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:
“Its a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was mine.”
—Jane Adams (20th century)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“Mothers often are too easily intimidated by their childrens negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)