In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
by
George Sudarshan
Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian American theoretical physicist and a professor at the University of Texas. Sudarshan has been credited with numerous contrib ...
. 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
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
, a quantum operation is called a
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. An example of classical information i ...
.
Note that some authors use the term "quantum operation" to refer specifically to
completely positive
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Definition
Let A and B be C*-algebras. A linea ...
(CP) and non-trace-increasing maps on the space of density matrices, and the term "
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. An example of classical information i ...
" to refer to the subset of those that are strictly trace-preserving.
Quantum operations are formulated in terms of the
density operator
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
description of a quantum mechanical system. Rigorously, a quantum operation is a
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
,
completely positive
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Definition
Let A and B be C*-algebras. A linea ...
map from the set of density operators into itself. In the context of quantum information, one often imposes the further restriction that a quantum operation
must be ''physical'', that is, satisfy
for any state
.
Some
quantum processes cannot be captured within the quantum operation formalism;
in principle, the density matrix of a quantum system can undergo completely arbitrary time evolution. Quantum operations are generalized by
quantum instrument
In physics, a quantum instrument is a mathematical abstraction of a quantum measurement, capturing both the classical and quantum outputs. It combines the concepts of measurement and quantum operation. It can be equivalently understood as a quant ...
s, which capture the classical information obtained during measurements, in addition to the
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
.
Background
The
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
provides a satisfactory account of
time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
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
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. 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 ''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
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
.
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 a 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 φ
1, ..., φ
''k'' with respective probabilities λ
1, ..., λ
''k''. 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
Karl Kraus, who relied on the earlier mathematical work of
Man-Duen 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
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
is a non-negative operator on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
with unit trace.
Mathematically, a quantum operation is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
Φ between spaces of
trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a Trace (linear algebra), trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the tra ...
operators on Hilbert spaces ''H'' and ''G'' such that
* If ''S'' is a density operator, Tr(Φ(''S'')) ≤ 1.
* Φ is
completely positive
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Definition
Let A and B be C*-algebras. A linea ...
, 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
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
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.
In the context of
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
, the quantum operations defined here, i.e. completely positive maps that do not increase the trace, are also called
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. An example of classical information i ...
s or ''stochastic maps''. The formulation here is confined to channels between quantum states; however, it can be extended to include classical states as well, therefore allowing quantum and classical information to be handled simultaneously.
Kraus operators
Kraus theorem (named after
Karl Kraus) characterizes
completely positive maps, that model quantum operations between quantum states. Informally, the theorem ensures that the action of any such quantum operation
on a state
can always be written as
, for some set of operators
satisfying
, where
is the identity operator.
Statement of the theorem
Theorem. Let
and
be Hilbert spaces of dimension
and
respectively, and
be a quantum operation between
and
. Then, there are matrices
mapping
to
such that, for any state
,
Conversely, any map
of this form is a quantum operation, provided
is satisfied.
The matrices
are called ''Kraus operators''. (Sometimes they are known as ''noise operators'' or ''error operators'', especially in the context of
quantum information processing
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
, where the quantum operation represents the noisy, error-producing effects of the environment.) The
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 ''A'' as a compositio ...
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.
Unitary equivalence
Kraus matrices are not uniquely determined by the quantum operation
in general. For example, different
Cholesky factorization
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...
s of the Choi matrix might give different sets of Kraus operators. The following theorem states that all systems of Kraus matrices representing the same quantum operation are related by a unitary transformation:
Theorem. Let
be a (not necessarily trace-preserving) quantum operation on a finite-dimensional Hilbert space ''H'' with two representing sequences of Kraus matrices
and
. 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 by unitary evolution after coupling a suitable
ancilla to the original system.
Remarks
These results can be also derived from
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 (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belav ...
, characterizing a completely positive finite-dimensional map by a unique Hermitian-positive density operator (
Choi matrix Choi may refer to:
* Choi (Korean surname), a Korean surname
* Choi, Macau Cantonese transliteration of the Chinese surname Cui (崔) and Xu (徐)
* Choi, Cantonese romanisation of Cai (surname) (蔡), a Chinese surname
* CHOI-FM, a radio station ...
) with respect to the trace. Among all possible Kraus representations of a given
channel
Channel, channels, channeling, etc., may refer to:
Geography
* Channel (geography), in physical geography, a landform consisting of the outline (banks) of the path of a narrow body of water.
Australia
* Channel Country, region of outback Austral ...
, there exists a canonical form distinguished by the orthogonality relation of Kraus operators,
. Such canonical set of orthogonal Kraus operators can be obtained by diagonalising the corresponding Choi matrix and reshaping its eigenvectors into square matrices.
There also exists an infinite-dimensional algebraic generalization of Choi's theorem, known as "Belavkin's Radon-Nikodym theorem for completely positive maps", which defines a density operator as a "Radon–Nikodym derivative" of a
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. An example of classical information i ...
with respect to a dominating completely positive map (reference channel). It is used for defining the relative fidelities and mutual informations for quantum channels.
Dynamics
For a non-relativistic quantum mechanical system, its
time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
is described by a
one-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi is in ...
of automorphisms
''t'' of ''Q''. This can be narrowed to unitary transformations: under certain weak technical conditions (see the article on
quantum logic
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The field takes as its starting point an observat ...
and the Varadarajan reference), there is a strongly continuous one-parameter group
''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
''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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of symmetries of ''Q'' satisfying the same weak continuity conditions, then the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of any element ''g'' of ''G'' is given by a unitary operator ''U'':
This mapping ''g'' → ''U''
''g'' is known as a
projective representation In the field of representation theory in mathematics, 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 group
\mathrm(V) = \mathrm(V) / F^*,
where GL(' ...
of ''G''. The mappings ''S'' → ''U''*
''g'' ''S'' ''U''
''g'' are reversible quantum operations.
Quantum measurement
Quantum operations can be used to describe the process of
quantum measurement
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what ...
. The presentation below describes measurement in terms of self-adjoint projections on a separable complex Hilbert space ''H'', that is, in terms of a PVM (
Projection-valued measure
In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are f ...
). In the general case, measurements can be made using non-orthogonal operators, via the notions of
POVM
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and ...
. The non-orthogonal case is interesting, as it can improve the overall efficiency of the
quantum instrument
In physics, a quantum instrument is a mathematical abstraction of a quantum measurement, capturing both the classical and quantum outputs. It combines the concepts of measurement and quantum operation. It can be equivalently understood as a quant ...
.
Binary measurements
Quantum systems may be measured by applying a series of ''yes–no questions''. This set of questions can be understood to be chosen from an
orthocomplemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b''&nbs ...
''Q'' of propositions in
quantum logic
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The field takes as its starting point an observat ...
. The lattice is equivalent to the space of self-adjoint projections on a separable complex Hilbert space ''H''.
Consider a system in some state ''S'', with the goal of determining 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
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
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
Here ''E'' can be understood to be a
projection operator
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
.
General case
In the general case, measurements are made on observables taking on more than two values.
When an observable ''A'' has a
pure point spectrum, it can be written in terms of an
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
basis of eigenvectors. That is, ''A'' has a spectral decomposition
where E
''A''(λ) is a family of pairwise orthogonal
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
s, each onto the respective eigenspace of ''A'' associated with the measurement value λ.
Measurement of the observable ''A'' yields an eigenvalue of ''A''. Repeated measurements, made on a
statistical ensemble
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
''S'' of systems, results in a probability distribution over the eigenvalue spectrum of ''A''. It is a
discrete probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, and is given by
Measurement of the statistical state ''S'' is given by the map
That is, immediately after measurement, the statistical state is a classical distribution over the eigenspaces associated with the possible values λ of the observable: ''S'' is a
mixed state.
Non-completely positive maps
Shaji and
Sudarshan argued in a Physical Review Letters paper that, upon close examination, complete positivity is not a requirement for a good representation of open quantum evolution. Their calculations show that, when starting with some fixed initial correlations between the observed system and the environment, the map restricted to the system itself is not necessarily even positive. However, it is not positive only for those states that do not satisfy the assumption about the form of initial correlations. Thus, they show that to get a full understanding of quantum evolution, non completely-positive maps should be considered as well.
See also
*
Quantum dynamical semigroup
In quantum mechanics, a quantum Markov semigroup describes the dynamics in a Markovian open quantum system. The axiomatic definition of the prototype of quantum Markov semigroups was first introduced by A. M. Kossakowski in 1972, and then develo ...
*
Superoperator
In physics, a superoperator is a linear operator acting on a vector space of linear operators.John Preskill, Lecture notes for Quantum Computation course at CaltechCh. 3
Sometimes the term refers more specially to a completely positive map which ...
References
*
*
*
*
* 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
{{DEFAULTSORT:Quantum Operation
Quantum mechanics