Research proof index

On the 1D Hilbert space, the optimal hypothesis testing rate is simply 1 - ε, since there's nothing to learn. (More generally this would hold whenever ρ=σ.) -

When the allowed Type I error ε is less than 1 (so, we have some limit on our errors), and the kernel of the state ρ contains the kernel of some element in S, then the optimal hypothesis rate is positive - there is some lower bound on the type II errors we'll see. In other words, under these conditions, we cannot completely avoid type II errors.

Lemma 3.10 of Hayashi's book "Quantum Information Theory - Mathematical Foundations". Also, Lemma 5 in https://arxiv.org/pdf/quant-ph/0107004. -- Used in (S60)

Lemma 3.10 of Hayashi's book "Quantum Information Theory - Mathematical Foundations". Also, Lemma 5 in https://arxiv.org/pdf/quant-ph/0107004. -- Used in (S60) -/ theorem pinching_bound (ρ σ : MState d) : ρ.M ≤ (↑(Fintype.card (spectrum ℝ σ.m)) : ℝ) • (pinching_map σ ρ).M := by rw [pinchingMap_apply_M] suffices ρ.M ≤ (Fintype.card (spectrum ℝ σ.m) : ℝ) •...

Exercise 2.8 of Hayashi's book "A group theoretic approach to Quantum Information".

The partial trace (left) of a positive definite matrix is positive definite.