# Measurements and preparations

== Measurements == Measurements extract classical information from quantum systems. They are channels (CP maps) $M : S\left(\mathcal\left\{H\right\}\right) \rightarrow \mathcal\left\{C_\left\{X\right\}\right\}$ mapping states $\varrho \in S\left(\mathcal\left\{H\right\}\right)$ on some Hilbert space $\mathcal\left\{H\right\}$ into a classical system $\mathcal\left\{C_\left\{X\right\}\right\}$. $\mathcal\left\{C_\left\{X\right\}\right\}$ denotes the space of functions on some (finite) set $X$, which we identify with the diagonal $|X| \times |X|$ matrices: $f \equiv \sum_\left\{x\right\} f\left(x\right) \, |x \rangle \langle x|$. Measurements are always of the form : $M\left(\varrho\right) = \sum_\left\{x\right\}^\left\{|X|\right\} tr\left(E_\left\{x\right\} \varrho\right) \, |x \rangle \langle x|$, where $E := \\left\{ E_x \\right\}_x \subset \mathcal\left\{B\right\} \left(\mathcal\left\{H\right\}\right)$ is a set of positive operators satisfying the normalization condition $\sum_x E_x = \mathbf\left\{1\right\}$. Such a set is sometimes called a '''positive operator valued measure (POVM)'''. If all $E_x$ are '''projections''', i.e., $E_\left\{x\right\}^\left\{\dagger\right\} E_\left\{x\right\}^\left\{\right\} = E_\left\{x\right\}^\left\{\right\}$, 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 $x \in X$ with probability $tr\left(E_\left\{x\right\} \varrho\right)$. In the Heisenberg representation measurements are completely positive and unital linear maps $M_\left\{*\right\} : \mathcal\left\{C_\left\{X\right\}\right\} \rightarrow \mathcal\left\{B\right\}\left(\mathcal\left\{H\right\}\right)$ of the form : $M_\left\{*\right\} \left(f\right) = \sum_\left\{x\right\}^\left\{|X|\right\} f_x \, E_x.$ == Preparations == Preparations encode classical information into quantum systems. They are channels (CP maps) $P : \mathcal\left\{C_\left\{X\right\}\right\} \rightarrow S\left(\mathcal\left\{H\right\}\right)$ mapping a classical probability distribution $f := \\left\{f_x \\right\}_x$ onto a set of quantum states $\\left\{ \varrho_x \\right\}_x$, and are always of the form : $P \left(f\right) = \sum_\left\{x\right\}^\left\{|X|\right\} f_x \, \varrho_x.$ 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 $P_\left\{*\right\}: \mathcal\left\{B\right\}\left(\mathcal\left\{H\right\}\right) \rightarrow \mathcal\left\{C_\left\{X\right\}\right\}$ of the form : $P_\left\{*\right\} \left(A\right) = \sum_\left\{x\right\}^\left\{|X|\right\} tr\left(\varrho_x A\right) \, |x \rangle \langle x|$. == 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; [http://arxiv.org/abs/quant-ph/0202122 quant-ph/0202122] == See also == * Channel (CP map) * The Church of the larger Hilbert space Category:Handbook of Quantum Information