Quantum Channel Conditioning and Measurement Models

If H₁ and H₂ are finite-dimensional Hilbert spaces, a channel from H₁ to H₂ is a completely positive, linear map I that takes the set of states S(H₁) for H₁ to the set of states S(H₂) for H₂. Corresponding to I there is a unique dual map I^* from the set of effects E(H₂) for H₂ to the set of effects E(H₁) for H₁. We call I^*(b) the effect b conditioned by I and the set Iᶜ = I^*(E(H₂)) the conditioned set of I. We point out that Iᶜ is a convex subeffect algebra of the effect algebra E(H₁). We extend this definition to the conditioning I^*(B) for an observable B on H₂ and say that an observable A is in Iᶜ if A=I^*(B) for some observable B. We show that Iᶜ is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.