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arXiv:1902.07699 · Leibniz international proceedings in informatics

Universality of EPR Pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion

Matthew Coudron, Aram W. Harrow

Published Jan 1, 2019• 1 citation
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Abstract

Entanglement assistance is known to reduce the quantum communication complexity of evaluating functions with distributed inputs. But does the type of entanglement matter, or are EPR pairs always sufficient? This is a natural question because in several other settings maximally entangled states are known to be less useful as a resource than some partially entangled state. These include non-local games, tasks with quantum communication between players and referee, and simulating bipartite unitaries or communication channels. By contrast, we prove that the bounded-error entanglement-assisted quantum communication complexity of a function cannot be improved by more than a constant factor by replacing maximally entangled states with arbitrary entangled states. In particular, we show that every quantum communication protocol using $Q$ qubits of communication and arbitrary shared entanglement can be $ε$-approximated by a protocol using $O(Q/ε+\log(1/ε)/ε)$ qubits of communication and only EPR pairs as shared entanglement. Our second result concerns an old question in quantum information theory: How much quantum communication is required to approximately convert one pure bipartite entangled state into another? We show that the communication cost of converting between two bipartite quantum states is upper bounded, up to a constant multiplicative factor, by a natural and efficiently computable quantity which we call the $\ell_{\infty}$-Earth Mover's Distance (EMD) between those two states. Furthermore, we prove a complementary lower bound on the cost of state conversion by the $ε$-smoothed $\ell_{\infty}$-EMD, which is a natural smoothing of the $\ell_{\infty}$-EMD that we will define via a connection with optimal transport theory.

DOI
10.4230/lipics.ccc.2019.20

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We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.
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It is known that general convertibility of bipartite entangled states is not possible to arbitrary error without some classical communication. While some tradeoffs between communication cost and conversion error have been proven, these bounds can be very loose. In particular, there are many cases in which tolerable error might be achievable using zero-communication protocols. In this work we address these cases by deriving the optimal fidelity of pure quantum state conversions under local unitaries as well as local operations and shared randomness. We also use these results to explore catalytic conversions between pure states using zero communication.
Distributed quantum computation connects multiple quantum processing units (QPUs) through quantum communication to jointly perform large-scale quantum computations. Since the number of qubits available at each QPU is limited, it is important to reduce quantum communication while keeping the qubit overhead small. To this end, we study an entanglement-assisted model, in which entanglement consumption serves as a measure of quantum communication cost. For exact deterministic implementations in this model, the operator Schmidt rank of the target bipartite unitary provides a general lower bound on entanglement cost when qubit overhead is unrestricted. However, it has remained unclear how closely this bound can be approached with constant qubit overhead. In this work, we show that this lower bound is attainable for every bipartite Clifford unitary using at most two auxiliary qubits per QPU. In addition, for non-Clifford unitaries specified by exact Clifford+TTT decompositions with TTT-count ttt, we construct implementations with the same qubit overhead whose entanglement cost is at most 2t2t2t Bell pairs above this lower bound.