== Measurements ==

Measurements extract classical information from quantum systems. They are channels (CP maps) $M : S(\mathcal{H}) \rightarrow \mathcal{C_{X}}$ mapping states $\varrho \in S(\mathcal{H})$ on some Hilbert space H into a classical system $\mathcal{C_{X}}$. $\mathcal{C_{X}}$ denotes the space of functions on some (finite) set *X*, which we identify with the diagonal ∣*X*∣ × ∣*X*∣ matrices: *f* ≡ ∑_{x}*f*(*x*) ∣*x*⟩⟨*x*∣. Measurements are always of the form

- $M(\varrho) = \sum_{x}^{|X|} tr(E_{x} \varrho) \, |x \rangle \langle x|$,

where *E* : = {*E*_{x}}_{x} ⊂ B(H) is a set of positive operators satisfying the normalization condition ∑_{x}*E*_{x} = **1**. Such a set is sometimes called a **positive operator valued measure (POVM)**. If all *E*_{x} are **projections**, i.e., *E*_{x}^{ † }*E*_{x}^{} = *E*_{x}^{}, then the set *E* is called a **projection-valued measure**.

The interpretation is straightforward: for a given input state $\varrho$, the measurement will result in the outcome *x* ∈ *X* with probability $tr(E_{x} \varrho)$.

In the Heisenberg representation measurements are completely positive and unital linear maps $M_{*} : \mathcal{C_{X}} \rightarrow \mathcal{B}(\mathcal{H})$ of the form

*M*_{ * }(*f*) = ∑_{x}^{∣X∣}*f*_{x}*E*_{x}.

### Preparations

Preparations encode classical information into quantum systems. They are channels (CP maps) $P : \mathcal{C_{X}} \rightarrow S(\mathcal{H})$ mapping a classical probability distribution *f* : = {*f*_{x}}_{x} onto a set of quantum states $\{ \varrho_x \}_x$, 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 $\varrho_x$ with probability *f*_{x}.

Dually, we may look at the preparation in Heisenberg picture as a completely positive and unital map $P_{*}: \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{C_{X}}$ 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