QuantumInfo.ResourceTheory.HypothesisTesting
Defines `OptimalHypothesisRate`, the optimal rate of distinguishing an `MState` ρ from a set of other mixed states `S`, with at most Type I error ε.
That is to say: take a projective measurement (a `POVM`) with elements `{T, 1-T}`, where measuring `T` will mean we conclude our unknown state was ρ, and measuring `1-T` will mean we think the state was something in `S`. We only accept T's such that `Tr[(1-T)ρ] ≤ ε`, that is, we have at most an ε probability of incorrectly concluding it was ρ. The Type II error associated to this T is then `max_{σ ∈ S} Tr[T σ]`, that is, the (worst possible, over possible states) chance of incorrectly concluding our state was in `S`. Optimize over `T` to get the lowest possible Type II error rate, and the resulting error rate is `OptimalHypothesisRate ρ ε S`.
We make this accessible through the notation `β_ ε(ρ‖S)`.
See [The tangled state of quantum hypothesis testing](https://doi.org/10.1038/s41567-023-02289-9) by Mario Berta et al. for a broader overview.
1 declaration
The set of valid measurement operators for is non-empty.
For a quantum mixed state and a Type I error tolerance , the set of Hermitian matrices satisfying (under the Loewner partial order) and the Type I error constraint is non-empty. This ensures that there is at least one valid measurement operator available to define the optimal hypothesis rate.