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Journal of Mathematical Physics

Maximizing the probability of detection for pure states

Andrzej Łuczak

Published May 1, 2009• 5 citations
Publisher

Abstract

We show that for an infinite number of linearly independent pure states there exists a uniquely determined simple measurement maximizing the probability of detection, generalizing thus the result known for a finite number of states. We also comment on the problem of the existence of an optimal measurement for a sequence of arbitrary normal states in the general Bayes approach.

DOI
10.1063/1.3123947

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We show that for an infinite number of linearly independent pure states there exists a uniquely determined simple measurement maximizing the probability of detection, generalizing thus the result known for a finite number of states. We also comment on the problem of the existence of an optimal measurement for a sequence of arbitrary normal states in the general Bayes approach.
We consider the problem of unambiguous (error-free) discrimination of NNN linearly independent pure quantum states with prior probabilities, where the goal is to find a measurement that maximizes the average probability of success. We derive an upper bound on the optimal average probability of success using a result on optimal local conversion between two bipartite pure states. We prove that for any $N\ensuremath{\ge}2$ an optimal measurement in general saturates our bound. In the exceptional cases we show that the bound is tight, but not always optimal.
We address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average correct probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used.
We consider the problem of determining the state of an unknown quantum sequence without error. The elements of the given sequence are drawn with equal probability from a known set of linearly independent pure quantum states with the property that their mutual inner products are all real and equal. This problem can be posed as an instance of unambiguous state discrimination where the states correspond to that of all possible sequences having the same length as the given one. We calculate the optimum probability by solving the optimality conditions of a semidefinite program. The optimum value is achievable by measuring individual members of the sequence, and no collective measurement is necessary.
Given nnn linearly independent pure states and their prior probabilities, we study the optimum unambiguous state discrimination problem. We derive the conditions for the optimum measurement strategy to achieve the maximum average success probability and establish two sets of equations that must be satisfied by the optimum solution in different situations. We also provide the detailed steps to find the optimum measurement strategy. The method and results we obtain are given a geometrical illustration with a numerical example. Furthermore, using these equations, we derive a formula which shows a clear analytical relation between the optimum solution and the nnn states to be discriminated. We also solve a generalized equal-probability measurement problem analytically. Finally, as another application of our result, the unambiguous discrimination problem of three pure states is studied in detail and analytical solutions are obtained for some interesting cases.
We propose a simple abstract formalisation of the act of observation, in which the system and the observer are assumed to be in a pure state and their interaction deterministically changes the states such that the outcome can be read from the state of the observer after the interaction. If the observer consistently realizes the outcome which maximizes the likelihood ratio that the outcome pertains to the system under study (and not to his own state), he will be called Bayes-optimal. We calculate the probability if for each trial of the experiment the observer is in a new state picked randomly from his set of states, and the system under investigation is taken from an ensemble of identical pure states. For classical statistical mixtures, the relative frequency resulting from the maximum likelihood principle is an unbiased estimator of the components of the mixture. For repeated Bayes-optimal observation in case the state space is complex Hilbert space, the relative frequency converges to the Born rule. Hence, the principle of Bayes-optimal observation can be regarded as an underlying mechanism for the Born rule. We show the outcome assignment of the Bayes-optimal observer is invariant under unitary transformations and contextual, but the probability that results from repeated application is non-contextual. The proposal gives a concise interpretation for the meaning of the occurrence of a single outcome in a quantum experiment as the unique outcome that, relative to the state of the system, is least dependent on the state of the observe at the instant of measurement.