Quantum operation

In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This formalism describes not only time evolution or symmetry transformations of isolated systems, but also transient interactions with an environment for purposes of measurement. This description is formulated in terms of the density operator description of a quantum mechanical system.

Background

In the following remarks, we will refer to the logical and statistical structure of quantum theory, in particular to the orthocomplemented lattice Q of propositions (or yes no questions); this is the space of self-adjoint projections on a separable complex Hilbert space H. Recall that a density operator is a non-negative operator on H of trace 1.

Mathematically, a quantum operation is a linear map γ on the space of trace class operators to itself such that

• If S is a density operator, γ(S) ≤ 1.
• γ 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.

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.

Theorem. Let γ be a quantum operation on the trace class operators of H. Then there is a sequence of bounded linear operators {B_i}_i on H such that

Conversely, any map γ of this form is a quantum operation provided

This theorem is a variant of the Stinespring factorization theorem and follows easily from a result of M. Choi. This is also proved in the Nielsen and Chuang reference, Theorem 8.1.

Examples

Dynamics

For a non-relativistic quantum mechanical system, its time evolution is described by a 1-parameter group of automorphisms {α_t}_t of Q. Moreover, under certain weak technical conditions (see the article on quantum logic and the Varadarajan reference) we can show there is a strongly continuous one-parameter group {U_t}_t of unitary transformations of the underlying Hilbert space such that the elements of Q evolve according to the formula:

The system time evolution can also be regarded dually as time evolution of the statistical state space. The evolution of the statistical state is given by a family of operators {β_t}_t such that

  .

Clearly, for each value of t, S → U*_t S U_t is a quantum operation. Moreover, this operation is reversible.

This can be easily generalized: If G is a connected Lie group of symmetries of Q satisfying the same weak continuity conditions , then any element g of G is given by a unitary operator U:

As it turns out the mapping g → U_g is a projective representation of G. The mappings S → U*_g S U_g are reversible quantum operations.

Measurement

Let us first consider quantum measurement of a system in the following narrow sense: We are given the system in some state S and we want to determine whether it has some property E, where E is an element of the lattice of quantum yes-no questions. Measurement in this context means submittting the system to some procedure to determine whether the state satisfies the property. The reference to system state in this discussion can be given an operational meaning by considering a statistical ensemble of systems. Each measurement yields some definite value 0 or 1; moreover application of the measurement process to the ensemble results in a predictable change of the statistical state. This transformation of the statistical state is given by the quantum operation

Measurement of a property is a special case of measurement of an observable A, which has an orthonormal basis of eigenvectors (such an observable is said to have pure point spectrum). Thus

Measurement of the observable A for a system in statistical state S has two outcomes:

• Determination of a sequence eigenvalues of A, which we can regard as determining a probability distribution of eigenvalues. This probability distribution will be discrete; in fact,

• Transformation of the statistical state S

References

• M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000

• M. Choi, Completely Positive Linear Maps on COmplex matrices, Linear Algebra and its Applications, 285-290, 1975

• K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory, Springer Verlag 1983

• W. F. Stinespring, Positive Functions on C*-algebras, Proceedings of the American Mathematical Society, 211-216, 1955

• V. Varadarajan, The Geometry of Quantum Mechanics vols 1 and 2, Springer-Verlag 1985