In this paper, the distinguishability of multipartite geometrically uniform quantum states obtained from a single reference state is studied in the symmetric subspace. We specially focus our attention on the unitary transformation in a way that the produced states remain in the symmetric subspace, so rotation group with Jy as the generator of rotation is applied. The optimal probability and measurements are obtained for the pure and some special mixed separable states and the results are compared with those obtained at the previous articles for the special cases. The results are valid for lin- early dependent states. The discrimination of these states is also investigated using the separable measurement. We introduce appropriate transformation to gain the optimal separable measurements equivalent to the optimal global measurements with the same optimal probability.
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。
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…
For any pair of quantum states (the hypotheses), the task of binary quantum hypotheses testing is to derive the tradeoff relation between the probability $p_{01}$ of rejecting the null hypothesis and $p_{10}$ of accepting the alternative hypothesis. The case when both hypotheses are explicitly given was solved in the pioneering work by Helstrom. Here, instead, for any given null hypothesis as a pure state, we consider the worst-case alternative hypothesis that maximizes $p_{10}$ under a constraint on the distinguishability of such hypotheses. Additionally, we restrict the optimization to separ…
We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system $XA$, corresponding to an ensemble of two quantum states, with $X$ being classical and $A$ being quantum. We study the resource theory for two different classes of free operations: $(i)$ ${\rm{CPTP}}_A$, which consists of quantum channels acting only on $A$, and $(ii)$ conditional doub…
We show that local PT-symmetric evolutions can lead to violation of the no-signaling principle for separable and even classically correlated bipartite shared quantum states. For classically correlated states, specially chosen PT-symmetric operations from a set of zero volume can also preserve the principle. The violations can be removed by using a CPT inner product instead of the traditional one.
Study of an N qubit mixed symmetric separable states is a long standing challenging problem as there exist no unique separability criterion. In this regard, we take up the N-qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer elegant mathematical analysis since the dimension of the Hilbert space reduces from 2^N to N + 1. Since there exists a one to one correspondence between spin-j system and an N-qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of spin density matrix. Further, we use g…
We analyse orthogonal bases in a composite $N\times N$ Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the $N^2$ reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case $N=2$ of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for $N=3$ and provide a general construction of $N^2$ states forming …
We introduce and explore two questions concerning spectra of operators that are of interest in the theory of entanglement in symmetric (i.e., bosonic) quantum systems. First, we investigate the inverse eigenvalue problem for symmetric entanglement witnesses -- that is, we investigate what their possible spectra are. Second, we investigate the problem of characterizing which separable symmetric quantum states remain separable after conjugation by an arbitrary unitary acting on symmetric space -- that is, which states are separable in every orthonormal symmetric basis. Both of these questions ha…
We prove that every symmetric separable state admits a convex decomposition into symmetric pure product states. While the result is not new in itself, here we focus on convex geometry. We discuss the decomposition in the context of numerical ranges and ground state problems of infinite bosonic systems.
Symmetry plays an important role in the field of quantum mechanics. In this paper, we consider a subclass of symmetric quantum states in the multipartite system $N^{\otimes d}$, namely, the completely symmetric states, which are invariant under any index permutation. It was conjectured by L. Qian and D. Chu [arXiv:1810.03125 [quant-ph]] that the completely symmetric states are separable if and only if it is a convex combination of symmetric pure product states. In this paper, we proved that this conjecture is true for both bipartite and multipartite cases. And we proved the completely symmetri…
In this paper, we consider a subclass of quantum states in the multipartite system, namely, the supersymmetric states. We investigate the problem whether they admit the symmetrically separable decomposition, i.e., each term in this decomposition is a supersymmetric pure product state $|x,x\rangle\langle x,x|$, which are called S-separable. We conjecture that any supersymmetric states are S-separable and we prove that this conjecture holds when the rank is less than or equal to 3 or $N$. Moreover, we propose another weaker conjecture that any separable supersymmetric states are S-separable. It …
The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in this paper, we only need one algorithm to solve the problem. We propose a tensor optimization method to check whether an $m$-partite quantum mixed state is separable or not and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Th…
Efficient fourth order symplectic integrators for near-harmonic separable Hamiltonian systems
提出了可分Hamilton系统H（p，q）=T（p）+V（q）的数值积分的有效四阶辛积分器.通过对简谐振子的高阶分解，得到了5 ~ 9级的对称分裂系数。对各种哈密顿系统的性能进行了评估：积分误差与S。Blanes et al.（2013），W. Kahan et al.（1999）和H. Yoshida（1990）.数值试验表明，对于一般的Hamilton系统，本文所得到的积分器的性能明显优于以往的积分器。
We explore sufficient conditions for inseparability in mixed states with a globally conserved charge, such as a particle number. We argue that even separable states may contain entanglement in fixed charge sectors, as long as the state can not be separated into charge conserving components. As a witness of symmetric inseparability we study the number entanglement (NE), $ΔS_m$, defined as the entropy change due to a subsystem's charge measurement. Whenever $ΔS_m > 0$, there exist inseparable charge sectors, having finite (logarithmic) negativity, even when the full state is either separable or …
