Helstrom Hovelo Theorem

You are given a quantum system in unknown state You are told is either in or with probability and The probability of correctly identifying the state by ANY measurement satisfies This bound is TIGHT

Proof

Do a 2-outcome measurement Probability of success

Properties s.t. then So only acts nontrivially on so eigenvalues are and Let corresponding eigenvalues be and and Spectral decomposition: Determine in terms of , Let ?????? Write in this basis

and we find the eigen values are

Let and be such that

Then for some with ,

Claim: Use and So