This paper proves that a set of orthogonal pure states are indistinguishable by restricted local projective measurement and classical communication if the sum of their Schmidt ranks is larger than the dimension of their joint Hilbert space. This result is useful in determining the local distinguishability of quantum states and is stronger in some respects than that of Hayashi et al [Phys. Rev. Lett. 96 , 040501]. In addition, it presents a new method to determine the local distinguishability of orthogonal states by projecting measurement operators into their subspaces.
This paper proves that a set of orthogonal pure states are indistinguishable by restricted local projective measurement and classical communication if the sum of their Schmidt ranks is larger than the dimension of their joint Hilbert space. This result is useful in determining the local distinguishability of quantum states and is stronger in some respects than that of Hayashi et al [Phys. Rev. Lett. 96 , 040501]. In addition, it presents a new method to determine the local distinguishability of orthogonal states by projecting measurement operators into their subspaces.
In this paper, 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, whereDis 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 operations and classical communication. Before our work, only a class of 3 otimes 3 nonlocal separable operations was known [Bennett , Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.
I consider the deterministic distinguishability of a set of orthogonal, bipartite states when only a single copy is available and the parties are restricted to local operations and classical communication, but with the additional requirement that entanglement must be preserved in the process. Several general theorems aimed at characterizing sets of states with which the parties can succeed in such a task are proven. These include (i) a maximum for the number of states when the Schmidt rank of every outcome must be at least a given minimum, (ii) an upper bound (equal to the dimension of Hilbert space if entanglement need not be preserved) for the sum over Schmidt ranks of the initial states when only one-way classical communication is allowed, and (iii) separately, a necessary and a sufficient condition on the states such that their original Schmidt ranks can always be preserved. Two additional theorems explicitly demonstrate a trade-off between the extent to which the set of states fill Hilbert space, as measured by their Schmidt ranks, and how refined the parties must make their measurements, an important factor in determining the Schmidt rank the state can retain after it has been identified. It is shown that our bound on the sum of Schmidt ranks can be exceeded if two-way communication is permitted, and this includes the case that entanglement need not be preserved, so that this sum can exceed the dimension of Hilbert space. Such questions, concerning how the various results are affected by the resources used by the parties (amount of classical communication and types of local operations), are addressed for each theorem. This subject is closely related to the problem of locally purifying an entangled state from a mixed state, which is of direct relevance to teleportation and dense coding using a mixed-state resource. In an appendix, I give an extremely simple and transparent proof of ``nonlocality without entanglement,'' a phenomenon originally discussed by Bennett and co-workers several years ago.
We consider different settings of the task to distinguish pure orthogonal quantum states under local operations and a limited amount of classical communication. In the first setting, the spatially separated parties are allowed to perform only local projective measurements without any classical communication during the measurements. Under such a restricted class of operations, if the states are indistinguishable, then within a second group of settings, the parties are allowed to use an additional resource for the distinguishing. The additional resource is either a pure entangled state or multiple identical copies of the given states. Comparisons between these two types of resources are also done in certain cases. Both probabilistic discrimination and perfect discrimination of the states are considered for the second group of settings. Within a third setting, the parties perform local projective measurements with a restriction on the availability of classical communication during the measurements. But in this setting the parties are allowed to use a maximally entangled state as a resource.
We study the local distinguishability of general multiqubit 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 multiqubit 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 2n. Employing these results, we show that any orthonormal basis of a subspace spanned by arbitrary three-qubit orthogonal unextendible product bases (UPB) cannot be exactly distinguishable by local operations and classical communication. This not only reveals another intrinsic property of three-qubit orthogonal UPB but also provides a class of locally indistinguishable subspaces with dimension 4. We also explicitly construct locally indistinguishable subspaces with dimensions 3 and 5, respectively. Similar to the bipartite case, these results on multipartite locally indistinguishable subspaces can be used to estimate the one-shot environment-assisted classical capacity of a class of quantum broadcast channels.
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 2n. When n=2 such a condition is also sufficient. Employing these results, we show that any orthonormal basis of a subspace spanned by arbitrary three-qubit orthogonal unextendible product bases (UPB) cannot be exactly distinguishable by local operations and classical communication. This not only reveals another intrinsic property of three-qubit orthogonal UPB, but also provides a class of locally indistinguishable subspaces with dimension 4. We also explicitly construct locally indistinguishable subspaces with dimensions 3 and 5, respectively. In particular, 3 is the minimal possible dimension of locally indistinguishable subspaces. Combining with the previous results, we conclude that any positive integer between 3 and 7 is the possible dimension of some three-qubit locally indistinguishable subspace.
We present a complete characterization for the local distinguishability of orthogonal $2\ensuremath{\bigotimes}3$ 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), but can be perfectly distinguished by LOCC. That indicates the ability of LOCC for discriminating $2\ensuremath{\bigotimes}3$ states is strictly more powerful than that of LPCC, which is strikingly different from the case of multiqubit states. We also show that classical communication plays a crucial role for local distinguishability by constructing a class of $m\ensuremath{\bigotimes}n$ states which require at least $2\phantom{\rule{0.2em}{0ex}}\mathrm{min}{m,n}\ensuremath{-}2$ rounds of classical communication in order to achieve a perfect local discrimination.