== Measurements ==
Measurements extract classical information from quantum systems. They are channels (CP maps) M : S(ℋ) → 𝒞𝒳 mapping states 𝜚 ∈ S(ℋ) on some Hilbert space ℋ into a classical system 𝒞𝒳. 𝒞𝒳 denotes the space of functions on some (finite) set X, which we identify with the diagonal |X| × |X| matrices: f ≡ ∑xf(x) |x⟩⟨x|. Measurements are always of the form
- M(ϱ)=∑|X|xtr(Exϱ)|x⟩⟨x|,
where E := {Ex}x ⊂ ℬ(ℋ) is a set of positive operators satisfying the normalization condition ∑xEx = 1. Such a set is sometimes called a positive operator valued measure (POVM). If all Ex are projections, i.e., Ex†Ex = Ex, then the set E is called a projection-valued measure.
The interpretation is straightforward: for a given input state 𝜚, the measurement will result in the outcome x ∈ X with probability tr(Ex𝜚).
In the Heisenberg representation measurements are completely positive and unital linear maps M* : 𝒞𝒳 → ℬ(ℋ) of the form
- M∗(f)=∑|X|xfxEx.
Preparations
Preparations encode classical information into quantum systems. They are channels (CP maps) P : 𝒞𝒳 → S(ℋ) mapping a classical probability distribution f := {fx}x onto a set of quantum states {𝜚x}x, and are always of the form
- P(f)=∑|X|xfxϱx.
Such a channel is an operation which prepares the state 𝜚x with probability fx.
Dually, we may look at the preparation in Heisenberg picture as a completely positive and unital map P* : ℬ(ℋ) → 𝒞𝒳 of the form
- P∗(A)=∑|X|xtr(ϱxA)|x⟩⟨x|.
References and further reading
- M. A. Nielsen, I. L. Chuang: Quantum Computation and Quantum Information; Cambridge University Press, Cambridge 2000; Ch. 8
- E. B. Davies: Quantum Theory of Open Systems; Academic Press, London 1976
- V. Paulsen: Completely Bounded Maps and Operator Algebras; Cambridge University Press, Cambridge 2002
- M. Keyl: Fundamentals of Quantum Information Theory; Phys. Rep. 369 (2002) 431-548; quant-ph/0202122