Quantum operation
Encyclopedia
In quantum mechanics
, a quantum operation (also known as quantum dynamical map or quantum process
) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan
. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel
.
Quantum operations are formulated in terms of the density operator
description of a quantum mechanical system. Rigorously, a quantum operation is a linear
, completely positive map from the set of density operators into itself.
Some quantum process
es cannot be captured within the quantum operation formalism; in principle, the density matrix of a quantum system can undergo completely arbitrary time evolution.
provides a satisfactory account of time evolution
of state for a quantum mechanical system under certain assumptions. These assumptions include
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:
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 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,..., φk with respective probabilities λ1,..., λk. The transition from a pure state to a mixed state is known as 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.
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
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.
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.
Kraus' theorem characterizes maps that model quantum operations between density operators of quantum state:
Theorem. 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
acting on G such that
Conversely, any map Φ of this form is a quantum operation provided
The matrices
are called Kraus operators. (Sometimes they are known as noise operators or error operators, especially in the context quantum information processing where the quantum operation represents the noisy, error-producing effects of the environment.) 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
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 {Bi}i≤ N and {Ci}i≤ N . Then there is a unitary operator matrix
such that
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 form
distinguished by the orthogonality relation of Kraus operators,
.
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.
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 E 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 {βt}t
such that
Clearly, for each value of t, S → U*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:
As it turns out the mapping g → Ug is a projective representation
of G. The mappings S → U*g S Ug are reversible quantum operations.
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, 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
where EA(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:
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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, a quantum operation (also known as quantum dynamical map or quantum process
Quantum process
In quantum mechanics, a quantum process is a somewhat ambiguous term which usually refers to the time evolution of an quantum system. Under very general assumptions, a quantum process is described by the quantum operation formalism , which is a linear, completely positive map from the set of...
) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan
George Sudarshan
Ennackal Chandy George Sudarshan , also E.C.G. Sudarshan, is a prominent Indian American physicist, author, and professor at The University of Texas at Austin.-Early life:...
. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel
Quantum channel
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit...
.
Quantum operations are formulated in terms of the density operator
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
description of a quantum mechanical system. Rigorously, a quantum operation is a linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
, completely positive map from the set of density operators into itself.
Some quantum process
Quantum process
In quantum mechanics, a quantum process is a somewhat ambiguous term which usually refers to the time evolution of an quantum system. Under very general assumptions, a quantum process is described by the quantum operation formalism , which is a linear, completely positive map from the set of...
es cannot be captured within the quantum operation formalism; in principle, the density matrix of a quantum system can undergo completely arbitrary time evolution.
Background
The Schrödinger pictureSchrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
provides a satisfactory account of time evolution
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
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
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the state via the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. 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 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
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
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,..., φk with respective probabilities λ1,..., λk. The transition from a pure state to a mixed state is known as 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.
Definition
Recall that a density operator is a non-negative operator on a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
with unit trace.
Mathematically, a quantum operation is a linear map Φ between spaces of trace class
Trace class
In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis....
operators on Hilbert spaces H and G such that
- If S is a density operator, Tr(Φ(S)) ≤ 1.
- Φ is completely positiveChoi's theorem on completely positive mapsIn mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional C*-algebras...
, 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.
Mathematical development
In the following remarks, we will refer to the logical and statistical structureQuantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
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.
Kraus' theorem characterizes maps that model quantum operations between density operators of quantum state:
Theorem. 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
acting on G such that
Conversely, any map Φ of this form is a quantum operation provided
The matrices
are called Kraus operators. (Sometimes they are known as noise operators or error operators, especially in the context quantum information processing where the quantum operation represents the noisy, error-producing effects of the environment.) The Stinespring factorization theoremStinespring factorization theorem
In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra as a composition of two completely positive maps each of which has a...
extends the above result to arbitrary separable Hilbert spaces H and G. There, S is replaced by a trace class operator and
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 {Bi}i≤ N and {Ci}i≤ N . Then there is a unitary operator matrix
such thatIn the infinite dimensional case, this generalizes to a relationship between two minimal Stinespring representations
Stinespring factorization theorem
In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra as a composition of two completely positive maps each of which has a...
.
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
Choi's theorem on completely positive maps
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional C*-algebras...
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 form
distinguished by the orthogonality relation of Kraus operators,
.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
Quantum channel
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit...
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.
Dynamics
For a non-relativistic quantum mechanical system, its time evolutionTime evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
is described by a one-parameter group
One-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G...
of automorphisms {αt}t of Q. Moreover, under certain weak technical conditions (see the article on quantum logic
Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
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 E 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 {βt}t
such that
Clearly, for each value of t, S → U*t S Ut is a quantum operation. Moreover, this operation is reversible.
This can be easily generalized: If G is a connected Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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 → Ug is a projective representation
Projective representation
In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
of G. The mappings S → U*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 (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 meaningOperational definition
An operational definition defines something in terms of the specific process or set of validation tests used to determine its presence and quantity. That is, one defines something in terms of the operations that count as measuring it. The term was coined by Percy Williams Bridgman and is a part of...
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, 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
where EA(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,
- Transformation of the statistical state S is given by
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.