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Optimum unambiguous discrimination between subsets of non-orthogonal states

Yuqing Sun, János A. Bergou, Mark S. Hillery

Published Dec 8, 2001• 0 citations
OpenAlex

Abstract

Unambiguous discrimination among non-orthogonal but linearly independent quantum states is possible with a certain probability of success. Here we consider a variant of that problem; rather than discriminating among all of the different states, we shall discriminate between two subsets of them. In particular, for the case of 3 states, we find the optimal strategy to distinguish one from the other two. This procedure has a higher probability of success than distinguishing all 3 states. We show how to construct generalized interferometers that provide an implementation of the optimal strategy.

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Optimum unambiguous discrimination between subsets of nonorthogonal quantum states

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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.

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arXiv:quant-ph/0403124•Physical Review A•Aug 5, 2004•64 citations
Unambiguous discrimination among non-orthogonal but linearly independent quantum states is possible with a certain probability of success. Here we consider a variant of that problem; rather than discriminating among all of the different states, we shall discriminate between two subsets of them. In particular, for the case of 3 states, we find the optimal strategy to distinguish one from the other two. This procedure has a higher probability of success than distinguishing all 3 states. We show how to construct generalized interferometers that provide an 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.
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.
Since the state of a quantum system is not an observable, it cannot be determined completely, in principle, if no a priori information is available. In this case, one needs to resort to various estimation strategies. The situation is completely different if a priori knowledge is available. Even unambigous discrimination is possible if the system is always prepared in a state that is selected from a known set of quantum states that are not orthogonal but linearly independent. This type of discrimination strategy has a higher probability of success by using generalised measurements than by using standard von Neumann measurements. We discuss here possible optical realizations, via generalized interferometers and single photon states, for the implementation of strategies that maximize the success rate. The implementations are based on exact analytical results. We also discuss possible applications for entanglement enhancement and quantum communication protocols.
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.
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 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.