Amplitude damping channel

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: ${\displaystyle H=-\sum _{\langle i,j\rangle }\hbar J_{ij}\left({\sigma }_{x}^{i}{\sigma }_{x}^{j}+{\sigma }_{y}^{i}{\sigma }_{y}^{j}+\gamma {\sigma }_{z}^{i}{\sigma }_{z}^{j}\right)-\sum _{i=1}^{N}\hbar B_{i}\sigma _{z}^{i}}$

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 ${\displaystyle |j\rangle }$, in which all spins are in the spin down state except for the j-th one, which is in the spin up state.

${\displaystyle |{j}\rangle \equiv \left|\downarrow \downarrow \cdots \downarrow \uparrow \downarrow \cdots \downarrow \right\rangle }$

The sender prepares his set of k input spins as:

${\displaystyle |\Psi \rangle _{A}\equiv \alpha \left|\Downarrow \right\rangle _{A}+\beta |\phi _{1}\rangle _{A}}$

where ${\displaystyle \left|\Downarrow \right\rangle }$ is the state where all positions have spin down, and ${\displaystyle |\phi _{1}\rangle }$ 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:

${\displaystyle \rho _{B}(t)=(|\alpha |^{2}+(1-\eta )|\beta |^{2})\left|\Downarrow \right\rangle _{B}\left\langle \Downarrow \right|+\eta |\beta |^{2}|\phi _{1}^{\prime }\rangle _{B}\langle \phi _{1}^{\prime }|+{\sqrt {\eta }}\alpha \beta ^{*}\left|\Downarrow \right\rangle _{B}\langle \phi _{1}^{\prime }|+{\sqrt {\eta }}\alpha ^{*}\beta |\phi _{1}^{\prime }\rangle _{B}\left\langle \Downarrow \right|}$

where ${\displaystyle \eta }$ is a constant defining the efficiency of the channel. If we represent the states in which one spin is up to be ${\displaystyle |1\rangle }$ and those where all spins are down to be ${\displaystyle |0\rangle }$, this becomes recognizable as the result of applying the amplitude damping channel ${\displaystyle {\mathcal {D}}_{n}}$, characterized by the following Kraus operators:

${\displaystyle A_{0}=|0\rangle \langle 0|+{\sqrt {\eta }}|1\rangle \langle 1|}$; ${\displaystyle A_{1}={\sqrt {1-\eta }}|0\rangle \langle 1|}$

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.

In order to calculate the quantum capacity, the map ${\displaystyle {\mathcal {D}}_{\eta }}$ is represented as follows:

${\displaystyle {\mathcal {D}}_{\eta }(\rho )\equiv {\mbox{Tr}}_{C}[V\left(\rho \otimes |0\rangle _{C}\langle 0|\right)V^{\dagger }]\;.}$

This representation of the map is obtained by adding an auxiliary Hilbert space ${\displaystyle {\mathcal {H}}_{C}}$ to that of ${\displaystyle {\mathcal {H}}_{A}}$. and introducing an operator V which operates on A and C. A complementary channel, ${\displaystyle {\tilde {\mathcal {D}}}_{\eta }}$ is also defined, where instead of tracing over C, we trace over A. A swapping operation S which transforms A into C is defined. Using this operation, as well as the rule for concatenation of amplitude damping channels, it is shown that for ${\displaystyle \eta \geqslant 0.5}$:

${\displaystyle {\tilde {\mathcal {D}}}_{\eta }(\rho )=S{\mathcal {D}}_{(1-\eta )/\eta }\left({\mathcal {D}}_{\eta }(\rho )\right)\;.}$

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:

${\displaystyle S({\mathcal {D}}_{\eta }(\rho ))=H_{2}(\left(1+{\sqrt {(1-2\,\eta \,p)^{2}+4\,\eta \,|\gamma |^{2}}}\right)/2)\;,}$

where ${\displaystyle p\in [0,1]}$ with state ${\displaystyle |1\rangle }$ and ${\displaystyle |\gamma |\leqslant {\sqrt {(1-p)p}}}$ is a coherence term. By looking at a purification of the state, it is found that:

${\displaystyle S(({\mathcal {D}}_{\eta }\otimes 1_{anc})(\Phi ))=H_{2}(\left(1+{\sqrt {(1-2\,(1-\eta )\,p)^{2}+4\,(1-\eta )\,|\gamma |^{2}}}\right)/2)}$

In order to maximize the quantum capacity, we choose that ${\displaystyle \gamma =0}$ (due to concavity of entropy, which yields the following as the quantum capacity:

${\displaystyle Q\equiv \max _{p\in [0,1]}\;{\Big \{}\;H_{2}(\eta \,p)-H_{2}((1-\eta )\,p)\;{\Big \}}\;}$

Finding the quantum capacity for ${\displaystyle \eta <0.5}$ is straightforward, as the quantum capacity vanishes as a direct result of the no-cloning theorem. The fact that channels can be composed in this fashion implies that quantum capacity of the channel must increase as a function of ${\displaystyle \eta }$.

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 ${\displaystyle \gamma =0}$. Thus, the entanglement assisted classical capacity is found to be

${\displaystyle C_{E}\equiv \max _{p\in [0,1]}\;{\Big \{}\;H_{2}(p)+H_{2}(\eta \,p)-H_{2}((1-\eta )\,p)\;{\Big \}}\;}$

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 ${\displaystyle \xi _{k}}$. The Holevo information is found to be:

${\displaystyle \chi \equiv H_{2}\left({\frac {1+{\sqrt {(1-2\,\eta \,p)^{2}+4\,\eta \,|\gamma |^{2}}}}{2}}\right)-\sum _{k}\xi _{k}H_{2}\left({\frac {1+{\sqrt {(1-2\,\eta \,p_{k})^{2}+4\,\eta \,|\gamma _{k}|^{2}}}}{2}}\right)\;}$

In this expression, ${\displaystyle p_{k}}$ and ${\displaystyle \gamma _{k}}$ are the population and a coherence term, as defined before, and ${\displaystyle p}$ and ${\displaystyle \gamma }$ are the average values of these.

In order to find C1, first an upper bound is found for C1, and then a set of ${\displaystyle p_{k},\gamma _{k},\xi _{k}}$ are found that satisfy this bound. As before, ${\displaystyle \gamma }$ is set to be 0 in order to maximize the first term of Holevo information. From here we use the fact that the binary entropy ${\displaystyle H_{2}(z)}$ is decreasing with respect to ${\displaystyle |1/2+z|}$ as well as the fact that ${\displaystyle H_{2}(1+{\sqrt {1-z^{2}}}/2)}$ is convex with respect to z to find the following inequality:

${\displaystyle \sum _{k}\xi _{k}H_{2}\left({\frac {1+{\sqrt {(1-2\,\eta \,p_{k})^{2}+4\,\eta \,|\gamma _{k}|^{2}}}}{2}}\right)\geqslant H_{2}\left({\frac {1+{\sqrt {1-4\,\eta \,(1-\eta )(\sum _{k}\xi _{k}p_{k})^{2}}}}{2}}\right)}$

By maximizing over all choices of p, the following upper bound for C1 is found:

${\displaystyle C_{1}\leqslant \max _{p\in [0,1]}{\Big \{}H_{2}\left(\eta \,p\right)-H_{2}\left({\frac {1+{\sqrt {1-4\,\eta \,(1-\eta )\,p^{2}}}}{2}}\right){\Big \}}\;}$

This upper bound is found to be the value for C1, and the parameters that realize this bound are ${\displaystyle \xi _{k}=1/d\,\!}$,${\displaystyle p_{k}=p\,\!}$, and ${\displaystyle \gamma _{k}=e^{2\pi ik/d}{\sqrt {(1-p)p}}}$.

From the expressions for the various capacities, it is possible to carry out a numerical analysis on them. For an ${\displaystyle \eta }$ of 1, the three capacities are maximized, which leads to the quantum and classical capacities both being 1, and the Entanglement assisted classical capacity being 2. As mentioned earlier, the quantum capacity is 0 for any ${\displaystyle \eta }$ less than 0.5, while the classical capacity and the entanglement assisted classical capacity reach 0 for ${\displaystyle \eta }$ of 0. When ${\displaystyle \eta }$ is less than 0.5, too much information is lost to the environment for quantum information to be sent to the receiving party.