Positive operator-valued measure
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We model a measurement with a POVM when we only care about the probability of obtaining a particular outcome and not about the post-measurement state.
- “Capacity for classical communication: We are merely concerned with minimizing the error probabilities of the classical transmission. The post-measurement state is no longer needed in the quantum information-processing.
variables (ℋ : Type) [complex_Hilbert_space ℋ]
/-- Positive operator-valued measure -/
class POVM (Λ : ι → (ℋ →ₗ[ℂ] ℋ)) :=
(ops_nonneg : ∀ i, Λ i ≥ 0)
(sum_ops_one : ∑ i, Λ i = 1)
The probability for obtaining outcome j is Tr {Λⱼρ}.”
- Generalize using measure theory and probability theory;