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Physical Review A

Minimal disturbance discrimination of symmetric pure states on the Bloch sphere’s equator

ShengLi Zhang, Xu‐Bo Zou, Ke Li, Chenhui Jin, Guang‐Can Guo

Published Apr 21, 2008• 5 citations
Publisher

Abstract

This paper addresses the minimal disturbance quantum discrimination among a set of $N$ pure states which are symmetrically distributed on a Bloch sphere equator. The optimal trade-off between the discrimination probability and the state disturbance is exactly derived. The minimal disturbance discrimination which saturates the trade-off turns out to be a convex combination of the minimum error discrimination and the identity maps.

DOI
10.1103/physreva.77.044302

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This paper addresses the minimal disturbance quantum discrimination among a set of NNN pure states which are symmetrically distributed on a Bloch sphere equator. The optimal trade-off between the discrimination probability and the state disturbance is exactly derived. The minimal disturbance discrimination which saturates the trade-off turns out to be a convex combination of the minimum error discrimination and the identity maps.
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.
A qubit pure state can be specified by a point on the Bloch sphere, and similarly certain quantum measurements can be specified by M points on a sphere which we refer to as an Etro sphere. A key objective of this paper is to compare the performance for discriminating between two qubit pure states using POVM designs based on the distribution of M points on the Etro sphere. We specifically address the case where the alignment of the M points relative to the coordinates of the sphere is unknown. Of particular interest is the insensitivity of the POVM designs as measured by the difference between the maximum and minimum probabilities of error over all alignments. We consider distributions of the M points corresponding to Platonic solids as well as optimal distributions with respect to several criteria including, for example, maximum nearest neighbor distance and minimum Riesz s-energy. We provide evidence through simulation of various performance tradeoffs such as the tradeoff between stability and best case performance.
The necessary and sufficient conditions for minimization of the generalized rate error for discriminating among NNN pure qubit states are reformulated in terms of Bloch vectors representing the states. For the direct optimization problem, an algorithmic solution to these conditions is indicated. A solution to the inverse optimization problem is given. General results are widely illustrated by particular cases of equiprobable states and N=2,3,4N=2,3,4N=2,3,4 pure qubit states given with different prior probabilities.
Using the convex optimization method and Helstrom family of ensembles introduced in Ref. [1], we have discussed optimal ambiguous discrimination in qubit systems. We have analyzed the problem of the optimal discrimination of N known quantum states and have obtained maximum success probability and optimal measurement for N known quantum states with equiprobable prior probabilities and equidistant from center of the Bloch ball, not all of which are on the one half of the Bloch ball and all of the conjugate states are pure. An exact solution has also been given for arbitrary three known quantum states. The given examples which use our method include: 1. Diagonal N mixed states; 2. N equiprobable states and equidistant from center of the Bloch ball which their corresponding Bloch vectors are inclined at the equal angle from z axis; 3. Three mirror-symmetric states; 4. States that have been prepared with equal prior probabilities on vertices of a Platonic solid. Keywords: minimum-error discrimination, success probability, measurement, POVM elements, Helstrom family of ensembles, convex optimization, conjugate states PACS Nos: 03.67.Hk, 03.65.Ta
We consider the problem of discrimination between a set of quantum states with the maximum probability of correct detection for a fixed rate of inconclusive results. We show that this type of measurement can be decomposed into two steps. The first step is the discrimination of whether the measurement result is inconclusive. The second step, which is performed in the case in which a conclusive result was obtained, is minimum error discrimination of output states of the first step. Using this result, we derive a closed-form analytical expression for an optimal measurement for both a geometrically uniform pure-state set and a set of binary doubly symmetric mixed states for any probability of an inconclusive result.