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.
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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
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
The sender prepares his set of k input spins as:
where
where
Evidently, the fact that an amplitude damping channel describes the transmission of quantum states across the spin chain stems from the fact that Hamiltonian of the system conserves energy. While energy can be spread out as the one-spin up state is transferred along the chain, it is not possible for spins in the down state to suddenly gain energy and become spin up states.
Capacities of the Amplitude Damping Channel
By describing the spin-chain as an amplitude damping channel, it is possible to calculate the various capacities associated with the channel. One useful property of this channel, which is used to find these capacities, is the fact that two amplitude damping channels with efficiencies
Quantum Capacity
In order to calculate the quantum capacity, the map
This representation of the map is obtained by adding an auxiliary Hilbert space
This relationship demonstrates that the channel is degradable, which guarantees that the coherent information of the channel is additive. This implies that the quantum capacity is achieved for a single channel use.
An amplitude damping mapping is applied to a general input state, and from this mapping, the von Neumann entropy of the output is found as:
where
In order to maximize the quantum capacity, we choose that
Finding the quantum capacity for
Entanglement Assisted Classical Capacity
To calculate the entanglement assisted capacity we must maximize the quantum mutual information. This is found by adding the input entropy of the message to the derived coherent information in the previous section. It is again maximized for
Classical Capacity
We now calculate C1, which is the maximum amount of classical information that can be transmitted by non-entangled encodings over parallel channel uses. This quantity acts as a lower bound for the classical capacity, C. To find C1, the classical capacity is maximized for n=1. We consider an ensemble of messages, each with probability
In this expression,
In order to find C1, first an upper bound is found for C1, and then a set of
By maximizing over all choices of p, the following upper bound for C1 is found:
This upper bound is found to be the value for C1, and the parameters that realize this bound are
Numerical Analysis of the Capacities
From the expressions for the various capacities, it is possible to carry out a numerical analysis on them. For an
Effectiveness of Spin-Chains as a Quantum Communication Channel
Having calculated the capacities for the amplitude damping channel as a function of the efficiency of the channel, it is possible to analyze the effectiveness of such a channel as a function of distance between the encoding site and the decoding site. Bose demonstrated that the efficiency drops as a function of
Future Study
Possibilities for future study in this field would include methods whereby spin-chain interactions could be used as a more effective channel. This would include the optimization of the values of