Abstract In classical information theory, channel capacity quantifies the maximum number of messages that can be reliably transmitted using shared information. An equivalent concept, termed uncommon information, represents the number of messages required to be exchanged to completely share all information in common. However, this equivalence does not extend to quantum information theory. Specifically, quantum uncommon information (QUI) is operationally defined as the minimal amount of entanglement required for the quantum communication task of quantum state exchange (QSE), where two parties exchange quantum states to share all quantum messages in common. Currently, an analytical closed-form expression for the QUI remains undetermined. In this work, by investigating underlying characterization of the QUI, we derive improved bounds on it. To obtain these bounds, we develop a subspace exchange strategy that leverages a common subspace of two parties to identify the unnecessary qubits for exchange. We also consider a referee-assisted exchange, wherein a referee aids two parties in efficiently performing the QSE. Our bounds provide more precise estimations for the QUI. Furthermore, we demonstrate that the subspace technique is a versatile tool for characterizing uncommon information not only in the bipartite scenario but also in various multi-partite ones.
Abstract In classical information theory, channel capacity quantifies the maximum number of messages that can be reliably transmitted using shared information. An equivalent concept, termed uncommon information, represents the number of messages required to be exchanged to completely share all information in common. However, this equivalence does not extend to quantum information theory. Specifically, quantum uncommon information (QUI) is operationally defined as the minimal amount of entanglement required for the quantum communication task of quantum state exchange (QSE), where two parties exchange quantum states to share all quantum messages in common. Currently, an analytical closed-form expression for the QUI remains undetermined. In this work, by investigating underlying characterization of the QUI, we derive improved bounds on it. To obtain these bounds, we develop a subspace exchange strategy that leverages a common subspace of two parties to identify the unnecessary qubits for exchange. We also consider a referee-assisted exchange, wherein a referee aids two parties in efficiently performing the QSE. Our bounds provide more precise estimations for the QUI. Furthermore, we demonstrate that the subspace technique is a versatile tool for characterizing uncommon information not only in the bipartite scenario but also in various multi-partite ones.
If two parties share an unknown quantum state, one can ask how much quantum communication is needed for party A to send her share to party B. Recently, it was found that the number of qubits which should be sent is given by the conditional entropy. This quantifies the notion of partial information, and it can even be negative. Here, we not only demand that A send her state to B, but additionally, B should send his state to A. Paradoxically, we find that requiring that the parties perform this additional task can lower the amount of quantum communication required. This primitive, which we call quantum state exchange, can be used to quantify the notion of uncommon information, since the two parties only need to send each other the parts of their state they don't hold in common. In the classical case, the concept of uncommon information follows trivially from the concept of partial information. We find that for quantum states, this is not so. We prove upper and lower bounds for the uncommon information and find optimal protocols for several classes of states.
We consider communication between two parties using a bipartite quantum operation, which constitutes the most general quantum mechanical model of two-party communication. We primarily focus on the simultaneous forward and backward communication of classical messages. For the case in which the two parties share unlimited prior entanglement, we give inner and outer bounds on the achievable rate region that generalize classical results due to Shannon. In particular, using a protocol of Bennett, Harrow, Leung, and Smolin, we give a one-shot expression in terms of the Holevo information for the entanglement-assisted one-way capacity of a two-way quantum channel. As applications, we rederive two known additivity results for one-way channel capacities: the entanglement-assisted capacity of a general one-way channel, and the unassisted capacity of an entanglement-breaking one-way channel. 1.
We consider communication between two parties using a bipartite quantum operation, which constitutes the most general quantum mechanical model of two-party communication. We primarily focus on the simultaneous forward and backward communication of classical messages. For the case in which the two parties share unlimited prior entanglement, we give inner and outer bounds on the achievable rate region that generalize classical results due to Shannon. In particular, using a protocol of Bennett, Harrow, Leung, and Smolin, we give a one-shot expression in terms of the Holevo information for the entanglement-assisted one-way capacity of a two-way quantum channel. As applications, we rederive two known additivity results for one-way channel capacities: the entanglement-assisted capacity of a general one-way channel, and the unassisted capacity of an entanglement-breaking one-way channel.
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols: the two parties may be given some number of quantum bits jointly prepared in a fixed superposition, prior to communicating with each other. The quantum channel is then said to be "entanglement-assisted." Shared randomness does not help in the transmission of information from one party to another. Moreover, it does not significantly reduce the classical complexity of computing functions vis-a-vis private-coin protocols. On the other hand, prior entanglement leads to startling phenomena such as "quantum teleportation" and "superdense coding." The problem of characterising the power of prior entanglement has baffled many researchers, especially in the setting of bounded-error protocols. It is open whether it leads to more than a factor of two savings (using superdense coding) or more than an additive O(log) savings (when used to create shared randomness). Few lower bounds are known for communication problems in this setting, and are all derived using sophisticated information theoretic techniques. In this paper, we focus on the most basic problem in the setting of communication over an entanglement-assisted quantum channel, that of communicating classical bits from one party to another. We derive optimal bounds on the number of quantum bits required for this task, for any given probability of error.
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as "quantum teleportation" and "superdense coding."The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as "quantum teleportation" and "superdense coding."The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.