We analyse a communication scenario over a particular causal broadcast channel whose state depends on a modulo sum. The receivers of the broadcast receive channel state information and collaborate to determine the channel state as to decode their private messages. Further, the receivers of the broadcast can collude up to the minimum non-collusion condition to determine state information of the other non-colluding receivers. We analyse three resource scenarios for the receivers: receivers can share entanglement without classically communicating, can just use classical communication, or have both entanglement and classical communication. Using results from secure multi-party communication, we find that when the receivers can share entanglement and communicate classically, they can receive messages from the sender at a non-zero rate with verifiable secure collaboration. In the entanglement only case a positive capacity is not possible. In the classical communication case, a non-zero rate of communication is achievable but the communication complexity overhead grows quadratically in the number of receivers versus linear in the number of receivers with entanglement.
We analyse a communication scenario over a particular causal broadcast channel whose state depends on a modulo sum. The receivers of the broadcast receive channel state information and collaborate to determine the channel state as to decode their private messages. Further, the receivers of the broadcast can collude up to the minimum non-collusion condition to determine state information of the other non-colluding receivers. We analyse three resource scenarios for the receivers: receivers can share entanglement without classically communicating, can just use classical communication, or have both entanglement and classical communication. Using results from secure multi-party communication, we find that when the receivers can share entanglement and communicate classically, they can receive messages from the sender at a non-zero rate with verifiable secure collaboration. In the entanglement only case a positive capacity is not possible. In the classical communication case, a non-zero rate of communication is achievable but the communication complexity overhead grows quadratically in the number of receivers versus linear in the number of receivers with entanglement.
We study the effects of quantum entanglement on the performance of two classical zero-error communication tasks among multiple parties. Both tasks are generalizations of the two-party zero-error channel-coding problem, where a sender and a receiver want to perfectly communicate messages through a one-way classical noisy channel. If the two parties are allowed to share entanglement, there are several positive results that show the existence of channels for which they can communicate strictly more than what they could do with classical resources. In the first task, one sender wants to communicate a common message to multiple receivers. We show that if the number of receivers is greater than a certain threshold then entanglement does not allow for an improvement in the communication for any finite number of uses of the channel. On the other hand, when the number of receivers is fixed, we exhibit a class of channels for which entanglement gives an advantage. The second problem we consider features multiple collaborating senders and one receiver. Classically, cooperation among the senders might allow them to communicate on average more messages than the sum of their individual possibilities. We show that whenever a channel allows single-sender entanglement-assisted advantage, then the gain extends also to the multisender case. Furthermore, we show that entanglement allows for a peculiar amplification of information which cannot happen classically, for a fixed number of uses of the channels.
Abstract We consider entanglement-assisted (EA) private communication over a quantum broadcast channel, in which there is a single sender and multiple receivers. We divide the receivers into two sets: the decoding set and the malicious set. The decoding set and the malicious set can either be disjoint or can have a finite intersection. For simplicity, we say that a single party Bob has access to the decoding set and another party Eve has access to the malicious set, and both Eve and Bob have access to the pre-shared entanglement with Alice. The goal of the task is for Alice to communicate classical information reliably to Bob and securely against Eve, and Bob can take advantage of pre-shared entanglement with Alice. In this framework, we establish a lower bound on the one-shot EA private capacity. When there exists a quantum channel mapping the state of the decoding set to the state of the malicious set, such a broadcast channel is said to be degraded. We establish an upper bound on the one-shot EA private capacity in terms of smoothed min- and max-entropies for such channels. In the limit of a large number of independent channel uses, we prove that the EA private capacity of a degraded quantum broadcast channel is given by a single-letter formula. Finally, we consider two specific examples of degraded broadcast channels and find their capacities. In the first example, we consider the scenario in which one part of Bob’s laboratory is compromised by Eve. We show that the capacity for this protocol is given by the conditional quantum mutual information of a quantum broadcast channel, and so we thus provide an operational interpretation to the dynamic counterpart of the conditional quantum mutual information. In the second example, Eve and Bob have access to mutually exclusive sets of outputs of a broadcast channel.
Communication over a quantum broadcast channel with cooperation between the receivers is considered. The first form of cooperation addressed is classical conferencing, where receiver 1 can send classical messages to receiver 2. Another cooperation setting involves quantum conferencing, where receiver 1 can teleport a quantum state to receiver 2. When receiver 1 is not required to recover information and its sole purpose is to help the transmission to receiver 2, the model reduces to the quantum primitive relay channel. The quantum conferencing setting is intimately related to quantum repeaters as the sender, receiver 1, and receiver 2 can be viewed as the transmitter, the repeater, and the destination receiver, respectively. We develop lower and upper bounds on the capacity region in each setting. In particular, the cutset upper bound and the decode-forward lower bound are derived for the primitive relay channel. Furthermore, we present an entanglement-formation lower bound, where a virtual channel is simulated through the conference link. At last, we show that as opposed to the multiple access channel with entangled encoders, entanglement between decoders does not increase the classical communication rates for the broadcast dual.
Communication over a quantum broadcast channel with cooperation between the receivers is considered. The first form of cooperation addressed is classical conferencing. Another cooperation setting involves quantum conferencing, where Receiver 1 can teleport a quantum state to Receiver 2. The conferencing setting is intimately related to quantum repeaters, as the sender, Receiver 1, and Receiver 2 can be viewed as the transmitter, the repeater, and the destination receiver, respectively. We develop lower and upper bounds on the capacity region in each setting. At last, we show that as opposed to the MAC with entangled encoders, entanglement between decoders does not increase the classical communication rates for the broadcast dual.
We investigate the secrecy capacity region of 2-receiver, 1-eavesdropper Broadcast Channel (BC) with two causal states and conferencing decoders. The encoder sends two messages, one of them for both legitimate receivers and the other one for the first legitimate receiver. It keeps these messages secret from the eavesdropper. Each state is causally available at the corresponding receiver, while both states are known causally at the encoder. Each receiver wishes to send the state which is not available at the other receiver, through a noiseless link with limited capacity. We find an inner bound on the secrecy capacity region of this channel. The achievability scheme employs block Markov coding and Shannon strategy for channels with state. In each block, we make use of the states to generate two keys which will be used in the next block. The idea is to utilize the cooperation of both receivers to recover both keys at each receiver while ensuring secrecy. We also establish an outer bound on the secrecy capacity region of the 2-rec, 1-eav. BC with Causal Channel State Information (CCSI) and conferencing. We prove that the proposed bounds coincide for a special case.
We consider a 1-to-K communication scenario, where a source transmits private messages to K receivers through a broadcast erasure channel, and the receivers feedback strictly, causally, and publicly their channel states after each transmission. We explore the achievable rate region when we require that the message to each receiver remains secret-in the information theoretical sense-from all the other receivers. We characterize the capacity of secure communication in all the cases where the capacity of the 1-to-K communication scenario without the requirement of security is known. As a special case, we characterize the secret-message capacity of a single receiver point-to-point erasure channel with public state-feedback in the presence of a passive eavesdropper. We find that in all the cases where we have an exact characterization, we can achieve the capacity using linear complexity two-phase schemes: in the first phase, we create appropriate secret keys, and in the second phase, we use them to encrypt each message. We find that the amount of key we need is smaller than the size of the message, and equal to the amount of encrypted message the potential eavesdroppers jointly collect. Moreover, we prove that a dishonest receiver that provides deceptive feedback cannot diminish the rate experienced by the honest receivers. We also develop a converse proof which reflects the two-phase structure of our achievability scheme. As a side result, our technique leads to a new outer bound proof for the nonsecure communication problem.