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 [5] and teleportation [2], 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 [7]. A good way to think of the super-dense coding protocol is that it is a statement of resource conversion [10]: 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 I (N ) ≡ max φAA I (A;B)ρ , where ρ ≡ NA→B(φAA), NA→B is the noisy channel connecting the sender to the receiver, and φAA is a pure, bipartite state prepared at the sender’s end of the channel. This result is the strongest statement 1 ar X iv :1 10 7. 13 47 v3 [ qu an tph ] 1 6 A ug 2 01 1 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 N and M [1]: I (N ⊗M) = I (N ) + I (M) , and the mutual information I (A;B) is concave in the input state when the channel is fixed [1] (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 [6]. 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 [35]. 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: 1. Sequential decoding for entanglement-assisted classical communication over a single-sender, singlereceiver quantum channel. 2. Sequential and successive decoding for entanglement-assisted classical communication over a quantum multiple access channel (a two-sender, single-receiver channel). 3. Simultaneous decoding for classical communication over an entanglement-assisted quantum multiple access channel. 4. Coherent simultaneous decoding for assisted and unassisted quantum communication over a quantum multiple access channel. 5. 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 [18] (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 [39] and Hsieh et al. [28] 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: