In
quantum information theory
Quantum information is the information of the quantum state, 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 re ...
, a quantum channel is a communication channel which can transmit
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 ...
, as well as classical information. An example of quantum information is the state of a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
. An example of classical information is a text document transmitted over the
Internet
The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, pub ...
.
More formally, quantum channels are
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) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a
quantum operation
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 discusse ...
viewed not merely as the
reduced dynamics of a system but as a pipeline intended to carry quantum information. (Some authors use the term "quantum operation" to also include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps.
)
Memoryless quantum channel
We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional.
The memoryless in the section title carries the same meaning as in classical
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
: the output of a channel at a given time depends only upon the corresponding input and not any previous ones.
Schrödinger picture
Consider quantum channels that transmit only quantum information. This is precisely a
quantum operation
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 discusse ...
, whose properties we now summarize.
Let
and
be the state spaces (finite-dimensional
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 ...
s) of the sending and receiving ends, respectively, of a channel.
will denote the family of operators on
. In 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 ...
, a purely quantum channel is a map
between
density matrices
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 in quantum mechanics, measurement ...
acting on
and
with the following properties:
#As required by postulates of quantum mechanics,
needs to be linear.
#Since density matrices are positive,
must preserve the
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
of positive elements. In other words,
is a
positive map.
#If an
ancilla of arbitrary finite dimension ''n'' is coupled to the system, then the induced map
, where ''I''
''n'' is the identity map on the ancilla, must also be positive. Therefore, it is required that
is positive for all ''n''. Such maps are called
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 ...
.
#Density matrices are specified to have trace 1, so
has to preserve the trace.
The adjectives completely positive and trace preserving used to describe a map are sometimes abbreviated CPTP. In the literature, sometimes the fourth property is weakened so that
is only required to be not trace-increasing. In this article, it will be assumed that all channels are CPTP.
Heisenberg picture
Density matrices acting on ''H
A'' only constitute a proper subset of the operators on ''H
A'' and same can be said for system ''B''. However, once a linear map
between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend
uniquely to the full space of operators. This leads to the adjoint map
, which describes the action of
in the
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, bu ...
:
The spaces of operators ''L''(''H''
''A'') and ''L''(''H''
''B'') are Hilbert spaces with the
Hilbert–Schmidt inner product. Therefore, viewing
as a map between Hilbert spaces, we obtain its adjoint
* given by
:
While
takes states on ''A'' to those on ''B'',
maps observables on system ''B'' to observables on ''A''. This relationship is same as that between the Schrödinger and Heisenberg descriptions of dynamics. The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa.
It can be directly checked that if
is assumed to be trace preserving,
is
unital, that is,
. Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel.
Classical information
So far we have only defined quantum channel that transmits only quantum information. As stated in the introduction, the input and output of a channel can include classical information as well. To describe this, the formulation given so far needs to be generalized somewhat. A purely quantum channel, in the Heisenberg picture, is a linear map Ψ between spaces of operators:
:
that is unital and completely positive (CP). The operator spaces can be viewed as finite-dimensional
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 ...
s. Therefore, we can say a channel is a unital CP map between C*-algebras:
:
Classical information can then be included in this formulation. The observables of a classical system can be assumed to be a commutative C*-algebra, i.e. the space of continuous functions
on some set
. We assume
is finite so
can be identified with the ''n''-dimensional Euclidean space
with entry-wise multiplication.
Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define
to include the relevant classical observables. An example of this would be a channel
:
Notice
is still a C*-algebra. An element
of a C*-algebra
is called positive if
for some
. Positivity of a map is defined accordingly. This characterization is not universally accepted; 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 ...
is sometimes given as the generalized mathematical framework for conveying both quantum and classical information. In axiomatizations of quantum mechanics, the classical information is carried in a
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice dualit ...
or
Frobenius category.
Examples
States
A state, viewed as a mapping from observables to their expectation values, is an immediate example of a channel.
Time evolution
For a purely quantum system, the time evolution, at certain time ''t'', is given by
:
where
and ''H'' is the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
and ''t'' is the time. Clearly this gives a CPTP map in the Schrödinger picture and is therefore a channel. The dual map in the Heisenberg picture is
:
Restriction
Consider a composite quantum system with state space
For a state
:
the reduced state of ''ρ'' on system ''A'', ''ρ''
''A'', is obtained by taking the
partial trace
In linear algebra and functional analysis, the partial trace is a generalization of the trace. Whereas the trace is a scalar valued function on operators, the partial trace is an operator-valued function. The partial trace has applications in q ...
of ''ρ'' with respect to the ''B'' system:
:
The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture. In the Heisenberg picture, the dual map of this channel is
:
where ''A'' is an observable of system ''A''.
Observable
An observable associates a numerical value
to a quantum mechanical ''effect''
.
's are assumed to be positive operators acting on appropriate state space and
. (Such a collection is called a
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 ...
.) In the Heisenberg picture, the corresponding ''observable map''
maps a classical observable
:
to the quantum mechanical one
:
In other words, one
integrate ''f'' against the POVM to obtain the quantum mechanical observable. It can be easily checked that
is CP and unital.
The corresponding Schrödinger map
takes density matrices to classical states:
:
where the inner product is the Hilbert–Schmidt inner product. Furthermore, viewing states as normalized
functionals, and invoking the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called the R ...
, we can put
:
Instrument
The observable map, in the Schrödinger picture, has a purely classical output algebra and therefore only describes measurement statistics. To take the state change into account as well, we define what is called a
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 ...
. Let
be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map
with pure quantum input
and with output space
:
:
Let
:
The dual map in the Heisenberg picture is
:
where
is defined in the following way: Factor
(this can always be done since elements of a POVM are positive) then
.
We see that
is CP and unital.
Notice that
gives precisely the observable map. The map
:
describes the overall state change.
Measure-and-prepare channel
Suppose two parties ''A'' and ''B'' wish to communicate in the following manner: ''A'' performs the measurement of an observable and communicates the measurement outcome to ''B'' classically. According to the message he receives, ''B'' prepares his (quantum) system in a specific state. In the Schrödinger picture, the first part of the channel
1 simply consists of ''A'' making a measurement, i.e. it is the observable map:
:
If, in the event of the ''i''-th measurement outcome, ''B'' prepares his system in state ''R
i'', the second part of the channel
2 takes the above classical state to the density matrix
:
The total operation is the composition
:
Channels of this form are called ''measure-and-prepare'' or in
Holevo form.
In the Heisenberg picture, the dual map
is defined by
:
A measure-and-prepare channel can not be the identity map. This is precisely the statement of the
no teleportation theorem
In quantum information theory, the no-teleportation theorem states that an arbitrary quantum state cannot be converted into a sequence of classical bits (or even an infinite number of such bits); nor can such bits be used to reconstruct the origin ...
, which says classical teleportation (not to be confused with
entanglement-assisted teleportation) is impossible. In other words, a quantum state can not be measured reliably.
In the
channel-state duality
In quantum information theory, the channel-state duality refers to the correspondence between quantum channels and quantum states (described by density matrices). Phrased differently, the duality is the isomorphism between completely positive map ...
, a channel is measure-and-prepare if and only if the corresponding state is
separable. Actually, all the states that result from the partial action of a measure-and-prepare channel are separable, and for this reason measure-and-prepare channels are also known as entanglement-breaking channels.
Pure channel
Consider the case of a purely quantum channel
in the Heisenberg picture. With the assumption that everything is finite-dimensional,
is a unital CP map between spaces of matrices
:
By
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 ...
,
must take the form
:
where ''N'' ≤ ''nm''. The matrices ''K''
''i'' are called
Kraus operator
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 discusse ...
s of
(after the German physicist
Karl Kraus, who introduced them). The minimum number of Kraus operators is called the Kraus rank of
. A channel with Kraus rank 1 is called pure. The time evolution is one example of a pure channel. This terminology again comes from the channel-state duality. A channel is pure if and only if its dual state is a pure state.
Teleportation
In
quantum teleportation, a sender wishes to transmit an arbitrary quantum state of a particle to a possibly distant receiver. Consequently, the teleportation process is a quantum channel. The apparatus for the process itself requires a quantum channel for the transmission of one particle of an entangled-state to the receiver. Teleportation occurs by a joint measurement of the sent particle and the remaining entangled particle. This measurement results in classical information which must be sent to the receiver to complete the teleportation. Importantly, the classical information can be sent after the quantum channel has ceased to exist.
In the experimental setting
Experimentally, a simple implementation of a quantum channel is
fiber optic
An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to ...
(or free-space for that matter) transmission of single
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s. Single photons can be transmitted up to 100 km in standard fiber optics before losses dominate. The photon's time-of-arrival (''time-bin entanglement'') or
polarization are used as a basis to encode quantum information for purposes such as
quantum cryptography
Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
. The channel is capable of transmitting not only basis states (e.g. , 0>, , 1>) but also superpositions of them (e.g. , 0>+, 1>). The
coherence
Coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference
* Coherence (units of measurement), a deriv ...
of the state is maintained during transmission through the channel. Contrast this with the transmission of electrical pulses through wires (a classical channel), where only classical information (e.g. 0s and 1s) can be sent.
Channel capacity
The cb-norm of a channel
Before giving the definition of channel capacity, the preliminary notion of the norm of complete boundedness, or cb-norm of a channel needs to be discussed. When considering the capacity of a channel
, we need to compare it with an "ideal channel"
. For instance, when the input and output algebras are identical, we can choose
to be the identity map. Such a comparison requires a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
between channels.
Since a channel can be viewed as a linear operator, it is tempting to use the natural
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
. In other words, the closeness of
to the ideal channel
can be defined by
:
However, the operator norm may increase when we tensor
with the identity map on some ancilla.
To make the operator norm even a more undesirable candidate, the quantity
:
may increase without bound as
The solution is to introduce, for any linear map
between C*-algebras, the cb-norm
:
Definition of channel capacity
The mathematical model of a channel used here is same as the
classical one.
Let
be a channel in the Heisenberg picture and
be a chosen ideal channel. To make the comparison possible, one needs to encode and decode Φ via appropriate devices, i.e. we consider the composition
:
where ''E'' is an encoder and ''D'' is a decoder. In this context, ''E'' and ''D'' are unital CP maps with appropriate domains. The quantity of interest is the ''best case scenario'':
:
with the infimum being taken over all possible encoders and decoders.
To transmit words of length ''n'', the ideal channel is to be applied ''n'' times, so we consider the tensor power
:
The
operation describes ''n'' inputs undergoing the operation
independently and is the quantum mechanical counterpart of
concatenation
In formal language, formal language theory and computer programming, string concatenation is the operation of joining character string (computer science), character strings wikt:end-to-end, end-to-end. For example, the concatenation of "sno ...
. Similarly, ''m invocations of the channel'' corresponds to
.
The quantity
:
is therefore a measure of the ability of the channel to transmit words of length ''n'' faithfully by being invoked ''m'' times.
This leads to the following definition:
:A non-negative real number ''r'' is an achievable rate of
with respect to
if
:For all sequences
where
and
, we have
:
A sequence
can be viewed as representing a message consisting of possibly infinite number of words. The limit supremum condition in the definition says that, in the limit, faithful transmission can be achieved by invoking the channel no more than ''r'' times the length of a word. One can also say that ''r'' is the number of letters per invocation of the channel that can be sent without error.
The channel capacity of
with respect to
, denoted by
is the supremum of all achievable rates.
From the definition, it is vacuously true that 0 is an achievable rate for any channel.
Important examples
As stated before, for a system with observable algebra
, the ideal channel
is by definition the identity map
. Thus for a purely ''n'' dimensional quantum system, the ideal channel is the identity map on the space of ''n'' × ''n'' matrices
. As a slight abuse of notation, this ideal quantum channel will be also denoted by
. Similarly, a classical system with output algebra
will have an ideal channel denoted by the same symbol. We can now state some fundamental channel capacities.
The channel capacity of the classical ideal channel
with respect to a quantum ideal channel
is
:
This is equivalent to the no-teleportation theorem: it is impossible to transmit quantum information via a classical channel.
Moreover, the following equalities hold:
:
The above says, for instance, an ideal quantum channel is no more efficient at transmitting classical information than an ideal classical channel. When ''n'' = ''m'', the best one can achieve is ''one bit per qubit''.
It is relevant to note here that both of the above bounds on capacities can be broken, with the aid of
entanglement. The
entanglement-assisted teleportation scheme allows one to transmit quantum information using a classical channel.
Superdense coding
In quantum information theory, superdense coding (also referred to as ''dense coding'') is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assum ...
. achieves ''two bit per qubit''. These results indicate the significant role played by entanglement in quantum communication.
Classical and quantum channel capacities
Using the same notation as the previous subsection, the classical capacity of a channel Ψ is
:
that is, it is the capacity of Ψ with respect to the ideal channel on the classical one-bit system
.
Similarly the quantum capacity of Ψ is
:
where the reference system is now the one qubit system
.
Channel fidelity
Another measure of how well a quantum channel preserves information is called channel fidelity, and it arises from
fidelity of quantum states
In quantum mechanics, notably in quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the ...
.
Bistochastic quantum channel
A bistochastic quantum channel is a quantum channel
which is
unital,
[John A. Holbrook, David W. Kribs, and Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction." ''Quantum Information Processing''. Volume 2, Number 5, p. 381-419. Oct 2003.] i.e.
.
See also
*
No-communication theorem
In physics, the no-communication theorem or no-signaling principle is a no-go theorem from quantum information theory which states that, during measurement of an entangled quantum state, it is not possible for one observer, by making a measureme ...
*
Amplitude damping channel In the theory of quantum communication, an amplitude damping channel is a quantum channel that models physical processes such as spontaneous emission. A natural process by which this channel can occur is a spin chain through which a number of spin ...
References
* M. Keyl and R. F. Werner, ''How to Correct Small Quantum Errors'', Lecture Notes in Physics Volume 611, Springer, 2002.
* .
{{DEFAULTSORT:Quantum Channel
Quantum information theory