Optimal discrimination of single-qubit mixed states

Abstract

We consider the problem of minimum-error quantum state discrimination for single-qubit mixed states. We present a method which uses the Helstrom conditions constructively and analytically; this algebraic approach is complementary to existing geometric methods, and solves the problem for any number of arbitrary signal states with arbitrary prior probabilities. It has long been known that the minimum error probability is given by the trace of the Lagrange operator, Γ. The remarkable feature of our approach is the central role played not by Γ, but by its inverse.

Optimal measurement strategies for the trine states with arbitrary prior probabilities

“…For δ = 0 our results agree with previous work [19,27]. As we have produced an analytic solution, it is also possible to use this to solve problems where state discrimination arises as a smaller part of a problem, as occurs when multiple copies are available [30].…”

“…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 .…”

Quantum state discrimination on reconfigurable noise-robust quantum networks

“…The smoothing parameter p accounts for the amount of coherent evolution given from H with respect to the irreversible evolution given by the Lindblad operators, and it allows to interpolate between a quantum walk (p = 0) and a classical random walk (p = 1). On the other side, quantum state discrimination has been one of the first problems faced in quantum information theory [41][42][43][44][45][46][47][48], but it is still a flourishing research field as demonstrated from recent theoretical [49][50][51][52][53][54] and experimental works [55][56][57][58][59][60], also considered in relation to machine learning approaches [61]. In its most general formulation, an observer wants to guess the quantum state of a system that is prepared in one of a set of feasible states, possibly by optimizing the measurement operators to apply on the system.…”

Optimal measurement preserving qubit channels

“…When it comes to qubit state discrimination, the symmetry operator K determines the possible measurement strategies. Specifically, the following cases exist [8]:…”