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arXiv:2601.20074

Local Distinguishability of Multipartite Orthogonal Quantum States: Generalized and Simplified

Ian George, Mohammad A. Alhejji

quant-phSubmitted Jan 27, 2026• Announced Jan 29, 2026• 0 citations
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Abstract

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 $\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}$, we strengthen this existence result by constructing an $O(d_A^2 d_B^2)$-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.

Comments
8 pages, 2 figures, comments welcome

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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.
In this work, we construct small sets of bipartite orthogonal pure states that cannot be perfectly distinguished by local operations and classical communication (LOCC). We mention that not all the states within the constructed sets are necessarily entangled. However, such a set contains at least one entangled state which cannot be conclusively identified by LOCC (with nonzero probability). Then, we show that the states of any such set can be perfectly distinguished by LOCC using a minimal-dimensional entangled resource state. Clearly, here the entangled resource state provides an advantage irrespective of the dimension of the given set. Using this result, we also prove that any pure entangled state is useful as a resource to distinguish the states of any present set unambiguously with nonzero probability under LOCC. These sets exist in all two-qudit Hilbert spaces. Furthermore, it is possible to decrease the average entanglement content of such a set or to increase the cardinality of the set without changing certain properties of the same.
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.
One-way local operations and classical communication (LOCC) can fail to distinguish four orthogonal maximally entangled states in every local dimension d≥4d\ge4d≥4. We show that every set of kkk pairwise orthogonal maximally entangled states in Cd⊗Cd\mathbb C^d\otimes\mathbb C^dCd⊗Cd is perfectly distinguishable by an A→B→AA\to B\to AA→B→A protocol whenever d≥Ck4[log⁡(2k)]νd\ge Ck^4[\log(2k)]^νd≥Ck4[log(2k)]ν, for universal constants C,ν>0C,ν>0C,ν>0. A single feedback message therefore raises the number of candidates that can always be distinguished from a constant to Ω ⁣(d1/4/(log⁡d)ν/4)Ω\!\left(d^{1/4}/(\log d)^{ν/4}\right)Ω(d1/4/(logd)ν/4), establishing a polynomial separation between one-way and two-way LOCC. In our protocol, Alice preserves exact orthogonality while reducing overlaps between Bob's conditional subspaces. Bob reduces the candidates to a singleton or an orthogonal pair and refines his outcome so that Alice can complete the discrimination. As a corollary, nine copies suffice for perfect two-way LOCC discrimination of any orthogonal maximally entangled set in all sufficiently large local dimensions.