On a conjecture regarding quantum hypothesis testing

TL;DR

This thesis investigates whether the classical result that the worst-case pair determines the asymptotic symmetric error exponent in hypothesis testing extends to the quantum setting, providing evidence supporting the conjecture.

Contribution

It introduces a new asymmetrical quantum hypothesis testing case and shows the conjecture holds, extending known special cases and providing further evidence.

Findings

Conjecture holds in the new asymmetrical case

Supports the conjecture's validity in quantum hypothesis testing

Provides analytical computation for a complex special case

Abstract

In this MSc thesis I consider the asymptotic behaviour of the symmetric error in composite hypothesis testing. In the classical case, when the null and alternative hypothesis are finite sets of states, the best achievable symmetric error exponent is simply the the best achievable symmetric error exponent corresponding to the "worst case pair". The conjecture -- raised several years ago -- is that this remains true in the quantum case, too. This is known to be true in some special case. However, all of the known special cases are in some sense "too nice", e.g., have certain symmetries. Hoping to find a counter-example, in my thesis I consider a new special case, which on one hand is as "asymmetrical" as possible, yet still analytically computable. However, as it turns out from some involved computation, the conjecture is also true in this case, and thus gives further evidence to this conjecture.