Now, the known ensembles of orthogonal states which are distinguishable by local operators and classical communication (LOCC) satisfy the condition that the sum of Schmidit numbers of the orthogonal states is not bigger than the dimensions of the whole space. A natural question is whether an arbitary ensembles of LOCC-distinguishable orthogonal states satisfies the condition. We first show that, in this paper, the answer is positive. Then we generalize it into multipartite systems, and show that a necessary condition for LOCC-distinguishability of multipartite orthogonal quantum states is that the sum of the least numbers of the product states (For bipartite system, the least number of product states is Schmidit number) of the orthogonal states is not bigger than the dimensions of the Hilbert space of the multipartite system. This necessary condition is very simple and general, and one can get many cases of indistinguishability by it. It means that the least number of the product states acts an important role in distinguishablity of states, and implies that the least number of the product states may be an good manifestion of quantum nonlocality in some sense. In fact, entanglement emphases the "amount" of nonlocality, but the least number of the product states emphases the types of nonlocality. For example, the known W states and GHZ states have different least number of the product states, and are different in type.
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）
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）
Decomposition of (finite-dimensional) operators in terms of orthogonal bases of matrices has been a standard method in quantum physics for decades. In recent years, it has become increasingly popular because of various methodologies applied in quantum information, such as the graph state formalism and the theory of quantum error correcting codes, but also due to the intensified research on the Bloch representation of quantum states. In this contribution we collect various interesting facts and identities that hold for finite-dimensional orthogonal matrix bases.
Distinguishability of Quantum States by Separable Operations
利用可分操作研究了多体量子态的可分性。我们首先给出了有限个正交量子态集合可通过可分操作区分的充分必要条件。这个条件的解析版本推导出的情况下，$(D-1)$纯态，其中$D$是考虑中的状态空间的总尺寸。这个结果的一些有趣的后果，然后仔细研究。值得注意的是，我们发现存在一个大类的$2\otimes 2$可分离的操作不被实现的本地操作和经典的通信。在我们的工作之前，只有一类$3\otimes 3$非局部可分运算是已知的[Bennett et al，Phys. Rev. A \textbf{59}，1070（1999）]。我们还证明了多体纯态的正交补的任何基都不能用可分操作区分当且仅当这个态不能是1或2个正交乘积态的叠加，即，的正交Schmidt数不小于3，从而推广了最近关于不可区分二部子空间的工作[Watrous，Phys.Rev.Lett. \textbf{95}，080505（2005）]。值得注意的是，我们通过考虑由两个qutrits组成的复合量子系统，得到了7维（或6维）不可区分子空间的显式构造。三个量子比特），这比先前已知的具有8维的不可区分的二分子空间稍好。
Recently, Halder \emph{et al.} [Phys. Rev. Lett. \textbf{122}, 040403 (2019)] proposed the concept strong nonlocality without entanglement: an orthogonal set of fully product states in multipartite quantum systems that is locally irreducible for every bipartition of the subsystems. As the difficulty of the problem, most of the results are restricted to tripartite systems. Here we consider a weaker form of nonlocality called local distinguishability based genuine nonlocality. A set of orthogonal multipartite quantum states is said to be genuinely nonlocal if it is locally indistinguishable for …
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…
Distinguishability theory is developed for quantum interference of the squeezed vacuum states on unitary linear interferometers. It is found that the entanglement of photon pairs over the Schmidt modes is one of the sources of distinguishability. The distinguishability is quantified by the symmetric part of the internal state of $n$ pairs of photons, whose normalization $q_{2n}$ is the probability that $2n$ photons interfere as indistinguishable. For two pairs of photons $q_{4}=(1+ 2\mathbb{P} )/3$, where $\mathbb{P} $ is the purity of the squeezed states ($K=1/\mathbb{P} $ is the Schmidt numb…
We prove a tight and close-to-optimal lower bound on the effectiveness of local quantum measurements (without classical communication) at discriminating any two bipartite quantum states. Our result implies, for example, that any two orthogonal quantum states of a $n_A\times n_B$ bipartite quantum system can be discriminated via local measurements with an error probability no larger than $\frac12 \left(1 - \frac{1}{c \min\{n_A, n_B\}} \right)$, where $1\leq c\leq 2\sqrt2$ is a universal constant, and our bound scales provably optimally with the local dimensions $n_A,n_B$. Mathematically, this i…
An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we construct a 7-qubit UPB of size 11. It solves an open problem in [Quantum Information Processing 19:185 (2020)]. Next, we graph-theoretically show that the UPB is locally indistinguishable in the bipartite systems of two qubits and five qubits, respectively. It turns out that the UPB corresponds to a complete graph with 11 vertices constructed by three sorts of nonisomorphic graphs. Taking the graphs as product vectors, we show that they ar…
From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as $[ϕ\rangle=λ_1 [00\rangle+λ_2 [11\rangle$ with $λ_1$ and $λ_2$ real numbers. For 3-qubits states, it is known, PRL 2000 85, that up to local gates every 3-qubit state can be written as $[ϕ\rangle=λ_1 [000\rangle+λ_2 e^{i θ}[100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\rangle$ with $λ_i\ge0$ and $0\le θ\le π$. In this paper, we show that no every 3-qubit state with real amplitudes can be transform in the form $[ϕ\rangle=λ_1 [000\rangle+λ_2 [100\rangle+λ_3[101\rangle+λ_4[110\rangle+λ_5[111\…
In this paper, we consider the LOCC distinguishability of product states. We employ polygons to analyze orthogonal product states in any system to show that with LOCC protocols, to distinguish seven orthogonal product states, one can exclude four states via a single copy. In bipartite systems, this result implies that N orthogonal product states are LOCC distinguishable if $\left \lceil \frac{N}{4} \right \rceil$ copies are allowed, where $\left \lceil l\right \rceil$ for a real number $l$ means the smallest integer not less than $l$. In multipartite systems, this result implies that N orthogo…
