Quantum-Kaniadakis entropy as a measure of quantum correlations through implicit bounds

The decompositions of separable Werner state, and also isotropic state, are well-known tough issues in quantum information theory, in this work we investigate them in the Bloch vector representation, exploring the symmetric informationally complete positive operator-valued measure (SIC-POVM) in the Hilbert space. We successfully get the decomposition for arbitrary $N\times N$ Werner state in terms of regular simplexes. Meanwhile, the decomposition of isotropic state is found to be related to the decomposition of Werner state via partial transposition. It is interesting to note that in the large $N$ limit, while the Werner states are either separable or non-steerably entangled, most of the isotropic states tend to be steerable.

Entanglement, and quantum correlation, are precious resources for quantum technologies implementation based on quantum information science, such as quantum communication, quantum computing, and quantum interferometry. Nevertheless, to our best knowledge, a directly or numerically computable measure for the entanglement of multipartite mixed states is still lacking. In this work, (i) we derive a measure of the degree of quantum correlation for mixed multipartite states. The latter possesses a closed-form expression valid in the general case unlike, to our best knowledge, all other known measures of quantum correlation. (ii) We further propose an entanglement measure, derived from this quantum correlation measure using a novel regularization procedure for the density matrix. Therefore, a comparison of the proposed measures, of quantum correlation and entanglement, allows one to distinguish between quantum correlation detached from entanglement and the one induced by entanglement and, hence, to identify separable but non-classical states. We have tested our quantum correlation and entanglement measures, on states well-known in literature: a general Bell diagonal state and the Werner states, which are easily tractable with our regularization procedure, and we have verified the accordance between our measures and the expected results for these states. Finally, we validate the two measures in two cases of multipartite states. The first is a generalization to three qubits of the Werner state, the second is a one-parameter three qubits mixed state interpolating between a bi-separable state and a genuine multipartite state, passing through a fully separable state.

Quantum discord is a prominent measure of quantum correlations, playing an\nimportant role in expanding its horizon beyond entanglement. Here we provide an\noperational meaning of (geometric) discord, which quantifies the amount of\nnon-classical correlation of an arbitrary quantum system in terms of its\nminimal distance from the set of classical states, in terms of teleportation\nfidelity for general two qubit and $d \\otimes d$ dimensional isotropic and\nWerner states. A critical value of the discord is found beyond which the two\nqubit state must violate the Bell inequality. This is illustrated by an open\nsystem model of a dissipative two qubit. For the $d \\otimes d$ dimensional\nstates the lower bound of discord is shown to be obtainable from an\nexperimentally measurable witness operator.\n

Entanglement does not describe all quantum correlations and several authors have shown the need to go beyond entanglement when dealing with mixed states. Various different measures have sprung up in the literature, for a variety of reasons, to describe bipartite and multipartite quantum correlations; some are known under the collective name {\it quantum discord}. Yet, in the same sprit as the criteria for entanglement measures, there is no general mechanism that determines whether a measure of quantum and classical correlations is a proper measure of correlations. This is partially due to the fact that the answer is a bit muddy. In this article we attempt tackle this muddy topic by writing down several criteria for a ``good" measure of correlations. We breakup our list into \emph{necessary}, \emph{reasonable}, and \emph{debatable} conditions. We then proceed to prove several of these conditions for generalized measures of quantum correlations. However, not all conditions are met by all measures; we show this via several examples. The reasonable conditions are related to continuity of correlations, which has not been previously discussed. Continuity is an important quality if one wants to probe quantum correlations in the laboratory. We show that most types of quantum discord are continuous but none are continuous with respect to the measurement basis used for optimization.

Quantum steering is an asymmetric correlation which occupies a place between entanglement and Bell nonlocality. In the paradigmatic scenario involving the protagonists Alice and Bob, the entangled state shared between them is said to be steerable from Alice to Bob if the steering assemblage on Bob's side does not admit a local hidden state (LHS) description. Quantum conditional entropies, on the other hand, provide for another characterization of quantum correlations. Contrary to our common intuition, conditional entropies for some entangled states can be negative, marking a significant departure from the classical realm. Quantum steering and quantum nonlocality, in general, share an intricate relation with quantum conditional entropies. In the present contribution, we investigate this relationship. For a significant class, namely the two-qubit Weyl states, we show that negativity of conditional R\'enyi 2-entropy and conditional Tsallis 2-entropy is a necessary and sufficient condition for the violation of a suitably chosen three settings steering inequality. With respect to the same inequality we find an upper bound for the conditional R\'enyi 2-entropy, such that the general two-qubit state is steerable. Moving from a particular steering inequality to local hidden-state descriptions, we show that some two-qubit Weyl states which admit a LHS model possess nonnegative conditional R\'enyi 2-entropy. However, the same does not hold true for some non-Weyl states. Our study further investigates the relation between nonnegativity of conditional entropy and LHS models in two-qudits for the isotropic and Werner states. There we find that, whenever these states admit a LHS model, they possess a nonnegative conditional R\'enyi 2-entropy. We then observe that the same holds true for a noisy variant of the two-qudit Werner state.

Quantum discord as a measure of the quantum correlations cannot be easily computed for most of density operators. In this paper, we present a measure of the total quantum correlations that is operationally simple and can be computed effectively for an arbitrary mixed state of a multipartite system. The measure is based on the coherence vector of the party whose quantumness is investigated as well as the correlation matrix of this part with the remainder of the system. Being able to detect the quantumness of multipartite systems, such as detecting the quantum critical points in spin chains, alongside with the computability characteristic of the measure, make it a useful indicator to be exploited in the cases which are out of the scope of the other known measures.

We introduce a geometric framework for studying Bell nonlocality in Hilbert space, where, for a given quantum state, nonlocality is quantified by the distance between the state and the set of local states. This approach applies to any Bell inequality and any measurement scenario. Whenever the local set is characterized, the proposed nonlocality measure can be computed explicitly. As a general result, we prove that for any scenario in arbitrary dimension the closest local state to a Werner state is itself a Werner state, and analogously, the closest local state to an isotropic state is again isotropic. In the two-qubit case, we further show that the closest local state to a Bell-diagonal state is Bell-diagonal as well. These structural results are independent of the specific Bell inequality considered, thus revealing intrinsic geometric features of these families of states and providing significant simplifications for computing the proposed measures. For the Clauser–Horne–Shimony–Holt inequality in two-qubit systems and the Collins-Gisin-Linden-Massar-Popescu inequality for two qudits of arbitrary finite dimension, we derive explicit geometric measures of nonlocality for Bell-diagonal, Werner, and isotropic states using various distance metrics, including the trace, Hellinger, Hilbert–Schmidt distances, and relative entropy. Furthermore, we prove in all generality that for all scenarios in which the local set is not fully characterized, the geometric measures provide rigorous lower bounds on nonlocality.

It is shown that for a given bipartite density matrix and by choosing a suitable separable set (instead of product set) on the separable-entangled boundary, optimal Lewenstein-Sanpera (L-S) decomposition can be obtained via optimization for a generic entangled density matrix. Based on this, We obtain optimal L-S decomposition for some bipartite systems such as $2\otimes 2$ and $2\otimes 3$ Bell decomposable states, generic two qubit state in Wootters basis, iso-concurrence decomposable states, states obtained from BD states via one parameter and three parameters local operations and classical communications (LOCC), $d\otimes d$ Werner and isotropic states, and a one parameter $3\otimes 3$ state. We also obtain the optimal decomposition for multi partite isotropic state. It is shown that in all $2\otimes 2$ systems considered here the average concurrence of the decomposition is equal to the concurrence. We also show that for some $2\otimes 3$ Bell decomposable states the average concurrence of the decomposition is equal to the lower bound of the concurrence of state presented recently in [Buchleitner et al, quant-ph/0302144], so an exact expression for concurrence of these states is obtained. It is also shown that for $d\otimes d$ isotropic state where decomposition leads to a separable and an entangled pure state, the average I-concurrence of the decomposition is equal to the I-concurrence of the state. Keywords: Quantum entanglement, Optimal Lewenstein-Sanpera decomposition, Concurrence, Bell decomposable states, LOCC} PACS Index: 03.65.Ud

We investigate some basic scenarios in which a given set of bipartite quantum\nstates may consistently arise as the set of reduced states of a global\nN-partite quantum state. Intuitively, we say that the multipartite state\n"joins" the underlying correlations. Determining whether, for a given set of\nstates and a given joining structure, a compatible N-partite quantum state\nexists is known as the quantum marginal problem. We restrict to bipartite\nreduced states that belong to the paradigmatic classes of Werner and isotropic\nstates in d dimensions, and focus on two specific versions of the quantum\nmarginal problem which we find to be tractable. The first is Alice-Bob,\nAlice-Charlie joining, with both pairs being in a Werner or isotropic state.\nThe second is m-n sharability of a Werner state across N subsystems, which may\nbe seen as a variant of the N-representability problem to the case where\nsubsystems are partitioned into two groupings of m and n parties, respectively.\nBy exploiting the symmetry properties that each class of states enjoys, we\ndetermine necessary and sufficient conditions for three-party joinability and\n1-n sharability for arbitrary d. Our results explicitly show that although\nentanglement is required for sharing limitations to emerge, correlations beyond\nentanglement generally suffice to restrict joinability, and not all unentangled\nstates necessarily obey the same limitations. The relationship between\njoinability and quantum cloning as well as implications for the joinability of\narbitrary bipartite states are discussed.\n

We investigate evolution of quantum correlations in ensembles of two-qubit\nnuclear spin systems via nuclear magnetic resonance techniques. We use discord\nas a measure of quantum correlations and the Werner state as an explicit\nexample. We first introduce different ways of measuring discord and geometric\ndiscord in two-qubit systems and then describe the following experimental\nstudies: (a) We quantitatively measure discord for Werner-like states prepared\nusing an entangling pulse sequence. An initial thermal state with zero discord\nis gradually and periodically transformed into a mixed state with maximum\ndiscord. The experimental and simulated behavior of rise and fall of discord\nagree fairly well. (b) We examine the efficiency of dynamical decoupling\nsequences in preserving quantum correlations. In our experimental setup, the\ndynamical decoupling sequences preserved the traceless parts of the density\nmatrices at high fidelity. But they could not maintain the purity of the\nquantum states and so were unable to keep the discord from decaying. (c) We\nobserve the evolution of discord for a singlet-triplet mixed state during a\nradio-frequency spin-lock. A simple relaxation model describes the evolution of\ndiscord, and the accompanying evolution of fidelity of the long-lived singlet\nstate, reasonably well.\n

Recently, local quantum uncertainty (LQU) as a measure of quantum correlations has been proposed by Girolami et al. We here have investigated the LQU dynamics in a pair of qubits system subjected to either local or collective classical phase noises. The explicit analytical expressions of LQU for the two-qubit system of interest initially in a Werner state under these dephasing noises have been derived. We compare the dynamics of LQU with that of entanglement as measured with concurrence. Our results show that LQU always decays asymptotically while the entanglement displays sudden death phenomenon. Besides, there is no simple relation between the LQU and entanglement since LQU may be smaller or larger than entanglement. Finally, we try to protect LQU against the dephasing noise by means of filtering operation.

Measurement-induced nonlocality (MIN), a quantum correlation measure for bipartite systems, is an indicator of maximal global effects due to locally invariant von Neumann projective measurements. It is originally defined as the maximal square of the Hilbert–Schmidt norm of the difference between pre- and post-measurement states. In this article, we propose a new form of MIN based on affinity. This quantity satisfies all the criteria of a bona fide measure of quantum correlation measures. This quantity is evaluated for both arbitrary pure and 2 × n dimensional (qubit–qudit) mixed states. The operational meaning of the proposed quantity is interpreted in terms of the interferometric power of the quantum state. We apply these results on two-qubit mixed states, such as the Werner, isotropic and Bell diagonal states.

In order to quantify the information content of quantum states in the presence of conserved quantities, Wigner and Yanase introduced the notion of skew information [Proc. Natl. Acad. Sci. USA 49, 910, (1963)], which was later identified as a paradigmatic version of quantum Fisher information. The skew information is quite different from, yet deeply connected with, the ubiquitous quantum entropies and has important applications in quantum theory. In this paper, pursuing further the original idea of Wigner and Yanase, we propose a measure for correlations in terms of the skew information, investigate its fundamental properties, and elucidate its characteristics. An appealing feature of this measure for correlations is that its evaluation does not involve any optimization, in sharp contrast to the entanglement and discord measures, and can be straightforwardly calculated. In particular, the algorithm and explicit formulas for general bipartite states are prescribed, and simple analytical expressions for some special states, including arbitrary bipartite pure states, the Bell-diagonal states, and some highly symmetric states such as the Werner states and the isotropic states, are obtained.

We investigate and compare three distinguished geometric measures of bipartite quantum correlations that have been recently introduced in the literature: the geometric discord, the measurement-induced geometric discord, and the discord of response, each one defined according to three contractive distances on the set of quantum states, namely the trace, Bures, and Hellinger distances. We establish a set of exact algebraic relations and inequalities between the different measures. In particular, we show that the geometric discord and the discord of response based on the Hellinger distance are easy to compute analytically for all quantum states whenever the reference subsystem is a qubit. These two measures thus provide the first instance of discords that are simultaneously fully computable, reliable (since they satisfy all the basic Axioms that must be obeyed by a proper measure of quantum correlations), and operationally viable (in terms of state distinguishability). We apply the general mathematical structure to determine the closest classical-quantum state of a given state and the maximally quantum-correlated states at fixed global state purity according to the different distances, as well as a necessary condition for a channel to be quantumness breaking.

A kind of new geometric measure of quantum correlations is formulated. The proposed formulation is in terms of the quantum Tsallis relative entropy and can naturally be viewed as a one-parameter extension quantum discordlike measure that satisfies all requirements of a good measure of quantum correlations. It is of an elegant analytic expression and contains several existing good quantum correlation measures as special cases. The partial coherence measure is also investigated.