Measurements and preparations

== 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 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)|xx|,

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., ExEx = 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)|xx|.

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

See also

Category:Handbook of Quantum Information

Last modified: 

Monday, October 26, 2015 - 17:56