Definition
Recall that a density operator is a non-negative operator on a Hilbert space with unit trace.
Mathematically, a quantum operation is a linear map Φ between spaces of trace class operators on Hilbert spaces H and G such that
- If S is a density operator, Tr(Φ(S)) ≤ 1.
- Φ is completely positive, that is for any natural number n, and any square matrix of size n whose entries are trace-class operators
and which is non-negative, then
is also non-negative. In other words, Φ is completely positive if is positive for all n, where denotes the identity map on the C*-algebra of matrices.
Note that by the first condition quantum operations may not preserve the normalization property of statistical ensembles. In probabilistic terms, quantum operations may be sub-Markovian. In order that a quantum operation preserve the set of density matrices, we need the additional assumption that it is trace-preserving.
Read more about this topic: Quantum Operation
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)