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arXiv:2504.20870 · IEEE International Symposium on Information Theory (ISIT)

Capacity Formulas for the Lossy Bosonic Compound Wiretap Channel

Florian Seitz, Janis Nötzel

Published Jun 22, 2025• 1 citation
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Abstract

We consider the bosonic compound wiretap channel. A pair of lossy channels connects a sender with both a (legitimate) receiver and an eavesdropper. The sender and receiver have only partial information about the actual state of the channels. In this situation, their task is to transmit the maximum amount of messages over an asymptotically large number of uses of their channel while guaranteeing at the same time that only an (asymptotically) negligible amount of information leaks to the eavesdropper. We prove capacity formulas for the case where both sender and receiver have the same information about the system and for the case where the sender has channel state information.

DOI
10.1109/isit63088.2025.11195468

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We consider the bosonic compound wiretap channel. A pair of lossy channels connects a sender with both a (legitimate) receiver and an eavesdropper. The sender and receiver have only partial information about the actual state of the channels. In this situation, their task is to transmit the maximum amount of messages over an asymptotically large number of uses of their channel while guaranteeing at the same time that only an (asymptotically) negligible amount of information leaks to the eavesdropper. We prove capacity formulas for the case where both sender and receiver have the same information about the system and for the case where the sender has channel state information.
Investigation of communication over a wiretap channel is one of the problems of information–theoretic security. The aim in the general wiretap channel model is to maximize the rate of the reliable communication from the source to the legitimate receiver, while keeping the confidential information as secret as possible from the wiretapper (eavesdropper). Here we consider the compound wiretap channel model, when the channels to the legitimate receiver and to the wiretapper depends on the number of possible states. We investigate the EEE -capacity–equivocation region which is the closure of the set of all achievable rate-reliability and equivocation pairs, where the rate-reliability function represents the optimal dependence of rate on the error probability exponent (reliability). Here the outer bound of this region is constructed in the case, when the states are not known to all terminals.
Compound channels model situations where a sender and receiver both have only partial information about the stationary channel they are transmitting over. They can be used to analyse the robustness of a communication system. With this work we provide a coding theorem for a class of practically relevant quantum compound channel models, where we consider the system uncertainty as due to a stationary but unknown jamming signal. Namely, we give an explicit formula for the case of the lossy Bosonic channel in the presence of a semi-classical attack utilizing the injection of coherent states into the transmission line. Mathematically, this is modelled by letting transmitter and jammer access the two different ports of a beam-splitter, and the receiver observe one of the output ports. While the transmitter can modulate their input, the jammer must transmit a constant (but unknown) sequence of states. We conjecture a new quantum entropy power inequality and show its relation to the capacity under study.
One of the problems of information - theoretic security concerns secure communication over a wiretap channel. The aim in the general wiretap channel model is to maximize the rate of the reliable communication from the source to the legitimate receiver, while keeping the confidential information as secret as possible from the wiretapper (eavesdropper). We introduce and investigate the E - capacity - equivocation region and the E - secrecy capacity function for the wiretap channel, which are, correspondingly, the generalizations of the capacity - equivocation region and secrecy - capacity studied by Csiszár and Körner (1978). The E - capacity equivocation region is the closure of the set of all achievable rate - reliability and equivocation pairs, where the rate - reliability function represents the optimal dependence of rate on the error probability exponent (reliability). By analogy with the notion of E - capacity, we consider the E - secrecy capacity function that for the given E is the maximum rate at which the message can be transmitted being kept perfectly secret from the wiretapper.
We consider the private classical capacity of a quantum wiretap channel, where the users (sender Alice, receiver Bob, and eavesdropper Eve) have access to the resource of a shared quantum state, additionally to their channel inputs and outputs. An extreme case is maximal entanglement or a secret key between Alice and Bob, both of which would allow for one-time padding the message. But here both the wiretap channel and the shared state are general. In the other extreme case that the state is trivial, we recover the wiretap channel and its private capacity [N. Cai, A. Winter and R. W. Yeung, Probl. Inform. Transm. 40(4):318-336, 2004]. We show how to use the given resource state to build a code for secret classical communication. Our main result is a lower bound on the assisted private capacity, which asymptotically meets the multi-letter converse and which encompasses all sorts of previous results as special cases.
We introduce and analyze an information theoretical task that we call the quantum multiple-access one-time pad. Here, a number of senders initially share a correlated quantum state with a receiver and an eavesdropper. Each sender performs a local operation to encode a classical message and sends their system to the receiver, who subsequently performs a measurement to decode the messages. The receiver will be able to decode the messages almost perfectly, while the eavesdropper must not be able to extract information about the messages even if they have access to the quantum systems transmitted. We consider a “conditional” scenario in which a portion of the receiver’s side information is also accessible to the eavesdropper. We investigate the maximum amount of classical information that can be encoded by each of the senders. We derive a single-letter characterization for the achievable rate region in an asymptotic limit of infinitely many copies and vanishingly small error.
It is a common belief that quantum key distribution systems are the one and only information-theoretically secure physical layer security protocol that enables secure data transmission without a need for the legitimate parties to have any channel knowledge. It is also known that this high security profile results in severe rate restrictions for the parties utilizing the quantum key distribution systems. This observation raises the question of whether quantifying the level of ignorance of the legitimate parties with regard to the channel parameters may enable us to navigate the large gray zone between insecure but highly performant systems on the one side and perfectly secure but highly non-performant systems on the other side. Indeed, by proving a capacity formula for the bosonic compound wiretap channel using the binary phase shift keying alphabet, we are able to quantify in this work exactly how channel uncertainty penalizes data transmission rates.