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 can be represented via an isometry as , and we say in this case that is the Stinespring representation of . In particular, the single-qubit amplitude damping channel has Stinespring representation given byAn alternative equivalent representation is given via Kraus operators. This means to represent the action of the channel in the formfor some set of operators such that . For the amplitude dampling channel, one choice of such representation readswithMore explicitly, we thus have
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 is prepared on A by first decoupling the spins on A from those on the remainder of the chain. After preparation, is allowed to interact with the state on the remainder of the chain, which initially has the state . The state of the spin chain as time progresses can be described by . From this relationship we can obtain the state of the spins belonging to register B by tracing away all other states of the chain.
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.
There are, however, a few issues with this description of the quantum channel. One of the assumptions involved with using such a channel is that we expect that the states of the chain are not disturbed. While it may be possible for a state to be encoded on A without disturbing the chain, a reading of the state from B will influence the states of the rest of the spin chain. Thus, any repeated manipulation of the registers A and B will have an unknown impact on the quantum channel. Given this fact, solving the capacities of this mapping would not be generally useful, since it will only apply when several copies of the chain are operating in parallel. In order to calculate meaningful values for these capacities, the simple model below allows for the capacities to be solved exactly.
Solvable Model
A spin chain, which is composed of a chain of particles with spin 1/2 coupled through a ferromagnetic Heisenberg interaction, is used, and is described by the Hamiltonian:
It is assumed that the input register, A and the output register B occupy the first k and last k spins along the chain, and that all spins along the chain are prepared to be in the spin down state in the z direction. The parties then use all k of their spin states to encode/decode a single qubit. The motivation for this method is that if all k spins were allowed to be used, we would have a k-qubit channel, which would be too complex to be completely analyzed. Clearly, a more effective channel would make use of all k spins, but by using this inefficient method, it is possible to look at the resulting maps analytically.
To carry out the encoding of a single bit using the k available bits, a one-spin up vector is defined , in which all spins are in the spin down state except for the j-th one, which is in the spin up state.
The sender prepares his set of k input spins as:
where is the state where all positions have spin down, and is the superposition of all possible one-spin up states. Using this input, it is possible to find a state which describes the whole chain at a given time t. From such a state, tracing out the N-k spins not belonging to the receiver, as we would have done with the earlier model, leaves the state on B: