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arXiv:1804.09693 · Physical Review A

State discrimination with postmeasurement information and incompatibility of quantum measurements

Claudio Carmeli, Teiko Heinosaari, Alessandro Toigo

Published Jul 19, 2018• 45 citations
arXivHTMLPDF

Abstract

We discuss the following variant of the standard minimum error state discrimination problem: Alice picks the state she sends to Bob among one of several disjoint state ensembles, and she communicates him the chosen ensemble only at a later time. Two different scenarios then arise: either Bob is allowed to arrange his measurement setup after Alice has announced him the chosen ensemble, or he is forced to perform the measurement before Alice's announcement. In the latter case, he can only postprocess his measurement outcome when Alice's extra information becomes available. We compare the optimal guessing probabilities in the two scenarios, and we prove that they are the same if and only if there exist compatible (i.e., jointly measurable) optimal measurements for all of Alice's state ensembles. When this is the case, postprocessing any of the corresponding joint measurements is Bob's optimal strategy in the postmeasurement information scenario. Furthermore, we establish a connection between discrimination with postmeasurement information and the standard state discrimination. By means of this connection and exploiting the presence of symmetries, we are able to compute the various guessing probabilities in many concrete examples.

DOI
10.1103/physreva.98.012126

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We discuss the following variant of the standard minimum error state discrimination problem: Alice picks the state she sends to Bob among one of several disjoint state ensembles, and she communicates him the chosen ensemble only at a later time. Two different scenarios then arise: either Bob is allowed to arrange his measurement setup after Alice has announced him the chosen ensemble, or he is forced to perform the measurement before Alice's announcement. In the latter case, he can only postprocess his measurement outcome when Alice's extra information becomes available. We compare the optimal guessing probabilities in the two scenarios, and we prove that they are the same if and only if there exist compatible (i.e., jointly measurable) optimal measurements for all of Alice's state ensembles. When this is the case, postprocessing any of the corresponding joint measurements is Bob's optimal strategy in the postmeasurement information scenario. Furthermore, we establish a connection between discrimination with postmeasurement information and the standard state discrimination. By means of this connection and exploiting the presence of symmetries, we are able to compute the various guessing probabilities in many concrete examples.
We demonstrate that quantum incompatibility can always be detected by means of a state discrimination task with partial intermediate information. This is done by showing that only incompatible measurements allow for an efficient use of premeasurement information in order to improve the probability of guessing the correct state. Thus, the gap between the guessing probabilities with pre- and postmeasurement information is a witness of the incompatibility of a given collection of measurements. We prove that all linear incompatibility witnesses can be implemented as some state discrimination protocol according to this scheme. As an application, we characterize the joint measurability region of two noisy mutually unbiased bases.
We demonstrate that quantum incompatibility can always be detected by means of a state discrimination task with partial intermediate information. This is done by showing that only incompatible measurements allow for an efficient use of premeasurement information in order to improve the probability of guessing the correct state. Thus, the gap between the guessing probabilities with pre- and postmeasurement information is a witness of the incompatibility of a given collection of measurements. We prove that all linear incompatibility witnesses can be implemented as some state discrimination protocol according to this scheme. As an application, we characterize the joint measurability region of two noisy mutually unbiased bases.
A set of pairwise non-orthogonal quantum states cannot be perfectly discriminated, and this remains true even when one bit of classical partial information is available prior to the measurement. Contrary to the usual intuition that earlier information is at least as valuable as later information, we show that the same bit can be more useful when it arrives after the measurement. We present a framework for state discrimination in which the sender provides classical information in response to a request from the receiver, which we refer to as a query. In some cases, pairwise non-orthogonal states can be perfectly discriminated when the query is allowed to depend on the measurement outcome. We give a general method for finding the optimal strategy in this setting, which turns out to be the standard minimum-error discrimination problem for an auxiliary ensemble.
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.
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.
Distinguishing different quantum states is a fundamental task having practical applications in information processing. Despite the effort devoted so far, however, strategies for optimal discrimination are known only for specific examples. In this paper we consider the problem of minimum-error quantum state discrimination where one attempts to minimize the average error. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalent class that share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.