Conclusive Exclusion of Quantum States

TL;DR

This paper formulates the quantum state exclusion problem as a Semidefinite Program, deriving conditions for optimal measurements, and explores implications for quantum foundations and hidden variable models.

Contribution

It introduces a general SDP framework for quantum state exclusion, providing necessary and sufficient conditions for optimality and bounds on error probabilities.

Findings

Derived conditions for optimal measurements in quantum state exclusion

Established bounds on the performance of hidden variable models in PBR-related tasks

Proved an analogue of Tsirelson's bound for the PBR experiment

Abstract

In the task of quantum state exclusion we consider a quantum system, prepared in a state chosen from a known set. The aim is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures can not have taken place. We ask what conditions the set of states must obey in order for this to be possible and how well we can complete the task when it is not. The task of quantum state discrimination forms a subclass of this set of problems. Within this paper we formulate the general problem as a Semidefinite Program (SDP), enabling us to derive sufficient and necessary conditions for a measurement to be optimal. Furthermore, we obtain a necessary condition on the set of states for exclusion to be achievable with certainty and give a construction for a lower bound on the probability of error. This task of conclusively excluding states has gained importance in the context of the foundations of quantum mechanics due to a result of Pusey, Barrett and Rudolph (PBR). Motivated by this, we use our SDP to derive a bound on how well a class of hidden variable models can perform at a particular task, proving an analogue of Tsirelson's bound for the PBR experiment and the optimality of a measurement given by PBR in the process. We also introduce variations of conclusive exclusion, including unambiguous state exclusion, and state exclusion with worst case error.