We show that quantum entanglement can be used as a substitute for communication when the goal is to compute a function whose input data are distributed among remote parties. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables one of them to learn the value of the function with only two bits of communication occurring among the parties, whereas, without quantum entanglement, three bits of communication are necessary. This result contrasts the well-known fact that quantum entanglement cannot be used to simulate communication among remote parties.
We show that quantum entanglement can be used as a substitute for communication when the goal is to compute a function whose input data are distributed among remote parties. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables one of them to learn the value of the function with only two bits of communication occurring among the parties, whereas, without quantum entanglement, three bits of communication are necessary. This result contrasts the well-known fact that quantum entanglement cannot be used to simulate communication among remote parties.
We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles in an entangled quantum state. We show that, although a prior quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the function with only three bits of communication occurring among the parties, whereas, without quantum entanglement, four bits of communication are necessary. We also show that, for a particular two-party probabilistic communication complexity problem, quantum entanglement results in less communication than is required with only classical random correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate communication among remote parties.
Quantum entanglement cannot be used to achieve direct communication between remote parties, but it can reduce the communication needed for some problems. Let each of k parties hold some partial input data to some fixed k-variable function f. The communication complexity of f is the minimum number of classical bits required to be broadcasted for every party to know the value of f on their inputs. We construct a function G such that for the one-round communication model and three parties, G can be computed with n+1 bits of communication when the parties share prior entanglement. We then show that without entangled particles, the one-round communication complexity of G is (3/2)n+1. Next we generalize this function to a function F. We show that if the parties share prior quantum entanglement, then the communication complexity of F is exactly k. We also show that, if no entangled particles are provided, then the communication complexity of F is roughly klog2k. These two results prove communication complexity separations better than a constant number of bits.
We consider a variation of the communication complexity scenario, where the parties are supplied with an extra resource: particles in an entangled quantum state. We note that "quantum nonlocality" can be naturally expressed in the language of communication complexity. These are communication complexity problems where the "output" is embodied in the correlations between the outputs of the individual parties. Without entanglement, the parties must communicate to produce the required correlations; whereas, with entanglement, no communication is necessary to produce the correlations. In this sense, nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication. We show how to transform examples of nonlocality into more traditional communication complexity problems, where the output is explicitly determined by each individual party. The resulting problems require communication with or without entanglement, but the required communication is less when entanglement is available. All these results are a noteworthy contrast to the well-known fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties.
Quantum computing resources have been known to provide speed-ups in computational complexity in many algorithms. The impact of these resources in communication, however, has not attracted much attention. We investigate the impact of \nquantum entanglement on communication complexity. We provide a positive result, \nby presenting a class of multi-party communication problems wherein the presence of \na suitable quantum entanglement lowers the classical communication complexity. We \nshow that, in evaluating certains function whose parameters are distributed among \nvarious parties, the presence of prior entanglement can help in reducing the required \ncommunication. We also present an outline of realizing the required entanglement \nthrough optical photon quantum computing. We also suggest the possible impact of \nour results on network information flow problems, by showing an instance of a lower \nbound which can be broken by adding limited power to the communication model.
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.