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 states, coupled by a time independent Hamiltonian, can be used to send a quantum state from one location to another. The resulting quantum channel ends up being identical to an amplitude damping channel, for which the quantum capacity, the classical capacity and the entanglement assisted classical capacity of the quantum channel can be evaluated.

Qubit Channel

We consider here the amplitude damping channel in the case of a single qubit.

Any quantum channel can be defined in several equivalent ways. For example, via Stinespring's dilation theorem, a channel ${\mathcal {D}}$ can be represented via an isometry $V$ as ${\mathcal {D}}(\rho )=\operatorname {tr} _{2}$ , and we say in this case that $V$ is the Stinespring representation of ${\mathcal {D}}$ .^[1] In particular, the single-qubit amplitude damping channel has Stinespring representation $V$ given by

V|0\rangle =|0,0\rangle ,\qquad V|1\rangle ={\sqrt {1-p}}|0,1\rangle +{\sqrt {p}}|1,0\rangle .

An alternative equivalent representation is given via Kraus operators. This means to represent the action of the channel in the form

{\mathcal {N}}(\rho )=\sum _{j}K_{j}\rho K_{j}^{\dagger }

for some set of operators

K_{j}

such that

\sum _{j}K_{j}^{\dagger }K_{j}=I

. For the amplitude dampling channel, one choice of such representation reads

{\mathcal {N}}(\rho )=K_{0}\rho K_{0}^{\dagger }+K_{1}\rho K_{1}^{\dagger }

with

K_{0}={\begin{pmatrix}1&0\\0&{\sqrt {1-p}}\end{pmatrix}},\qquad K_{1}={\begin{pmatrix}0&{\sqrt {p}}\\0&0\end{pmatrix}}\;.

More explicitly, we thus have

{\cal {N}}_{p}\left={\begin{pmatrix}\rho _{00}+p\rho _{11}&{\sqrt {1-p}}\rho _{01}\\{\sqrt {1-p}}\rho _{10}&(1-p)\rho _{11}\end{pmatrix}}\;.

Model for a Spin Chain Quantum Channel

The main construct of the quantum channel based on spin chain correlations is to have a collection of N coupled spins. At either side of the quantum channel, there are two groups of spins and we refer to these as quantum registers, A and B. A message is sent by having the sender of the message encode some information on register A, and then, after letting it propagate over some time t, having the receiver later retrieve it from B. The state $\rho _{A}$ is prepared on A by first decoupling the spins on A from those on the remainder of the chain. After preparation, $\rho _{A}$ is allowed to interact with the state on the remainder of the chain, which initially has the state $\sigma _{0}$ . The state of the spin chain as time progresses can be described by $R(t)=U(t)(\rho _{A}\otimes \sigma _{0})U^{\dagger }(t)$ . From this relationship we can obtain the state of the spins belonging to register B by tracing away all other states of the chain.

$\rho _{B}(t)={\mbox{Tr}}^{(B)}$

This gives the mapping below, which describes how the state on A is transformed as a function of time as it is transmitted over the quantum channel to B. U(t) is just some unitary matrix which describes the evolution of the system as a function of time.

$\rho _{A}\rightarrow {\mathcal {M}}(\rho _{A})\equiv \rho _{B}(t)={\mbox{Tr}}^{(B)}$

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