Classical capacity of a qubit depolarizing memory channel with Markovian correlated noise
We study the classical capacity of a forgetful quantum memory channel that switches between two qubit depolarizing channels according to an ergodic Markov chain. The capacity of this quantum memory channel depends on the parameters of the two depolarizing channels and the memory of the ergodic Markov chain. When the number of input qubit’s is two, we show that depending on channel parameters either the maximally entangled input states or product input states achieve the classical capacity. Our conjecture based on numerics is that as the number of input qubits are increased the classical capacity approaches the product state capacity for all values of the parameters.
Keywords: Quantum memory channel, classical capacity, hidden Markov model, entropy rate
Reliable transmission of classical information over quantum channels is an important problem in quantum information theory. The maximum amount of classical information that can be reliably transmitted over a quantum channel is called the classical capacity of the channel. Quantum memory channels model many physical situations where the noise effects are correlated. An unmodulated spin chain  and a micro maser  have been proposed as physical models of quantum channels with memory effects. In recent years there has been a lot of interest in quantum channels with memory . Capacity formulas for various classes of quantum memory channels were derived in . As in the memoryless case, an important question is whether entangled inputs enhance the communication capacity of a quantum memory channel. For Pauli channels with memory, it was shown that below a certain threshold value of the noise correlation parameter, the maximally entangled states provide the optimal two-use classical capacity . An experimental demonstration of enhancement of classical information by entangling qubits for correlated Pauli channels was provided in . In bosonic continuous variable memory channels also it is known that entangled inputs enhance capacity .
A subclass of quantum memory channels are forgetful channels where the effects of the initial memory configuration are forgotten over time. In this paper we look at a forgetful quantum memory channel where the noise correlations come from a classical ergodic Markov chain. Depending on the state of the Markov chain, one of the two depolarizing channels gets applied to the input qubit. This channel was first studied in  and the product state capacity was derived in terms of the entropy rate of a hidden Markov process . In the current work we investigate whether the classical capacity of this channel is enhanced if entangled inputs are allowed and give a partial answer to this question. We show that when the input consists of two qubits the channel capacity is achieved by either the maximally entangled input states or product input states depending on an explicit function of the parameters of the channel. Secondly, our numerical results point out that as the number of inputs qubits are increased the classical capacity approaches the product state capacity. In section 2 we give a background on quantum memory channel, show the channel construction and review the result on the product state capacity of the channel. In section 3 show that for two qubits the classical information carrying capacity of maximally entangled states is better than the product states. Finally in section 4 we present numerical evidence to support that the product state capacity of this channel is equal to its classical capacity.
2 Construction of the channel and its product state capacity
A quantum channel is a completely positive trace preserving map where and are the observable algebras of the input and output systems, respectively. Memoryless channels are channels where the noise acts independently on each input state. Multiple uses of a memoryless channel is given by the tensor product . The classical capacity of a quantum channel is the maximum rate at which classical information can be transmitted over the channel. The one shot classical capacity of a quantum channel is given by the Holevo capacity .
where is the Von Neumann entropy of the state and the maximum is taken over all possible input ensembles. This is the capacity of the quantum channel when only product state encoding is allowed and is also known as the product state capacity. The n-use classical capacity by allowing entangled inputs is the amount of classical information that can be reliably transmitted per channel use is given by
The classical capacity of the channel is given by
Proving the additivity of the Holevo capacity was one of the most important problems in quantum information theory over the past decade. Additivity implies that entangled inputs cannot enhance the rate of information transmission. Additivity of the Holevo capacity has been shown to be true in depolarizing, unital and entanglement breaking channels . The superadditivity of the Holevo capacity in certain higher dimensional quantum channels was show by Hastings in  thus disproving the additivity conjecture.
When a tensor product structure of the multi use of the channel does not hold we fall in the regime of channels with memory. Amongst the physically relevant models, we follow the non-anticipatory model of quantum memory channel given in . In this model of quantum memory channels, besides the input and output, there is a third system which represents the state of the memory. The channel operates on the input state and the state of the memory resulting in an output state and a new state of the memory. Thus a quantum memory channel is represented by a CPTP map . The action of n successive uses of this channel is given by the channel where
The final output state can be determined by performing a partial trace over the memory system. A quantum channel is called forgetful if the memory behavior does not depend on the initial memory configuration. That is, if for any input state and there exists an such that for all
for any pair of initial memory , and is the trace distance
A depolarizing channel is a quantum channel that retains its state with probability and moves to the completely mixed state with probability
This map is completely positive for . In this paper we look at a special case of quantum memory channels where the state of the memory transits between state ‘0’ and state ‘1’ according to an ergodic Markov chain with transition matrix . If the memory state is ‘0’ then a depolarizing channel is applied and if the memory state is ‘1’ then a depolarizing channel is applied where
The action of this quantum memory channel a state can be written as
Successive application of this channel results in the n qubit channel
It was shown in  that the product states of the form
with minimize the output entropy and the product state capacity is given by the following theorem
Theorem 2.1 ().
The product state capacity of the quantum memory channel is given by
Moreover, the output state has the following eigen decomposition
where the eigen values are given by
From 4 we see that the limit is equal to the entropy rate of a hidden Markov process with stationary measure . The observed process is whether a bit was flipped and the hidden process is which of the two depolarizing channel was used. Efficient computation of the entropy rate of a hidden Markov process is a longstanding problem in classical information theory. If one of the two depolarizing channels is a perfect channel then the capacity of this quantum memory channel can be computed efficiently by the techniques of algebraic measures which was shown in [14, 15].
3 Entanglement enhanced communication with two inputs
In this section we focus on the question whether the capacity given by equation (2) is enhanced by using entangled inputs when the input consists of two qubits. The channel we study comes under the category of Pauli channels. A single qubit Pauli channel is given by the mapping
where and are the Pauli matrices given by
A memoryless two qubit Pauli channel consists of two independent uses of the single qubit channel. In a quantum memory channel the Pauli rotations are not independent. An example of a well studied two qubit quantum memory channel is the following
In this channel with probability the same Pauli rotation is applied to the two successive input and with probability the two rotations are independent. It was shown  that there exists threshold value of the memory above which maximally entangled input states maximize the two qubit capacity of this channel whereas below the threshold value the product states maximize the capacity.
In the quantum memory channel that we study in this paper the channel transition probabilities are Markov and for simplicity we look at the special case with the transition matrix where . This quantum memory channel can be parameterized by three parameters: the memory of the Markov chain, and and henceforth we refer to it as . To find the 2-qubit capacity of the channel we look at the action of the channel on a general pure input state of two qubit’s
The channel is a covariant channel , that is, we have the relation
We also have the identity
Thus for equiprobable ensemble of states we can see that from the capacity formula (2) the first term is equal to due to equation (5)and the second term due to equation (5) is equal to . Thus we get that for a two qubit channel, 2-use capacity is given by
We show that entangled inputs indeed enhance the capacity of the channel . In fact, similar to results of , depending on the parameters we either have the maximally entangled input states or the product input states that maximize capacity for the channel with two inputs. Define
Our main result is
For any channel , if , then the two qubit capacity of the quantum memory channel is achieved by the maximally entangled input states otherwise it is achieved at the product input state.
According to equation (7) to find the capacity of the channel we need to look for input states that minimize the output entropy. The action of on the input pure states is given by
The diagonal elements and are given by
where , , and are given by equation (4) applied to the case . That is,
To determine the off diagonal element we consider the action of the tensor product of depolarizing channels and
From this we get
The four eigenvalues of the output state counting multiplicities are
If then is maximized when else if then it is maximized when . Correspondingly if the output entropy is minimized (and hence channel capacity is maximized) when (product states) and if then output entropy is minimized (and hence channel capacity is maximized) when (maximally entangled states). A simple calculation gives that the condition is the same as . ∎
The output eigenvalues that correspond to the minimum output entropy when are obtained by substituting in equation (12) and as expected we get the eigenvalues to be . The two qubit capacity if (product state inputs) is
A straightforward calculation shows that
and hence when the output eigenvalues are
The two qubit capacity if (maximally entangled inputs) is
4 Classical capacity of the channel
In this section we provide numerical evidence that the product state capacity of the quantum memory channel is equal to the classical capacity. For quantum memory channels that include a periodic channel with depolarizing branches and a convex combination of depolarizing channels the classical capacity was shown to be equal to the product state capacity . We conjecture that for the quantum memory channel studied in this paper
It is clear that . Although we are unable to show the reverse inequality we give the following bound
where is the entropy rate of the Markov process with transition matrix .
From the capacity formula of equation (2) it is clear that
Define the quantities
For an ensemble we have the inequality 
where . Using this inequality we get
Now we claim that
Proof of Claim 4.3
Firstly due to the additivity of the depolarizing channel  we have that
We also have that
Also, for any compound channel that is a tensor product of depolarizing channels, the minimum output entropy is obtained on the product states of the form given by equation (4), thus states of the form right and left side of the equation (19) are equal and hence
Finally it is also clear that
We provide figures (3),(4) and (5) in support conjecture 4.1. In these we look at the mutual information for a variety of entangled states including the W-state, GHZ-state and the maximally entangled state and compare it with the mutual information of the product state. For there is a small range of parameters (see figure) where the mutual information of the maximally entangled states is higher than that of the product states. By varying the parameters , and we find that for and the product states have more mutual information than the entangled states.
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