Multipartite quantum and classical correlations in symmetric n-qubit mixed states

We discuss how to calculate genuine multipartite quantum and classical correlations in symmetric, spatially invariant, mixed $n$-qubit density matrices. We show that the existence of symmetries greatly reduces the amount of free parameters to be optimized in order to find the optimal measurement that minimizes the conditional entropy in the discord calculation. We apply this approach to the states exhibited dynamically during a thermodynamic protocol to extract maximum work. We also apply the symmetry criterion to a wide class of physically relevant cases of spatially homogeneous noise over multipartite entangled states. Exploiting symmetries we are able to calculate the nonlocal and genuine quantum features of these states and note some interesting properties.


Introduction
How quantum systems can be correlated is typically discussed within three categories: non-local, non-separable (entangled), or non-classical (discordant). Their formal differences aside, one common aspect of them all is the difficulty in extending any definition to mixed multipartite systems. Quantum discord was originally defined as the difference between two quantum analogues of the classical mutual information. 1,2 The interest around such a quantumness quantifier is justified considering the fact that there are examples in mixed-state quantum computation where it appears clear that entanglement in not the only meaningful indicator. 4 Even if the role of discord in quantum computation has not yet been fully clarified, there are many contexts where its use has been productive. 5 The original definition of discord, being based on the separation between system and apparatus, 1 applies naturally to bipartite systems. As in the case of entanglement 3 , the extension to the multipartite scenario is not trivial. A series of postulates any good measure of multipartite correlations should obey was given in Ref. 6 by Bennett et al.. Three main generalizations of quantum discord to the multipartite case have been proposed: Rulli and Sarandy introduced the so called global discord (GD) which is a natural extension of a symmetrized version of the QD and is based on a collective measurement; 7 Modi and co-workers proposed a unified view of correlations that applies in both the bipartite and in the multipartite scenarios based on the use of the relative entropy to quantify the "distance" between states; 8 in Ref. 9, relative entropy was also employed to give a measure of genuine total, classical, and quantum correlations. Genuine correlations were defined as the amount of correlation that cannot be accounted for considering any of the possible subsystems. It was explicitly described how to implement the general definition to the case of three-qubit pure states. However, due to consistency problems, it is not clear to what states that definition can be easily applied.
Recently, people started investigating the possibility of exploiting the degree of quantumness of a system in order to achieve advantage in thermodynamic processes. More specifically, the role of entanglement was studied in Refs. 10, 11 in connection to the maximal work extraction problem in finite quantum systems under cyclic transformations. 12 Considering the same physical process, in Ref. 13, we analyzed the behavior of multipartite quantum discord, using both the definition of global discord 7,14 and the measure of genuinely multipartite correlations. 9 The protocol consists of a series of swap operations between different eigenstates of a density matrix that is diagonal with respect to the Hamiltonian basis both at the beginning and at the end of the cycle. As the state conserves its spatial invariance during the cycle, it is possible to apply symmetry considerations. In this paper, after reviewing the general definition of genuine quantum and classical correlations, we will show how to calculate them for the family of states generated during the thermodynamic process described here.
An equally important line of inquiry has been the dynamics of quantum correlations in open systems. In particular, understanding how quantum correlations behave under adverse effects is interesting from both a pragmatic viewpoint regarding the utility of such correlations, and in understanding the differences and similarities arising when studying different types of quantum correlations. Only recently has the study of multipartite non-local 17,18 and non-classical 19,20 correlations in such adverse situations been studied since the calculation of these quantities is typically extremely difficult. We show for some widely applicable noisy processes that we can again efficiently calculate both the non-local and genuinely quantum correlations for a class of symmetric states.

Genuine multipartite correlations
In this section we briefly review the definition of genuine quantum and classical correlations given in Ref. 9. Given an n-partite density matrix and and its reduced states j (j = 1, . . . , n), genuine correlations can be defined starting from the generalization of mutual information to n parties T ( ) as a measure of total correlations: where S(.) is the von Neumann entropy. Genuine correlations represent the amount of correlations that cannot be accounted for considering any of the possible reduced subsystems: an n-partite state has genuine n-partite correlations if it is non-product in every bipartite cut (this definition is in agreement with the general criteria given in Ref. 6). According to this criterion, genuine total correlations T (n) ( ) coincide with the distance, measured through the relative entropy, between and the closest state with no n-partite correlations, that is, the closest state which is product at least along a bipartite cut. 8 This definition implies that T (n) coincides with the minimum bipartite mutual information present in the system. Because of its bipartite character, T (n) can be divided into its quantum (D (n) ) and classical (J (n) ) parts as if the system were a true bipartite one. As shown in Ref. 9, a consistent set of definitions for any level of separability can be done for pure states of three qubits |ψ 123 . It was found that When it comes to the general case of mixed states of any dimension, it is not always possible to apply this definition. Even in the tripartite case, it can happen that the sum of tripartite and bipartite quantum (or classical) correlations exceeds its total value. This is due to the fact that the different sub-parts are in general different to each other, which causes the different combinations of correlations not to sum as desired. This problem is obviously not present if the state under investigation is spatial symmetric, which will be the case for the states we discuss in this work.

Global discord
The global discord (GD) is a multipartite extension of the original definition of the bipartite discord when collective measurements are applied. 7 It is defined p l l , and p stands for the string of indices (p 1 . . . p n ). Dif-ferently to the measure presented in the previous section, the GD is not sensitive to genuine correlations, i.e. states that are quantumly correlated but are nevertheless separable over some bipartition. However, it is a reliable indicator of quantumness in a given state, a non-zero GD guarantees that at least some of the subsystems exhibit quantum correlations.
In general this quantity is difficult to calculate again due to the required minimization in Eq. (3), however symmetries can help simplify the calculation. An alternative formulation that reduces the computational effort was given in Ref. 14, where the multi-qubit projections were expressed in terms of local multi-qubit rotations,R i (θ i , φ i ) = cos θ i1 1+i sin θ i cos ϕ iσy +i sin θ i sin ϕ iσx applied to the separable eigenstates of ⊗ n i=1σ z . The minimization of Eq. (3) of an arbitrary state is then a minimization over the 2n angles associated with the rotations. When the state is symmetric, as we shall show, it is possible to reduce the number of parameters to minimize over.

Multipartite non-locality
The final indicator we will assess is the multipartite non-locality based on the violation of a Bell-type inequality. There exist a wide variety of such inequalities, each with its own merits, we will restrict ourselves to extension of the tripartite Svetlichy inequality 15 to N -partite systems given by Collins et al 16 , wherein an iterative means to to construct multipartite Bell inequalities for dichotomic observables was given. Taking o j and O j as the two outcomes of a measurement performed over one of them. Setting m 1 = o 1 (M 1 = O 1 ), the polynomials are given by We can then define the generalized Svetlichny polynomials N n = m n (n even) The bound imposed on N n by local hidden-variable models is 1, while quantum mechanically an n-qubit GHZ state achieves the maximum value of √ 2 n−1 and √ 2 n−2 for an even and odd number of particles, respectively. These inequalities are particularly useful as they detect genuine multipartite non-locality similar to the genuine correlations outlined in Sec. 2.1. Additionally, depending on the degree of violation we can determine if there exists a k-separable hidden variable model to reproduce the correlations in a given state. While formally there is no way to 'quantify' the non-locality in a state, an intuitive picture can be obtained by considering a states resilience to noise before it no longer violates a given inequality.

Genuine correlations of symmetric states
Given a symmetric n-partite state, genuine quantum (classical) correlations are, unambiguously, the quantum (classical) part of the minimum bipartite correlation in the state. Following the same criterion also m-partite correlations (m < n) can be calculated. Despite the existence of a clear definition, as calculating discord requires a minimization procedure, it is in general hard to find analytical results. However, the presence of symmetries can greatly simplify the calculation.
Our goal is to calculate any {n − k : k} discord of an n-partite state, that is, the discord between the first n−k and last k sub-parts of the state (or any other spatial combination). The elements of a complete set of orthogonal projectors representing the measurement process on k sub-parts can be generically written as P i = |ψ i ψ i |, with where |i is any of the states of the k-partite basis. By definition of bipartite discord, where {k} is the reduced density matrix over the k sub-parts of the state, p i = Tr { ψ i | |ψ i }, and where the measurement process minimizing the conditional entropy has been already found. Let us assume the existence of a symmetry operator acting on the k qubits to be measured under the action of which the total density matrix is left unchanged: If we put Eq. (8) into Eq. (7) we get where |ψ i = U {k} |ψ i . Notice that the coefficients p i are not modified. As a consequence, the comparison between Eq. (7) to Eq. (9) leads us to the conclusion that the optimal basis {|ψ i } is made by eigenstates of U {k} , that is, |ψ i = e iφi |ψ i .
where I j is the identity operator in the j-excitation subspace. This state can be obtained starting from (p 0 |0 0| + p 1 |1 1|) ⊗n by coherently mixing |0 ⊗n to |1 ⊗n to the end of reordering these eigenvalues of the state at the end of the cycle. For this density matrix, two symmetry operators can be identified: the translation operator T (here, for the sake of clarity, with the word translation we mean any spatial manipulation of the state), which embodies the spatial invariance of the state, and parity-related operator (the parity P tells us if in the state there is an even or odd number of 0s). Let us consider the form of the symmetric eigenstates explicitly for different values of k (starting from k = 2, as we need a multipartite basis for the measurement).
• k = 2 The symmetries are T and the parity P . The family of common eigenvectors of these operators is |ψ 1 = cos θ|00 + sin θ|11 |ψ 2 = − sin θ|00 + cos θ|11 So, the minimization necessary to calculate discord is reduced to finding the optimal value of a single parameter θ (which is found to be π/4).Then, the Bell basis is optimal. • k = 3 In this case, any translation operator acting on the system commutes with , while the parity does not. Nevertheless, a symmetry can be found by multiplying P by an operator, acting on the three qubits, such that |0 → |0 and |1 → e iπ/3 |1 (let us call this symmetry operator P (3) ). Eigenstates of P (3) invariant under translation are |000 and |111 with eigenvalue +1, |W = (|001 + |010 + |100 )/ √ 3 and two orthogonal combinations more |W and |W (one of them could be, for instance, (|001 + e i2π/3 |010 + e −i2π/3 |100 )/ Actually, given that when evaluating ψ i | |ψ i phases do not matter, we can safely use |ψ 1 , |ψ 2 , three times |W , and three times |W . As in the k = 2 case, the optimal value of θ is found to be π/4. The optimal basis then consists of maximally entangled states. • k > 3 The generalization to any k is straightforward. A symmetry operator P (k) always exists and is the product of the parity P by a second operator such that |0 → |0 and |1 → e iπ/k |1 if k is odd and |1 → e 2iπ/k |1 if k is even. The optimal set of measurements is represented by |GHZ ± = (|0 ⊗k ± |1 ⊗k )/ √ k and by the generalized |W state (Fourier mode) in any of the j-excitation subspaces (j = 1, . . . , k − 1) with multiplicity k j .
So, despite the multipartite nature of the state a number of operations of size k is sufficient to calculate any bipartite discord between many-qubit parts. In order to get the true n-partite discord a quantitative comparison among all the possible cuts is finally required. In Fig. 1 we show the behavior of D (n) as a function of the state parameter p 0 for different values of n. Any of the lines plotted has been obtained taking the minimum over all the possible {n − k : k} partitions.

Global discord
For Eq. (10) we can also efficiently calculate analytically the global discord. The first simplification is noting that the final term in Eq. (3) corresponds to a relative entropy for each individual qubit, which due to the fully symmetric nature of the state we can fully determine this term by calculating it for any single qubit and taking n times this. The most difficult part comes from calculating the relative entropy for the total state given by the first term in Eq. (3), since for an arbitrary n qubit state the minimization requires 2n angles. However, since the state is fully symmetric it is clear that only a single angle is required, as it is the same applied to each qubit. For the state at hand we find the angle required to minimize Eq. is θ = π 2 . Finally we find the GD for Eq. (10) In Fig. 1 (b) we show the behavior of G n as a function of the state parameter p 0 for different values of n.

Multipartite correlations in open systems
We next assess the multipartite measures and indicators in open systems. In particular we study the broadly applicable amplitude and phase damping channels applied to the generalized n-qubit GHZ multipartite entangled states. The amplitude damping (AD) channel describes the probability of losing an excitation to the surrounding environment. By modeling the environment, E, as a qubit interacting with the system qubit, S, the action of the amplitude damping channel is The phase damping (PD) channel acts in a similar manner, the only difference being this affects only the coherence present in the system and leaves the populations (i.e. the energy) unchanged. The phase damping channel acts as We apply these channels locally to each qubit of the multipartite states (assuming the same damping rate for all qubits). By then tracing over the all environment degrees of freedom we determine the density matrices for the various n-qubit noisy states. Starting from the generalized n-qubit GHZ state Multipartite quantum and classical correlations in symmetric mixed states 9 we find the corresponding multipartite noisy states take the form

Genuine correlations
The same derivation described in Sec. 3 can be applied to the case of a multipartite GHZ state under amplitude damping. In fact, AD nGHZ is both T -and P (k) -symmetric and all the considerations made before are valid. The optimal value of θ, however, will not be generally equal to π/4 any more.
As for the case of P D GHZ , the solution is even simpler, given that the state is rank 2, and, irrespective of the cut considered, it is always possible to purify it adding an external qubit as ancilla. 21 Then, the Koashi-Winter formula can be used to find an analytical expression for the multipartite discord. 22 For instance, in the tripartite case, we have where a denotes the ancilla and E(.) is the entanglement of formation, which can be calculated analytically, since in the jk part only two levels are populated.

Nonlocality
Determining the non-locality is again a difficult task because in order to search for a violation of the inequality we are required to maximize over the two chosen measurement settings. This is a state-dependent issue, meaning that for a given state we need to be careful precisely what measurement settings we choose, and regardless, as with the calculation of discord, there is an optimization required. Each of the different elements entering the Svetlichny inequalities Eq. (5) corresponds to an n-partite correlation function determined by applying local rotations to each qubit. For the damped GHZ state Eq. (15) we apply are the local rotation R(θ ki i ) = cos(θ ki i )σ x +sin(θ ki i )σ y to each of the n qubits with k i = 1 or 2 being the two possible measurement settings. We find the correlation function takes a simple form C(θ k1 1 . . . θ kn n ) = Tr R(θ k1 1 . . . θ kn n ) AD GHZ = 2(1 − λ) n 2 α 1 α 2 cos θ k1 1 + · · · + θ kn n , with k i = 1 or 2. All that remains is to iteratively determine the corresponding Svetlichny polynomial for a given n to study the non-locality. For clarity, let us explicitly determine this for n = 2. We have then from Eq. (5) with each term inside the brackets corresponding to a correlation function. Using Eq. (18) with n = 2 we have which is the well known CHSH inequality for a 2 qubit generalized Bell state. Larger n then follows iteratively applying Eq. (5) in the same fashion, and we find there are 2 n distinct correlation functions appearing in the inequality. It is interesting to notice that if we calculate the same quantities for the PD affected GHZ state, Eq. (16), we find precisely the same result. This means that the non-locality is not sensitive the type of noise the state is undergoing. In Fig. 2 we show the non-locality for a n-qubit GHZ state undergoing either AD or PD. In panel (a) α 1 = 1 √ 2 , i.e. we start from a perfect GHZ state. A few interesting features to note, the larger the system the more noise it can withstand before it no longer violates the classical bound of 1 (lowest solid horizontal line). The dashed horizontal lines in the plots at 2 and 4, correspond to the bounds for 1 : (n − 1) separability for 4 or 5 and 6 or 7 qubits respectively. Values below these lines for a general state means we would not be certain the violation of Eq. (5) was due to full n-partite or n − 1-partite correlations, however given the initial form of the state we know in this special instance that our violation arises due to full n-partite correlations. In panel (b) we , this state is initially much less correlated. A qualitatively similar behavior holds, however, we never see large violations of the inequalities and for very small systems n = 2, 3 there is no violation at all.

Conclusions
Even if calculating correlations in multipartite states is a hard and generally unsolved problem, the existence of symmetries greatly simplifies the task. Inspired by maximal work extraction protocol from a cyclic transformation (ergotropy), we have shown how to calculate genuine and global discord under eigenstate swap. Using the same symmetry considerations, we have also studied the behavior of multipartite nonlocality, together with genuine and global discord, in the presence of local noise. Beyond the quantification of the correlations in this instance, we also noted a stark difference between the non-locality and discord; while the discord was sensitive to the type of lossy channel applied, the nonlocality was unable to discriminate which noisy process the state was undergoing.
Note Added: On completion of this work we became aware of Ref. 23 where the authors present a complementary study where they provide analytical forms for quantifying the quantum correlations in a class of symmetric tripartite states related to those states studied here.