We revisit the problem of conveying classical messages by transmitting quantum states, and derive new, optimal bounds on the number of quantum bits required for this task. Much of the previous work on this problem, and on other communication tasks in the setting of bounded error entanglement-assisted communication, is based on sophisticated information theoretic arguments. Our results are derived from first principles, using a simple linear algebraic technique. A direct consequence is a tight lower bound for the Inner Product function that has found applications to privacy amplification in quantum key distribution protocols.
We revisit the problem of conveying classical messages by transmitting quantum states, and derive new, optimal bounds on the number of quantum bits required for this task. Much of the previous work on this problem, and on other communication tasks in the setting of bounded error entanglement-assisted communication, is based on sophisticated information theoretic arguments. Our results are derived from first principles, using a simple linear algebraic technique. A direct consequence is a tight lower bound for the Inner Product function that has found applications to privacy amplification in quantum key distribution protocols.
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.
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.
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.
Consider a two-party quantum communication protocol for computing some function f : {0, 1}/sup n/ /spl times/ {0, 1}/sup n/ /spl rarr/ Z. We show that the first message of P can be compressed to 0(k) classical bits using prior entanglement if it carries at most k bits of information about the sender's input. This implies a general direct sum result for one-round and simultaneous quantum protocols. It also implies a new round elimination lemma in quantum communication, which allows us to extend recent classical lower bounds on the cell probe complexity of some data structure problems, e.g. approximate nearest neighbor searching on the Hamming cube {0, 1}/sup n/, to the quantum setting. We then show an optimal tradeoff between the privacy losses of Alice and Bob in computing f in terms of the one-round quantum communication complexity of f with prior entanglement. This tradeoff is independent of the number of rounds of communication. The above message compression and privacy tradeoff results use a lot of qubits of prior entanglement, leading one to wonder how much prior entanglement is really required by a quantum protocol. We show that Newman's [1991] technique of reducing the number of public coins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of black-box reduction of prior entanglement that subsumes Newman's technique. Intuitively, a black-box reduction does not change the unitary transforms of Alice and Bob; it only decreases the amount of entanglement of the prior entangled state. We prove that such a black-box reduction is impossible for quantum protocols by exhibiting a particular one-round quantum protocol for the equality function where the black-box technique fails to reduce the amount of prior entanglement by more than a constant factor.