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LOCC Distinguishability of Orthogonal Product States via Multiply Copies

Hao Shu

Published Oct 26, 2020• 0 citations
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Abstract

In this paper, we consider the LOCC distinguishability of product states. We use polygons to analyse 7 orthogonal product states in any system to show that using LOCC protocol to distinguish 7 orthogonal product states in a bipartite or multipartite system, one can exclude 4 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 which is not less than $l$. In multipartite systems, this result implies that N orthogonal product states are LOCC distinguishable if $\left \lceil \frac {N}{4}\right \rceil +1$ copies are allowed. we also give a theorem to show how many states can be excluded by using a single copy if we are distinguishing n orthogonal product states by LOCC protocol in a bipartite system.

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In this paper, we consider the LOCC distinguishability of product states. We use polygons to analyse 7 orthogonal product states in any system to show that using LOCC protocol to distinguish 7 orthogonal product states in a bipartite or multipartite system, one can exclude 4 states via a single copy. In bipartite systems, this result implies that N orthogonal product states are LOCC distinguishable if ⌈N4⌉\left \lceil \frac {N}{4}\right \rceil⌈4N​⌉ copies are allowed, where ⌈l⌉\left \lceil l\right \rceil⌈l⌉ for a real number lll means the smallest integer which is not less than lll. In multipartite systems, this result implies that N orthogonal product states are LOCC distinguishable if ⌈N4⌉+1\left \lceil \frac {N}{4}\right \rceil +1⌈4N​⌉+1 copies are allowed. we also give a theorem to show how many states can be excluded by using a single copy if we are distinguishing n orthogonal product states by LOCC protocol in a bipartite system.
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 these states. In this paper, I propose a framework for the distinguishability by one-way LOCC (111-LOCC) of sets of orthogonal bipartite states in a dA⊗dBd_A \otimes d_BdA​⊗dB​ bipartite system, where dA,dBd_A, d_BdA​,dB​ are the dimensions of both subsytems, labelled as AAA and BBB. I show that if the iii-th party (where i=A,Bi=A,Bi=A,B) can initiate a 111-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 111-LOCC protocol lies in a subspace of di×did_i \times d_idi​×di​ hermitian matrices, denoted by $\Tb^{(i)}$, and that the method to extract this information (of the existence of this 111-LOCC protocol) from $\Tb^{(i)}$ depends on the value of $dim \Tb^{(i)}$. In this way one can give sweeping results for the 111-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $dim \Tb^{(i)}$. Thus I propose that the value of $dim \Tb^{(i)}$ gives the common underlying property based on which sweeping results for the 111-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
It is necessary to investigate the local distinguishability of orthogonal quantum state sets, as their adoption in protocol design helps diminish quantum state transmission and cut operational costs. In this paper, we explore the local distinguishability of six orthogonal product states (OPSs) on any bipartite quantum system. We classify different sets of six bipartite OPSs into eight categories by using the vectors of the numbers of pairwise orthogonality relations, where any two states are orthogonal on only one subsystem within each set. We find that these eight categories contain a total of 78 distinct cases, all but five of which are perfectly distinguishable via local operations and classical communication (LOCC). Furthermore, we discuss the local distinguishability of those five distinct cases in detail. Our work explicitly characterizes the local distinguishability of six bipartite OPSs.
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.
In this paper, we give the local distinguishability of four orthogonal product states (OPSs) on multipartite and high-dimension systems. To characterize the different orthogonal relationships between four multipartite OPSs, a new concept, i.e., the vector of the numbers of pairwise orthogonal relations, is proposed. We classify four tripartite OPSs, any two of which are orthogonal only on one subsystem, into three categories by using the vectors of the numbers of pairwise orthogonal relations. We prove that any four tripartite OPSs belonging to two of the three categories can be perfectly distinguished by local operations and classical communication (LOCC). Meanwhile, we find an interesting phenomenon that two sets of four tripartite OPSs belonging to the remaining one of the three categories have different local distinguishability even if they have the same orthogonal graph. This means that the orthogonal graph is not a sufficient condition for determining the local distinguishability of a set of OPSs. Additionally, for the remaining category, we specify the conditions under which four tripartite OPSs can be locally distinguished and the conditions under which four tripartite OPSs cannot be perfectly distinguished by LOCC. On the other hand, we prove that any four tripartite OPSs with seven or more pairwise orthogonal relations can be perfectly distinguished by LOCC. Furthermore, we prove that any four OPSs with six or more pairwise orthogonal relations can be perfectly distinguished by LOCC on four-, five-, and six-partite systems.
Local distinguishability of orthogonal quantum states can effectively reduce the consumption of quantum resources and lower economic costs in quantum protocols. Although numerous achievements have been made regarding local distinguishability of orthogonal quantum states, some fundamental issues have not been effectively addressed. For example, the local distinguishability of five orthogonal product states (OPSs) is still unknown up to now. In this paper, we give the properties of local distinguishability of five OPSs on bipartite and tripartite quantum systems. Firstly, to characterize the structure of a set of bipartite OPSs, we propose the concept of the vector of orthogonal relations for a set of bipartite OPSs. Secondly, we classify the structures of five bipartite OPSs into six categories by this concept and prove that five of these six categories can be perfectly distinguished by local operations and classical communication (LOCC). Thirdly we show that the local distinguishability of each case of the sixth category singly. On the other hand, we first divide the structures of five tripartite OPSs into eight categories by the vectors of orthogonal relations of five tripartite OPSs. Then we give the local distinguishability of each category. Our work enriches the research results of quantum nonlocality and will provide a clear understanding of the local distinguishability of five OPSs.
For general quantum systems, many sets of locally indistinguishable orthogonal quantum states have been constructed so far. However, it is interesting how much entanglement resources are sufficient and/or necessary to distinguish these states by local operations and classical communication (LOCC). Here we first present a method to locally distinguish a set of orthogonal product states in $5\ensuremath{\bigotimes}5$ by using two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Then we generalize the distinguishing method for a class of orthogonal product states in $d\ensuremath{\bigotimes}d$ (ddd is odd). Furthermore, for a class of nonlocality of orthogonal product states in $d\ensuremath{\bigotimes}d$ ($d\ensuremath{\ge}5$), we prove that these states can be distinguished by LOCC with two copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. Finally, for some multipartite orthogonal product states, we also present a similar method to locally distinguish these states with multiple copies of $2\ensuremath{\bigotimes}2$ maximally entangled states. These results also reveal the phenomenon of less nonlocality with more entanglement.