Trade-off capacities of the quantum Hadamard channels Kamil Br´ adler, Patrick Hayden, Dave Touchette, and Mark M. Wilde School of Computer Science, McGill University, Montreal, Qu´ebec, H3A 2A7, Canada (Dated: November 10, 2018) Coding theorems in quantum Shannon theory express the ultimate rates at which a sender can transmit information over a noisy quantum channel. More often than not, the known formulas expressing these transmission rates are intractable, requiring an optimization over an infinite number of uses of the channel. Researchers have rarely found quantum channels with a tractable classical or quantum capacity, but when such a finding occurs, it demonstrates a complete understanding of that channel’s capabilities for transmitting classical or quantum information. Here, we show that the three-dimensional capacity region for entanglement-assisted transmission of classical and quantum information is tractable for the Hadamard class of channels. Examples of Hadamard channels include generalized dephasing channels, cloning channels, and the Unruh channel. The generalized dephasing channels and the cloning channels are natural processes that occur in quantum systems through the loss of quantum coherence or stimulated emission, respectively. The Unruh channel is a noisy process that occurs in relativistic quantum information theory as a result of the Unruh effect and bears a strong relationship to the cloning channels. We give exact formulas for the entanglement-assisted classical and quantum communication capacity regions of these channels. The coding strategy for each of these examples is superior to a na¨ıve time-sharing strategy, and we introduce a measure to determine this improvement. PACS numbers: 03.67.Hk, 03.67.Pp, 04.62.+v Keywords: quantum Shannon theory, trading resources, entanglement-assisted classical and quantum com- munication, Hadamard channel, cloning channel, Unruh channel One of the aims of quantum Shannon theory is to char- acterize the ultimate limits on the transmission of infor- mation over a noisy quantum channel. Holevo, Schu- macher, and Westmoreland contributed the first seminal result in this direction by providing a lower bound for the ultimate limit of a noisy quantum channel to transmit classical information [1, 2], a result now known as the HSW coding theorem. Lloyd, Shor, and Devetak then contributed increasingly rigorous proofs of the quantum channel coding theorem [3–5], now known as the LSD coding theorem, that provides a lower bound on the ul- timate limit for a noisy quantum channel to transmit quantum information. Other expository proofs appeared later, providing insight into the nature of quantum cod- ing [6–10]. Bennett et al . [11] and Barnum et al . [12] also showed that the capacity of a quantum channel for transmitting quantum information is the same as that channel’s capacity for generating shared entanglement between sender and receiver. These three results form the core of the dynamic, single-resource quantum Shannon theory, where a sender exploits a noisy quantum channel to establish a single noiseless resource, namely, classical communication, quantum communication, or shared en- tanglement, with a receiver. A formula for the capacity of a channel gives a “single- letter” characterization if the computation of the capac- ity requires an optimization over only a single use of the channel, and the formula gives a “multi-letter” charac- terization otherwise. A single-letter characterization im- plies that the computation of the capacity is tractable for a fixed input dimension of the channel, whereas a Thus, in hindsight, we might now say that any chan- nel with a single-letter capacity formula is a “rare gem” in quantum Shannon theory, given that the known for- mulas for capacities generally give multi-letter charac- terizations. In fact, researchers once conjectured that the HSW formula for the classical capacity would gen- erally give a single-letter characterization [20, 21], un- til the recent result of Hastings [14]. Researchers have found several examples of these gems for the classical capacity: the identity channel [22], unital qubit chan- nels [23], erasure channels [24], Hadamard channels [25], entanglement-breaking channels [26], depolarizing chan- multi-letter characterization typically requires an opti- mization over an infinite number of uses of the channel and is therefore intractable in this case. This “single- letterization” issue does not play a central role in clas- sical information theory for the most basic task of com- munication over a noisy classical channel, because single- letterization occurs naturally in Shannon’s original anal- ysis for classical, memoryless channels [13]. But this is- sue plays a prominent role in the domain of quantum Shannon theory even for the most basic communication tasks. Our knowledge so far indicates that the computa- tion of the classical capacity is intractable in the general case [14–17], with the same seeming to hold generally for the quantum capacity [18, 19]. These results un- derscore our incomplete understanding of the nature of quantum information, but they also have the surprising and “uniquely quantum” respective consequences that the strong correlations present in entangled uses of a quantum channel boost the classical capacity and that the degeneracy property of quantum codes can enhance the quantum capacity. I. INTRODUCTION arXiv:1001.1732v2 [quant-ph] 9 Apr 2010