Recent work has precisely characterized the achievable trade-offs between three key information processing tasks-classical communication (generation or consumption), quantum communication (generation or consumption), and shared entanglement (distribution or consumption), measured in bits, qubits, and ebits per channel use, respectively. Slices and corner points of this three-dimensional region reduce to well-known protocols for quantum channels. A trade-off coding technique can attain any point in the region and can outperform time sharing between the best-known protocols for accomplishing each information processing task by itself. Previously, the benefits of trade-off coding that had been found were too small to be of practical value (viz., for the dephasing and the universal cloning machine channels). In this Letter, we demonstrate that the associated performance gains are in fact remarkably high for several physically relevant bosonic channels that model free-space or fiber-optic links, thermal-noise channels, and amplifiers. We show that significant performance gains from trade-off coding also apply when trading photon-number resources between transmitting public and private classical information simultaneously over secret-key-assisted bosonic channels.
Recent work has precisely characterized the achievable trade-offs between three key information processing tasks-classical communication (generation or consumption), quantum communication (generation or consumption), and shared entanglement (distribution or consumption), measured in bits, qubits, and ebits per channel use, respectively. Slices and corner points of this three-dimensional region reduce to well-known protocols for quantum channels. A trade-off coding technique can attain any point in the region and can outperform time sharing between the best-known protocols for accomplishing each information processing task by itself. Previously, the benefits of trade-off coding that had been found were too small to be of practical value (viz., for the dephasing and the universal cloning machine channels). In this Letter, we demonstrate that the associated performance gains are in fact remarkably high for several physically relevant bosonic channels that model free-space or fiber-optic links, thermal-noise channels, and amplifiers. We show that significant performance gains from trade-off coding also apply when trading photon-number resources between transmitting public and private classical information simultaneously over secret-key-assisted bosonic channels.
The trade-off capacity region of a quantum channel characterizes the optimal net rates at which a sender can communicate classical, quantum, and entangled bits to a receiver by exploiting many independent uses of the channel, along with the help of the same resources. Similarly, one can consider a trade-off capacity region when the noiseless resources are public, private, and secret-key bits. We identified [see Wilde, Hayden, and Guha, Phys. Rev. Lett. 108, 140501 (2012)] these trade-off rate regions for the pure-loss bosonic channel and proved that they are optimal provided that a long-standing minimum-output entropy conjecture is true. Additionally, we showed that the performance gains of a trade-off coding strategy when compared to a time-sharing strategy can be quite significant. In this paper, we provide detailed derivations of the results announced there, and we extend the application of these ideas to thermal-noise and amplifying bosonic channels. We also derive a ``rule of thumb'' for trade-off coding, which determines how to allocate photons in a coding strategy if a large mean photon number is available at the channel input. Our results on the amplifying bosonic channel also apply to the ``Unruh channel'' considered in the context of relativistic quantum information theory.
We consider the problem of communicating quantum states by simultaneously making use of a noiseless classical channel, a noiseless quantum channel, and shared entanglement. We specifically study the version of the problem in which the sender is given knowledge of the state to be communicated. In this setting, a trade-off arises between the three resources, some portions of which have been investigated previously in the contexts of the quantum-classical trade-off in data compression, remote state preparation, and superdense coding of quantum states, each of which amounts to allowing just two out of these three resources. We present a formula for the triple resource trade-off that reduces its calculation to evaluating the data compression trade-off formula. In the process, we also construct protocols achieving all the optimal points. These turn out to be achievable by trade-off coding and suitable time sharing between optimal protocols for cases involving two resources out of the three mentioned above.
In this paper, we give tradeoffs between classical communication, quantum communication, and entanglement for processing information in the Shannon-theoretic setting. We first prove a “unit-resource” capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation. The optimal strategy mixes the three fundamental protocols of teleportation, superdense coding, and entanglement distribution. We then provide an achievable rate region and a matching multiletter converse for the “direct-static” capacity theorem. This theorem applies to the scenario where a large number of copies of a noisy bipartite state are available (in addition to consumption or generation of the above three noiseless resources). Our coding strategy involves a protocol that we name theclassically assisted state redistribution protocoland the three fundamental protocols. We finally provide an achievable rate region and a matching multiletter converse for the “direct-dynamic” capacity theorem. This theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources. Our coding strategy combines theclassically enhanced father protocolwith the three fundamental unit protocols.
This chapter unifies all of the channel coding theorems that we have studied in this book. One of the most general information-processing tasks that a sender and receiver can accomplish is to transmit classical and quantum information and generate entanglement with many independent uses of a quantum channel and with the assistance of classical communication, quantum communication, and shared entanglement.1 The resulting rates for communication are net rates that give the generation rate of a resource less its consumption rate. Since we have three resources, all achievable rates are rate triples ( C,Q,E ) that lie in a three-dimensional capacity region, where C is the net rate of classical communication, Q is the net rate of quantum communication, and E is the net rate of entanglement consumption/generation. The capacity theorem for this general scenario is known as the quantum dynamic capacity theorem, and it is the main theorem that we prove in this chapter. All of the rates given in the channel coding theorems of previous chapters are special points in this three-dimensional capacity region. The proof of the quantum dynamic capacity theorem comes in two parts: the direct coding theorem and the converse theorem. The direct coding theorem demonstrates that the strategy for achieving any point in the three-dimensional capacity region is remarkably simple: we just combine the protocol from Corollary 22.5.2 for entanglement-assisted classical and quantum communication with the three unit protocols of teleportation, super-dense coding, and entanglement distribution. The interpretation of the achievable rate region is that it is the unit resource capacity region from Chapter 8 translated along the points achievable with the protocol from Corollary 22.5.2. In the proof of the converse theorem, we analyze the most general protocol that can consume and generate classical communication, quantum communication, and entanglement along with the consumption of many independent uses of a quantum channel, and we show that the net rates for such a protocol are bounded by the regularization of the achievable rate region.
This chapter unifies all of the channel coding theorems that we have studied in this book. One of the most general information-processing tasks that a sender and receiver can accomplish is to transmit classical and quantum information and generate entanglement with many independent uses of a quantum channel and with the assistance of classical communication, quantum communication, and shared entanglement. The resulting rates for communication are net rates that give the generation rate of a resource less its consumption rate. Since we have three resources, all achievable rates are rate triples ( C, Q, E ) that lie in a three-dimensional capacity region, where C is the net rate of classical communication, Q is the net rate of quantum communication, and E is the net rate of entanglement consumption/generation. The capacity theorem for this general scenario is known as the quantum dynamic capacity theorem, and it is the main theorem that we prove in this chapter. All of the rates given in the channel coding theorems of previous chapters are special points in this three-dimensional capacity region. The proof of the quantum dynamic capacity theorem comes in two parts: the direct coding theorem and the converse theorem. The direct coding theorem demonstrates that the strategy for achieving any point in the three-dimensional capacity region is remarkably simple: we just combine the protocol from Corollary 21.5.3 for entanglement-assisted classical and quantum communication with the three unit protocols of teleportation, super-dense coding, and entanglement distribution.