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arXiv:1704.01409 · Physical Review A

Optimal sequential measurements for bipartite state discrimination

Sarah Croke, Stephen Mark Barnett, Graeme Weir

Published May 3, 2017• 18 citations
arXivHTMLPDF

Abstract

State discrimination is a useful test problem with which to clarify the power and limitations of different classes of measurement. We consider the problem of discriminating between given states of a bipartite quantum system via sequential measurement of the subsystems, with classical feed-forward of measurement results. Our aim is to understand when sequential measurements, which are relatively easy to implement experimentally, perform as well, or almost as well, as optimal joint measurements, which are in general more technologically challenging. We construct conditions that the optimal sequential measurement must satisfy, analogous to the well-known Helstrom conditions for minimum error discrimination in the unrestricted case. We give several examples and compare the optimal probability of correctly identifying the state via global versus sequential measurement strategies.

DOI
10.1103/physreva.95.052308

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State discrimination is a useful test problem with which to clarify the power and limitations of different classes of measurement. We consider the problem of discriminating between given states of a bipartite quantum system via sequential measurement of the subsystems, with classical feed-forward of measurement results. Our aim is to understand when sequential measurements, which are relatively easy to implement experimentally, perform as well, or almost as well, as optimal joint measurements, which are in general more technologically challenging. We construct conditions that the optimal sequential measurement must satisfy, analogous to the well-known Helstrom conditions for minimum error discrimination in the unrestricted case. We give several examples and compare the optimal probability of correctly identifying the state via global versus sequential measurement strategies.
We investigate an optimization problem of finding quantum sequential measurements, which forms a wide class of state discrimination problems with the restriction that only local operations and one-way classical communication are allowed. Sequential measurements from Alice to Bob on a bipartite system are considered. Using the fact that the optimization problem can be formulated as a problem with only Alice's measurement and is convex programming, we derive its dual problem and necessary and sufficient conditions for an optimal solution. Our results are applicable to various practical optimization criteria, including the Bayes criterion, the Neyman-Pearson criterion, and the minimax criterion. In the setting of the problem of finding an optimal global measurement, its dual problem and necessary and sufficient conditions for an optimal solution have been widely used to obtain analytical and numerical expressions for optimal solutions. Similarly, our results are useful to obtain analytical and numerical expressions for optimal sequential measurements. Examples in which our results can be used to obtain an analytical expression for an optimal sequential measurement are provided.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
We study sequential state discrimination measurements performed on the same qubit by subsequent observers. Specifically, we focus on the case when the observers perform a kind of a minimum-error type state discriminating measurement where the goal of the observers is to maximize their joint probability of successfully guessing the state that the qubit was initially prepared in. We call this the Joint Best Guess strategy. In this scheme, Alice prepares a qubit in one of two possible states. The qubit is first sent to Bob, who measures it, and then on to Charlie, and so on to altogether N consecutive receivers who all perform measurements on it. The goal for all observers is to determine which state Alice sent. In the Joint Best Guess strategy, every time a system is received the observer is required to make a guess, aided by the measurement, about its state. The price to pay for this requirement is that errors must be permitted, the guess can be correct or in error. There is a nonzero probability for all the receivers to successfully identify the initially prepared state, and we maximize this joint probability of success. This work is a step toward developing a theory of nondestructive sequential quantum measurements and could be useful in multiparty quantum communication schemes based on communicating with single qubits, particularly in schemes employing continuous variable states. It also represents a case where subsequent observers can probabilistically and optimally get around both the collapse postulate and the no-broadcasting theorem.
Generalized contextuality is known to be required in optimal strategies for quantum state discrimination protocols. More recently, sequential discrimination tasks have been studied; n players attempt to determine in which state a qubit was prepared, in such a way that they all have a finite probability of success. We consider the extent to which contextuality plays a role in sequential versions of both unambiguous and minimum error discrimination. In the standard nonsequential case where n = 1, we use the COPE formalism to demonstrate that the presence of contextuality is guaranteed not only for the optimal measurement, but for a specific set of nonoptimal measurements as well. In the sequential case n > 1, we show that the presence of contextuality depends on which states are prepared, and on the protocol (unambiguous or minimum error) selected.
We study an optimal quantum measurement for generalized quantum state discrimination problems, which include the problem of finding an optimal measurement maximizing the average success probability with and without a fixed rate of inconclusive results and the problem of finding an optimal measurement in the Neyman-Pearson strategy. We show that for any generalized quantum state discrimination problem, there is a corresponding minimum error discrimination problem sharing the same optimal measurement. We propose an approach in which the optimal measurement is obtained by solving the corresponding minimum error discrimination problem, and clarify the relationship between optimal solutions to the original and the corresponding problems, with which one can obtain an optimal solution to the original problem in some cases. Moreover, as an example of application of our approach, we present an algorithm for numerically obtaining optimal solutions to generalized quantum state discrimination problems.
We investigate unambiguous discrimination between given quantum states with a sequential measurement, which is restricted to local measurements and one-way classical communication. If the given states are binary or each of their individual state spaces is two dimensional, then it is in some cases known whether a sequential measurement achieves a globally optimal unambiguous measurement. In contrast, for more than two states each of whose individual systems is more than two dimensional, the problem becomes extremely complicated. This paper focuses on symmetric ternary separable pure states each of whose individual systems is three dimensional, which include phase shift keyed (PSK) optical coherent states and a lifted version of ``double trine'' states. We provide a necessary and sufficient condition for an optimal sequential measurement to be globally optimal for the bipartite case. A sufficient condition of global optimality for multipartite states is also presented. One can easily judge whether these conditions hold for given states. Some examples are given, which demonstrate that, despite the restriction to local measurements and one-way classical communication, a sequential measurement can be globally optimal in quite a few cases.