubscriptket𝑎ket𝑏subscript𝐼𝑄subscriptitalic-ϕ𝐴𝐵subscript𝜌𝐴𝐵\displaystyle=\max_\\keta\,\\ketb\I_Q(\phi_AB(\rho_AB))= roman_max start_POSTSUBSCRIPT start_ARG italic_a end_ARG ⟩ , start_ARG italic_b end_ARG ⟩ end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) (14)



=maxa⟩,S(ϕ(ρA))+S(ϕ(ρB))-S(ϕ(ρAB))absentsubscriptket𝑎ket𝑏𝑆italic-ϕsubscript𝜌𝐴𝑆italic-ϕsubscript𝜌𝐵𝑆italic-ϕsubscript𝜌𝐴𝐵\displaystyle=\max_\\keta\,\\ketb\S(\phi(\rho_A))+S(\phi(\rho_B)% )-S(\phi(\rho_AB))= roman_max start_POSTSUBSCRIPT start_ARG italic_a end_ARG ⟩ , start_ARG italic_b end_ARG ⟩ end_POSTSUBSCRIPT italic_S ( italic_ϕ ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) ) + italic_S ( italic_ϕ ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) ) - italic_S ( italic_ϕ ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ) (15)



=max,b⟩∑ah(pa)+∑bh(pb)-∑a,bh(pab)absentsubscriptket𝑎ket𝑏subscript𝑎ℎsubscript𝑝𝑎subscript𝑏ℎsubscript𝑝𝑏subscript𝑎𝑏ℎsubscript𝑝𝑎𝑏\displaystyle=\max_\\keta\,\\ketb\\sum_ah(p_a)+\sum_bh(p_b)-% \sum_a,bh(p_ab)= roman_max start_POSTSUBSCRIPT , start_ARG italic_b end_ARG ⟩ end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) + ∑ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) (16) The symmetric QD is



δS(A;B)=IQ(A;B)-JS(A;B).subscript𝛿𝑆𝐴𝐵subscript𝐼𝑄𝐴𝐵subscript𝐽𝑆𝐴𝐵\delta_S(A;B)=I_Q(A;B)-J_S(A;B).italic_δ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) = italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_A ; italic_B ) - italic_J start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) . (17) The interpretation of the symmetric QD is that it is the smallest loss in quantum MI after local projective measurements on A𝐴Aitalic_A and B𝐵Bitalic_B.



The asymmetric CC and symmetric CC are related by the inequality



JS(A;B)≤J(A|B).subscript𝐽𝑆𝐴𝐵𝐽conditional𝐴𝐵J_S(A;B)\leq J(A|B).italic_J start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) ≤ italic_J ( italic_A | italic_B ) . (18) This inequality results from the fact that the quantum mutual information of a state cannot increase under a measurement of one subsystem. Hence, the correlations after measurement reduce (or remain the same) if Bob does a measurement in addition to Alice.



II.2.2 Extended symmetric quantum discord



An alternative way of defining the classical correlations is using the classical mutual information Piani et al. (2008); Wu et al. (2009); Terhal et al. (2002). This turns out to be equivalent to symmetric QD but is defined when the measurement is any POVM.



Let 𝒜𝒜\mathcalAcaligraphic_A be the random variable that describes measurement outcomes a𝑎aitalic_a on state A𝐴Aitalic_A, and ℬℬ\mathcalBcaligraphic_B be the random variable that describes measurement outcomes b𝑏bitalic_b on state B𝐵Bitalic_B. The classical MI between 𝒜𝒜\mathcalAcaligraphic_A and ℬℬ\mathcalBcaligraphic_B is



IC(𝒜,ℬ)=H(𝒜)+H(ℬ)-H(𝒜ℬ),subscript𝐼𝐶𝒜ℬ𝐻𝒜𝐻ℬ𝐻𝒜ℬI_C(\mathcalA,\mathcalB)=H(\mathcalA)+H(\mathcalB)-H(\mathcalAB),italic_I start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( caligraphic_A , caligraphic_B ) = italic_H ( caligraphic_A ) + italic_H ( caligraphic_B ) - italic_H ( caligraphic_A caligraphic_B ) , (19) where H(X)𝐻𝑋H(X)italic_H ( italic_X ) is the Shannon entropy of variable X𝑋Xitalic_X. It is defined by



H(X)=∑xh(px),𝐻𝑋subscript𝑥ℎsubscript𝑝𝑥H(X)=\sum_xh(p_x),italic_H ( italic_X ) = ∑ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) , (20) where pxsubscript𝑝𝑥p_xitalic_p start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is the probability that X=x𝑋𝑥X=xitalic_X = italic_x.



We introduce a new measure of the classical correlations JESsubscript𝐽𝐸𝑆J_ESitalic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT, we call the extended symmetric CC. It is defined as the maximum classical mutual information between the measurement outcomes made on A𝐴Aitalic_A and B𝐵Bitalic_B, maximized over all POVM measurements.



JES(A;B)subscript𝐽𝐸𝑆𝐴𝐵\displaystyle J_ES(A;B)italic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) =maxΠa,ΠbIC(𝒜,ℬ)absentsubscriptsubscriptΠ𝑎subscriptΠ𝑏subscript𝐼𝐶𝒜ℬ\displaystyle=\max_\\Pi_a\,\\Pi_b\I_C(\mathcalA,\mathcalB)= roman_max start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , roman_Π start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( caligraphic_A , caligraphic_B ) (21)



=maxΠa,Πb∑ah(pa)+∑bh(pb)-∑a,bh(pab)absentsubscriptsubscriptΠ𝑎subscriptΠ𝑏subscript𝑎ℎsubscript𝑝𝑎subscript𝑏ℎsubscript𝑝𝑏subscript𝑎𝑏ℎsubscript𝑝𝑎𝑏\displaystyle=\max_\\Pi_a\,\\Pi_b\\sum_ah(p_a)+\sum_bh(p_b)-% \sum_a,bh(p_ab)= roman_max start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , roman_Π start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) + ∑ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) (22)



=maxΠa,Πb∑bh(pb)-∑apa∑bh(pb|a)absentsubscriptsubscriptΠ𝑎subscriptΠ𝑏subscript𝑏ℎsubscript𝑝𝑏subscript𝑎subscript𝑝𝑎subscript𝑏ℎsubscript𝑝conditional𝑏𝑎\displaystyle=\max_\\Pi_a\,\\Pi_b\\sum_bh(p_b)-\sum_ap_a\sum% _bh(p_a)= roman_max start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , roman_Π start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT ) (23) This quantity is symmetric, because classical mutual information is symmetric. We then define the extended symmetric QD as



δES(A;B)=IQ(A;B)-JES(A;B).subscript𝛿𝐸𝑆𝐴𝐵subscript𝐼𝑄𝐴𝐵subscript𝐽𝐸𝑆𝐴𝐵\delta_ES(A;B)=I_Q(A;B)-J_ES(A;B).italic_δ start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) = italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_A ; italic_B ) - italic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) . (24)



The same quantity is being maximised in Eqs. (16) and (22). This implies two things. Firstly, there is an equivalent interpretation of the symmetric CC as the classical MI between measurement outcomes on A𝐴Aitalic_A and B𝐵Bitalic_B, maximised over local projective measurements. Secondly, we have the inequality,



JES(A;B)≥JS(A;B),subscript𝐽𝐸𝑆𝐴𝐵subscript𝐽𝑆𝐴𝐵J_ES(A;B)\geq J_S(A;B),italic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) ≥ italic_J start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) , (25) because projective measurements are a subset of POVMs. The extended symmetric CC can be viewed as an extension of the symmetric CC, in that it extends the definition of symmetric CC to general POVMs.



II.3 Comparison of CC measures



Figure 1 shows the interpretations of the three different CC quantities. Suppose Alice and Bob share n𝑛nitalic_n copies of a bipartite state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. Alice measures each copy separately using the same measurement. Let 𝒜𝒜\mathcalAcaligraphic_A be the random variable that describes her measurement outcomes. Bob does the same thing, and ℬℬ\mathcalBcaligraphic_B describes his measurement outcomes. The maximum classical MI between 𝒜𝒜\mathcalAcaligraphic_A and ℬℬ\mathcalBcaligraphic_B is the symmetric CC if Alice and Bob can perform projective measurements, extended CC if they can do any POVM measurements.



Now suppose Bob is allowed to interact all his copies before he measures them, as in Figure 1(c). Can he gain any more information about Alice’s measurements outcomes? The answer is yes, provided Alice sends Bob some additional classical information. The maximum information Bob can obtain about Alice’s measurement outcome subtracting the additional classical information Alice sends is equal to the asymmetric CC. A protocol that achieves this rate is described in Ref. Devetak and Winter (2003).



These quantities are related by



min(J(A|B),J(B|A))≥JES(A;B)≥JS(A;B).𝐽conditional𝐴𝐵𝐽conditional𝐵𝐴subscript𝐽𝐸𝑆𝐴𝐵subscript𝐽𝑆𝐴𝐵\min\left(J(A|B),J(B|A)\right)\geq J_ES(A;B)\geq J_S(A;B)\;.roman_min ( italic_J ( italic_A | italic_B ) , italic_J ( italic_B | italic_A ) ) ≥ italic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) ≥ italic_J start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_A ; italic_B ) . (26)



II.4 Multipartite quantum discord



The extended symmetric CC and QD can be defined for multipartite states Piani et al. (2008), using multipartite extensions of the classical MI and quantum MI Watanabe (1960). Let a multipartite state ρA→subscript𝜌→𝐴\rho_\vecAitalic_ρ start_POSTSUBSCRIPT over→ start_ARG italic_A end_ARG end_POSTSUBSCRIPT be distributed to n𝑛nitalic_n parties, where A→=[A1,A2,…,An]→𝐴subscript𝐴1subscript𝐴2…subscript𝐴𝑛\vecA=[A_1,A_2,\ldots,A_n]over→ start_ARG italic_A end_ARG = [ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ]. Let Aisubscript𝐴𝑖A_iitalic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT denote the subsystem received by i𝑖iitalic_i-th party. Each party measures their subsystem, and 𝒜isubscript𝒜𝑖\mathcalA_icaligraphic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT denotes the random variable that describes measurement outcomes aisubscript𝑎𝑖a_iitalic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT on subsystem Aisubscript𝐴𝑖A_iitalic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. The multipartite classical MI is



IC(𝒜1;𝒜2;…;𝒜n)=∑i=1nH(𝒜i)-H(𝒜1𝒜2…𝒜n),subscript𝐼𝐶subscript𝒜1subscript𝒜2…subscript𝒜𝑛superscriptsubscript𝑖1𝑛𝐻subscript𝒜𝑖𝐻subscript𝒜1subscript𝒜2…subscript𝒜𝑛I_C(\mathcalA_1;\mathcalA_2;\ldots;\mathcalA_n)=\sum_i=1^nH(% \mathcalA_i)-H(\mathcalA_1\mathcalA_2\ldots\mathcalA_n),italic_I start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; caligraphic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_H ( caligraphic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_H ( caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … caligraphic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) , (27) The n𝑛nitalic_n-partite extended symmetric CC of state ρA→subscript𝜌→𝐴\rho_\vecAitalic_ρ start_POSTSUBSCRIPT over→ start_ARG italic_A end_ARG end_POSTSUBSCRIPT is



JES(A1;A2;…;An)subscript𝐽𝐸𝑆subscript𝐴1subscript𝐴2…subscript𝐴𝑛\displaystyle J_ES(A_1;A_2;\ldots;A_n)italic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )



=\displaystyle== maxΠa1,Πa2,…,ΠanIC(𝒜1;𝒜2;…;𝒜n),subscriptsubscriptΠsubscript𝑎1subscriptΠsubscript𝑎2…subscriptΠsubscript𝑎𝑛subscript𝐼𝐶subscript𝒜1subscript𝒜2…subscript𝒜𝑛\displaystyle\max_\\Pi_a_1\,\\Pi_a_2\,\ldots,\\Pi_a_n\I_C% (\mathcalA_1;\mathcalA_2;\ldots;\mathcalA_n),roman_max start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; caligraphic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) , (28) The maximization is over local POVM measurements ΠaisubscriptΠsubscript𝑎𝑖\\Pi_a_i\ roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT performed on subsystems Aisubscript𝐴𝑖A_iitalic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.



H(𝒜i)=∑aih(pai),𝐻subscript𝒜𝑖subscriptsubscript𝑎𝑖ℎsubscript𝑝subscript𝑎𝑖H(\mathcalA_i)=\sum_a_ih(p_a_i),italic_H ( caligraphic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_h ( italic_p start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) , (29) where pai=Tr(ρAiΠai)subscript𝑝subscript𝑎𝑖tracesubscript𝜌subscript𝐴𝑖subscriptΠsubscript𝑎𝑖p_a_i=\Tr\rho_A_i\Pi_a_iitalic_p start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_Tr ( start_ARG italic_ρ start_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ).



H(𝒜→)=∑a1∑a2…∑anh(p(a1,a2,…,an)),𝐻→𝒜subscriptsubscript𝑎1subscriptsubscript𝑎2…subscriptsubscript𝑎𝑛ℎ𝑝subscript𝑎1subscript𝑎2…subscript𝑎𝑛H(\vec\mathcalA)=\sum_a_1\sum_a_2\ldots\sum_a_nh(p(a_1,a_2% ,\ldots,a_n)),italic_H ( over→ start_ARG caligraphic_A end_ARG ) = ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_h ( italic_p ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ) , (30) where p(a1,a2,…,an)=Tr(ρA→Πa1Πa2…Πan)𝑝subscript𝑎1subscript𝑎2…subscript𝑎𝑛tracesubscript𝜌→𝐴subscriptΠsubscript𝑎1subscriptΠsubscript𝑎2…subscriptΠsubscript𝑎𝑛p(a_1,a_2,\ldots,a_n)=\Tr\rho_\vecA\Pi_a_1\Pi_a_2\ldots\Pi_% a_nitalic_p ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = roman_Tr ( start_ARG italic_ρ start_POSTSUBSCRIPT over→ start_ARG italic_A end_ARG end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ).



Similarly, the multipartite quantum MI is given by



IQ(A1;A2;…;An)=∑i=1nS(Ai)-S(A1A2…An).subscript𝐼𝑄subscript𝐴1subscript𝐴2…subscript𝐴𝑛superscriptsubscript𝑖1𝑛𝑆subscript𝐴𝑖𝑆subscript𝐴1subscript𝐴2…subscript𝐴𝑛I_Q(A_1;A_2;\ldots;A_n)=\sum_i=1^nS(A_i)-S(A_1A_2\ldots A_n% ).italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_S ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - italic_S ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) . (31) This allows us to define the multipartite extended symmetric QD by



δES(A1;A2;…;An)=IQ(A1;A2;…;An)-JES(A1;A2;…;An).subscript𝛿𝐸𝑆subscript𝐴1subscript𝐴2…subscript𝐴𝑛subscript𝐼𝑄subscript𝐴1subscript𝐴2…subscript𝐴𝑛subscript𝐽𝐸𝑆subscript𝐴1subscript𝐴2…subscript𝐴𝑛\delta_ES(A_1;A_2;\ldots;A_n)=I_Q(A_1;A_2;\ldots;A_n)\\ -J_ES(A_1;A_2;\ldots;A_n).start_ROW start_CELL italic_δ start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - italic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ; italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; … ; italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) . end_CELL end_ROW (32) Rulli and Sarandy defined a similar quantity called the global quantum discord Rulli and Sarandy (2011), but the measurements are restricted to projective measurements. The multipartite extended symmetric QD can be viewed as an extension of the global quantum discord to general POVM measurements.



III Gaussian multipartite CC and QD



Let us define the Gaussian multipartite CC JG(A→)subscript𝐽𝐺→𝐴J_G(\vecA)italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( over→ start_ARG italic_A end_ARG ) to be the maximum classical MI achievable when the measurement on each subsystem are restricted to Gaussian measurements. Hence JG(A→)subscript𝐽𝐺→𝐴J_G(\vecA)italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( over→ start_ARG italic_A end_ARG ) is equivalent to Eq. (II.4) except the maximization is over Gaussian POVMs, rather than all POVMs.



We introduce the Gaussian multipartite QD given by



δG(A→)=IQ(A→)-JG(A→).subscript𝛿𝐺→𝐴subscript𝐼𝑄→𝐴subscript𝐽𝐺→𝐴\delta_G(\vecA)=I_Q(\vecA)-J_G(\vecA).italic_δ start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( over→ start_ARG italic_A end_ARG ) = italic_I start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( over→ start_ARG italic_A end_ARG ) - italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( over→ start_ARG italic_A end_ARG ) . (33) Suppose n𝑛nitalic_n parties each receive one mode of an n𝑛nitalic_n-partite Gaussian state. We now describe how to calculate the Gaussian multipartite QD in this situation. Each party performs a Gaussian measurement on their subsystem. A Gaussian measurement of a single-mode Gaussian state can be described by a phase shift θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT followed by a beam splitter with transmissivity 0≤ti≤10subscript𝑡𝑖10\leq t_i\leq 10 ≤ italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ 1 and orthogonal quadrature measurements Qisubscript𝑄𝑖Q_iitalic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and Pisubscript𝑃𝑖P_iitalic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT on the outputs of the beam splitter. Figure 2 shows a diagram of the measurements performed for the case of a tripartite Gaussian state.



The Gaussian multipartite CC of a bipartite state is



JG(A1,A2)=maxθ1,t1,θ2,t2IC(Q2;Q1P1)+IC(P2|Q2;Q1P1)=maxθ1,t1,θ2,t2[G(Q2)-G(Q2|Q1P1)+G(P2|Q2)-G(P2|Q1P1Q2)],subscript𝐽𝐺subscript𝐴1subscript𝐴2subscriptsubscript𝜃1subscript𝑡1subscript𝜃2subscript𝑡2subscript𝐼𝐶subscript𝑄2subscript𝑄1subscript𝑃1subscript𝐼𝐶conditionalsubscript𝑃2subscript𝑄2subscript𝑄1subscript𝑃1subscriptsubscript𝜃1subscript𝑡1subscript𝜃2subscript𝑡2𝐺subscript𝑄2𝐺|subscript𝑄2subscript𝑄1subscript𝑃1𝐺|subscript𝑃2subscript𝑄2𝐺|subscript𝑃2subscript𝑄1subscript𝑃1subscript𝑄2J_G(A_1,A_2)=\max_\theta_1,t_1,\theta_2,t_2I_C(Q_2;Q_1P_% 1)+I_C(P_2|Q_2;Q_1P_1)\\ =\max_\theta_1,t_1,\theta_2,t_2\big[G(Q_2)-G(Q_2|Q_1P_1)+G% (P_2|Q_2)\\ -G(P_2|Q_1P_1Q_2)\big]\;,start_ROW start_CELL italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = roman_max start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_I start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL = roman_max start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_G ( italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_G ( italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_G ( italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL - italic_G ( italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ] , end_CELL end_ROW (34) where G(Q)𝐺𝑄G(Q)italic_G ( italic_Q ) is the differential entropy of Q𝑄Qitalic_Q. The measurement outcome of a Gaussian measurement performed on a Gaussian state are normally distributed. Consider a random variable X𝑋Xitalic_X that is normally distributed. The probability density of X𝑋Xitalic_X is



px=12πVXexp(-(x-μX)22VX),subscript𝑝𝑥12𝜋subscript𝑉𝑋superscript𝑥subscript𝜇𝑋22subscript𝑉𝑋p_x=\frac1\sqrt2\pi V_X\exp(-\frac(x-\mu_X)^22V_X),italic_p start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 italic_π italic_V start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT end_ARG end_ARG roman_exp ( start_ARG - divide start_ARG ( italic_x - italic_μ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_V start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT end_ARG end_ARG ) , (35) where μXsubscript𝜇𝑋\mu_Xitalic_μ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT is the mean of X𝑋Xitalic_X and VXsubscript𝑉𝑋V_Xitalic_V start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT is the variance of X𝑋Xitalic_X. The differential entropy of X𝑋Xitalic_X is



G(VX)𝐺subscript𝑉𝑋\displaystyle G(V_X)italic_G ( italic_V start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) =-∫-∞∞pxlog2pxdxabsentsuperscriptsubscriptsubscript𝑝𝑥subscript2subscript𝑝𝑥𝑑𝑥\displaystyle=-\int_-\infty^\inftyp_x\log_2p_x\;dx= - ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_d italic_x (36)



=12log2(2πeVX),absent12subscript22𝜋𝑒subscript𝑉𝑋\displaystyle=\frac12\log_2(2\pi eV_X),= divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 2 italic_π italic_e italic_V start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) , (37) Note that the differential entropy of X𝑋Xitalic_X does not depend on the mean of X𝑋Xitalic_X.



There is an additional property that allows us to simplify Eq. (34). The conditional variances do not depend on other measurement outcomes. For example, the variance of Q2subscript𝑄2Q_2italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT conditioned on the measurement outcomes of Q1subscript𝑄1Q_1italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and P1subscript𝑃1P_1italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, denoted VQ2|Q1P1subscript𝑉conditionalsubscript𝑄2subscript𝑄1subscript𝑃1V_Q_2italic_V start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, will be a constant independent of the measurement outcomes of Q1subscript𝑄1Q_1italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and P1subscript𝑃1P_1italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Therefore, Eq. (34) becomes



JG(A1,A2)=maxθ1,t1,θ2,t2[G(VQ2)+G(VP2|Q2)-G(VQ2|Q1P1)-G(VP2|Q1P1Q2)].subscript𝐽𝐺subscript𝐴1subscript𝐴2subscriptsubscript𝜃1subscript𝑡1subscript𝜃2subscript𝑡2𝐺subscript𝑉subscript𝑄2𝐺subscript𝑉conditionalsubscript𝑃2subscript𝑄2𝐺subscript𝑉conditionalsubscript𝑄2subscript𝑄1subscript𝑃1𝐺subscript𝑉conditionalsubscript𝑃2subscript𝑄1subscript𝑃1subscript𝑄2J_G(A_1,A_2)=\max_\theta_1,t_1,\theta_2,t_2\Big[G(V_Q_2)% \\ +G(V_P_2)-G(V_Q_2)-G(V_P_2)\Big].start_ROW start_CELL italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = roman_max start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_G ( italic_V start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL + italic_G ( italic_V start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) - italic_G ( italic_V start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) - italic_G ( italic_V start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ] . end_CELL end_ROW (38) The extension to n𝑛nitalic_n-partite states is



JG(A→)=subscript𝐽𝐺→𝐴absent\displaystyle J_G(\vecA)=italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( over→ start_ARG italic_A end_ARG ) = maxθ1,t1,θ2,t2,…,θn,tn[fragmentssubscriptsubscript𝜃1subscript𝑡1subscript𝜃2subscript𝑡2…subscript𝜃𝑛subscript𝑡𝑛[\displaystyle\ \max_\theta_1,t_1,\theta_2,t_2,\ldots,\theta_n,t_n% \big[roman_max start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_θ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT [



H(𝒜2)-H(𝒜2|𝒜1)𝐻subscript𝒜2𝐻conditionalsubscript𝒜2subscript𝒜1\displaystyle\ H(\mathcalA_2)-H(\mathcalA_2|\mathcalA_1)italic_H ( caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_H ( caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )



+\displaystyle++ H(𝒜3)-H(𝒜3|𝒜2𝒜1)𝐻subscript𝒜3𝐻conditionalsubscript𝒜3subscript𝒜2subscript𝒜1\displaystyle\ H(\mathcalA_3)-H(\mathcalA_3|\mathcalA_2\mathcalA% _1)italic_H ( caligraphic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) - italic_H ( caligraphic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )



⋮⋮\displaystyle\vdots⋮



+\displaystyle++ H(𝒜n)-H(𝒜n|𝒜n-1𝒜n-2…A1)],fragmentsHfragments(subscript𝒜𝑛)Hfragments(subscript𝒜𝑛|subscript𝒜𝑛1subscript𝒜𝑛2…subscript𝐴1)],\displaystyle\ H(\mathcalA_n)-H(\mathcalA_n|\mathcalA_n-1\mathcal% A_n-2\ldots A_1)\big],italic_H ( caligraphic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) - italic_H ( caligraphic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | caligraphic_A start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT caligraphic_A start_POSTSUBSCRIPT italic_n - 2 end_POSTSUBSCRIPT … italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] , (39) where



H(𝒜i)=G(VQi)+G(VPi|Qi)𝐻subscript𝒜𝑖𝐺subscript𝑉subscript𝑄𝑖𝐺subscript𝑉conditionalsubscript𝑃𝑖subscript𝑄𝑖H(\mathcalA_i)=G(V_Q_i)+G(V_Q_i)italic_H ( caligraphic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_G ( italic_V start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) + italic_G ( italic_V start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) (40) and



H(𝒜i|𝒜i-1…𝒜1)=G(VQi|Qi-1Pi-1…Q1P1)+G(VPi|QiQi-1Pi-1…Q1P1).𝐻conditionalsubscript𝒜𝑖subscript𝒜𝑖1…subscript𝒜1𝐺subscript𝑉conditionalsubscript𝑄𝑖subscript𝑄𝑖1subscript𝑃𝑖1…subscript𝑄1subscript𝑃1𝐺subscript𝑉conditionalsubscript𝑃𝑖subscript𝑄𝑖subscript𝑄𝑖1subscript𝑃𝑖1…subscript𝑄1subscript𝑃1H(\mathcalA_i|\mathcalA_i-1\ldots\mathcalA_1)=G(V_Q_i-1P_% i-1\ldots Q_1P_1)\\ +G(V_P_i).start_ROW start_CELL italic_H ( caligraphic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | caligraphic_A start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT … caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_G ( italic_V start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT … italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL + italic_G ( italic_V start_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_Q start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT … italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) . end_CELL end_ROW (41) The calculation of Eq. (III) involves an optimization over 2n2𝑛2n2 italic_n variables. We will demonstrate the calculation of Eq. (III) for some states.



III.1 Properties



We state and prove several desirable properties of the the Gaussian multipartite CC and QD.



1. Gaussian multipartite CC is symmetric. This is true because the classical mutual information is symmetric.



2. The Gaussian multipartite QD is zero for product states. This follows from the nonnegativity of the Gaussian multipartitie QD and the fact that quantum mutual information is zero for product states.



3. The Gaussian multipartite CC does not increase under local Gaussian operations. This is because local Gaussian operations can be considered part of the measurements.



III.2 Example: two-mode EPR state



Consider a two-mode Einstein-Podolsky-Rosen (EPR) state. The covariance of the quadratures of the EPR state is Weedbrook et al. (2012)



VEPR=(cosh2r0sinh2r00cosh2r0-sinh2rsinh2r0cosh2r00-sinh2r0cosh2r).subscript𝑉𝐸𝑃𝑅matrix2𝑟02𝑟002𝑟02𝑟2𝑟02𝑟002𝑟02𝑟V_EPR=\beginpmatrix\cosh 2r&0&\sinh 2r&0\\ 0&\cosh 2r&0&-\sinh 2r\\ \sinh 2r&0&\cosh 2r&0\\ 0&-\sinh 2r&0&\cosh 2r\endpmatrix.italic_V start_POSTSUBSCRIPT italic_E italic_P italic_R end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL roman_cosh 2 italic_r end_CELL start_CELL 0 end_CELL start_CELL roman_sinh 2 italic_r end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL roman_cosh 2 italic_r end_CELL start_CELL 0 end_CELL start_CELL - roman_sinh 2 italic_r end_CELL end_ROW start_ROW start_CELL roman_sinh 2 italic_r end_CELL start_CELL 0 end_CELL start_CELL roman_cosh 2 italic_r end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - roman_sinh 2 italic_r end_CELL start_CELL 0 end_CELL start_CELL roman_cosh 2 italic_r end_CELL end_ROW end_ARG ) . (42) The measurement that attains the Gaussian multipartite CC will have phase shifts of zero, i.e. θ1=θ2=0subscript𝜃1subscript𝜃20\theta_1=\theta_2=0italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0. In fact, the phase shifts will be zero for any quadrature covariance matrix that has zero covariance between Q𝑄Qitalic_Q and P𝑃Pitalic_P quadratures. Values of the other parameters t1subscript𝑡1t_1italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and t2subscript𝑡2t_2italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT were found by performing the optimisation analytically. The optimum occurs when t1=t2=0subscript𝑡1subscript𝑡20t_1=t_2=0italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 or 1111, giving JG(A1,A2)=log2(cosh2r)subscript𝐽𝐺subscript𝐴1subscript𝐴2subscript22𝑟J_G(A_1,A_2)=\log_2(\cosh 2r)italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( roman_cosh 2 italic_r ). This corresponds to performing homodyne measurements on each subsystem.



If r=1𝑟1r=1italic_r = 1 then JG=1.912subscript𝐽𝐺1.912J_G=1.912italic_J start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT = 1.912. Gaming news If instead we consider the extended symmetric CC, where the measurements are not restricted to Gaussian measurements we obtain JES=2.337subscript𝐽𝐸𝑆2.337J_ES=2.337italic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT = 2.337. This is obtained when both parties measure in the Fock number state basis. By restricting to Gaussian measurements we reduce the amount of classical correlations that can be seen. The calculation of Gaussian classical correlations however, is much simpler. In general, it is nontrivial to find the measurement that optimises JESsubscript𝐽𝐸𝑆J_ESitalic_J start_POSTSUBSCRIPT italic_E italic_S end_POSTSUBSCRIPT. Additionally, Gaussian measurements have the added bonus of being easy to do experimentally, requiring only linear optical elements and homodyne measurements.



III.3 Example: noisy EPR state



We calculated the multipartite Gaussian CC and QD for an EPR state subjected to three different types of noise, which is plotted in Fig. 3(a,b,c). A useful result, derived by Ref. Mišta Jr. and Tatham (2016) for the calculation of Gaussian intrinsic entanglement, is that for a state with quadrature covariance matrix



(a0cx00a0cpcx0b00cp0b)matrix𝑎0subscript𝑐𝑥00𝑎0subscript𝑐𝑝subscript𝑐𝑥0𝑏00subscript𝑐𝑝0𝑏\beginpmatrixa&0&c_x&0\\ 0&a&0&c_p\\ c_x&0&b&0\\ 0&c_p&0&b\endpmatrix( start_ARG start_ROW start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_b end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_b end_CELL end_ROW end_ARG ) (43) with cx≥|cp|≥0subscript𝑐𝑥subscript𝑐𝑝0c_x\geq|c_p|\geq 0italic_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ≥ | italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT | ≥ 0, the Gaussian multipartite CC of this state is obtained by a homodyne measurements of the Q𝑄Qitalic_Q quadratures (corresponding to the measurement when t1=t2=1subscript𝑡1subscript𝑡21t_1=t_2=1italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1) if



ab+ba+1ab-ab-cx2≥0.𝑎𝑏𝑏𝑎1𝑎𝑏𝑎𝑏superscriptsubscript𝑐𝑥20\sqrt\fracab+\sqrt\fracba+\frac1\sqrtab-\sqrtab-c_x^% 2\geq 0.square-root start_ARG divide start_ARG italic_a end_ARG start_ARG italic_b end_ARG end_ARG + square-root start_ARG divide start_ARG italic_b end_ARG start_ARG italic_a end_ARG end_ARG + divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_a italic_b end_ARG end_ARG - square-root start_ARG italic_a italic_b - italic_c start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ 0 . (44) Since the two-mode states we consider are symmetric in the Q𝑄Qitalic_Q and P𝑃Pitalic_P quadratures, homodyne measurements of the P𝑃Pitalic_P quadratures (corresponding to the measurement when t1=t2=0subscript𝑡1subscript𝑡20t_1=t_2=0italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0) gives the same classical MI. When the above inequality is not satisfied, numerical optimisation revealed that for all two mode states we considered the optimal measurement is a heterodyne measurement of both modes (corresponding to t1=t2=1/2subscript𝑡1subscript𝑡212t_1=t_2=1/2italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 / 2).



As is typical of quantum discord quantities, we observe that when noise is increased sufficiently such that the state becomes separable, determined using Duan’s inseparability criterion Duan et al. (2000), there is still a nonzero amount of Gaussian multipartite QD.



We also calculate the asymmetric CC and QD. For the states we consider, the asymmetric QD is equal to the asymmetric Gaussian QD, and additionally this is obtained by a heterodyne measurement on one of the subsystems Pirandola et al. (2014). We are unaware of any simple means of calculating the symmetric QD or extended symmetric QD for Gaussian states, so we chose not to calculate these quantities.



III.3.1 Uncorrelated noise



Firstly let us consider the case in which uncorrelated quadrature noise is added to each mode of the EPR state. The quadrature covariance matrix of the resulting state is VEPR+vI4subscript𝑉𝐸𝑃𝑅𝑣subscript𝐼4V_EPR+vI_4italic_V start_POSTSUBSCRIPT italic_E italic_P italic_R end_POSTSUBSCRIPT + italic_v italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT where I4subscript𝐼4I_4italic_I start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is the 4-by-4 identity matrix, and v≥0𝑣0v\geq 0italic_v ≥ 0 is a parameter that controls the amount of noise. A plot of correlation is shown in Fig. 3(a). The total correlations, as measured by the quantum MI, decreases as v𝑣vitalic_v increases. The Gaussian multipartite CC, Gaussian multipartite QD, assymetric CC, and symmetric CC also all decrease as the noise increases.



III.3.2 Multiplicative noise



Now consider the case in which the quadrature covariance is multiplied by a factor v≥1𝑣1v\geq 1italic_v ≥ 1, so the quadrature covariance matrix is vVEPR𝑣subscript𝑉𝐸𝑃𝑅vV_EPRitalic_v italic_V start_POSTSUBSCRIPT italic_E italic_P italic_R end_POSTSUBSCRIPT. This type of noise is realised if the EPR state is generated by mixing on a beam splitter two squeezed states that are impure. Then v𝑣vitalic_v is equal to multiplication of the squeezed state quadrature variances. The information quantities are shown in Fig. 3(b). The total correlations, as measured by the quantum MI, decreases as v𝑣vitalic_v increases. Despite this, the Gaussian multipartite CC and asymmetric CC do increase, however this is at the expense of the Gaussian multipartite QD and asymmetric QD, which decrease.



For v𝑣vitalic_v less than some value, the Gaussian multipartite CC is constant. This is the region in which a homodyne measurement is optimal.



III.3.3 Correlated noise



The third case we consider is adding classically correlated noise to Q𝑄Qitalic_Q quadratures of each mode, and classically anticorrelated noise to the P𝑃Pitalic_P quadratures. The quadrature covariances of the resulting state is



VEPR+(v0v00v0-vv0v00-v0v).subscript𝑉𝐸𝑃𝑅matrix𝑣0𝑣00𝑣0𝑣𝑣0𝑣00𝑣0𝑣V_EPR+\beginpmatrixv&0&v&0\\ 0&v&0&-v\\ v&0&v&0\\ 0&-v&0&v\endpmatrix.italic_V start_POSTSUBSCRIPT italic_E italic_P italic_R end_POSTSUBSCRIPT + ( start_ARG start_ROW start_CELL italic_v end_CELL start_CELL 0 end_CELL start_CELL italic_v end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_v end_CELL start_CELL 0 end_CELL start_CELL - italic_v end_CELL end_ROW start_ROW start_CELL italic_v end_CELL start_CELL 0 end_CELL start_CELL italic_v end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - italic_v end_CELL start_CELL 0 end_CELL start_CELL italic_v end_CELL end_ROW end_ARG ) . (45) The information quantities are shown in Fig. 3(c). Unsurprisingly, the Gaussian multipartite CC and the asymmetric CC increase as a function of v𝑣vitalic_v, because we are adding classically correlated noise.



Adding correlated noise initially reduces the asymmetric QD and Gaussian multipartite QD, which also results in a dip in the quantum MI at the start. The heterodyne measurement is much better for detecting the added classical correlations, so when the heterodyne measurement is optimal the Gaussian multipartite QD is almost constant. For large v𝑣vitalic_v, the asymmetric QD and Gaussian multipartite QD appear almost constant but they are in fact slowly increasing.



It is perhaps counterintuitive that classically correlated noise can increase quantum discord. This can be more easily seen in Fig. 3(d), where classically correlated noise is added to a vacuum state. The state initially has zero correlations, but when the noise is added, all the correlation measures increase, including Gaussian multipartite QD and asymmetric QD. Generating a state with nonzero assymetric QD in this manner was done experimentally by Gu et al. (2012).



III.4 Example: noisy Gaussian tripartite GHZ state



The tripartite Gaussian state equivalent to the three-qubit GHZ and W states is a state with quadrature covariance matrix given by Adesso et al. (2006)



VGHZ=(a0c+0c+00a0c-0c-c+0a0c+00c-0a0c-c+0c+0a00c-0c-0a)subscript𝑉𝐺𝐻𝑍matrix𝑎0superscript𝑐0superscript𝑐00𝑎0superscript𝑐0superscript𝑐superscript𝑐0𝑎0superscript𝑐00superscript𝑐0𝑎0superscript𝑐superscript𝑐0superscript𝑐0𝑎00superscript𝑐0superscript𝑐0𝑎V_GHZ=\beginpmatrixa&0&c^+&0&c^+&0\\ 0&a&0&c^-&0&c^-\\ c^+&0&a&0&c^+&0\\ 0&c^-&0&a&0&c^-\\ c^+&0&c^+&0&a&0\\ 0&c^-&0&c^-&0&a\\ \endpmatrixitalic_V start_POSTSUBSCRIPT italic_G italic_H italic_Z end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_a end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_c start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_a end_CELL end_ROW end_ARG ) (46) where



c±=a2-1±(a2-1)(9a2-1)4a.superscript𝑐plus-or-minusplus-or-minussuperscript𝑎21superscript𝑎219superscript𝑎214𝑎c^\pm=\fraca^2-1\pm\sqrt(a^2-1)(9a^2-1)4a.italic_c start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT = divide start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ± square-root start_ARG ( italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) ( 9 italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 ) end_ARG end_ARG start_ARG 4 italic_a end_ARG . (47) Like for the two-mode case, we calculate the Gaussian multipartite CC and QD for the state subjected to three different types of noise. Figure 4 shows our results. To determine whether a state is separable, we use the method of Giedke et al. (2001).



III.4.1 Uncorrelated noise



Consider a three-mode GHZ state with uncorrelated quadrature noise added to each of the three modes. The resulting state has a quadrature covariances VGHZ+vI6subscript𝑉𝐺𝐻𝑍𝑣subscript𝐼6V_GHZ+vI_6italic_V start_POSTSUBSCRIPT italic_G italic_H italic_Z end_POSTSUBSCRIPT + italic_v italic_I start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT where I6subscript𝐼6I_6italic_I start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT is the 6-by-6 identity matrix. Figure 4(a) is a plot of the information quantities. Just as in the two mode case, the quantum MI, Gaussian multipartite CC and QD all decrease as v𝑣vitalic_v increases.



III.4.2 Multiplicative noise



The information quantities for a three-mode GHZ state with multiplicative noise, i.e. a state with covariance matrix vVGHZ𝑣subscript𝑉𝐺𝐻𝑍vV_GHZitalic_v italic_V start_POSTSUBSCRIPT italic_G italic_H italic_Z end_POSTSUBSCRIPT, are shown in Fig. 4(b). Similar to the two made case, this type of noise reduces the total correlations (quantum MI) and Gaussian multipartite QD, while at the same time increasing the Gaussian multipartite CC when v𝑣vitalic_v is large. Just as in the two-mode case, homodyne measurements on each mode give the a classical mutual information that does not depend on v𝑣vitalic_v. Hence, the Gaussian multipartite CC is constant when the optimal measurement consists of homodyne measurements, which for a=2𝑎2a=2italic_a = 2, is when v<3.082𝑣3.082v<3.082italic_v <3.082.



III.4.3 Correlated noise



Now we consider the case in which correlated noise is added to the Q𝑄Qitalic_Q quadratures of each mode and anticorrelated noise is added to the P𝑃Pitalic_P quadratures. The resulting state has quadrature covariances



VGHZ+v(101010010-0.50-0.51010100-0.5010-0.51010100-0.50-0.501).subscript𝑉𝐺𝐻𝑍𝑣matrix1010100100.500.510101000.50100.510101000.500.501V_GHZ+v\beginpmatrix1&0&1&0&1&0\\ 0&1&0&-0.5&0&-0.5\\ 1&0&1&0&1&0\\ 0&-0.5&0&1&0&-0.5\\ 1&0&1&0&1&0\\ 0&-0.5&0&-0.5&0&1\endpmatrix.italic_V start_POSTSUBSCRIPT italic_G italic_H italic_Z end_POSTSUBSCRIPT + italic_v ( start_ARG start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL - 0.5 end_CELL start_CELL 0 end_CELL start_CELL - 0.5 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - 0.5 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL - 0.5 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - 0.5 end_CELL start_CELL 0 end_CELL start_CELL - 0.5 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARG ) . (48) Note that the matrix contains -0.50.5-0.5- 0.5 terms. This is the largest anticorrelation that three classical variables with variance of 1 can have.



The information quantities for this state are shown in Fig. 4(c). We notice three properties that are the same as the two-mode case. (1) Initially there is a dip in the quantum MI and Gaussian multipartite QD. (2) The Gaussian mulitpartite CC increase as a function of v𝑣vitalic_v. (3) While homodyne measurements are optimal, after the initial dip, the multipartite QD increases as a function of v𝑣vitalic_v. When homodyne measurements are not optimal, the mulitpartite QD appears almost constant but is in fact slowly increasing.



Figure 4(d) shows the correlations present when the correlated noise is added to a three-mode vacuum state. Just as in the two-mode case, the Gaussian multipartite QD discord is nonzero for v≥0𝑣0v\geq 0italic_v ≥ 0, despite the fact that the state is separable.



III.4.4 Measurements



In all of the cases described above, for small v𝑣vitalic_v, the measurement that attains the Gaussian multipartite CC and QD consists of homodyne measurements of the Q𝑄Qitalic_Q quadrature on each mode (t1=t2=t3=1subscript𝑡1subscript𝑡2subscript𝑡31t_1=t_2=t_3=1italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 1). Homodyne measurements of the P𝑃Pitalic_P quadratures does not give the same classical MI because there is an asymmetry in the Q𝑄Qitalic_Q and P𝑃Pitalic_P quadratures of the Gaussian GHZ state; the anticorrelations of the P𝑃Pitalic_P quadratures are less than the correlations of the Q𝑄Qitalic_Q quadratures.



For large v𝑣vitalic_v, the optimal measurement consists of beam splitter transmisivities t1=t2=t3=tsubscript𝑡1subscript𝑡2subscript𝑡3𝑡t_1=t_2=t_3=titalic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_t where t<1𝑡1t<1italic_t <1. In stark contrast to the two-mode case, this value of t𝑡titalic_t depends on v𝑣vitalic_v. A plot the relationship between v𝑣vitalic_v and t𝑡titalic_t is shown in Fig. 5. There is a discontinuity at the point where homodyne measurements are no longer optimal; the value of t𝑡titalic_t abruptly changes from 1 to some value that is less than 1. Note that there is also a discontinuity in the two made case, in which t𝑡titalic_t changes from 1 or 0 to 1/2121/21 / 2.



We have introduced a new measure of the classical correlations of a multipartite Gaussian state, defined as the maximum classical MI between Gaussian measurement outcomes performed on each subsystem. We introduce a new measure of multipartite Gaussian QD defined by subtracting the multipartite Gaussian CC from the multiparite quantum MI of the state. The Gaussian multipartite CC is easy to calculate, requiring an optimisation over at most 2n2𝑛2n2 italic_n variables for an n𝑛nitalic_n-mode Gaussian state. We envisage this measure being relevant in Gaussian quantum information experiments that do not use any non-Gaussian measurements.



We calculated the Gaussian multipartite CC and QD for a two-mode EPR state and a three-mode Gaussian GHZ state subjected to different types of noise.



This research is supported by the Australian Research Council (ARC) under the Centre of Excellence for Quantum Computation and Communication Technology (CE110001027). We acknowledge funding from the Defence Science and Technology group. We would like to thank Mile Gu for discussions on the paper.

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