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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
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
-
$M(\varrho) = \sum_{x}^{|X|} tr(E_{x} \varrho)
\, |x \rangle \langle x|$,
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
-
$M_{*} (f) = \sum_{x}^{|X|} f_x \,
E_x.$
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
-
$P (f) = \sum_{x}^{|X|} f_x \,
\varrho_x.$
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
-
$P_{*} (A) = \sum_{x}^{|X|} tr(\varrho_x A) \,
|x \rangle \langle x|$.
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
See also
Category:Handbook
of Quantum Information
Last modified:
Monday, October 26, 2015 - 17:56
