In the topic of distinguishment of orthogonal states in dd bipartite system,with the way of judging dimension of there corresponding orthogonal complement,and then determine whether the orthogonal complement space contains a maximally Abelian subspace,at last propose a new algorithm to distinguish states based on one-way LOCC.For generalized orthogonal Bell states in 44 bipartite system,by studying their distinguishment,this paper propose a faster algorithm based on one way LOCC.Experimental results show that the algorithm in ddbipartite system can be applies in distinguishing any orthogonal bipartite states.What’s more,the algorithm in 44 bipartite system is less complicated.
Local distinguishability of quantum states is a basic problem in quantum information theory. For an arbitrary group of entangled states in d ⊗ d bipartite system, Singal presented a subspace related to these states, and then he determined that the group of states can be distinguished by one-way local operations and classical communication(1-LOCC) if this subspace contains a maximally Abelian subspace. In this paper according to Singal's results, we present an algorithm to distinguish orthogonal bipartite states in d ⊗ d system based on 1-LOCC; we gives a sufficient condition to judge whether generalized orthogonal quantum states in 4 ⊗ 4 can be distinguished. Finally, we show an corresponding example to verify the conclusion.
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 (
1-LOCC) of sets of orthogonal bipartite states in a
dA⊗dB bipartite system, where
dA,dB are the dimensions of both subsytems, labelled as
A and
B. I show that if the
i-th party (where
i=A,B) can initiate a
1-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a
1-LOCC protocol lies in a subspace of
di×di hermitian matrices, denoted by $\Tb^{(i)}$, and that the method to extract this information (of the existence of this
1-LOCC protocol) from $\Tb^{(i)}$ depends on the value of $dim \Tb^{(i)}$. In this way one can give sweeping results for the
1-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
1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
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 Rapid Communication, I propose a framework for the distinguishability by one-way local operations and classical communication (1-LOCC) of sets of orthogonal bipartite states in a ${d}_{A}\ensuremath{\bigotimes}{d}_{B}$ bipartite system, where
dA,dB are the dimensions of both subsytems, labeled as
A and
B. I show that if the
ith party (where
i=A,B) can initiate a 1-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 1-LOCC protocol lies in a subspace of ${d}_{i}\ifmmode\times\else\texttimes\fi{}{d}_{i}$ Hermitian matrices, denoted by ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$, and that the method to extract this information (of the existence of this 1-LOCC protocol) from ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ depends on the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. In this way one can give sweeping results for the 1-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. Thus I propose that the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ gives the common underlying property based on which sweeping results for the 1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
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
⌈4N⌉ copies are allowed, where
⌈l⌉ 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
⌈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 a seminal work [PRL85.4972], Walgate, Short, Hardy, and Vedral prove in finite dimensions that for every pair of pure multipartite orthogonal quantum states, there exists a one-way local operations and classical communication (LOCC) protocol that perfectly distinguishes the pair. We extend this result to infinite dimensions with a simpler proof. For states on
CdA×dA⊗CdB×dB, we strengthen this existence result by constructing an
O(dA2dB2)-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.
Abstract It is an interesting topic how much entanglement resources are necessary to distinguish locally indistinguishable orthogonal quantum states by local operations and classical communication (LOCC). Using only one ebit of entanglement some locally indistinguishable orthogonal product bases can be distinguished by LOCC [2019 Phys. Rev. A 99 012343]. However, it is still an open question whether one ebit of entanglement is sufficient for distinguishing a locally indistinguishable set of orthogonal maximally entangled states. We focus on the generalized Bell states (GBSs), and firstly find that any four GBSs in a 4 ⊗ 4 quantum system are always distinguished by one-way LOCC using one ebit of entanglement. Then we consider a general case, i.e., d ⊗ d quantum system ( d is even), and show that the set L of any GBSs can be distinguished by one-way LOCC using one ebit of entanglement if S A \ S L is nonempty, where S A is the admissible solutions set of L and S L is the genuinely nonadmissible solutions set of L . This criterion is very simple and convenient since only some congruence equations are calculated. Finally, we find a curious and counterintuitive case, i.e., in a 3 ⊗ 3 quantum system there exist four GBSs which cannot be distinguished by one-way LOCC using one ebit of entanglement, although the dimension of the subsystem is less than four. These results can help us better understand the role of entanglement as a resource in local state discrimination problems.