# Measurements and preparations

== 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 ≡ ∑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 $\varrho$, 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) = ∑xXfxEx.

### 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 :  = {fx}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 fx.

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

$P_{*} (A) = \sum_{x}^{|X|} tr(\varrho_x A) \, |x \rangle \langle x|$.