Measurements and preparations

== Measurements == Measurements extract classical information from quantum systems. They are [[channel (CP map) | channels (CP maps)]] $M\; :\; S(\backslash mathcal\{H\})\; \backslash rightarrow\; \backslash mathcal\{C\_\{X\}\}$ mapping [[states]] $\backslash varrho\; \backslash in\; S(\backslash mathcal\{H\})$ on some Hilbert space $\backslash mathcal\{H\}$ into a classical system $\backslash mathcal\{C\_\{X\}\}$. $\backslash mathcal\{C\_\{X\}\}$ denotes the space of functions on some (finite) set $X$, which we identify with the diagonal $|X|\; \backslash times\; |X|$ matrices: $f\; \backslash equiv\; \backslash sum\_\{x\}\; f(x)\; \backslash ,\; |x\; \backslash rangle\; \backslash langle\; x|$. Measurements are always of the form : $M(\backslash varrho)\; =\; \backslash sum\_\{x\}^\{|X|\}\; tr(E\_\{x\}\; \backslash varrho)\; \backslash ,\; |x\; \backslash rangle\; \backslash langle\; x|$, where $E\; :=\; \backslash \{\; E\_x\; \backslash \}\_x\; \backslash subset\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\})$ is a set of positive operators satisfying the normalization condition $\backslash sum\_x\; E\_x\; =\; \backslash mathbf\{1\}$. Such a set is sometimes called a '''positive operator valued measure (POVM)'''. If all $E\_x$ are '''projections''', i.e., $E\_\{x\}^\{\backslash dagger\}\; E\_\{x\}^\{\}\; =\; E\_\{x\}^\{\}$, then the set $E$ is called a '''projection-valued measure'''. The interpretation is straightforward: for a given input state $\backslash varrho$, the measurement will result in the outcome $x\; \backslash in\; X$ with probability $tr(E\_\{x\}\; \backslash varrho)$. In the Heisenberg representation measurements are completely positive and unital linear maps $M\_\{*\}\; :\; \backslash mathcal\{C\_\{X\}\}\; \backslash rightarrow\; \backslash mathcal\{B\}(\backslash mathcal\{H\})$ of the form : $M\_\{*\}\; (f)\; =\; \backslash sum\_\{x\}^\{|X|\}\; f\_x\; \backslash ,\; E\_x.$ == Preparations == Preparations encode classical information into quantum systems. They are [[channel (CP map) | channels (CP maps)]] $P\; :\; \backslash mathcal\{C\_\{X\}\}\; \backslash rightarrow\; S(\backslash mathcal\{H\})$ mapping a classical probability distribution $f\; :=\; \backslash \{f\_x\; \backslash \}\_x$ onto a set of quantum states $\backslash \{\; \backslash varrho\_x\; \backslash \}\_x$, and are always of the form : $P\; (f)\; =\; \backslash sum\_\{x\}^\{|X|\}\; f\_x\; \backslash ,\; \backslash varrho\_x.$ Such a channel is an operation which prepares the state $\backslash varrho\_x$ with probability $f\_x$. Dually, we may look at the preparation in Heisenberg picture as a completely positive and unital map $P\_\{*\}:\; \backslash mathcal\{B\}(\backslash mathcal\{H\})\; \backslash rightarrow\; \backslash mathcal\{C\_\{X\}\}$ of the form : $P\_\{*\}\; (A)\; =\; \backslash sum\_\{x\}^\{|X|\}\; tr(\backslash varrho\_x\; A)\; \backslash ,\; |x\; \backslash rangle\; \backslash langle\; x|$. == 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]]