Foundational object in quantum communication theory
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 general dynamics of a qubit. An example of classical information is a text document transmitted over the Internet.
Terminologically, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a quantum operation 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 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: 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, whose properties we now summarize.
Let formula_1 and formula_2 be the state spaces (finite-dimensional Hilbert spaces) of the sending and receiving ends, respectively, of a channel. formula_3 will denote the family of operators on formula_4 In the Schrödinger picture, a purely quantum channel is a map formula_5 between density matrices acting on formula_1 and formula_2 with the following properties:
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 formula_5 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 "HA" only constitute a proper subset of the operators on "HA" and same can be said for system "B". However, once a linear map formula_5 between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend formula_5 uniquely to the full space of operators. This leads to the adjoint map formula_17, which describes the action of formula_5 in the Heisenberg picture:
The spaces of operators "L"("H""A") and "L"("H""B") are Hilbert spaces with the Hilbert–Schmidt inner product. Therefore, viewing formula_19 as a map between Hilbert spaces, we obtain its adjoint formula_5* given by
formula_21
While formula_5 takes states on "A" to those on "B", formula_17 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 formula_5 is assumed to be trace preserving, formula_17 is unital, that is,formula_26. 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 a 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:
formula_27
that is unital and completely positive (CP). The operator spaces can be viewed as finite-dimensional C*-algebras. Therefore, we can say a channel is a unital CP map between C*-algebras:
formula_28
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 formula_29 on some set formula_30. We assume formula_30 is finite so formula_29 can be identified with the "n"-dimensional Euclidean space formula_33 with entry-wise multiplication.
Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define formula_34 to include the relevant classical observables. An example of this would be a channel
formula_35
Notice formula_36 is still a C*-algebra. An element formula_37 of a C*-algebra formula_38 is called positive if formula_39 for some formula_40. Positivity of a map is defined accordingly. This characterization is not universally accepted; the quantum instrument 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 or Frobenius category.
Examples.
Time evolution.
For a purely quantum system, the time evolution, at certain time "t", is given by
formula_41
where formula_42 and "H" is the Hamiltonian and "t" is the time. This gives a CPTP map in the Schrödinger picture and is therefore a channel. The dual map in the Heisenberg picture is
formula_43
Restriction.
Consider a composite quantum system with state space formula_44 For a state
formula_45
the reduced state of "ρ" on system "A", "ρ""A", is obtained by taking the partial trace of "ρ" with respect to the "B" system:
formula_46
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
formula_47
where "A" is an observable of system "A".
Observable.
An observable associates a numerical value formula_48 to a quantum mechanical "effect" formula_49. formula_49's are assumed to be positive operators acting on appropriate state space and formula_51. (Such a collection is called a POVM.) In the Heisenberg picture, the corresponding "observable map" formula_52 maps a classical observable
formula_53
to the quantum mechanical one
formula_54
In other words, one integrates "f" against the POVM to obtain the quantum mechanical observable. It can be easily checked that formula_52 is CP and unital.
The corresponding Schrödinger map formula_56 takes density matrices to classical states:
formula_57
where the inner product is the Hilbert–Schmidt inner product. Furthermore, viewing states as normalized functionals, and invoking the Riesz representation theorem, we can put
formula_58
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. Let formula_59 be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map formula_60 with pure quantum input formula_61 and with output space formula_62:
formula_63
Let
formula_64
The dual map in the Heisenberg picture is
formula_65
where formula_66 is defined in the following way: Factor formula_67 (this can always be done since elements of a POVM are positive) then formula_68.
We see that formula_52 is CP and unital.
Notice that formula_70 gives precisely the observable map. The map
formula_71
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 formula_51 simply consists of "A" making a measurement, i.e. it is the observable map:
formula_73
If, in the event of the "i"-th measurement outcome, "B" prepares his system in state "Ri", the second part of the channel formula_52 takes the above classical state to the density matrix
formula_75
The total operation is the composition
formula_76
Channels of this form are called "measure-and-prepare" or "entanglement-breaking."
In the Heisenberg picture, the dual map formula_77 is defined by
formula_78
A measure-and-prepare channel can not be the identity map. This is precisely the statement of the no teleportation theorem, 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, 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, which is why measure-and-prepare channels are also known as entanglement-breaking channels.
Pure channel.
Consider the case of a purely quantum channel formula_52 in the Heisenberg picture. With the assumption that everything is finite-dimensional, formula_52 is a unital CP map between spaces of matrices
formula_81
By Choi's theorem on completely positive maps, formula_52 must take the form
formula_83
where "N" ≤ "nm". The matrices "K""i" are called Kraus operators of formula_52 (after the German physicist Karl Kraus, who introduced them). The minimum number of Kraus operators is called the Kraus rank of formula_52. 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 (or free-space for that matter) transmission of single photons. 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. The channel is capable of transmitting not only basis states (e.g. formula_86, formula_87) but also superpositions of them (e.g. formula_88). The coherence 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 formula_60, we need to compare it with an "ideal channel" formula_90 . For instance, when the input and output algebras are identical, we can choose formula_90 to be the identity map. Such a comparison requires a metric between channels.
Since a channel can be viewed as a linear operator, it is tempting to use the natural operator norm. In other words, the closeness of formula_60 to the ideal channel formula_90 can be defined by
formula_94
However, the operator norm may increase when we tensor formula_60 with the identity map on some ancilla.
To make the operator norm even a more undesirable candidate, the quantity
formula_96
may increase without bound as formula_97 The solution is to introduce, for any linear map formula_60 between C*-algebras, the cb-norm
formula_99
Definition of channel capacity.
The mathematical model of a channel used here is same as the classical one.
Let formula_100 be a channel in the Heisenberg picture and formula_101 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
formula_102
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":
formula_103
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
formula_104
The formula_105 operation describes "n" inputs undergoing the operation formula_106 independently and is the quantum mechanical counterpart of concatenation. Similarly, "m invocations of the channel" corresponds to formula_107.
The quantity
formula_108
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 formula_52 with respect to formula_106 if
For all sequences formula_111 where formula_112 and formula_113, we have
formula_114
A sequence formula_115 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 formula_52 with respect to formula_106, denoted by formula_118 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 formula_34, the ideal