Unital qubit channels are widely accepted models for decoherence in the communication medium as well as in the communication buffer. Though the former mode of decoherence has been the main topic of interest in quantum Shannon theory, recently researchers have started to appreciate the impact of the latter on the design of a practical quantum communication system [3, 4, 5, 1]. For quantifying the effective capacity in the presence of decoherence in the transmission buffer, quantum additive queue-channels were introduced and analyzed in . The queue-channel model can also capture decoherence during serial processing of a sequence of qubits by a quantum processor.
A general upper-bound on the capacity of additive queue-channels were derived in  and it was shown to be achieved in the case of the erasure channel, depolarizing channel  and symmetric generalized amplitude damping channel . In this paper, we show that the same upper-bound is achievable for a large class of unital qubit queue-channels using product decoding, a.k.a. non-entangled measurement.
The i.i.d. unital channel has been studied extensively and its classical capacity has been characterized. The classical capacity is known to be additive and is achieved by non-entangled (product) encoding. However, to the best of our knowledge, the following questions have not been explicitly resolved: (a) can product decoding achieve the classical capacity of the channel, and (b) if so, is there an explicit POVM that achieves the capacity? These questions are important with or without decoherence due to buffering because entangled measurement (non-product decoding) requires a reliable quantum processor, which may not be available in the near future. Motivated by the practical issue of decoherence during buffering, we further ask: what is the impact of decoherence at the transmission buffer on the classical capacity and does it change the answers to questions (a) and (b)?
I-a Our Contributions
We show that the upper-bound on the classical capacity of additive queue-channel is achievable for any unital qubit queue-channel if the encoder has non-causal side information regarding the waiting times of the qubits. In the absence of this side information, we show that for a large class of unital qubit queue-channels, the same upper-bound can be achieved. In both cases, we show that non-entangled projective measurements can achieve the capacity and provide explicit descriptions of the encoders and the projective measurements. As an intermediate result, we derive a capacity achieving non-entangled projective measurement for any i.i.d. unital qubit channel. To the best of our knowledge, this result has not been discussed in the literature before and thus, may be of independent interest.
Ii Unital Qubit Queue-channels
A unital qubit channel is a completely positive trace preserving map from the space of all linear operators on a two dimensional Hilbert space to itself and satisfies . In quantum Shannon theory, the classical capacity of an i.i.d. unital channel is a well studied topic. In the i.i.d. setting, if a classical message is transmitted as a density state in , then the receiver receives the density state . It is well known that the classical capacity of i.i.d. unital channel is additive  and can be achieved using product encoding .
A unital qubit queue-channel is a unital channel where qubits experience different unital channels parametrized by the random time they spend in the buffer before transmission. In this case, for the transmitted state , the output state would be , if the waiting times are known at the receiver and the output state would be , if the waiting times are not known at the receiver. Example of unital qubit queue-channels are the depolarizing queue-channels  and the symmetric generalized amplitude damping queue-channels .
The waiting times come from the queuing dynamics of the buffer. The queuing dynamics and the notion of capacity for the above queue-channel are the same as in . For the sake of completeness, we state the queuing model and a few relevant concepts verbatim from .
"The buffering process is modeled as a continuous-time single-server queue. To be specific, the single-server queue is characterised by (i) A server that processes the qubits in the order in which they arrive, that is in a First Come First Served (FCFS) fashion***The FCFS assumption is not required for our results to hold, but it helps the exposition., and (ii) An "unlimited buffer" — that is, there is no limit on the number of qubits that can wait to be transmitted. We denote the time between preparation of the th and th qubits by , where
are i.i.d. random variables. Theses are viewed as inter-arrival times of a point process of rate where The "service time," or the time taken to transmit qubit is denoted by , where are also assumed to be i.i.d. random variables, independent of the inter-arrival times . The "service rate" of the qubits is denoted by We assume that (i.e., mean transmission time is strictly less than the mean preparation time) to ensure stability of the queue. Qubit has a waiting time . The waiting times of the other qubits can be obtained using the well known Lindley’s recursion:
In queuing parlance, the above system describes a continuous-time queue. Under mild conditions, the sequence for a stable queue is ergodic, and reaches a stationary distribution We assume that the waiting times of the qubits are available at the receiver during decoding.
An important difference between the queue-channel introduced above and the usual i.i.d. channels is that this channel is a part of continuous time dynamics. Hence, the usual notion of capacity per channel use for i.i.d. channels is not pertinent here. As mentioned before, the above channel model is closely related to quantum queue-channels studied in . So, we first do a short review of the notion of capacity per unit time and some relevant capacity results in .
Ii-a Classical capacity of unital quantum queue-channels
A rate is called an achievable rate for a quantum queue-channel if there
exists a sequence of quantum codes with probability of
quantum codes with probability of erroras and .
The information capacity of the queue-channel is the supremum of all achievable rates for a given arrival and service process, and is measured in bits per unit time.
Note that the information capacity of the queue-channel depends on the arrival process, the service process, and the noise model."
As discussed in Sec. I, in this paper, we derive the capacity of this channel and show that product encoding and product decoding achieve that capacity. Towards this, an important intermediate step of (possibly) independent interest is to design an explicit product encoding and product decoding strategy for i.i.d. unital qubit channels.
Iii Product Encoding/Decoding for i.i.d. Untial Qubit Channels
It is well known that product encoding achieves the classical capacity of an i.i.d. unital qubit channel, which is equal to the Holevo information . In this section, we show that product decoding is sufficient to achieve that capacity and provide an explicit capacity achieving product encoding and decoding strategy. To the best of our knowledge, this explicit result is not available in the current literature.
The classical capacity of an i.i.d. unital qubit channel is given by the Holevo information 
Iii-a Product encoding and decoding
For a unital qubit channel , let
is the operator norm (which equals the largest eigenvalue of the operator). Letbe the state that achieves the supremum in (2) and
Consider the classical binary symmetric channel (BSC) with cross-over probability and choose any capacity achieving encoder and decoder. For example, one can choose the well known random coding and typical decoding, or an appropriate polar code and the corresponding decoder.
For sending a message over the unital channel, first map the message to an appropriate classical binary codeword from the chosen classical codebook. Then map symbol to state and symbol to , and transmit over the unital channel.
At the receiver, use the projection measurements and obtain a sequence of and . Then, use the classical decoder chosen for the BSC.
The above product encoding and decoding strategy for the unital qubit channel achieves the capacity in (1).
First, we prove that the above quantum encoding and decoding induces a classical i.i.d. BSC on the i.i.d. unital qubit channel. Rest follows using the fact that  and is the Shannon capacity of BSC.
The probability that bit is decoded as bit is equal to the probability that the projection measurements on gives . Similarly, the probability that bit is decoded as bit is same as the probability of the event that the projection measurement on gives .
The second probability is given by
This expression, however, is exactly equal to first probability, which in turn is given by
This completes the proof. ∎
The main insight from the above theorem is summarized in the following remark.
Every untial qubit channel has an induced binary symmetric channel whose Shannon capacity equals the classical capacity of the unital qubit channel.
Next, building on the above insight and Theorem 4 in , we study unital qubit queue-channels.
Iv Capacity of Unital Qubit Queue-channels
We start with the capacity upper-bound in , which is applicable to any additive queue-channel. We assume that the waiting times of the qubits are available at the receiver during decoding.
Theorem 2 (, Theorem 1).
The classical capacity of a unital qubit queue-channel is upper-bounded by , irrespective of whether the encoder knows the waiting times or does not know the waiting times.
The case where waiting times are not known at the encoder is a direct re-statement from . The other case also follows directly from the same result by considering the fact that in deriving the upper-bound in , the encoder is allowed the additional side information regarding the waiting times. ∎
We study encoding and decoding strategies that achieve the above bound in both settings. We start with the simpler setting where encoder knows the waiting times and later we study the more practical setting, where the encoder does not know the waiting times.
Iv-a Encoder knows waiting times
In the Shannon theory literature, the knowledge of the future parameters of a time-varying channel at the receiver is called non-causal side information. This is not practical when the channel variation is fast and unpredictable (i.i.d. like). However, as the waiting times result into a Markov process, such an assumption is not so impractical. In certain slowly varying queues, the waiting times can be predicted within a reasonable accuracy.
In this setting, the product encoding and product decoding strategy is similar to the one considered in Sec. III. However, some modifications are necessary to address the non-i.i.d. nature of the queue-channel.
First, we introduce a modified version of (1). For an unital qubit channel parametrized by waiting time , let
Message to classical bits: Pick any capacity achieving encoder and decoder for the classical binary symmetric queue-channel . A detailed discussion on this channel can be found in .
Product encoding and decoding of qubits: Map the th classical bit to or , depending on whether it is or , respectively. At the decoder, use the projection measurements .
The above product encoding and product decoding strategy for unital qubit queue-channel achieves the capacity upper-bound in Theorem 2.
Iv-B Encoder does not know waiting times
Here we consider the setting where the queue evolution cannot be predicted and hence, the encoder has no knowledge of . This is a more prevalent setting in quantum communication. In fact, in many practical quantum communication systems, the encoding and the decoding has to be chosen at time zero, and cannot be adapted according to the queue evolution. We show that, in this setting, again a simple product encoder and product decoder achieves capacity for a large class of untial qubit queue-channels.
Let us consider a family of i.i.d. unital qubit channels , parametrized by a non-negative real number . This means that the channel acts on any joint state as
, where the parameter determines the map. As discussed in Theorem 1, the classical capacity of this channel is achieved by the product encoding and product decoding
and is equal to the Shannon capacity of the binary symmetric channel with crossover probability .
It is well known that any state can be expressed as linear combination of the Pauli matrices , , and . This leads to three natural induced channels for any qubit channel: map to and
, projectors onto the two eigenvectors of, and measure using these same projectors. For the i.i.d. unital qubit channel , parametrized by , this leads to three induced binary symmetric channels , .
A family of i.i.d. unital qubit channels and an unital qubit queue-channel are closely related. In a unital qubit queue-channel, each qubit sees a different unital qubit channel depending on its waiting time . This can be captured by an appropriate choice of the family , such that the channel seen by any qubit is for . Clearly, the family of channels corresponding to a queue-channel is dictated by the environment of the buffer and the nature of the interactions.
We call a unital qubit queue-channel time-ordered if the ordering of the Shannon capacities of the induced channels , and of the corresponding family of unital qubit channels is to -invariant.
Encoding and decoding: Pick a capacity achieving encoder and decoder for the classical binary symmetric queue-channel . Irrespective of the sequence number, map bit to and to and use projection measurements .
The above product encoding and product decoding strategy achieves the capacity upper-bound in Theorem 2 for time-ordered unital qubit queue-channels.
The following lemma is crucial to the proof of this theorem.
For time-ordered untial qubit queue-channel and for any , i.e., the optimal encoding and decoding strategy does not change with waiting time.
This lemma implies that the above product encoding and decoding strategy converts a time-ordered unital qubit queue-channel into a binary symmetric queue-channel with crossover probabilities
Rest follows from Theorem 4 in . ∎
Proof of Lemma 5.
First, note that upto local unitaries at the channel input and output, any unital channel can be written as a convex combination of Pauli channels :
where are the Pauli matrices (see discussion between Prop. 6.41 and Ex. 6.43 in ).
Thus, any can be equivalently represented by the three probabilities , where
We prove Lemma 5 using the following lemma, which gives and for a unital qubit channel in terms of the Pauli matrices. Hence, this result may be of independent interest.
For a unital channel , given by ,
Lemma 6 is applicable to any parametrized by . Thus, if the orderings of remain unchanged with , and remain unchanged with .
Finally, note that the ordering of are the same as the ordering of the Shannon capacities of . To see this, let us first find the crossover probability of . Using simple trace calculations, it turns out that the crossover probability of the BSC is . Thus, its Shannon capacity is . Note that at and increases monotonically with . Thus, the capacity of is monotonic in and hence, ordering of remains unchanged with for a time-ordered unital qubit queue-channel, which completes the proof of this lemma. ∎
Proof of Lemma 6.
For any state , where ,
Thus, , where and , the channel output
where , , and denotes entrywise dot product. Thus,
The last step follows from the fact that is a pure state.
After doing the matrix products and using some linear algebra involving linearity of trace, and the facts that , , and for , , one obtains
It is evident that the supremum is obtained when and for . Using the definition of , it follows that .
The notion of time-ordered unital qubit queue-channels is directly connected to the binary i.i.d. classical channels induced by the Pauli matrices. This gives a physical interpretation of the conditions under which the statement in Theorem 4 holds true. However, a result like Theorem 4, holds for a broader class of channels.
Let be the class of i.i.d. unital qubit channels corresponding to the unital qubit queue-channels. We call the queue-channel to have time invariant encoding if corresponding to the family is -invariant.
An example of such a channel is the symmetric generalized amplitude damping channel studied in . This class of unital qubit queue-channels include time-ordered queue-channels. It follows from Lemma 5.
Encoding and decoding: Pick a capacity achieving code for the queue-channel BSC() and map to and and decode them using .
The above product encoding and product decoding strategy achieves the capacity upper-bound in Theorem 2 for unital qubit queue-channels with time invariant encoding.
From the definition of time invariant encoding, it follows that and from the encoding/decoding it is clear that the decoder uses the optimal product decoder corresponding to . This result is equivalent to Lemma 5 in the proof of Theorem 4.
Rest follows exactly as the proof of Theorem 4. ∎
Though Theorem 7 is more general than the time-ordered case, the result in the time-ordered case are of interest in its own right. Note that Lemma 6 gives a simple closed form encoder and decoder for any i.i.d. unital qubit channel, which is of independent interest. Also, in practice, we could not find any realistic physical scenario which would lead to a unital qubit queue-channel which has time invariant encoding but is not time-ordered.
We derived an explicit non-entangled projective measurement strategy for i.i.d unital channel and proved that it achieves the classical capacity. Building on this insight, we showed that non-entangled projective measurements achieve the classical capacity of a broad class of unital qubit queue-channels that includes the well known unital qubit queue-channels like depolarizing channels, Pauli channels and symmetric generalized amplitude damping channels.
VS gratefully acknowledges support from NSF CAREER Award CCF 1652560 and NSF grant PHY 1915407. The work of AC was supported by the Department of Science and Technology, Government of India under Grant SERB/SRG/2019/001809 and Grant INSPIRE/04/2016/001171. PM and KJ acknowledge the Metro Area Quantum Access Network (MAQAN) project, supported by the Ministry of Electronics and Information Technology, India vide sanction number 13(33)/2020-CC&BT. The authors thank Mark Wilde for useful discussions.
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