Measurements and preparations

Contents

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 \mathcal{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| \times |X| matrices: f \equiv \sum_{x} f(x) \, |x \rangle \langle 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 \subset \mathcal{B} (\mathcal{H}) is a set of positive operators satisfying the normalization condition \sum_x E_x = \mathbf{1} . Such a set is sometimes called a positive operator valued measure (POVM). If all Ex are projections, i.e., E_{x}^{\dagger} 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 \in 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) = \sum_{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: = {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|.


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

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