A note on the inner products of pure states and their antidistinguishability

TL;DR

This paper investigates the relationship between inner products of pure quantum states and their antidistinguishability, providing a counterexample to a conjecture using semidefinite programming techniques.

Contribution

It introduces a semidefinite programming-based method to certify antidistinguishability and disproves a conjecture relating small inner products to antidistinguishability for four states.

Findings

Counterexample to the conjecture for d=4

Semidefinite programming effectively certifies antidistinguishability

Small inner products do not guarantee antidistinguishability in all cases

Abstract

A set of d quantum states is said to be antidistinguishable if there exists a d-outcome POVM that can perfectly identify which state was not measured. A conjecture by Havl\'i\v{c}ek and Barrett states that if a set of d pure states has small pair-wise inner products, then the set must be antidistinguishable. In this note we provide a certificate of antidistinguishability via semidefinite programming duality and use it to provide a counterexample to this conjecture when d = 4.