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

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 of the passage of "t" units of time on the state of an isolated system S is given by a unitary operator "U"_"t" 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 "U"_"t" "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 those 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,..., φ_"k" with respective probabilities λ₁,..., λ_"k". 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 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 $Phi\; otimes\; I\_n$ is positive for all "n", where $I\_n$ denotes the identity map on the C*-algebra of $n\; imes\; n$ 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.

Kraus' theorem characterizes maps that model quantum operations between density operators of quantum state:

Theorem [This theorem is proved in the Nielsen and Chuang reference, Theorems 8.1 and 8.3. ] . Let "H" and "G" be Hilbert spaces of dimension "n" and "m" respectively, and Φ be a quantum operation taking the density matrices acting on "H" to those acting on "G". Then there are matrices :$\{\; B\_i\; \}\_\{1\; leq\; i\; leq\; nm\}$ acting on "G" such that:$Phi(S)\; =\; sum\_i\; B^*\_i\; S\; B\_i.$Conversely, any map Φ of this form is a quantum operation provided:$sum\_i\; B\_i\; B^*\_i\; leq\; 1.$

The matrices $\{\; B\_i\; \}$ are called "Kraus operators". The Stinespring factorization theorem extends the above result to arbitrary separable Hilbert spaces "H" and "G". There, "S" is replaced by a trace class operator and $\{\; B\_i\; \}$ by a sequence of bounded operators.

Kraus matrices are not uniquely determined by the quantum operation Φ in general. For example, different Cholesky factorizations of the Choi matrix might give different sets of Kraus operators. The following theorem states that all systems of Kraus matrices which represent the same quantum operation are related by a unitary transformation:

Theorem. Let Φ be a (not necessarily trace preserving) quantum operation on a finite dimensional Hilbert space "H" with two representing sequences of Kraus matrices {"B"_"i"}_{"i"≤ N} and {"C"_"i"}_{"i"≤ N} . Then there is a unitary operator matrix $;\; (u\_\{ij\})\_\{ij\}$ such that:$C\_i\; =\; sum\_\{j\}\; u\_\{ij\}\; B\_j\; .\; quad$

In the infinite dimensional case, this generalizes to a relationship between two minimal Stinespring representations.

It is a consequence of Stinespring's theorem that all quantum operations can be implemented via unitary evolution after coupling a suitable ancilla to the original system.

These results can be also derived from Choi's theorem on completely positive maps characterizing a completely positive finite-dimensional map by a unique Hermitian-positive density operator (Choi matrix) with respect to the trace. Among all possible Kraus representations of a given channel there exists a canonical formdistinguished by the orthogonality relation of Kraus operators, $\{\; m\; Tr\}\; A^\{dagger\}\_i\; A\_j\; sim\; delta\_\{ij\}$.Such a canonical set of orthogonal Kraus operators can be obtained by diagonalising the corresponding Choi matrix and reshaping its eigenvectors into square matrices.

There exists also an infinite dimensional algebraic generalization of Choi's theorem [Belavkin's Radon-Nikodym theorem for completely positive maps] which defines a density operator as a "Radon-Nikodym derivative" of a quantum channel with respect to a dominating completely positive map (reference channel). It is used for defining the relative fidelities and mutual informations for quantum channels.

In the context of quantum information, quantum operations as defined above, i.e. completely positive maps that do not increase the trace, are also called "quantum channels" or stochastic maps. In the above discussion, we have confined ourselves to channels between quantum states. In other words, both the input and output spaces consist of quantum states. This formulation can be extended to include classical states as well, therefore allowing us to handle quantum and classical information simultaneously.

Examples

Dynamics

For a non-relativistic quantum mechanical system, its time evolution is described by a one-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 "E" of "Q" evolve according to the formula::$alpha\_t(E)\; =\; U^*\_t\; E\; U\_t.$ 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"::$g\; cdot\; E\; =\; U\_g\; E\; U\_g^*.\; quad$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 (v. sup.) 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 yieldssome 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:$S\; mapsto\; E\; S\; E\; +\; (I\; -\; E)\; S\; (I\; -\; E).$Measurement of a property is a special case of measurement of an observable "A", so let us turn to this more general case.

Consider an observable "A" having an orthonormal basis of eigenvectors (such an observable is said to have pure point spectrum). "A" now has a spectral decomposition:$A\; =\; sum\_lambda\; lambda\; operatorname\{E\}\_A(lambda)$where E_"A"(λ) is a family of pairwise orthogonal projections (each onto the respective eigenspace of "A" associated with the measurement value λ, of course). "Repeated" measurement of the observable "A" for an a system in statistical state "S" has the following results:

*Determination of eigenvalues of "A", which we can regard as determining a probability distribution of eigenvalues. This probability distribution will be discrete; in fact,::$operatorname\{Pr\}(lambda)\; =\; operatorname\{Tr\}(S\; operatorname\{E\}\_A(lambda)).$

*Transformation of the statistical state "S" is given by ::$S\; mapsto\; sum\_lambda\; operatorname\{E\}\_A(lambda)\; S\; operatorname\{E\}\_A(lambda)$which means that immediately "after" measurement the statistical state is a classical distribution over the eigenspaces associated with the possible values λ of the observable.

See also

*Quantum channel

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

* V. P. Belavkin, P. Staszewski, Radon-Nikodym Theorem for Completely Positive Maps, Reports on Mathematical Physics, v.24, No 1, 49-55, 1986.

* 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