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.
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.
We show optimal Direct Sum result for the one-way entanglement-assisted quantum com- munication complexity for any relation fX × Y × Z. We show: Q 1,pub (f ⊕m ) = (m · Q 1,pub (f)), where Q 1,pub (f), represents the one-way entanglement-assisted quantum communication complexity of f with error at most 1/3 and f ⊕m represents m-copies of f. Similarly for the one-way public-coin classical communication complexity we show: R 1,pub (f ⊕m ) = (m · R 1,pub (f)), where R 1,pub (f), represents the one-way public-coin classical communication complexity of f with error at most 1/3. We show similar optimal Direct Sum results for the Simultaneous Message Passing (SMP) quantum and classical models. For two-party two-way protocols we present optimal Privacy Trade-off results leading to a Weak Direct Sum result for such protocols. We show our Direct Sum and Privacy Trade-off results via message compression arguments. These arguments also imply a new round elimination lemma in quantum communication, which allows us to extend classical lower bounds on the cell probe complexity of some data structure problems, e.g. Approximate Nearest Neighbor Searching (ANN) on the Hamming cube {0,1} n and Predecessor Search to the quantum setting. In a separate result we show that Newman's (New91) 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. 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.
We show optimal Direct Sum result for the one-way entanglement-assisted quantum communication complexity for any relation f subset of X x Y x Z. We show: Q^{1,pub}(f^m) = Omega(m Q^{1,pub}(f)), where Q^{1,pub}(f), represents the one-way entanglement-assisted quantum communication complexity of f with error at most 1/3 and f^m represents m-copies of f. Similarly for the one-way public-coin classical communication complexity we show: R^{1,pub}(f^m) = Omega(m R^{1,pub}(f)), where R^{1,pub}(f), represents the one-way public-coin classical communication complexity of f with error at most 1/3. We show similar optimal Direct Sum results for the Simultaneous Message Passing quantum and classical models. For two-way protocols we present optimal Privacy Trade-off results leading to a Weak Direct Sum result for such protocols. We show our Direct Sum and Privacy Trade-off results via message compression arguments which also imply a new round elimination lemma in quantum communication. This allows us to extend 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}^n and Predecessor Search to the quantum setting. In a separate result we show that Newman's 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. We prove that such a black-box reduction is impossible for quantum protocols. In the final result in the theme of message compression, we provide an upper bound on the problem of Exact Remote State Preparation.
The private simultaneous messages model is a non-interactive version of the multiparty secure computation, which has been intensively studied to examine the communication cost of the secure computation. We consider its quantum counterpart, the private simultaneous quantum messages (PSQM) model, and examine the advantages of quantum communication and prior entanglement of this model. In the PSQM model, k parties P1,…,Pk initially share a common random string (or entangled states in a stronger setting), and they have private classical inputs x1,…,xk. Every Pi generates a quantum message from the private input xi and the shared random string (entangled states), and then sends it to the referee R. Receiving the messages, R computes F(x1,…,xk). Then, R learns nothing except for F(x1,…,xk) as the privacy condition. We obtain the following results for this PSQM model. (1) We demonstrate that the privacy condition inevitably increases the communication cost in the two-party PSQM model as well as in the classical case presented by Applebaum, Holenstein, Mishra, and Shayevitz. In particular, we prove a lower bound (3−o(1))n of the communication complexity in two-party PSQM protocols with a shared random string for random Boolean functions of 2n-bit input, which is larger than the trivial upper bound 2n of the communication complexity without the privacy condition. (2) We demonstrate a factor two gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings by designing a multiparty PSQM protocol with shared entangled states for a total function that extends the two-party equality function. (3) We demonstrate an exponential gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings for a two-party partial function.
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.
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.
Private simultaneous quantum messages (PSQM) provide a minimal model of noninteractive secure multiparty quantum computation, yet their communication complexity for concrete functions remains poorly understood. We show that familiar total functions already exhibit both genuine quantum advantages and strong structural limitations. For three-party equality on n-bit inputs, which determines whether the players' n-bit strings are identical, we construct an entanglement-assisted protocol with asymptotic communication (1.633…)n qubits, whereas every perfectly correct protocol without prior entanglement requires at least 3n qubits. For three-bit majority, one shared GHZ state and one transmitted qubit per player suffice for perfect correctness and perfect privacy, reducing the communication from the optimal classical cost of six bits to three qubits. Three-bit AND behaves differently. No perfectly correct and perfectly private protocol exists when the players share an arbitrary three-qubit state, apply arbitrary local unitary encodings, and transmit one qubit each. However, increasing the local dimension overcomes this obstruction: one shared three-qutrit GHZ state and one transmitted qutrit per player yield a one-shot protocol with total communication 3log23. Together with the six-bit classical lower bound, this gives a strict quantum communication advantage. More generally, for every k≥3, we construct a perfectly correct and perfectly private one-shot protocol for k-bit AND using one qudit of dimension at most k per player, and hence at most klog2k qubits in total.