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arXiv:2206.08313 · Physical Review A

Inner products of pure states and their antidistinguishability

Vincent Russo, Jamie Sikora

Published Mar 27, 2023• 8 citations
arXivHTMLPDF

Abstract

A set of d quantum states is said to be antidistinguishable if there exists a d-outcome POVM that can perfectly identify which state was not measured. A conjecture by Havlíček and Barrett states that if a set of d pure states has small pair-wise inner products, then the set must be antidistinguishable. In this note we provide a certificate of antidistinguishability via semidefinite programming duality and use it to provide a counterexample to this conjecture when d = 4.

DOI
10.1103/physreva.107.l030202

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A set of d quantum states is said to be antidistinguishable if there exists a d-outcome POVM that can perfectly identify which state was not measured. A conjecture by Havlíček and Barrett states that if a set of d pure states has small pair-wise inner products, then the set must be antidistinguishable. In this note we provide a certificate of antidistinguishability via semidefinite programming duality and use it to provide a counterexample to this conjecture when d = 4.
A set of pure quantum states is said to be antidistinguishable if upon sampling one at random, there exists a measurement to perfectly determine some state that was not sampled. We show that antidistinguishability of a set of n pure states is equivalent to a property of its Gram matrix called ( n − 1 ) -incoherence, thus establishing a connection with quantum resource theories that lets us apply a wide variety of new tools to antidistinguishability. As a particular application of our result, we present an explicit formula (not involving any semidefinite programming) that determines whether or not a set with a circulant Gram matrix is antidistinguishable. We also show that if all inner products are smaller than ( n − 2 ) / ( 2 n − 2 ) then the set must be antidistinguishable, and we show that this bound is tight when n ≤ 4 . We also give a simpler proof that if all the inner products are strictly larger than ( n − 2 ) / ( n − 1 ) , then the set cannot be antidistinguishable, and we show that this bound is tight for all n .
Abstract The Pusey–Barrett–Rudolph theorem has recently provoked a lot of discussion regarding the reality of the quantum state. In this article we focus on a property called antidistinguishability, which is a main component in constructing the proof for the PBR theorem. In particular we study algebraic conditions for a set of pure quantum states to be antidistinguishable, and a novel sufficient condition is presented. We also discuss a more general criterion which can be used to show that the sufficient condition is not necessary. Lastly, we consider how many quantum states needs to be added into a set of pure quantum states in order to make the set antidistinguishable. It is shown that in the case of qubit states the answer is one, while in the general but finite dimensional case the answer is at most n , where n is the size of the original set.
A set of quantum states is said to be antidistinguishable if, upon being given a randomly chosen state from the set, it is possible to exclude with certainty a state that was definitely not prepared. We study quantum nonlocality within the framework of local state exclusion and prove that any set of multipartite pure states containing a state that is orthogonal to all the others is locally antidistinguishable. Next, we introduce the task of local state antimarking\textit{local state antimarking}local state antimarking, where, instead of a single state, a non-repetitive sequence of states from a known set of multipartite states is randomly selected and distributed to spatially separated parties, whose objective is to identify, using LOCC, at least one sequence that was not supplied. We present an ensemble of product states that is not globally antidistinguishable; yet, by choosing states from it without replacement, one can construct sequences that can be perfectly excluded globally but, importantly, not locally. This reveals a hitherto unexplored form of `nonlocality without entanglement'. Finally, for a given ensemble, we compare its local antidistinguishability and antimarkability with its conclusive local distinguishability and markability. We demonstrate that no strict hierarchy exists between these paradigms: there exist product-state ensembles that permit one task while strictly forbidding the other, and vice versa.
Abstract Antidistinguishability, a key concept in quantum information, describes quantum states that can be excluded by null measurement outcomes, thus enabling indirect state certification through incompatible measurements. This work systematically explores the fundamental properties of quantum state antidistinguishability. First, we present several essential characteristics of antidistinguishable quantum states. Leveraging the Bloch representation, we establish a rigorous correspondence between quantum states and positive-operator-valued measures (POVMs). Through this approach, we prove that in qubit systems, real quantum states can be antidistinguished only by real POVMs, while complex states necessitate complex POVMs. However, the scenario becomes significantly more intricate in high-dimensional quantum systems. Finally, we derive the general forms for a three-state antidistinguishable set through real and complex POVMs, respectively, achieving a complete solution to the antidistinguishability problem within qubit systems.
We investigate antidistinguishability of states within the framework of general probabilistic theories (GPTs). We formulate antidistinguishability, strong and equal antidistinguishability as refined notions that imposed additional constraint on the measurement effects. We establish general results relating these notions of antidistinguishability and derive an upper bound on the cardinality of equally antidistinguishable sets in terms of the affine dimension of the state space. We then study antidistinguishability in polygonal theories, obtaining conditions for antidistinguishability of a set of states. In consequence, we show that the set of all pure states in a polygon model is antidistinguishable. Additionally, we identify broad families of strongly and equally antidistinguishable states. Finally, using Random Exclusion Codes, whose success probability is governed by the antidistinguishability of different sets of encoding states, we probe the nonclassicality of polygon theories. We find that certain polygon models can outperform the optimal quantum value, while their optimal performance converges to the quantum limit in the large-polygon limit.