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