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Fortschritte der Physik

Minimum‐error strategy for discriminating between subsets of nonorthogonal quantum states

Ulrike Herzog, János A. Bergou

Published Feb 26, 2003• 1 citation
Publisher

Abstract

Abstract We consider a quantum system that is prepared, with a given a priori probability, in a pure state that belongs to a known set of N nonorthogonal quantum states. We study a minimum‐error measurement for assigning the state of the system to one or the other of two complementary subsets of the set of the given states. For the case that the N states span a Hilbert space that is only two‐dimensional, a simple analytical solution is derived for the minimum error probability and the optimum measurement strategy. If one of the subsets contains only a single state, the measurement is referred to as quantum state filtering. Our general result is applied to investigate minimum‐error quantum state filtering of three arbitrary linearly dependent states. Moreover, we discuss a generalized measurement for performing minimum‐error filtering of three special linearly independent states.

DOI
10.1002/prop.200310017

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Abstract We consider a quantum system that is prepared, with a given a priori probability, in a pure state that belongs to a known set of N nonorthogonal quantum states. We study a minimum‐error measurement for assigning the state of the system to one or the other of two complementary subsets of the set of the given states. For the case that the N states span a Hilbert space that is only two‐dimensional, a simple analytical solution is derived for the minimum error probability and the optimum measurement strategy. If one of the subsets contains only a single state, the measurement is referred to as quantum state filtering. Our general result is applied to investigate minimum‐error quantum state filtering of three arbitrary linearly dependent states. Moreover, we discuss a generalized measurement for performing minimum‐error filtering of three special linearly independent states.
A measurement strategy is developed for a different kind of hypothesis testing. It assigns, with minimum probability of error, the state of a quantum system to one or the other of two complementary subsets of a set of N given nonorthogonal quantum states occurring with given a priori probabilities. A general analytical solution is obtained for N states that are restricted to a two-dimensional subspace of the Hilbert space of the system. The result for the special case of three arbitrary but linearly dependent states is applied to a variety of sets of three states that are symmetric and equally probable. It is found that, in this case, the minimum-error probability for distinguishing one of the states from the other two is only about half as large as the minimum-error probability for distinguishing all three states individually.
Abstract Unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a new variant of that problem. Instead of discriminating among all of the N different states, we now ask for less. We want to unambiguously assign the state to one of two complementary subsets of the set of N given non‐orthogonal quantum states, each occurring with given a priori probabilities. We refer to the special case when one subset contains only one state and the other contains the remaining N‐1 states as unambiguous quantum state filtering. We present an optimal analytical solution for the special case of N=3, and discuss the optimal strategy to unambiguously distinguish |ψ1〉 from the set {|ψ2〉,|ψ3〉}. For unambiguous filtering the subsets need not be linearly independent. We briefly discuss how to construct generalized interferometers (multiports) which provide a fully linear optical implementation of the optimal strategy.
It is known that unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of the different states, we shall only discriminate between two subsets of them. In particular, for the case of three nonorthogonal states, ${|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},$ we show that the optimal strategy to distinguish $|{\ensuremath{\psi}}_{1}〉$ from the set ${|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉}$ has a higher success rate than if we wish to discriminate among all three states. Somewhat surprisingly, for unambiguous discrimination the subsets need not be linearly independent. A fully analytical solution is presented, and we also show how to construct generalized interferometers (multiport) which provide an optical implementation of the optimal strategy.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.
We provide an exact solution of finding the optimum measurement strategy for distinguishing three mixed quantum states with mirror symmetry. The signal states reside in two-dimensional Hilbert space. By mirror symmetry, two of the three states have the same prior probabilities and transform into each other under the mirror symmetry. We find that the optimum measurement strategy may consist of three, two, or a single nonzero probability operator measure elements depending upon the prior probabilities of the quantum signals.
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.