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arXiv:2503.02568 · Physical Review Research

Quantum state exclusion for group-generated ensembles of pure states

A. Diebra, S. Llorens, E. Bagán, Gael Sentís, R. Muñoz-Tapia

Published Nov 23, 2025• 1 citation
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Abstract

Quantum state exclusion is the task of determining which states from a given set a system was not prepared in. We provide a complete solution to optimal quantum state exclusion for arbitrary sets of pure states generated by finite groups, establishing necessary and sufficient conditions for perfect (zero-error conclusive) exclusion. When perfect exclusion is impossible, we introduce two natural extensions: minimum-error and unambiguous exclusion. For both, we derive the optimal protocols and present analytical expressions for the corresponding failure probabilities and measurements, providing additional insight into how quantum states encode information.

DOI
10.1103/2k5d-bprn

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Quantum state exclusion is the task of determining which states from a given set a system was not prepared in. We provide a complete solution to optimal quantum state exclusion for arbitrary sets of pure states generated by finite groups, establishing necessary and sufficient conditions for perfect (zero-error conclusive) exclusion. When perfect exclusion is impossible, we introduce two natural extensions: minimum-error and unambiguous exclusion. For both, we derive the optimal protocols and present analytical expressions for the corresponding failure probabilities and measurements, providing additional insight into how quantum states encode information.
In the task of quantum state exclusion, we consider a quantum system prepared in a state chosen from a known set. The aim is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures cannot have taken place. We ask what conditions the set of states must obey in order for this to be possible and how well we can complete the task when it is not. The task of quantum state discrimination forms a subclass of this set of problems. Within this paper, we formulate the general problem as a semidefinite program (SDP), enabling us to derive sufficient and necessary conditions for a measurement to be optimal. Furthermore, we obtain a necessary condition on the set of states for exclusion to be achievable with certainty, and we give a construction for a lower bound on the probability of error. This task of conclusively excluding states has gained importance in the context of the foundations of quantum mechanics due to a result from Pusey, Barrett, and Rudolph (PBR). Motivated by this, we use our SDP to derive a bound on how well a class of hidden variable models can perform at a particular task, proving an analog of Tsirelson's bound for the PBR experiment and the optimality of a measurement given by PBR in the process. We also introduce variations of conclusive exclusion, including unambiguous state exclusion, and state exclusion with worst-case error.
Retrieving classical information from quantum systems is central to quantum information processing. As a more general task than quantum state discrimination, which focuses on identifying the exact state, quantum state exclusion only requires ruling out options, revealing fundamental limits of information extraction from quantum systems. In this work, we study the conclusive exclusion of quantum states generated by group actions, establishing explicit criteria for when such exclusion is possible. For systems with complex symmetries, including finite and compact Lie groups, we derive a sufficient condition for conclusive exclusion based on the initial state's amplitudes and the group's structure. As applications to special groups such as Abelian groups, we establish necessary and sufficient conditions for conclusive state exclusion and generalize the Pusey-Barrett-Rudolph result to a wider range of scenarios. Finally, we explore zero-error communication via conclusive exclusion of quantum states and derive a lower bound on the feedback-assisted and non-signalling-assisted zero-error capacity of classical-quantum channels generated by group actions.
Quantum state exclusion is the task of identifying at least one state from a known set that was not used in the preparation of a quantum system. A set of quantum states is said to admit state exclusion if there exists a measurement whose outcomes can be put in one-to-one correspondence with the states in the set, such that each outcome rules out its corresponding state with certainty (while possibly also ruling out other states), and each outcome occurs with nonzero probability for at least one state in the set. State exclusion, however, is not always possible in the single-copy setting. In this paper, we investigate whether access to multiple identical copies of the system enables state exclusion. We prove that for any set of three or more pure states, state exclusion becomes possible with a finite number of copies. Moreover, we show that the number of copies required may be arbitrarily large: in particular, for every natural number NNN, we construct sets of states for which state exclusion remains impossible with NNN or fewer copies.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
Abstract Consider a quantum system prepared in a state chosen from a known set, the aim of quantum state discrimination is to perform a measurement on the system which can determine that a subset of the possible preparation procedures have been prepared. However, the aim of quantum state exclusion is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures have not taken place. For classical states, probabilities of error in discrimination and exclusion are always equal. For quantum states, we point out that probabilities of error in discrimination and exclusion are still equal in two-state ensembles, however they are no longer equal in multiple-state ensembles. More specifically, the error-probability of discrimination is not less than the error-probability of exclusion. We also show that the error-probability of exclusion is continuous which is similar with the error-probability of discrimination.
We investigate the unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of NNN pure states, and find an optimal solution for some parameter range when N=3N=3N=3. In both cases we use the reduction method for the corresponding problem of mixed-state discrimination, as introduced by Raynal et al., which reduces the problem to the discrimination of two pure states only for N=2N=2N=2. Finally, we provide a necessary and sufficient condition for unambiguous comparison of mixed states to be possible.