Sequential, successive, and simultaneous decoders for entanglement-assisted classical communication
Bennett et al. showed that allowing shared entanglement between a sender and receiver before communication begins dramatically simplifies the theory of quantum channels, and these results suggest that it would be worthwhile to study other scenarios for entanglement-assisted classical communication. In this vein, the present paper makes several contributions to the theory of entanglement-assisted classical communication. First, we rephrase the Giovannetti-Lloyd-Maccone sequential decoding argument as a more general “packing lemma” and show that it gives an alternate way of achieving the entanglement-assisted classical capacity. Next, we show that a similar sequential decoder can achieve the Hsieh-Devetak-Winter region for entanglement-assisted classical communication over a multiple access channel. Third, we prove the existence of a quantum simultaneous decoder for entanglement-assisted classical communication over a multiple access channel with two senders. This result implies a solution of the quantum simultaneous decoding conjecture for unassisted classical communication over quantum multiple access channels with two senders, but the three-sender case still remains open (Sen recently and independently solved this unassisted two-sender case with a different technique). We then leverage this result to recover the known regions for unassisted and assisted quantum communication over a quantum multiple access channel, though our proof exploits a coherent quantum simultaneous decoder. Finally, we determine an achievable rate region for communication over an entanglement-assisted bosonic multiple access channel and compare it with the Yen-Shapiro outer bound for unassisted communication over the same channel.
Shared entanglement between a sender and receiver leads to surprises such as super-dense coding  and teleportation , and these protocols were the first to demonstrate that entanglement, classical bits, and quantum bits can interact in interesting ways. For this reason, one could argue that these protocols and their noisy generalizations [10, 29, 30] make quantum information theory [31, 37] richer than its classical counterpart . A good way to think of the super-dense coding protocol is that it is a statement of resource conversion : one noiseless qubit channel and one noiseless ebit are sufficient to generate two noiseless bit channels between a sender and receiver.
Bennett et al. explored a generalization of the super-dense coding protocol in which a sender and receiver are given noiseless entanglement in whatever form they wish and access to many independent uses of a noisy quantum channel, and the goal is to determine how many asymptotically perfect noiseless bit channels that the sender and receiver can simulate with the aforementioned resources [3, 4, 25]. The entanglement-assisted classical capacity theorem provides a beautiful answer to this question. The optimal rate at which they can communicate classical bits in the presence of free entanglement is equal to the mutual information of the channel [4, 25], defined as
where , is the noisy channel connecting the sender to the receiver, and is a pure, bipartite state prepared at the sender’s end of the channel. This result is the strongest statement that quantum information theorists have been able to make in the theory of quantum channels, because the above channel mutual information is additive as a function of any two channels and :
and the mutual information is concave in the input state when the channel is fixed  (these two properties imply that we can actually calculate the entanglement-assisted classical capacity of any quantum channel). Furthermore, this information measure is particularly robust in the sense that a quantum feedback channel from receiver to sender does not increase it—Bowen showed that the classical capacity of a quantum channel in the presence of unlimited quantum feedback communication is equal to the entanglement-assisted classical capacity . For these reasons, the entanglement-assisted classical capacity of a quantum channel is the best formal analogy of Shannon’s classical capacity of a classical channel .
The simplification that shared entanglement brings to the theory of quantum channels suggests that it might be fruitful to explore other scenarios in which communicating parties share entanglement, and this is precisely the goal of the present paper. Indeed, we explore five different scenarios for entanglement-assisted classical communication:
Sequential decoding for entanglement-assisted classical communication over a single-sender, single-receiver quantum channel.
Sequential and successive decoding for entanglement-assisted classical communication over a quantum multiple access channel (a two-sender, single-receiver channel).
Simultaneous decoding for classical communication over an entanglement-assisted quantum multiple access channel.
Coherent simultaneous decoding for assisted and unassisted quantum communication over a quantum multiple access channel.
Entanglement-assisted classical communication over a bosonic multiple access channel.
We briefly overview each of these scenarios in what follows.
Our first contribution is a sequential decoder for entanglement-assisted classical communication, meaning that the receiver performs a sequence of measurements with “yes/no” outcomes in order to determine the message that the sender transmits (the receiver performs these measurements on the channel outputs and his share of the entanglement). The idea of this approach is the same as the recent Giovannetti-Lloyd-Maccone (GLM) sequential decoder for unassisted classical communication  (which in turn bears similarities to the Feinstein approach [15, 32, 38]). In fact, our approach for proving that the sequential method works for the entanglement-assisted case is to rephrase their argument as a more general “packing lemma” [28, 37] and exploit the entanglement-assisted coding scheme of Hsieh et al. [28, 37].
Our next contribution is to extend this sequential decoding argument to a quantum multiple access channel. Winter  and Hsieh et al.  have already shown that successive decoding works well for unassisted and assisted transmission of classical information over a quantum multiple access channel, respectively. (Here, successive decoding means that the receiver first decodes one sender’s message and follows by decoding the other sender’s message). We show that a receiver can exploit a sequence of measurements with “yes/no” outcomes to determine the first sender’s message, followed by a different sequence of “yes/no” measurements to determine the second sender’s message. Thus, our decoder here is both sequential and successive and generalizes the GLM sequential decoding scheme.
Our third contribution is to prove that the receiver of an entanglement-assisted quantum multiple access channel can exploit a quantum simultaneous decoder to detect two messages sent by two respective senders. A simultaneous decoder is different from a successive decoder—it can detect the two senders’ messages asymptotically faithfully as long as their transmission rates are within the pentagonal rate region of the multiple access channel [13, 39, 28]. A simultaneous decoder is more powerful than a successive decoder for two reasons:
A simultaneous decoder does not require the use of time-sharing in order to achieve the rate region of the multiple access channel (whereas a successive decoder requires the use of time-sharing). Thus, the technique should generalize well to the setting of “one-shot” information theory , where time-sharing does not apply because that theory is concerned with what is possible with a single use of a quantum channel.
Nearly every proof in classical network information theory exploits a simultaneous decoder . Thus, a quantum simultaneous decoder would be of broad interest for a network theory of quantum information. In particular, the strategy for achieving the best known achievable rate region of the classical interference channel exploits a simultaneous decoder [22, 13]. (An interference channel has two senders and two receivers, and each sender is interested in communicating with one particular receiver.)
We should mention that Fawzi et al. could prove the existence of a quantum simultaneous decoder for certain quantum channels , but a proof for the general case remained missing and they did not address the entanglement-assisted case. Though, the results of this paper and recent work of Sen  give a quantum simultaneous decoder for unassisted communication over a two-sender multiple access channel and solve the conjecture from Ref.  for the two-sender case. It remains unclear how to prove the conjecture for the case of three senders. The results of this work might be useful for establishing an achievable rate region for a quantum interference channel setting in which sender-receiver pairs share entanglement before communication begins, but this remains the topic of future work.
We then leverage the above result to recover the known regions for assisted and unassisted quantum communication over a quantum multiple access channel [27, 40, 28]. We call the decoder a coherent quantum simultaneous decoder because we construct an isometry from the above simultaneous decoding POVM, and the isometry is what enables quantum communication between both senders and the receiver.
Our final contribution is to determine an achievable rate region for entanglement-assisted classical communication over the multiple access bosonic channel studied in Ref. . This channel is simply a beamsplitter with two input ports, where the receiver obtains one output port and the environment of the channel obtains the other output port. The beamsplitter is a simplified model for light-based free-space communication in a multiple-access setting. In order to calculate the rate region for this setting, we apply the theorem of Hsieh et al. in Ref.  with both senders sharing a two-mode squeezed vacuum state  with the receiver. Since this state achieves the entanglement-assisted capacity of the single-mode lossy bosonic channel [20, 19, 26], we might suspect that it should do well in the multiple access setting. Though, it still remains open to determine whether this strategy is optimal.
1 Packing Argument for a Sequential Decoder
Giovannetti, Lloyd, and Maccone (GLM) offered a scheme for transmitting classical information over a quantum channel that exploits a sequential decoder . In their sequential decoding scheme, the receiver tries to distinguish the transmitted message from a list of all possible messages one by one until the correct one is identified, by performing a sequence of projective measurements. We recast this procedure as a general packing argument in this section, and the next section demonstrates that the sequential decoding scheme works well for entanglement-assisted classical communication.
Theorem 1 (Sequential Packing)
Let be an ensemble of states indexed by letters in an alphabet . Each state has the following spectral decomposition:
and the expected density operator of the ensemble is as follows:
Suppose there exists a code subspace projector and codeword subspace projectors such that the following properties hold for some , , and for all :
Then corresponding to a message set , we can construct a random code with such that the receiver can reliably distinguish between the states by performing a sequence of projective measurements using the projectors and . More precisely, suppose that our performance measure is the expectation of the average success probability where the expectation is with respect to all possible random choices of codes. Then we can bound this performance measure from below (as long as is positive):
implying that the performance measure becomes arbitrarily close to one if is large, , and is arbitrarily small.
Proof. The proof of this lemma is similar to the GLM proof, and we thus place it in Appendix A.
2 Sequential Decoding for Entanglement-Assisted Communication
In this section, we show an application of the GLM sequential decoding scheme to entanglement-assisted classical communication by exploiting the coding approach of Hsieh et al. . The approach thus gives another way of achieving the entanglement-assisted classical capacity of a quantum channel.
Theorem 2 (Entanglement-Assisted Sequential Decoding)
The sequential decoding scheme can achieve the entanglement-assisted classical capacity of a quantum channel.
Proof. Suppose that a quantum channel connects Alice to Bob and that they share many copies of an arbitrary entangled pure state :
where Alice has access to the system and Bob has access to the system . Alice chooses a message from her message set uniformly at random, applies a corresponding encoder to her shares of the entanglement, and sends the systems to Bob. Later in the analysis, we would like to be able to “pull” these encoding operations through the channel so that they are equivalent to some other operator acting at Bob’s end. In order to do this, we can write the many copies of the shared entanglement as a direct sum of maximally entangled states [28, 37]. Starting from the Schmidt decomposition for one copy of the state
is a type class, is the dimension of a type class subspace , is a representative sequence for the type class , and each is maximally entangled on the type class subspace specified by (see Refs. [28, 37] for more details on this approach). Thus, applying an operator acting on type class subspaces at Alice’s end is equivalent to applying the transpose of the same operator at Bob’s end. As in Refs. [28, 37], Alice constructs her encoders using the Heisenberg-Weyl set of operators that act on each of the type class subspaces
where determines a phase that is applied to the operators in each subspace. We denote this unitary by where is some vector that contains all the needed indices , and . Let denote the set of all such possible vectors. We construct a random code where is a vector chosen uniformly at random from and the corresponding set of encoders is then . Since the “transpose trick” holds for each of these unitaries, we have that
The induced ensemble at Bob’s end is then
Let denote the expected state of the ensemble:
We give Bob the following code subspace projector:
and the codeword subspace projectors:
where , , and are the -typical projectors for many copies of the states , and , respectively.
At this point we would like to apply our packing argument from Theorem 1 and we would like to have the following conditions hold:
where the function goes to zero as and . The first three conditions are shown in Refs. [28, 37]. The fourth condition follows from the equipartition property of typical subspaces  and the fact that for any unitary operator . The fifth condition follows from the fact that the projector commutes with the density operator . By our packing argument in Theorem 1 that gives a bound on the expectation of the average success probability, there exists a particular code, with which Alice can transmit messages from her set and Bob can detect the transmitted state by performing a series of projective measurements, with its average success probability being greater than
Therefore, Alice can pick the size of to be , and the rate of communication is then
with the average success probability becoming greater than
Thus, for sufficiently large , the sequential decoding scheme achieves the entanglement-assisted classical capacity with arbitrarily high success probability.
As a final note, we should clarify a bit further: there is a codebook for Alice with entanglement-assisted quantum codewords of the following form:
If Alice sends message , Bob performs a sequence of measurements in the following order (assuming a correct sequence of events):
3 Packing Argument for Sequential and Successive Decoding over a Multiple Access Channel
We now extend the packing argument from Section 1 to a multiple-access setting, in which there are two senders and one receiver. The resulting scheme is both sequential and successive—sequential in the above sense where the receiver linearly tests one codeword at a time and successive in the sense that the receiver first decodes one sender’s message and follows by decoding the other sender’s message. After doing so, we then briefly remark how this argument achieves the known strategies for both unassisted  and assisted classical communication .
Theorem 3 (Sequential and Successive Decoding)
Suppose there exists a doubly-indexed ensemble of quantum states, where two independent distributions generate the different indices and :
Averaging with the distributions and leads to the following states:
Suppose that there exist projectors , , , and such that
Suppose that is large, , is large, , and is arbitrarily small. Then there exists a sequential and successive decoding scheme for the receiver that succeeds with high probability, in the sense that the expectation of the average success probability is arbitrarily high:
with chosen so that
Proof. The random construction of the code is similar to that in the proof of Theorem 1. Given a message set , we construct a code for Alice randomly such that each takes a value with probability . Similarly, given a message set , we construct a code for Bob randomly such that each takes a value with probability . Using this code, Alice chooses a message from the message set , Bob chooses a message from the message set , and they encode their messages in the quantum codeword .
Suppose that the first sender Alice transmits message and the second sender Bob transmits message . Without loss of generality, the receiver first tries to recover the message that Alice transmits. In order to do so, he measures followed by to determine if the transmitted message corresponds to the first codeword of Alice, with corresponding to the outcome YES and corresponding to the outcome NO. Suppose that the outcome is NO. He then measures to project the state back into the large subspace. Assuming a correct sequence of events, the receiver continues and measures and for until getting to the correct outcome . Thus, the sequence of projectors measured is as follows, under the assumption of a correct sequence of events:
After receiving a YES outcome from , the receiver assumes that the first sender transmitted message . The receiver then tries to determine the codeword that Bob transmitted by exploiting the projectors and . He does this in a similar fashion as above, proceeding in the following order (again under the assumption of a correct sequence of events):
The POVM corresponding to the above measurement strategy is as follows:
The average success probability of any particular code is
and the expectation of the average success probability is
Observe that we can rewrite the success probability in (3.22) as follows:
We can then obtain the following lower bound on (3.26):
by exploiting the following inequality:
which holds for all positive operators , , and that have spectrum less than one. So it remains to show that both Tr is arbitrarily close to one and is arbitrarily small when averaging over all codewords and taking the expectation over random codes. We can apply Theorem 1 to obtain the following inequality:
with chosen as given in the statement of the theorem. We can then apply the Gentle Operator Lemma for ensembles (Lemma 9.4.3 in Ref. ) to prove the following inequality:
Invoking Theorem 1 one more time gives us the following lower bound:
and this completes the proof of the theorem, by combining the above two inequalities with the lower bound in (3.29).
It is straightforward to apply this packing argument to either unassisted or assisted transmission of classical information over a quantum multiple access channel. For the unassisted case, one could exploit Winter’s coding scheme with conditionally typical projectors , and we would pick the parameters as
so that we would have
For the entanglement-assisted case, one could exploit the coding structure of Hsieh et al.  that we have discussed throughout this article, and we would pick the parameters as
so that we would have
4 Entanglement-Assisted Quantum Simultaneous Decoding
In this section, we prove the existence of a simultaneous decoder for entanglement-assisted classical communication over a quantum multiple access channel with two senders. A simultaneous decoder differs from a successive decoder in the sense that such a decoder allows for the receiver to reliably detect the messages of both senders with a single measurement as long as the rates are within the pentagonal rate region specified by Theorem 6 of Ref.  and Theorem 4 below (it might also be helpful to consult Ref.  to see the difference between classical successive and simultaneous decoders). The advantage of a simultaneous decoder over a successive decoder is that there is no need to invoke time-sharing in order to achieve the Hsieh-Devetak-Winter rate region of the entanglement-assisted multiple access channel in Ref. . Also, an analogous classical decoder is required in order to achieve the Han-Kobayashi rate region for the classical interference channel  (though it requires a simultaneous decoder for three senders).
Concerning the quantum interference channel, Fawzi et al. made progress towards demonstrating that a quantized version of the classical Han-Kobayashi rate region is achievable for classical communication over a quantum interference channel , though they were only able to prove this result up to a conjecture regarding the existence of a quantum simultaneous decoder for general channels. The importance of this conjecture stems not only from the fact that it would allow for a quantization of the Han-Kobayashi rate region, but also more broadly from the fact that many coding theorems in classical network information theory exploit the simultaneous decoding technique . Thus, having a general quantum simultaneous decoder for an arbitrary number of senders should allow for the wholesale import of much of classical network information theory into quantum network information theory.
Our result below applies only to channels with two senders, and the technique unfortunately does not generally extend to channels with three senders. Thus, this important case still remains open as a conjecture. Sen independently arrived at the results here by exploiting both the proof structure outlined below and a different technique as well .
Theorem 4 (Entanglement-Assisted Simultaneous Decoding)
Suppose that Alice and Charlie share many copies of an entangled pure state where Alice has access to the system and Charlie has access to the system . Similarly, let Bob and Charlie share many copies of an entangled pure state . Let be a multiple access channel that connects Alice and Bob to Charlie, and let
Then there exists an entanglement-assisted classical communication code with a corresponding quantum simultaneous decoder, such that the following rate region is achievable for :
where the entropies are with respect to the state in (4.1).
Proof. Suppose that Alice has a message set and Bob has a message set from which they will each choose a message and uniformly at random to send to Charlie. They construct random codes and in the same way as explained in the proof of Theorem 2. Both of them encode their messages by applying unitary encoders to their respective shares of the entanglement, giving rise to the following states after applying the transpose trick to each type class [28, 37]:
Then they both send their share of the state to Charlie over the multiple access channel , giving rise to a state at Charlie’s receiving end:
Charlie decodes with a simultaneous decoding POVM , defined as follows:
The projectors , , , , , and are -typical projectors for the state onto the specified systems after tracing out all other systems.
The average error probability when Alice and Bob choose their messages independently and uniformly at random is
We can upper bound this error probability from above111We are indebted to Pranab Sen for this observation  (c.f., versions 1 and 2 of this paper). as
where we define
The first inequality follows from the inequality
for any operators (Corollary 9.1.1 of Ref. ). The second inequality follows from the properties of quantum typicality, the Gentle Operator Lemma (Lemma 9.4.2 of Ref. ), and the inequality proved in Ref. . We now recall the Hayashi-Nagaoka operator inequality  which holds for any positive operator and such that and :
and applying the Hayashi-Nagaoka operator inequality, we obtain the following upper bound on the error probability:
Considering the first term , we can prove that
where approaches zero when becomes large. This inequality follows from the following inequalities
(which can be proved with the methods of Ref. ) and by applying “measurement on approximately close states” (Corollary 9.1.1 of Ref. ) and the Gentle Operator Lemma (Lemma 9.4.2 of Ref. ) several times.
In order to analyze the second term , we need to take the expectation over all random codes and make several observations about the behavior of the codeword states under the expectation. Note that the encoding unitaries after the transpose trick and the channel commute because they act on different systems, so we can apply the encoding unitaries first. To simplify the calculation, we first consider only applying a random encoding unitary to the system :
where is the maximally mixed state on the type subspace . To see why the last equality holds, we note that when , averaging over all elements in gives rise to the state ; when , it can be shown that the whole expression sums up to zero [28, 37]. Now we can append the other state at Bob’s side and send the overall state through the channel. Therefore, we have that
Now consider the above state sandwiched between the projectors :
At this point, we note that , for a typical type , and