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Proceedings of the Steklov Institute of Mathematics

Increasing the Distinguishability of Quantum States with an Arbitrary Success Probability

Dmitry Anatolevich Kronberg

Published Jul 1, 2021• 12 citations
Publisher

Abstract

Abstract It is well known that nonorthogonal quantum states cannot be reliably distinguished; however, for a number of sets of quantum states, the operation of unambiguous discrimination is possible, which either provides full information or yields an inconclusive result. In this paper, a generalization of such a transformation is constructed that has an increased success probability and makes the states more distinguishable. It is shown that after this transformation the states can be reliably distinguished without loss of the total success probability.

DOI
10.1134/s0081543821020115

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Abstract It is well known that nonorthogonal quantum states cannot be reliably distinguished; however, for a number of sets of quantum states, the operation of unambiguous discrimination is possible, which either provides full information or yields an inconclusive result. In this paper, a generalization of such a transformation is constructed that has an increased success probability and makes the states more distinguishable. It is shown that after this transformation the states can be reliably distinguished without loss of the total success probability.
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 give a unified treatment of quantum operations which transform one set of pure states into a another pure set. Extremal cases are deterministic transformations, which have unit success probability, and unambiguous state discrimination, which reveals the state and has the lowest success probability for a fixed initial ensemble. We obtain necessary and, for linearly independent initial states, sufficient conditions for the realisability of a given transformation with a fixed probability of success for each state. We then examine the relationship between deterministic transformations, majorization and distinguishability.
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.
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.
We present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.