In quantum information theory, a quantum channel is a communication channel that 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 H A {\displaystyle H_{A}} and H B {\displaystyle H_{B}} be the state spaces (finite-dimensional Hilbert spaces) of the sending and receiving ends, respectively, of a channel. L ( H A ) {\displaystyle L(H_{A})} will denote the family of operators on H A . {\displaystyle H_{A}.} In the Schrödinger picture, a purely quantum channel is a map Φ {\displaystyle \Phi } between density matrices acting on H A {\displaystyle H_{A}} and H B {\displaystyle H_{B}} with the following properties:
As required by postulates of quantum mechanics, Φ {\displaystyle \Phi } needs to be linear. Since density matrices are positive, Φ {\displaystyle \Phi } must preserve the cone of positive elements. In other words, Φ {\displaystyle \Phi } is a positive map. If an ancilla of arbitrary finite dimension n is coupled to the system, then the induced map I n ⊗ Φ , {\displaystyle I_{n}\otimes \Phi ,} where In is the identity map on the ancilla, must also be positive. Therefore, it is required that I n ⊗ Φ {\displaystyle I_{n}\otimes \Phi } is positive for all n. Such maps are called completely positive. Density matrices are specified to have trace 1, so Φ {\displaystyle \Phi } 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 Φ {\displaystyle \Phi } 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 Φ {\displaystyle \Phi } between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend Φ {\displaystyle \Phi } uniquely to the full space of operators. This leads to the adjoint map Φ ∗ {\displaystyle \Phi ^{*}} , which describes the action of Φ {\displaystyle \Phi } in the Heisenberg picture: The spaces of operators L(HA) and L(HB) are Hilbert spaces with the Hilbert–Schmidt inner product. Therefore, viewing Φ : L ( H A ) → L ( H B ) {\displaystyle \Phi :L(H_{A})\rightarrow L(H_{B})} as a map between Hilbert spaces, we obtain its adjoint Φ {\displaystyle \Phi } * given by
⟨ A , Φ ( ρ ) ⟩ = ⟨ Φ ∗ ( A ) , ρ ⟩ . {\displaystyle \langle A,\Phi (\rho )\rangle =\langle \Phi ^{*}(A),\rho \rangle .}
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