One-Shot Capacity Bounds on the Simultaneous Transmission of Classical and Quantum Information

Abstract

We study the communication capabilities of a quantum channel under the most general channel model known as the one-shot model. Unlike classical channels that can only be used to transmit classical information (bits), a quantum channel can be used for transmission of classical information, quantum information (qubits) and simultaneous transmission of classical and quantum information. In this work, we investigate the one-shot capabilities of a quantum channel for simultaneously transmitting of bits and qubits. This problem was studied in the asymptotic regime for a memoryless channel and a regularized characterization of the capacity region was reported. It is known that the transmission of private classical information is closely related to the problem of quantum information transmission. We resort to this idea and find achievable and converse bounds on the simultaneous transmission of the public and private classical information. then by shifting the classical private rate to the quantum information rate, the obtained rate regions will be translated into rate regions of thThis in turn, leads to a rate region for simulttaneous transmission of classical and quantum information. In the case of asymptotic i.i.d. setting, our one-shot result is evaluated to the known results in the literature. Our main tools used in the achievability proofs are position-based decoding and convex-split lemma.

Publicness, Privacy and Confidentiality in the Single-Serving Quantum Broadcast Channel

“…We believe that our definitions in this paper are more reasonable. The broad scope of the rate region developed in this paper enables us to recover not only the classical result of Watanabe and Oohama [7], but also the case of simultaneous transmission of public and private information [23], the simultaneous transmission of the classical and quantum information [11], [24] and the capacity region of the quantum broadcast channel by Yard-Hayden-Devetak [26].…”

“…From the theorems above, the recent result of the current authors on the simultaneous transmission of classical and quantum information can be recovered. The slight difference between the results stems from the fact that in [24], there is a single criterion for the error probability and secrecy while in this work separate criteria are considered.…”

“…In order to show that the results also hold without assistance of shared randomness, we need to derandomize the code. The derandomization is a standard procedure which can be done by expanding the states and corresponding POVMS and using a property of the trace distance given by the equality in (3) (see [24], [16], [38]). The only point that might be needed to be made here is the structure of the test operators in Bob's decoders (as well as that of Charlie).…”

“…The current authors with their colleagues in a former work [23], unified the problems of one-shot transmission of public and confidential information over quantum channels and proposed a protocol for simultaneously achieviable public and confidential rates as well as tight converse bounds. Latter, following the Devetak's proof of the quantum capacity [10], they proved a one-shot result for simultaneous transmission of the classical and quantum information [24].…”

“…The former leads to an operational interpretation of a recently-defined mutual information-like quantity and the latter, which is proved and should be considered as an independent lemma on its own right, gives rise to a new mutual-information like quantity as well as its operational meaning. We note that in a former work of the current authors [24], different definitions and approaches was taken to address the problem. We believe that our definitions in this paper are more reasonable.…”