Petal-shape probability areas: complete quantum state discrimination

TL;DR

This paper explores the geometric representation of quantum state inner products as petal-shaped areas and analyzes the optimal discrimination strategies for sets of equally separated pure quantum states, considering both unambiguous and ambiguous protocols.

Contribution

It introduces the petal-shaped allowed areas for inner products of quantum states and studies their impact on the complete discrimination of equi-separated states using combined strategies.

Findings

Success probabilities depend on the inner product's magnitude and phase.

Maximal unambiguous discrimination occurs when the second protocol yields no ambiguous information.

The geometric petal shape characterizes the linear independence of quantum state sets.

Abstract

We find the allowed complex numbers associated with the inner product of N equally separated pure quantum states. The allowed areas on the unitary complex plane have the form of petals. A point inside the petal-shape represents a set of N linearly independent (LI) pure states, and a point on the edge of that area represents a set of N linearly dependent (LD) pure states. For each one of those LI sets we study the complete discrimination of its N equi-separated states combining sequentially the two known strategies: first the unambiguous identification protocol for LI states, followed, if necessary, by the error-minimizing measurement scheme for LD states. We find that the probabilities of success for both unambiguous and ambiguous discrimination procedures depend on both the module and the phase of the involved inner product complex number. We show that, with respect to the phase-parameter, the maximal probability of discriminating unambiguously the N non-orthogonal pure states holds just when there no longer be probability of obtaining ambiguously information about the prepared state by applying the second protocol if the first one was not successful.