We show that a set of linearly independent quantum states $\{(U_{m,n}\otimes I)ρ^{AB}(U_{m,n}^{\dagger}\otimes I)\}_{m,n=0}^{d-1}$, where $U_{m,n}$ are generalized Pauli matrices, cannot be discriminated deterministically or probabilistically by local operations and classical communications (LOCC). On the other hand, any $l$ maximally entangled states from this set are locally distinguishable if $l(l-1)\le 2d$. The explicit projecting measurements are obtained to locally discriminate these states. As an example, we show that four Werner states are locally indistinguishable.
In this paper, we study the one-way local operations and classical communication (LOCC) problem. In $\mathbb{C}^d\otimes\mathbb{C}^d$ with $d\geq4$, we construct a set of $3\lceil\sqrt{d}\rceil-1$ one-way LOCC indistinguishable maximally entangled states which are generalized Bell states. Moreover, we show that there are four maximally entangled states which cannot be perfectly distinguished by one-way LOCC measurements for any dimension $d\geq 4$.
In this paper, we consider the problem of discriminating quantum states by local operations and classical communication (LOCC) when an arbitrarily small amount of error is permitted. This paradigm is known as asymptotic state discrimination, and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic LOCC. We use this new criterion to prove a gap in the LOCC and separable distinguishability norms. We then turn to the operational advantage of using two-way classical communication over one-way communication in LOCC processing. With…
We explore the intersection of studies in quantum error correction and quantum local operations and classical communication (LOCC). We consider one-way LOCC measurement protocols as quantum channels and investigate their error correction properties, emphasizing an operator theory approach to the subject, and we obtain new applications to one-way LOCC state distinguishability as well as new derivations of some established results. We also derive conditions on when states that arise through the stabilizer formalism for quantum error correction are distinguishable under one-way LOCC.
In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of …
In this article, we show a sufficient and necessary condition for locally distinguishable bipartite states via one-way local operations and classical communication (LOCC). With this condition, we present some minimal structures of one-way LOCC indistinguishable quantum state sets. As long as an indistinguishable subset exists in a state set, the set is not distinguishable. We also list several distinguishable sets as instances.
We bring together in one place some of the main results and applications from our recent works in quantum information theory, in which we have brought techniques from operator theory, operator algebras, and graph theory for the first time to investigate the topic of distinguishability of sets of quantum states in quantum communication, with particular reference to the framework of one-way local quantum operations and classical communication (LOCC). We also derive a new graph-theoretic description of distinguishability in the case of a single qubit sender.
We study the problem of distinguishing maximally entangled quantum states by using local operations and classical communication (LOCC). A question of fundamental interest is whether any three maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are distinguishable by LOCC. In this paper, we restrict ourselves to consider the generalized Bell states. And we prove that any three generalized Bell states in $\mathbb{C}^d\otimes\mathbb{C}^d (d\geq 4)$ are locally distinguishable.
In this paper, we mainly study the local distinguishable multipartite quantum states by local operations and classical communication (LOCC) in $m_1\otimes m_2\otimes\ldots\otimes m_n$ , where the quantum system $m_1$ belongs to Alice, $m_2$ belongs to Bob, \ldots and $m_n$ belongs to Susan. We first present the pure tripartite distinguishable orthogonal quantum states by LOCC in $m_1\otimes m_2\otimes m_3$. With the conclusion in $m_1\otimes m_2\otimes m_3$, we prove distinguishability or indistinguishability of some quantum states. At last, we give the $n$-party distinguishable quantum states…
In the general bipartite quantum system $m \otimes n$, Wang \emph{et al.} [Y.-L Wang \emph{et al.}, Phys. Rev. A \textbf{92}, 032313 (2015)] presented $3(m+n)-9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\otimes n $. First, in $3\otimes n (3< n)$ quantum system, we construct $3n-2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\otimes n (4\leq m\leq n)$, we present $3n+m…
Suppose a set of $m$-partite, $m\geq3$, pure orthogonal fully separable states is given. We consider the task of distinguishing these states perfectly by local operations and classical communication (LOCC) in different $k$-partitions, $1<k<m$. Based on this task, it is possible to classify the sets of product states into different classes. For tripartite systems, a classification of the sets with explicit examples is presented. Few important cases related to the aforesaid task are also studied when the number of parties, $m\geq4$. These cases never appear for a tripartite system. However, to d…
The are substantial studies on distinguishabilities, especially local distinguishability, of quantum states. It is shown that a necessary condition of a local distinguishable state set is the total Schmidt rank not larger than the system dimension. However, if we view states in a larger system, the restriction will be invalid. Hence, a nature problem is that can indistinguishable states become distinguishable by viewing them in a larger system without employing extra resources. In this paper, we consider this problem for (perfect or unambiguous) LOCC$_{1}$, PPT and SEP distinguishabilities. We…
Graph Structures for Local Distinguishability of Quantum Product States
Department of Mathematics & Statistics, University of Guelph（圭尔夫大学数学与统计系）；Department of Mathematics & Computer Science, Brandon University（布兰登大学数学与计算机科学系）；Department of Mathematical Sciences, Bentley University（本特利大学数学科学系）
Global versus Local Discrimination of Locally Implementable Multipartite Unitaries
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；Indian Institute of Technology Bhubaneswar（印度技术学院布巴内斯瓦尔分校）；S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）
We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator systems, operator algebras, and Hilbert C*-modules all naturally arise in this setting, and we make use of these structures to derive new results and new derivations of some established results in the study of LOCC. We also show that perfect distinguishability under one-way LOCC and under arbitrary operations is equivalent for several families of operators tha…
We study the problem of transforming a set of pure bipartite states into another using deterministic LOCC (local operations and classical communication). Necessary conditions for the existence of such a transformation are obtained using LOCC constraints on state transformation, entanglement, and distinguishability. These conditions are shown to be independent but not sufficient. We discuss their satisfiability and classify all possible input-output pairs of sets accordingly. We also prove that strict inclusions hold between LOCC, separable, and positive partial transpose operations for set tra…
We provide a necessary condition that a quantum measurement can be implemented by the class of protocols known as Local Operations and Classical Communication, or LOCC, including when an error is allowed but must vanish in the limit of an infinite number of rounds, a case referred to as asymptotic LOCC. Our condition unifies, extends, and provides an intuitive, geometric justification for previous results on asymptotic LOCC. We use our condition to answer a variety of long-standing, unsolved problems, including for distinguishability of certain sets of states by LOCC. These include various cla…
Local distinguishability of six bipartite orthogonal product states
College of Computer Science and Engineering, Shandong University of Science and Technology（山东科技大学计算机科学与工程学院）；College of Cyberspace Science and Technology, Beijing University of Technology（北京工业大学网络空间技术学院）；College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）
It is known that quantum interference can disappear with the mere possibility of distinguishability without actually performing the act. We create such distinguishability in an unbalanced SU(1,1) interferometer and indeed observe no interference in the direct photodetection of the outputs. On the other hand, such distinguishability can be erased with a projective measurement. Here, we report a method of homodyne detection that can also recover interference effect. We find that it is the indistinguishability in amplitude measurement that leads to the recovery of interference, and the quantum na…
Local distinguishability of five orthogonal product states on bipartite and tripartite quantum systems
College of Computer Science and Engineering, Shandong University of Science and Technology（山东科技大学计算机科学与工程学院）；Faculty of Information Technology, Beijing University of Technology（北京工业大学信息学部）；College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）
Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
固定量子系统上的每一个足够丰富的测量集合通过最佳偏差定义了该系统状态的统计范数，该最佳偏差可以在使用来自该集合的测量区分状态时实现（假设相等的先验）。Holevo-Helstrom定理说，对于所有测量的集合，这个范数是迹范数。对于有限维，任何范数的下限和上限都是由迹范数的常数（尽管依赖于维数）的倍数来限定的，因此我们为自己设定了计算或限定最佳可能的“支配常数”的任务，这些常数对应于各种受限的测量集合，从而确定这些集合相对于所有测量集合的最坏情况和最好情况性能。我们看看的情况下，允许的集合由一个单一的测量，即均匀随机连续POVM及其近似2-设计和4-设计分别。在这里，我们发现渐近紧的优势常数的界限。此外，我们分析了多方设置与任何LOCC测量允许。在两个当事人的情况下，我们表明，较低的控制常数是相同的张量积的局部均匀随机POVMs（常数）。这肯定地回答了一个关于二分数据隐藏的（接近）最优性的公开问题：LOCC在区分d x d二分系统的两个正交状态时可以实现的偏差是Omega（1/d），这是已知的紧。最后，我们使用我们的分析，以获得任何这样的测量的确定性关系（在桑切斯-鲁伊斯的意义上）和下界的局部可访问的信息的二分系统。
We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of $k \leq d$ orthogonal maximally entangled states in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012) -- arXiv:1107.3224 [quant-ph]] gives an affirmative answer for the case $k = d$. We give, for the first time, a proof that such sets of states indeed exist even in the case $k < d$. Our result is constructive …
