Quantum operation

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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.

Contents

Background

The Schrödinger picture provides a satisfactory account of time evolution of state for a quantum mechanical system under certain assumptions. These assumptions include

  • The system is non-relativistic
  • The system is isolated.

The Schrödinger picture for time evolution has several mathematically equivalent formulations. One such formulation expresses the time rate of change of the state via the Schrödinger equation. A more suitable formulation for this exposition is expressed as follows:

The effect on the state isolated system S of the passage of t units of time is given by a unitary operator Ut on the Hilbert space H associated to S.

This means that if the system is in a state corresponding to v ∈ H at an instant of time s, then the state after t units of time will be Ut v. For relativistic systems, there is no universal time parameter, but we can still formulate the effect of certain reversible transformations on the quantum mechanical system. For instance, state transformations relating observers in different frames of reference are given by unitary transformations. In any case, these state transformations carry pure states into pure states; this is often formulated by saying that in this idealized framework, there is no decoherence.

For interacting (or open systems) such as systems undergoing measurement, the situation is entirely different. To begin with, the state changes experienced by such systems cannot be accounted for exclusively by a transformation on the set of pure states (that is those associated to vectors of norm 1 in H). After such an interaction, a system in pure state φ may no longer be in the pure state φ. In general it will be in a statistical mix of a sequence of pure states φ1,..., φk with respective probabilities λ1,..., λk. This state of affairs is sometimes expressed by saying that the system experiences decoherence.

Numerous mathematical formalisms have been established to handle the case of an interacting system. The quantum operation formalism emerged around 1983 from work of K. Kraus, who relied on the earlier mathematical work of M. D. Choi. It has the advantage that it expresses operations such as measurement as a mapping from density states to density states. In particular, the effect of quantum operations stays within the set of density states.

Mathematical formalism

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 on H to itself such that

  • If S is a density operator, Tr(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
<math> \begin{bmatrix} S_{11} & \cdots & S_{1 n}\\ \vdots & \ddots & \vdots \\ S_{n 1} & \cdots & S_{n n}\end{bmatrix} <math>

and which is non-negative, then

<math> \begin{bmatrix} \gamma(S_{11}) & \cdots & \gamma(S_{1 n})\\ \vdots & \ddots & \vdots \\ \gamma(S_{n 1}) & \cdots & \gamma(S_{n n})\end{bmatrix} <math>

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 {Bi}iN on H such that

<math> \gamma(S) = \sum_{i \in \mathbb{N}} B^*_i S B_i. <math>

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

<math> \sum_i B_i B^*_i \leq 1. <math>

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.

In case H has finite dimension n, the sequence can be assumed to have only n2 non-zero entries.

The operators Bi are referred to by physicists as Kraus matrices (or more accurately as Kraus operators). Kraus matrices are not uniquely determined by the quantum operation γ, although all systems of Kraus matrices which represent the same quantum operation are related by a unitary transformation:

Theorem. Let γ be a quantum operation on the trace class operators of H with two representing sequences of Kraus matrices {Bi}iN and {Ci}iN. Then there is an infinite scalar unitary matrix ui j such that

<math> C_i = \sum_{i, j} u_{ij} B_j \quad <math>

Examples

Dynamics

For a non-relativistic quantum mechanical system, its time evolution is described by a one-parameter group of automorphisms {αt}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 {Ut}t of unitary transformations of the underlying Hilbert space such that the elements of Q evolve according to the formula:

<math> \alpha_t(E) = U^*_t E U_t <math>

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 {βt}t such that

<math> \operatorname{Tr}(\beta_t(S) E) = \operatorname{Tr}(S \alpha_{-t}(E)) = \operatorname{Tr}(S U _t E U^*_t )=\operatorname{Tr}( U^*_t S U _t E )<math>.

Clearly, for each value of t, SU*t S Ut 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:

<math> g \cdot E = U_g E U_g^* \quad <math>

As it turns out the mapping gUg is a projective representation of G. The mappings SU*g S Ug 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 submitting 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

<math> S \mapsto E S E + (I - E) S (I - E). <math>

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 A has a spectral decomposition

<math> A = \sum_\lambda \lambda \operatorname{E}_A(\lambda). <math>

where EA(λ) is a family of pairwise orthogonal projections. Measurement of the observable A for a system in statistical state S has the following results:

  • 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,
<math> \operatorname{Pr}(\lambda) = \operatorname{Tr}(S \operatorname{E}_A(\lambda))<math>
  • Transformation of the statistical state S
<math> S \mapsto \sum_\lambda \operatorname{E}_A(\lambda) S \operatorname{E}_A(\lambda) <math>

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
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