I consider the problem of deterministically distinguishing the state of a multipartite system, from a set of $N\ge d+1$ orthogonal states, where $d$ is the dimension of each party's subsystem. It is shown that if the set of orthogonal states is chosen at random, then there is a vanishing probability that this set will be perfectly distinguishable under the restriction that the parties use only local operations on their subsystems and classical communication amongst themselves.
Recently, Zhang et al [Phys. Rev. A 92, 012332 (2015)] presented $4d-4$ orthogonal product states that are locally indistinguishable and completable in a $d\otimes d$ quantum system. Later, Zhang et al. [arXiv: 1509.01814v2 (2015)] constructed $2n-1$ orthogonal product states that are locally indistinguishable in $m\otimes n$ ($3\leq m \leq n$). In this paper, we construct a locally indistinguishable and completable orthogonal product basis with $4p-4$ members in a general $m\otimes n$ ($3\leq m \leq n$) quantum system, where $p$ is an arbitrary integer from $3$ to $m$, and give a very simple …
We give a explicit construction of $d$ locally indistinguishable orthogonal maximally entangled states in $\mathbb{C}^d\otimes\mathbb{C}^d$ for any $d\geq 4$. This gives an answer to the conjecture proposed by S. Bandyopadhyay in 2009. Thus it reflects the nonlocality of the fundamental feature of quantum mechanics.
A subspace of a multipartite Hilbert space is called \textit{locally indistinguishable} if any orthogonal basis of this subspace cannot be perfectly distinguished by local operations and classical communication. Previously it was shown that any $m\otimes n$ bipartite system such that $m>2$ and $n>2$ has a locally indistinguishable subspace. However, it has been an open problem since 2005 whether there is a locally indistinguishable bipartite subspace with a qubit subsystem. We settle this problem by showing that any $2\otimes n$ bipartite subspace is locally distinguishable in the sense it con…
In this paper, we construct $2n-1$ locally indistinguishable orthogonal product states in $\mathbb{C}^n\otimes\mathbb{C}^{4}~(n>4)$ and $\mathbb{C}^n\otimes\mathbb{C}^{5}~(n\geq 5)$ respectively. Moreover, a set of locally indistinguishable orthogonal product states with $2(n+2l)-8$ elements in $\mathbb{C}^n\otimes\mathbb{C}^{2l}~(n\geq 2l>4)$ and a class of locally indistinguishable orthogonal product states with $2(n+2k+1)-7$ elements in $\mathbb{C}^n\otimes\mathbb{C}^{2k+1}~(n\geq 2k+1>5)$ are also constructed respectively. These classes of quantum states are then shown to be distinguishabl…
We explicitly exhibit a set of four ququad-ququad orthogonal maximally entangled states that cannot be perfectly distinguished by means of local operations and classical communication. Before our work, it was unknown whether there is a set of $d$ locally indistinguishable $d\otimes d$ orthogonal maximally entangled states for some positive integer $d$. We further show that a $2\otimes 2$ maximally entangled state can be used to locally distinguish this set of states without being consumed, thus demonstrate a novel phenomenon of "Entanglement Discrimination Catalysis". Based on this set of stat…
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…
We show a simple semidefinite program whose optimal value is equal to the maximum probability of perfectly distinguishing orthogonal maximally entangled states using any PPT measurement (a measurement whose operators are positive under partial transpose). When the states to be distinguished are given by the tensor product of Bell states, the semidefinite program simplifies to a linear program. In [Phys. Rev. Lett. 109, 020506 (2012) -- arXiv:1107.3224v1], Yu, Duan and Ying exhibit a set of 4 maximally entangled states in $C^4 \otimes C^4$, which is distinguishable by any PPT measurement only w…
Recently, Halder \emph{et al.} [S. Halder \emph{et al.}, Phys. Rev. Lett. \textbf{122}, 040403 (2019)] present two sets of strong nonlocality of orthogonal product states based on the local irreducibility. However, for a set of locally indistinguishable orthogonal entangled states, the remaining question is whether the states can reveal strong quantum nonlocality. Here we present a general definition of strong quantum nonlocality based on the local indistinguishability. Then, in $2 \otimes 2 \otimes 2$ quantum system, we show that a set of orthogonal entangled states is locally reducible but l…
We study the constructions of nonlocal orthogonal product states in multipartite systems that cannot be distinguished by local operations and classical communication. We first present two constructions of nonlocal orthogonal product states in tripartite systems $\mathcal{C}^{d}\otimes\mathcal{C}^{d}\otimes\mathcal{C}^{d}~(d\geq3)$ and $\mathcal{C}^d\otimes \mathcal{C}^{d+1}\otimes \mathcal{C}^{d+2}~(d\geq 3)$. Then for general tripartite quantum system $\mathcal{C}^{n_{1}}\otimes\mathcal{C}^{n_{2}}\otimes\mathcal{C}^{n_{3}}$ $(3\leq n_{1}\leq n_{2}\leq n_{3})$, we obtain $2(n_{2}+n_{3}-1)-n_{1…
An orthogonal product basis (OPB) of a finite-dimensional Hilbert space $H=H_1\otimes H_2\otimes\cdots\otimes H_n$ is an orthonormal basis of $H$ consisting of product vectors $x_1\otimes x_2\otimes\cdots\otimes x_n$. We show that the problem of classifying the OPBs of an $n$-qubit system can be reduced to a purely combinatorial problem. We solve this combinatorial problem in the case of four qubits and obtain 33 multiparameter families of OPBs. Each OPB of four qubits is equivalent, under local unitary operations and qubit permutations, to an OPB belonging to at least one of these families.
We present a complete characterization for the local distinguishability of orthogonal $2\otimes 3$ pure states except for some special cases of three states. Interestingly, we find there is a large class of four or three states that are indistinguishable by local projective measurements and classical communication (LPCC) can be perfectly distinguishable by LOCC. That indicates the ability of LOCC for discriminating $2\otimes 3$ states is strictly more powerful than that of LPCC, which is strikingly different from the case of multi-qubit states. We also show that classical communication plays a…
We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of $\mathbb{C}^d\otimes\mathbb{C}^d$, where $d$ is odd, Zhang \emph{et al} have constructed $d^2$ orthogonal product basis quantum states which are locally indistinguishable in [Phys. Rev. A. {\bf 90}, 022313(2014)]. We find a subset contains with $6d-9$ orthogonal product states which are still locally indistinguishable. Then we generalize our method to arbitrary bipartite quantum system $\mathbb{C}^m\otimes\mathbb{C}^n$. We present a small …
By incorporating the asymmetry of local protocols, i.e., some party has to start with a nontrivial measurement, into an operational method of detecting the local indistinguishability proposed by Horodecki {\it et al.} [Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to efficiently detect the local indistinguishability of maximally entangled states. Locally indistinguishable sets of $d$ maximally entangled states in a $d\otimes d$ system are systematically constructed for all $d\ge 4$ as an application. Furthermore, by exploiting the fact that local protocols are necessarily …
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…
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 …
We study the local distinguishability of general multi-qubit states and show that local projective measurements and classical communication are as powerful as the most general local measurements and classical communication. Remarkably, this indicates that the local distinguishability of multi-qubit states can be decided efficiently. Another useful consequence is that a set of orthogonal $n$-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension $2^n$. When $n=2$ such a condition is also sufficient. Employing these resu…
Bennett et al. \cite{BDF+99} identified a set of orthogonal {\em product} states in the $3\otimes 3$ Hilbert space such that reliably distinguishing those states requires non-local quantum operations. While more examples have been found for this counter-intuitive ``nonlocality without entanglement'' phenomenon, a complete and computationally verifiable characterization for all such sets of states remains unknown. In this Letter, we give such a characterization for the $3\otimes 3$ space.
The (im)possibility of local distinguishability of orthogonal multipartite quantum states still remains an intriguing question. Beyond $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$, the problem remains unsolved even for maximally entangled states (MES). So far, the only known condition for the local distinguishability of states is the well-known orthogonality preservation (OP). Using an upper bound on the locally accessible information for bipartite states, we derive a very simple necessary condition for any set of pairwise orthogonal MES in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$ to be perfectly local…
In this paper, we investigate the construction of orthogonal product states with strong nonlocality in multiparty quantum systems. Firstly, we focus on the tripartite system and propose a general set of orthogonal product states exhibiting strong nonlocality in $d\otimes d\otimes d$ quantum system, which contains $6{{\left( d-1 \right)}^{2}}$ states. Secondly, we find that the number of the sets constructed in this way could be further reduced. Then using $4\otimes 4\otimes 4$ and $5\otimes 5\otimes 5$ quantum systems as examples, it can be seen that when d increases, the reduced quantum state…
Nonlocal sets of orthogonal product states (OPSs) are widely used in quantum protocols owing to their good property. Thus a lot of attention are paid to how to construct a nonlocal set of orthogonal product states though it is a difficult problem. In this paper, we propose a novel general method to construct a nonlocal set of orthogonal product states in $\mathbb{C}^{d} \otimes \mathbb{C}^{d}$ for $d\geq3$. We give an ingenious proof for the local indistinguishability of those product states. The set of product states, which are constructed by our method, has a very good structure. Subsequentl…
The unextendible product bases (UPBs) are interesting members from the family of orthogonal product states. In this paper, we investigate the construction of 3-qubit UPB with strong nonlocality of different sizes. First, a UPB set in ${{C}^{3}}\otimes {{C}^{3}}\otimes {{C}^{3}}$ of size 12 is presented based on the Shifts UPB, the structure of which is described by mapping the system to a $3\times 3\times 3$ Rubik's Cube. After observing the orthogonal graph of each qubit, we provide a general method of constructing UPB in ${{C}^{d}}\otimes {{C}^{d}}\otimes {{C}^{d}}$ of size ${{\left( d-1 \ri…
A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving local measurement can be performed to this set. In general, it is difficult to show that such an orthogonality-preserving local measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully show the existence of unexten…
In this paper, we generalize the concept of strong quantum nonlocality from two aspects. Firstly in $\mathbb{C}^d\otimes\mathbb{C}^d\otimes\mathbb{C}^d$ quantum system, we present a construction of strongly nonlocal quantum states containing $6(d-1)^2$ orthogonal product states, which is one order of magnitude less than the number of basis states $d^3$. Secondly, we give the explicit form of strongly nonlocal orthogonal product basis in $\mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3$ quantum system, where four is the largest known number of subsystems in which there …
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.
We study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of $(D-1)$ pure states, where $D$ is the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of $2\otimes 2$ separable operations not being realizable by local opera…
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…
Recently, much attention have been paid to the constructions of nonlocal multipartite orthogonal product states. Among the existing results, some are relatively complex in structure while others have many constraint conditions. In this paper, we firstly give a simple method to construct a nonlocal set of orthogonal product states in $\otimes_{j=1}^{n}\mathbb{C}^{d}$ for $d\geq 2$. Then we give an ingenious proof for local indistinguishability of the set constructed by our method. According to the characteristics of this construction method, we get a new construction of nonlocal set with fewer …
Construction of Sets of Orthogonal Quantum States with Minimal Nonlocality in Bipartite and Tripartite Systems of Unequal Local Dimensions
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 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）；College of Cyberspace Science and Technology, Beijing University of Technology（北京工业大学网络空间科学技术学院）
In this paper, we consider the orthogonal product set (OPS) with strong quantum nonlocality. Based on the decomposition of plane geometry, we present a sufficient condition for the triviality of orthogonality-preserving POVM on fixed subsystem and partially answer an open question given by Yuan et al. Phys. Rev. A \textbf{102}, 042228 (2020)}. The connection between the nonlocality and the plane structure of OPS is established. We successfully construct a strongly nonlocal OPS in $\mathcal{C}^{d_{A}}\otimes \mathcal{C}^{d_{B}}\otimes \mathcal{C}^{d_{C}}$ $(d_{A,B,C}\geq 4)$, which contains few…
In quantum state discrimination, the observers are given a quantum system and aim to verify its state from the two or more possible target states. In the local quantum state marking as an extension of quantum state discrimination, there are N composite quantum systems and N possible orthogonal target quantum states. Distant Alice and Bob are asked to correctly mark the states of the given quantum systems via local operations and classical communication. Here we investigate the local state marking with N 4 ${\otimes}$ 4 systems, N=4, 5, 6, and 7. Therein, Alice and Bob allow for three local ope…
Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit "quantum nonlocality without entanglement" if the states are locally indistinguishable, i.e. it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the no…
Recently using entanglement as resource to distinguish orthogonal product states by local operations and classical communication (LOCC) has been studied intensively. Zhang. et al. presented protocols to use entanglement to distinguish certain classes of orthogonal product states in $\mathbb{C}^m\otimes \mathbb{C}^n$\cite{Zhang016}. In this paper, we study local distinguishability of multipartite orthogonal product states and provide a practical solution. Our method relies upon a special class of locally indistinguishable multipartite product states introduced by Wang et. al. to build a protoco…
A set of multipartite orthogonal product states is strongly nonlocal if it is locally irreducible in every bipartition, which shows the phenomenon of strong quantum nonlocality without entanglement. It is known that unextendible product bases (UPBs) can show the phenomenon of quantum nonlocality without entanglement. Thus it is interesting to investigate the strong quantum nonlocality for UPBs. Most of the UPBs with the minimum size cannot demonstrate strong quantum nonlocality. In this paper, we construct a series of UPBs with different large sizes in $d_A\otimes d_B\otimes d_C$ and $d_A\otim…
We study the distinguishability of bipartite quantum states by Positive Operator-Valued Measures with positive partial transpose (PPT POVMs). The contributions of this paper include: (1). We give a negative answer to an open problem of [M. Horodecki $et. al$, Phys. Rev. Lett. 90, 047902(2003)] showing a limitation of their method for detecting nondistinguishability. (2). We show that a maximally entangled state and its orthogonal complement, no matter how many copies are supplied, can not be distinguished by PPT POVMs, even unambiguously. This result is much stronger than the previous known on…
Motivated by the separability problem in quantum systems $2\otimes4$, $3\otimes3$ and $2\otimes2\otimes2$, we study the maximal (proper) faces of the convex body, $S_1$, of normalized separable states in an arbitrary quantum system with finite-dimensional Hilbert space $H=H_1\otimes H_2\otimes\cdots\otimes H_n$. To any subspace $V$ of $H$ we associate a face $F_V$ of $S_1$ consisting of all states $ρ\in S_1$ whose range is contained in $V$. We prove that $F_V$ is a maximal face if and only if $V$ is a hyperplane. If $V$ is the hyperplane orthogonal to a product vector, we prove that $\dim F_V=…
The construction of nonlocal sets of quantum states has attracted much attention in recent years. We first introduce two Lemmas related to the triviality of orthogonality-preserving local measurements. Then we propose a general construction of nonlocal set of $n(d-1)+1$ orthogonal product states in $(\mathbb{C}^{d})^{\otimes n}$. The sets of nonlocal orthogonal product states are also put forward for the multipartite quantum systems with arbitrary dimensions. Our novel construction gives rise to nonlocal sets of orthogonal product states with much less members and thus reveals the phenomenon o…
The common use in literature of orthogonal measurements in obtaining quantum discord for two-qubit states is discussed and compared with more general measurements. We prove the optimality of orthogonal measurements for rank 2 states. While for rank 3 and 4 mixed states they are not optimal, we present strong numerical evidence showing that they give the correct quantum discord up to minimal corrections. Based on the connection, through purification with an ancilla, between discord and entanglement of formation (EoF), we give a tight upper bound for the EoF of a $2\otimes N$ mixed state of rank…
Quite recently, Cai et al. [arXiv:2006.07165v1] proposed a new concept "absolutely entangled set" for bipartite quantum systems: for any possible choice of global basis, at least one state of the set is entangled. There they presented a minimum example with a set of four states in two qubit systems and they proposed a quantitative measure for the absolute set entanglement. In this work, we derive two necessity conditions for a set of states to be an absolutely entangled set. In addition, we give a series constructions of absolutely entangled bases on $\mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ …
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 …
In this paper, we mainly study the local indistinguishability of multipartite product states. Firstly, we follow the method of Z.-C. Zhang \emph{et al}[Phys. Rev. A 93, 012314(2016)] to give another more concise set of $2n-1$ orthogonal product states in $\mathbb{C}^m\otimes\mathbb{C}^n(4\leq m\leq n)$ which can not be distinguished by local operations and classical communication(LOCC). Then we use the 3 dimension cubes to present some product states which give us an intuitive view how to construct locally indistinguishable product states in tripartite quantum system. At last, we give an expli…
A set of multipartite orthogonal quantum states is strongly nonlocal if it is locally irreducible for every bipartition of the subsystems [Phys. Rev. Lett. 122, 040403 (2019)]. Although this property has been shown in three-, four- and five-partite systems, the existence of strongly nonlocal sets in $N$-partite systems remains unknown when $N\geq 6$. In this paper, we successfully show that a strongly nonlocal set of orthogonal entangled states exists in $(\mathbb{C}^d)^{\otimes N}$ for all $N\geq 3$ and $d\geq 2$, which for the first time reveals the strong quantum nonlocality in general $N$-…
