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arXiv:2610.16138

Perfect discrimination of an orthogonal $2\otimes2$ basis may require collective separable measurements on two copies

Somshubhro Bandyopadhyay

quant-phSubmitted Oct 12, 2026• Announced Oct 13, 2026• 0 citations
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Abstract

We consider an orthogonal basis of $\mathbb C^2\otimes\mathbb C^2$ with three entangled states and one product state. This basis is not perfectly distinguishable by LOCC or even by separable measurements when a single copy of the unknown state is available. We show that with two copies no adaptive separable protocol, which measures the copies one at a time, distinguishes the basis perfectly. We then show that a collective separable measurement on the two copies does. Whether two copies suffice for LOCC remains open.

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11 pages, 2 tables

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Perfect discrimination of an orthogonal 2⊗22\otimes22⊗2 basis may require collective separable measurements on two copies
We consider an orthogonal basis of C2⊗C2\mathbb C^2\otimes\mathbb C^2C2⊗C2 with three entangled states and one product state. This basis is not perfectly distinguishable by LOCC or even by separable measurements when a single copy of the unknown state is available. We show that with two copies no adaptive separable protocol, which measures the copies one at a time, distinguishes the basis perfectly. We then show that a collective separable measurement on the two copies does. Whether two copies suffice for LOCC remains open.
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.
Abstract It is a central fact in quantum mechanics that non-orthogonal states cannot be distinguished perfectly. In general, the optimal measurement for distinguishing such states is a collective measurement. However, to the best our knowledge, collective measurements have not been used to enhance quantum state discrimination to date. One of the main reasons for this is the fact that, in the usual state discrimination setting with equal prior probabilities, at least three copies of a quantum state are required to be measured collectively to outperform separable measurements. This is very challenging experimentally. In this work, by considering unequal prior probabilities, we propose and experimentally demonstrate a protocol for distinguishing two copies of single qubit states using collective measurements which achieves a lower probability of error than can be achieved by any non-entangling measurement. Additionally, we implemented collective measurements on three and four copies of the unknown state and found they performed poorly.
More than two decades ago, Bennett \emph{et al.} [Phys. Rev. A \textbf{59}, 1070 (1999)] asked whether perfect local discrimination of orthogonal quantum states can require more than two copies. This question was subsequently answered for adaptive protocols that process the copies separately [Phys. Rev. Lett. \textbf{126}, 210505 (2021)], but remained open when each laboratory is allowed to process its local copies collectively. Here we resolve this stronger setting and identify collective access across repeated local inputs as a distinct resource. For every fixed odd-prime local dimension, there exist complete maximal-stabilizer eigenbases whose copy complexity under individual-copy separable measurements diverges with system size. Under collective processing this behavior changes sharply: every maximal-stabilizer eigenbasis in odd-prime local dimension is perfectly decoded by one-round collective LOCC using at most three copies. Collective processing, however, does not remove multicopy hardness in general. For every fixed local dimension d≥2d\ge2d≥2, we prove the existence, within an explicit phase family, of complete bases whose copy complexity remains unbounded even under collective separable measurements, with a square root of the number of subsystems as lower-bound scale. Thus sample number, spatial measurement power, and coherent access across repeated local inputs are distinct resources in distributed quantum measurement.
We study the discrimination of a pair of orthogonal quantum states in the many-copy setting. This is not a problem when arbitrary quantum measurements are allowed, as then the states can be distinguished perfectly even with one copy. However, it becomes highly nontrivial when we consider states of a multipartite system and locality constraints are imposed. We hence focus on the restricted families of measurements such as local operation and classical communication (LOCC), separable operations (SEP), and the positive-partial-transpose operations (PPT) in this paper. We first study asymptotic discrimination of an arbitrary multipartite entangled pure state against its orthogonal complement using LOCC/SEP/PPT measurements. We prove that the incurred optimal average error probability always decays exponentially in the number of copies, by proving upper and lower bounds on the exponent. In the special case of discriminating a maximally entangled state against its orthogonal complement, we determine the explicit expression for the optimal average error probability, thus establishing the associated Chernoff exponent. Our technique is based on the idea of using PPT operations to approximate LOCC. Then, we show an infinite asymptotic separation between SEP and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB): they can be distinguished perfectly by PPT measurements, while the optimal error probability using SEP measurements admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is UPB.
In $2\ensuremath{\bigotimes}2$, more than two orthogonal Bell states with a single copy can never be discriminated with certainty if only local operations and classical communication (LOCC) are allowed. We show here that more than ddd numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$, which are in canonical form, used by Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)], can never be discriminated with certainty by LOCC, when single copies of the states are provided. Interestingly we show here that all orthogonal maximally entangled states, which are in canonical form, can be discriminated with certainty by LOCC if and only if two copies of each of the states are provided. We provide here a conjecture regarding the highly nontrivial problem of local distinguishability of any ddd or fewer numbers of pairwise orthogonal maximally entangled states in $d\ensuremath{\bigotimes}d$ (in the single copy case).
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.