Memory Effects in Spin-Chain Channels for Information Transmission
We investigate the multiple use of a ferromagnetic spin chain for quantum and classical communications without resetting. We find that the memory of the state transmitted during the first use makes the spin chain a qualitatively different quantum channel during the second transmission, for which we find the relevant Kraus operators. We propose a parameter to quantify the amount of memory in the channel and find that it influences the quality of the channel, as reflected through fidelity and entanglement transmissible during the second use. For certain evolution times, the memory allows the channel to exceed the memoryless classical capacity (achieved by separable inputs) and in some cases it can also enhance the quantum capacity.
pacs:03.67.Hk, 03.65.-w, 03.67.-a, 03.65.Ud.
Recently spin chains have been proposed as potential channels for short distance
quantum communications (See, for example, Refs.boseCP (); danielthesis ()).
The basic idea is to simply place the state to be transmitted at one
end of a spin chain initially in its ground state, allow it to propagate for a specific amount of time, and then receive it at the other end.
Generically, while propagating, the information will also inevitably disperse in the chain, and even when a transmission is considered complete
(i.e., the state is considered to have been received with some fidelity/probability),
some information of the state lingers in the channel. It
is thus assumed that a reset of the spin chain to its ground state is made after each transmission Giovannetti (). If, on the other hand, a
second transmission is performed through the channel without resetting, then the memory of the first transmission should affect the second
transmission. A spin chain channel without resetting is thus an
interesting physical model of a channel with memory Kretschmann ().
In this paper, we show that a ferromagnetic spin chain used without resetting is a very different channel than those studied so far in the extensive literature of quantum channels with memory macchiavello (); Daems (); macchiavello2 (); Arshed (); memarzadeh (); Datta (); BDW+04 (); Hamada (); Daffer (); Bowen (); Kretschmann (); Bowen2 (); plenio (); Arrigo (). Firstly, the channels usually studied are those with the noise during multiple uses being correlated with each other macchiavello (); Daems (); macchiavello2 (); Arshed (); memarzadeh (); Datta (); BDW+04 (), but being independent of the transferred states. In our model, however, the state transmitted during the first use modifies the type of noise during the second use. Secondly, the noise is most often assumed as Markovian correlated macchiavello (); Daems (); macchiavello2 (); Arshed (); memarzadeh (); BDW+04 (), while this is not the case for us. Thirdly, and most importantly, the channel noises in our case stem from a physical model described by a Hamiltonian. This should stimulate activity in calculating its capacities. To this end, we also introduce a memory parameter to quantify the amount of memory. This parameter depends on the distance between the Kraus operators of the second use of the channel with and without memory, so this method can be used to quantify the amount of memory for those channels that admit a description in terms of separate Kraus operators on different uses.
There is also a very important practical issue which motivates our work. The standard way of resetting the chain requires its interaction with a zero temperature environment Giovannetti2 () and this may open up unnecessary avenues for decoherence. Thus one either resets actively by performing a cooling sequence at the chain ends danielthesis () or uses it several times without resetting which automatically raises the question of the effect of memory of one transmission on a subsequent transmission. Multiple usage of a chain of two spins has been studied in rossini () to compute the rate of information transmission, but using the swap operators on both spins, a chain of length removes the memory effects. We will compare and contrast our results for the ferromagnetic channel without resetting with some results that have emerged in the recent literature macchiavello (); Daems (); macchiavello2 (); Arshed (); memarzadeh (); Datta (); BDW+04 (); Hamada (); Daffer (); Bowen (); Kretschmann (); Bowen2 (); plenio (); Arrigo ().
Let us consider a communication system like that of fig.
1(a) which has a set of sender and receiver registers to
store the quantum input and output states respectively and a
ferromagnetic open spin chain as a quantum channel. The registers
are isolated from the channel, and the Hamiltonian of the
chain commutes with (total spin in the direction) so the
number of the excitations in the channel is preserved through the
dynamics. Specifically we are going to consider a Heisenberg chain
with spins coupled by the Hamiltonian
where and are the
coupling and magnetic field, respectively, and
is the vector of
Pauli operators at site .
To transfer a quantum state from the register
(we will restrict our attention to two uses of the channel, so ) to the register we put
the state in the channel by applying a swap operator which
exchanges the state of the register and the first spin of the
then we leave the spin chain to evolve for time and finally
the transmission is completed by applying another swap operator
which exchanges the state of site with the register
. The total operator to transfer the quantum state from the
sender register to the receiver register is
. The initial state of the system is
where is an arbitrary initial density matrix of the
sender registers, is the
ground state of the chain and contains all receiver
register spins in the states . In the numerical analysis for
this paper we have used . In fact, for the
operations and exclude any memory effects rossini () while for the
quality of transmission is low boseCP ().
After the first transmission the total state is which is the case studied in boseCP (). The received state in the register is , where ’s are the following operators:
with the index going from to . In the above equation, , where represents one flipped spin in site of the channel and all the other spins in . The operator is a zero matrix which is included here for comparison with the memory case later on. The effect of the operators () can be combined into one operator, to show that the chain acts as an amplitude damping channel boseCP (). Except the case of perfect transfer, some information of the first state remains in the state of the channel and the effect of channel is no longer described by the Kraus operators (1). We assume that in the first transmission the state of the sender register is a general pure state , but it is easy to generalize the results to mixed input states. After the first transmission the state of the channel can be calculated by tracing out the state of the registers from . We obtain
The state (2) shows that with probability the channel is in the state and acts like an amplitude damping channel but there are some corrections with probability due to the state . To find the Kraus operators of the channel with the state one can consider a general density matrix in where the channel is in the state . By applying the operator on the state of whole system, the state of the register is transferred to the register (albeit with a certain fidelity), so the Kraus operators can be easily derived. We will write down the Kraus operators in a certain way (for simplicity and interpretation), though ours may not be the only way to write the Kraus operators for the channel. Two of the Kraus operators of the channel with the initial state are as in (1) multiplied by the coefficient and the others are some matrices that we shall soon introduce. Thus we can describe the effect of the channel with initial state as a probabilistic effect, which means that with probability the channel affects the inputs like an amplitude damping channel with Kraus operators (1) and with the probability the effect of the channel is specified by the following Kraus operators:
where the index goes from 1 to ,
the two excitation amplitude transition with , and means
all the spins of the channel are in except the sites and
. Notice that includes physical interaction
(scattering) between the first and second state.
In order to get a complete description of the channel for the second use we know that with probability the state of the channel is (the spin chain is an amplitude damping channel) and with probability the state of the chain is but acts as an amplitude damping channel. Thus with total probability the spin chain is an amplitude damping channel, otherwise with the probability the channel is in the state and its effect is specified by the Kraus operators (5). Therefore, we have
where is the amplitude damping evolution (1) and is the evolution
with Kraus operators (5).
If we consider the memory as a deviation of the channel effect from the memoryless case, then to find a distance between the two evolutions we can consider the distance between the Kraus operators in the two cases. Thus, to quantify this deviation, the following memory parameter is suggested:
Notice that we have multiplied the summation of the distances in Eq. (9) by which is the probability that this evolution takes place. By substituting the exact form of the operators in (9) for the case , we arrive at
It is clear that the memory parameter is dependent on the first input of the chain as well as the channel parameter . The largest deviation from the memoryless case is given for , corresponding to the transmission of on the first use. In this case the maximum of is . For we have perfect transfer, and for the first state is swapped out by the sender into .
To compare the quality of transmission we can compare the average fidelities. The average fidelity in the th use of the channel is where is the fidelity of the th transmission and the integration performed over the surface of the Bloch sphere for all pure input states . The total description of the channel in the second use, Eq. (8) helps to compute the average fidelity for the second transmission. It is easy to show that,
where is the average fidelity for memoryless case, and we have used the identity that to simplify the final result. In fig. 2(a) the average fidelities for the second use of the channel has been plotted for equal time evolutions (setting ). In this figure the average fidelity for the memoryless case has been compared with the case where the state has been transferred in the first use and with the case of average inputs in the first transmission. When the average fidelity of the first transmission has a peak, which means almost perfect transmission, the next transmission is also good. In non-optimal times when the first transmission is not good the memory effect can improve the quality of transmission. In fig. 2(b) the parameter (we have used this parameter instead of just for simplicity) and the average fidelity for the second transmission after sending the state in the first use, have been plotted together. When take its minimum it means that the amount of the memory in the channel is high, so the average fidelity in the second transmission has a low value because the state of the channel is highly mixed and there is information from the previous transmission in it. In the other case when the parameter has a peak it means that after the first transmission the channel has been nearly reset to the initial ground state. But in this case the average fidelity for the second transmission is not necessarily high because the average fidelity also depends from the time evolution . For example in fig. 2(b) for the memory has a low value but the average fidelity is not high because of the non-optimal . In this non-optimal time, has a large value, which means that the information is packed in the first spin and swapped out to the sender register, so the chain reset to its ground state. The same happens for the second transmission, so that the average fidelity is low.
Another problem that can be compared for different uses of the
channel is the entanglement distribution. In this case the sender
registers are a set of pair registers like fig. 1(b). Dual
registers (k=1,2) contain a maximally entangled state. In
the first transmission the state is transferred to the
register to create an entangled pair (not necessarily maximal)
between . In the second transmission, without resetting the
chain, the state of is transferred to the register to
create the entanglement between . In fig. 2(c) the
concurrence as a measure of entanglement Wootters () for the
states (memoryless) and (memory
case) has been plotted. It shows that the effect of memory is always
destructive. The peaks of
entanglement are located at times where nearly perfect transmission happens.
Let us now discuss the dependence of the fidelity on . As
shown above the quality of state transmission in the second use of
the channel depends on the time evolution as well. We chose
a range of such that the memory parameter is
increasing for the case that the state is transferred in the
first use. For each value of we have compared the maximum
average fidelity in a long range of . In fig. 3 we
have plotted this maximum value of the average fidelity
in the second transmission versus . Figure 3 is
very interesting because it shows that the average fidelity is
decreasing when is increased. This shows that the remaining
probability amplitude in the chain has a destructive effect on the
of transfer in the second use of the chain.
Finally we investigate whether the memory effect (taking equal evolution times for simplicity) can enhance either the quantum capacity or the single-shot classical capacity which are both known for the memoryless (amplitude damping) channel Giovannetti (). As we will show below, such enhancement is indeed possible, and can be demonstrated even without explicitly calculating the capacities. We compare the Holevo bound for a special equiprobable bipartite input states in memory channel with the classical capacity of separable input states in memoryless channels Giovannetti (). Assume that all the four possible equiprobable classical input data are encoded into a special kind of input states,
where and all these sates vary
from separable states (=0) to the maximally entangled one
(). In fig. 1(a) one can prepare any of
states in registers and and by applying the
operator this state is received as the state
in registers and . The Holevo bound for input states
(14) per use is
where and is the Von
Neumann entropy of the state . To find the optimal input
states one can maximize over the parameter
. Surprisingly, the maximum
is always achieved by
separable states (). In fig. 4(a) we have plotted
the and also the real capacity of memoryless channel
with separable input states Giovannetti () in terms of .
The memory helps to increase the classical capacity in non-optimal
times. These results for spin chains
are analogous to those of memory dephasing channel Arrigo ().
The coherent information as a lower bound for quantum capacity is , where is the input and is a purification of . In fig. 1(b) consider two maximally entangled states in registers and so the states of unprimed sender registers are . These two states are transferred through the chain by and we can consider two maximally entangled states in registers and as a purification of transferred states. In fig. 4(b) we compare the quantum capacity of Giovannetti () with the coherent information per use in our model. We see that though the effect is small, there are certain memory channels ( i.e., certain ) for which even a lower bound to the true quantum capacity exceeds the memoryless quantum capacity.
In conclusion, we have given a characterization of the behavior of a spin chain without resetting. It provides an interesting example of a quantum memory channel, where the memory of the state transmitted during the first use produces a qualitatively different channel in the second use.
We have found the relevant Kraus operators for this model and we have introduced a parameter to quantify the amount of memory in the channel which has broader applicability even outside the domain of spin chain channels.
We have shown that the memory effect can enable one to exceed the known classical capacity for separable inputs and the quantum capacity of the memoryless channel. Our study might pave the way for the computation of the full capacities of such a spin chain channel with memory.
S.B. is supported by the EPSRC, through which a part of the stay of A.B. at UCL is funded, the QIP IRC (GR/S82176/01), the Wolfson Foundation, and the Royal Society of Science and Technology. A.B. thanks the British Council in Iran for financial support. D.B. and S.M. are grateful to S.B. for the hospitality at UCL.
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