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arXiv:2302.08882 · Communications Physics

Discriminating mixed qubit states with collective measurements

Lorcán O. Conlon, Falk X. Eilenberger, Ping Koy Lam, Syed M. Assad

Published Nov 22, 2023• 16 citations
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Abstract

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.

DOI
10.1038/s42005-023-01454-z

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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.
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.
We theoretically investigate schemes to discriminate between two nonorthogonal quantum states given multiple copies. We consider a number of state discrimination schemes as applied to nonorthogonal, mixed states of a qubit. In particular, we examine the difference that local and global optimization of local measurements makes to the probability of obtaining an erroneous result, in the regime of finite numbers of copies NNN, and in the asymptotic limit as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Five schemes are considered: optimal collective measurements over all copies, locally optimal local measurements in a fixed single-qubit measurement basis, globally optimal fixed local measurements, locally optimal adaptive local measurements, and globally optimal adaptive local measurements. Here an adaptive measurement is one in which the measurement basis can depend on prior measurement results. For each of these measurement schemes we determine the probability of error (for finite NNN) and the scaling of this error in the asymptotic limit. In the asymptotic limit, it is known analytically (and we verify numerically) that adaptive schemes have no advantage over the optimal fixed local scheme. Here we show moreover that, in this limit, the most naive scheme (locally optimal fixed local measurements) is as good as any noncollective scheme except for states with less than 2% mixture. For finite NNN, however, the most sophisticated local scheme (globally optimal adaptive local measurements) is better than any other noncollective scheme for any degree of mixture.
We show a geometric formulation for minimum-error discrimination of qubit states that can be applied to arbitrary sets of qubit states given with arbitrary a priori probabilities. In particular, when qubit states are given with equal a priori probabilities, we provide a systematic way of finding optimal discrimination and the complete solution in a closed form. This generally gives a bound to cases when prior probabilities are unequal. Then it is shown that the guessing probability does not depend on detailed relations among the given states, such as the angles between them, but on a property that can be assigned by the set of given states itself. This also shows how a set of quantum states can be modified such that the guessing probability remains the same. Optimal measurements are also characterized accordingly, and a general method of finding them is provided.
One of the fundamental tenets of quantum mechanics is that nonorthogonal states cannot be distinguished perfectly. When distinguishing multiple copies of a mixed quantum state, a collective measurement, which generates entanglement between the different copies of the unknown state, can achieve a lower error probability than nonentangling measurements. The error probability that can be attained using a collective measurement on a finite number of copies of the unknown state is given by the Helstrom bound. In the limit where we can perform a collective measurement on asymptotically many copies of the quantum state, the quantum Chernoff bound gives the attainable error probability. It is natural to ask, then, what strategies can be employed to reach these two bounds---is entanglement across all available modes always necessary or can other, experimentally more simple measurements saturate these bounds? In this work we address this question. We find analytic expressions for the Helstrom bound for arbitrarily many copies of the unknown state in simple qubit examples. Using these analytic expressions, we investigate whether the quantum Chernoff bound can be saturated by repeatedly implementing the MMM-copy Helstrom measurement. We also investigate the necessary conditions to saturate the MMM-copy Helstrom bound. It is known that a collective measurement on all MMM copies of the unknown state is always sufficient to saturate the MMM-copy Helstrom bound. However, general conditions for when such a measurement is necessary to saturate the Helstrom bound remain unknown. We investigate specific measurement strategies which involve entangling operations on fewer than all MMM copies of the unknown state. For many regimes we find that a collective measurement on all MMM copies of the unknown state is necessary to saturate the MMM-copy Helstrom bound.
Quantum measurements can sometimes extract more information from multiple copies of a state than from each copy separately. One of the simplest settings is the discrimination of three qubit states when two identical copies are available and one is allowed to perform an arbitrary joint measurement. Recent numerical work identified the double-trine ensemble as the apparent optimal benchmark in this setting, but an analytic proof of its optimality remained open. In this paper, we prove that the double-trine ensemble is indeed globally optimal. More precisely, among all triples of equally likely qubit states, no choice of states can give a larger optimal two-copy discrimination success probability. The result holds in full generality, whether the signal states are pure or mixed, real or complex. This exactly solved case serves as a foundation for a broader analytic understanding of optimal collective measurements.
Quantum nonlocality, a form of nonclassicality, serves as a crucial resource in quantum information processing. Nonlocality in quantum systems is exemplified by the perfect discrimination of orthogonal quantum states through single joint measurements, which is unattainable using local operations and classical communication (LOCC). Bandyopadhyay [S. Bandyopadhyay, Phys. Rev. Lett. 106, 210402 (2011)] established that, with LOCC alone, at most $N\ensuremath{-}1$ copies of NNN orthogonal quantum states are required for perfect discrimination. Here, we present the experimental verification of orthogonal quantum state discrimination employing LOCC. Our results demonstrate that up to three copies suffice for the perfect distinction of four orthogonal two-qubit states via LOCC. The verified method of perfectly distinguishing orthogonal quantum state sets offers potential applications in quantum information processes, including quantum secret sharing and quantum teleportation.