Channel (CP map)

== Introduction == Any device taking classical or quantum systems of a certain type as input and (possibly different) classical or quantum systems as output is a '''channel'''. This definition covers any processing step in information theory, from preparations to free and controlled time evolution to measurements. Channels are thus among the central concepts of classical and quantum information science. [[Classical channels]] are those that can transmit or store only classical information, like electrical wires or the Royal Mail. [[Quantum channels]] can transmit both classical and quantum information. Physical realizations of quantum channels include everything from optical fibers or coupled spin chains for quantum communication, to shielded atoms in optical traps for quantum storage. Classical (resp. quantum) [[capacity]] measures the amount of classical (quantum) information that can be sent undistorted through the channel. == Definition and Properties == Mathematically, a channel is represented by a map $T\; :\; S(\backslash mathcal\{H\}\_\{1\})\; \backslash rightarrow\; S(\backslash mathcal\{H\}\_\{2\})$ mapping [[states]] (''i.e.'' density operators) on some [[Hilbert space]] $\backslash mathcal\{H\}\_\{1\}$ to states on some (possibly different) Hilbert space $\backslash mathcal\{H\}\_\{2\}$. Classical channels are included in this setup if we interpret classical functions $f$ as diagonal matrices, $f\; \backslash equiv\; \backslash sum\_\{x\}\; f(x)\; \backslash ,\; |x\; \backslash rangle\; \backslash langle\; x|$. Which such maps $T$ qualify as channels? The channel should respect convex mixtures, and hence be '''linear'''. It should preserve the normalization of states, and thus be '''trace-preserving (TP)''': $tr(T(\backslash varrho))\; =\; tr(\backslash varrho)$. We also require that $T$ map positive operators to positive operators, so that $T(\backslash varrho)$ is a valid density operator for any input density operator $\backslash varrho$. Finally, these properties should remain true if the operation is only applied to part of a larger system. So we require that $T\; \backslash otimes\; id\_\{n\}$ be positive for all $n\; \backslash in\; \backslash mathbb\{N\}$, where $id\_\{n\}$ denotes the identity operation on the $n\; \backslash times\; n$ matrices. A map with this property is usually called '''completely positive'''. Obviously, complete positivity of $T$ implies positivity. The converse holds if the input or output system is classical. However, in the quantum case there are maps which are positive, but not completely positive. The matrix transpose operation $\backslash varrho\; \backslash mapsto\; \backslash varrho^\{t\}$ is a prominent example of such an unphysical operation. :''In summary, a channel is a completely positive and trace-preserving (CPTP) map $T\; :\; S(\backslash mathcal\{H\}\_\{1\})\; \backslash rightarrow\; S(\backslash mathcal\{H\}\_\{2\})$ between state spaces $S(\backslash mathcal\{H\}\_\{i\})$ associated with physical systems.'' The above approach to quantum channels is ''axiomatic'', i.e. based on a set of postulates, which are required by the statistical interpretation of quantum mechanics. An alternative way to characterize the possible channels is ''constructive'': we allow just those channels which can be built from the basic operations of (1) tensoring with a second system in a specified state, (2) unitary transformation, and (3) reduction to a subsystem. Luckily, the two approaches agree: the above three types of maps are CPTP, and by '''Stinespring's dilation theorem''' (known in the community under the jargon [[The Church of the larger Hilbert space|Church of the larger Hilbert space]]), every completely positive map can be decomposed into a product of the above three operations. == Heisenberg vs. Schrödinger == We have so far described channels as operations on states: when the system is initially in the state $\backslash varrho\; \backslash in\; S(\backslash mathcal\{H\}\_\{1\})$ and send through the channel $T\; :\; S(\backslash mathcal\{H\}\_\{1\})\; \backslash rightarrow\; S(\backslash mathcal\{H\}\_\{2\})$, the expectation value of the measurement of the observable $A\; \backslash in\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\}\_\{2\})$ at the output side of the channel is $tr(T(\backslash varrho)\; A)$. Since the dynamics is carried by the states (while the observables are static), this approach is usually called the '''Schrödinger representation''' of the channel. Alternatively, we may describe the dynamics in the '''Heisenberg representation''' as a transformation on observables. The corresponding channel $T^\{*\}:\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\}\_\{2\})\; \backslash rightarrow\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\}\_\{1\})$ is defined in terms the '''duality relation''' :$tr(T\_\{\}(\backslash varrho)\; \backslash ,\; A)\; =\; tr(\backslash varrho\; \backslash ,\; T^\{*\}(A))$ for all states $\backslash varrho\; \backslash in\; S\; (\backslash mathcal\{H\}\_\{1\})$ and observables $A\; \backslash in\; \backslash mathcal\{B\}\; (\backslash mathcal\{H\}\_\{2\})$. The duality relation guarantees that all expectation values coincide, so both descriptions are completely equivalent. The Schrödinger dual $T\_\{\}$ is linear, completely positive and trace-preserving iff $T^\{*\}\; \backslash ,$ is linear, completely positive and unit-preserving (or ''unital''), i.e., $T^\{*\}(\backslash mathbf\{1\})\; =\; \backslash mathbf\{1\}$. == Special cases == [[States]] are channels with one-dimensional input space $\backslash mathbb\{C\}$. '''Measurements''' are channels $M\; :\; S(\backslash mathcal\{H\}\_\{1\})\; \backslash rightarrow\; \backslash mathcal\{C\}$ with classical range algebra $\backslash mathcal\{C\}$, while '''preparations''' are channels $P\; :\; \backslash mathcal\{C\}\; \backslash rightarrow\; S(\backslash mathcal\{H\}\_\{2\})$ with classical domain (cf. [[Measurements and preparations]]). == References and further reading == * M. A. Nielsen, I. L. Chuang: ''Quantum Computation and Quantum Information''; Cambridge University Press, Cambridge 2000; Ch. 8 * E. B. Davies: ''Quantum Theory of Open Systems''; Academic Press, London 1976 * V. Paulsen: ''Completely Bounded Maps and Operator Algebras''; Cambridge University Press, Cambridge 2002 * M. Keyl: ''Fundamentals of Quantum Information Theory''; Phys. Rep. '''369''' (2002) 431-548; [http://arxiv.org/abs/quant-ph/0202122 quant-ph/0202122] == See also == * [[Classical channel capacity]] * [[Quantum channel capacity]] * [[Measurements and preparations]] * [[The Church of the larger Hilbert space]] [[Category:Handbook of Quantum Information]]