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Journal of Modern Optics

Strategies for discriminating between non-orthogonal quantum states

Anthony Chefles, Stephen Mark Barnett

Published Jun 1, 1998• 112 citations
Publisher

Abstract

Any measurement strategy designed to discriminate between nonorthogonal quantum states will have a finite probability of yielding erroneous or inconclusive results. The lower bounds on these probabilities for conclusive and error-free strategies are given by the Helstrom and Ivanovic–Feres limits respectively. We consider intermediate measurements which interpolate between these extremes and derive a general lower bound on the combination of errors and inconclusive results. We also describe a family of measurement strategies which attains this limit.

DOI
10.1080/09500349808230919

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Any measurement strategy designed to discriminate between nonorthogonal quantum states will have a finite probability of yielding erroneous or inconclusive results. The lower bounds on these probabilities for conclusive and error-free strategies are given by the Helstrom and Ivanovic–Feres limits respectively. We consider intermediate measurements which interpolate between these extremes and derive a general lower bound on the combination of errors and inconclusive results. We also describe a family of measurement strategies which attains this limit.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
Summary form only given. It is not possible to discriminate infallibly between non-orthogonal states. Attempts to do so will inevitably result in incorrect or inconclusive results, or both. One limiting case is the Helstrom measurement, which always gives a result, but not necessarily the correct one. The other is the Ivanovic-Dieks-Peres case, which sometimes gives an inconclusive result, but when it provides a discrimination between the states it is always correct. Here we present an experimental demonstration of measurement of this second type. The results were in excellent agreement with the theoretical predictions and demonstrated significant advantages over previous experimental techniques. We used a free-space interferometer constructed from four polarising beam splitters to perform this task. Two nonorthogonal linear polarisations of heavily attenuated pulses of mode-locked laser light formed the states to be discriminated.
Abstract The impossibility of deterministic and error-free discrimination among nonorthogonal quantum states lies at the core of quantum theory and constitutes a primitive for secure quantum communication. Demanding determinism leads to errors, while demanding certainty leads to some inconclusiveness. One of the most fundamental strategies developed for this task is the optimal unambiguous measurement. It encompasses conclusive results, which allow for error-free state retrodictions with the maximum success probability, and inconclusive results, which are discarded for not allowing perfect identifications. Interestingly, in high-dimensional Hilbert spaces the inconclusive results may contain valuable information about the input states. Here, we theoretically describe and experimentally demonstrate the discrimination of nonorthogonal states from both conclusive and inconclusive results in the optimal unambiguous strategy, by concatenating a minimum-error measurement at its inconclusive space. Our implementation comprises four- and nine-dimensional spatially encoded photonic states. By accessing the inconclusive space to retrieve the information that is wasted in the conventional protocol, we achieve significant increases of up to a factor of 2.07 and 3.73, respectively, in the overall probabilities of correct retrodictions. The concept of concatenated optimal measurements demonstrated here can be extended to other strategies and will enable one to explore the full potential of high-dimensional nonorthogonal states for quantum communication with larger alphabets.
Quantum measurement and control science provides a toolkit for implementing quantum information protocols, overcoming noisy operation and minimizing the use of costly quantum resources. The importance of finding optimal measurement and control strategies is revealed in the task of quantum state discrimination, a characteristic feature of quantum mechanics and a primitive for quantum information science and technology. When a quantum system is prepared in one of two known nonorthogonal quantum states, no measurement can deterministically tell which state the system is in. Two strategies can be applied to discriminate non-orthogonal states efficiently. Unambiguous state discrimination (USD) is strategy that provides a guess which is either correct, or inconclusive [1]. Alternatively, minimum error discrimination (MED) strategy will always provide a conclusive answer, which can sometimes be incorrect [2]. Perfect nonorthogonal state discrimination is still impossible even when multiple copies of the system are available, although different strategies increase the chance of having conclusive (for USD), or correct (for MED) result. Multiple-copy measurement strategies can be either collective, when a single measurement involves all the copies of the system, or local, when each copy of the system is measured separately. The latter can be further divided into fixed, where the measurement applied to each copy is the same, or adaptive, where the measurement on each subsequent copy depends on the outcomes of the previous measurements [3,4]. A multiple-copy MED strategy is defined by its goal of minimizing the average error for fixed resources, i.e. the number of copies of the system. An alternative approach we consider, is to minimize the average resources required, while keeping errors below a given bound [5]. This approach is central for fault-tolerant quantum computing and has been applied to a number of quantum control strategies [6,7]. We call the corresponding state discrimination task the guaranteed bounded error discrimination (GBED) task. Intuitively, one may assume that the multiple-copy strategies optimal for MED would be also optimal for GBED. We experimentally apply two known local non-adaptive strategies, previously considered for the MED task, to the two-state GBED problem [5]. We then derive and experimentally demonstrate a new local strategy, designed specifically for the GBED task that outperforms other strategies. Moreover, it performs usually better than, and in the regime of small error, scales better than, the theoretical performance of the optimal adaptive strategy for the MED task. The discovered reversal in the performance of schemes when swapping the task definition from performance-maximization to resource-minimization, is similar to that previously observed in state purification [6], suggesting that this phenomenon is a generic one. [1] I. D. Ivanovic, Phys. Lett. A 123, 257 (1987). [2] C. W. Helstrom, Quantum detection and estimation theory (1976). [3] A. Acín, et al., Phys. Rev. A 71, 032338 (2005). [4] B. L. Higgins, et al., Phys. Rev. Lett. 103, 220503 (2009). [5] S. Slussarenko, et al., PRL 118, 030502 (2017). [6] H. M. Wiseman and J. F. Ralph. New J. Phys. 8, 90 (2006). [7] J. Combes, et al., Phys. Rev. Lett. 100, 160503 (2008).
The minimum error probability for distinguishing between two quantum states is bounded by the Helstrom limit, derived under the assumption that measurement strategies are restricted to positive operator-valued measurements. We explore scenarios in which the error probability for discriminating two quantum states can be reduced below the Helstrom bound under some constrained access of resources, indicating the use of measurement operations that go beyond the standard positive operator-valued measurements framework. We refer to such measurements as non-positive operator-valued measurements. While existing literature often associates these measurements with initial entanglement between the system and an auxiliary, followed by joint projective measurement and discarding the auxiliary, we demonstrate that initial entanglement between system and auxiliary is not necessary for the emergence of such measurements in the context of state discrimination. Interestingly, even initial product states can give rise to effective non-positive measurements on the subsystem, and achieve sub-Helstrom discrimination error when discriminating quantum states of the subsystem.
Research in nonorthogonal-state discrimination has given rise to two conventional optimal strategies: unambiguous discrimination (UD) and minimum error discrimination. We explore the experimentally relevant range of measurement strategies between the two, where the rate of inconclusive results is minimized for a bounded-error rate. We first provide some constraints on the problem that apply to generalized measurements [positive-operator-valued measurements (POVMs)]. We then provide the theory for the optimal projective measurement in this range. Through analytical and numerical results we investigate this family of projective, bounded-error strategies and compare it to the POVM family as well as to experimental implementation of UD using POVMs. We also discuss a possible application of these bounded-error strategies to quantum key distribution.