Optimal measurements for the discrimination of quantum states with a fixed rate of inconclusive results

Abstract

We study the discrimination of N mixed quantum states in an optimal measurement that maximizes the probabihty of correct results while the probability of inconclusive results is fixed at a given value. After considering the discrimination of N states in a d -dimensional Hilbert space, we focus on the discrimination of qubit states. We develop a method to determine an optimal measurement for discriminating arbitrary qubit states, taking into account that often the optimal measurement is not unique and the maximum probabihty of correct results can be achieved by several different measurements. Analytical results are derived for a number of examples, mostly for the discrimination between qubit states which possess a partial symmetry, but also for discriminating N equiprobable qubit states and for the dicrimination between a pure and a uniformly mixed state in d dimensions.In the special case where the fixed rate of inconclusive results is equal to zero, our method provides a treatment for the minimum-error discrimination of arbitrary qubit states which differs from previous approaches.

Optimal discrimination of single-qubit mixed states

“…Thus if we define our z-axis to be such that the z-component of p j ρ j is the same for each of the three signal states identified, then Γ also has the same z-component, and the optimal measurement operators lie in the equatorial plane. Note that a similar discussion may be found in [26].…”

“…The study of quantum state discrimination has a long history, beginning with the pioneering work of Helstrom, Holevo, and others in the 1960s and 1970s [7,8], who sought to understand the fundamental limits imposed by quantum theory on optical communications. Since then there has been much fur- * g.weir.2@research.gla.ac.uk ther development, with the construction of strategies based on various figures of merit [4][5][6][7][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], as well as ones that interpolate between these [26,27]. We are interested here in the minimum error strategy, based on perhaps the most natural figure of merit: what is the highest possible probability of correctly identifying the state?…”

Optimal measurement strategies for the trine states with arbitrary prior probabilities

“…By writing Γ −1 in the form 1 2 (a1 + b ·σ), we find three linear equations in three unknowns. As described in [27] and [37], we may assume from symmetry that the optimal POVM will be in the same plane as the states, so b z = 0, and hence find a, b x , b y . Thus we can find Γ and hence P Corr , the optimal probability of correctly identifying the state which was sent, as P Corr = k p k Tr(ρ k π k ) = Tr(Γ) = 4a a 2 −|b| 2 .…”

An optimal discrimination of two mixed qubit states with a fixed rate of inconclusive results

“…In fact q (0) 0 and q (1) 0 are the key to solve FRIR of two qubit states. In fact, Nakahira et al [37] and Herzog [38] could not give a solution to the FRIR of two mixed qubit states. In this paper we provide a solution to the FRIR of two mixed qubit states.…”