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Register a new userNecessary and sufficient conditions for local discrimination of generalized Bell states: finding out all locally indistinguishable sets of generalized Bell states
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In general, for a bipartite quantum system $mathbb{C}^{d}otimesmathbb{C}^{d}$ and an integer $k$ such that $4leq kle d$,there are few necessary and sufficient conditions for local discrimination of sets of $k$ generalized Bell states (GBSs) and it is difficult to locally distinguish $k$-GBS sets.In this paper, we consider the local discrimination of GBS sets and the purpose is to completely solve the problem of local discrimination of GBS sets in some bipartite quantum systems,specifically, we show some necessary and sufficient conditions for local discrimination of GBS sets by which the local discrimination of GBS sets can be quickly determined.Firstly some sufficient conditions are given, these sufficient conditions are practical and effective.Fan$^{,}$s and Wang et al.$^{,}$s results (Phys Rev Lett 92:177905, 2004: Phys Rev A 99:022307, 2019) can be deduced as special cases of these conditions.Secondly in $mathbb{C}^{4}otimesmathbb{C}^{4}$, a necessary and sufficient condition for local discrimination of GBS sets is provided,all locally indistinguishable 4-GBS sets are found,and then we can quickly determine the local discriminability of an arbitrary GBS set.In $mathbb{C}^{5}otimesmathbb{C}^{5}$, a concise necessary and sufficient condition for one-way local discrimination of GBS sets is obtained,which gives an affirmative answer to the case $d=5$ of the problem proposed by Wang et al. (Phys Rev A 99:022307, 2019).
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In this paper, we study the local unitary classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we firstly introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all of pairs in $dotimes d$ quantum system into $prod_{j=1}^{n} (k_{j} + 1) $ LU-inequivalent pairs when the prime factorization of $d=prod_{j=1}^{n}p_j^{k_j}$. Moreover, all of triples in $p^alphaotimes p^alpha$ quantum system for prime $p$ can be partitioned into $frac{(alpha + 3)}{6}p^{alpha} + O(alpha p^{alpha-1})$ LU-inequivalent triples, especially, when $alpha=2$ and $p>2$, there are exactly $lfloor frac{5}{6}p^{2}rfloor + lfloor frac{p-2}{6}+(-1)^{lfloorfrac{p}{3}rfloor}frac{p}{3}rfloor + 3$ LU-inequivalent triples.
In this paper, we mainly consider the local indistinguishability of the set of mutually orthogonal bipartite generalized Bell states (GBSs). We construct small sets of GBSs with cardinality smaller than $d$ which are not distinguished by one-way local operations and classical communication (1-LOCC) in $dotimes d$. The constructions, based on linear system and Vandermonde matrix, is simple and effective. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs, and greatly improve previous results in [Zhang emph{et al.}, Phys. Rev. A 91, 012329 (2015); Wang emph{et al.}, Quantum Inf. Process. 15, 1661 (2016)]. The case that $d$ is odd of the results also shows that the set of 4 GBSs in $5otimes 5$ in [Fan, Phys. Rev. A 75, 014305 (2007)] is indeed a 1-LOCC indistinguishable set which can not be distinguished by Fans method.
Entanglement is the defining feature of quantum mechanics, and understanding the phenomenon is essential at the foundational level and for future progress in quantum technology. The concept of steering was introduced in 1935 by Schrodinger as a generalization of the Einstein-Podolsky-Rosen (EPR) paradox. Surprisingly, it has only recently been formalized as a quantum information task with arbitrary bipartite states and measurements, for which the existence of entanglement is necessary but not sufficient. Previous experiments in this area have been restricted to the approach of Reid [PRA 40, 913], which followed the original EPR argument in considering only two different measurement settings per side. Here we implement more than two settings so as to be able to demonstrate experimentally, for the first time, that EPR-steering occurs for mixed entangled states that are Bell-local (that is, which cannot possibly demonstrate Bell-nonlocality). Unlike the case of Bell inequalities, increasing the number of measurement settings beyond two--we use up to six--dramatically increases the robustness of the EPR-steering phenomenon to noise.
For a bipartite entangled state shared by two observers, Alice and Bob, Alice can affect the post-measured states left to Bob by choosing different measurements on her half. Alice can convince Bob that she has such an ability if and only if the unnormalized postmeasured states cannot be described by a local-hidden-state (LHS) model. In this case, the state is termed steerable from Alice to Bob. By converting the problem to construct LHS models for two-qubit Bell diagonal states to the one for Werner states, we obtain the optimal models given by Jevtic textit{et al.} [J. Opt. Soc. Am. B 32, A40 (2015)], which are developed by using the steering ellipsoid formalism. Such conversion also enables us to derive a sufficient criterion for unsteerability of any two-qubit state.
We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In the simplest nontrivial scenario of two projective measurements, we solve this problem completely by virtue of the connection between the steering problem and the joint-measurement problem. A necessary and sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that a Bell-diagonal state is steerable by two projective measurements iff it violates the Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast with the strict hierarchy expected between steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the volume of the steering ellipsoid. In particular, we determine the maximal concurrence and ellipsoid volume of Bell-diagonal states that are not steerable by two projective measurements. Finally, we explore the steerability of Bell-diagonal states under three projective measurements. A simple sufficient criterion is derived, which can detect the steerability of many states that are not steerable by two projective measurements. Our study offers valuable insight on steering of Bell-diagonal states as well as the connections between entanglement, steering, and Bell nonlocality.
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