Abstract
Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially with the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given bound and is restricted to classical communication.
Highlights
Bob share, in addition to the state ρAB to be measured, an auxiliary entangled state ηA B
Two spacelike separated observers Alice (A) and Bob (B) sharing a bipartite system AB aim to determine a certain non-local property of their joint state ρAB = ρAB(t0) at a specific time t0. This property is described by a non-local positive operator valued measure (POVM)
We prove the security of certain protocols assuming that the adversarial players have less entanglement than a given linear bound and are restricted to classical communication
Summary
It is instructive to see that while teleportation is not directly applicable in the setting of instantaneous measurement, it can be used advantageously. We show that in order to implement a bipartite unitary U = UAB applied to a state | = | AB ∈ (C2)⊗n ⊗ (C2)⊗n, it suffices to provide a protocol P with the following properties: it proceeds by application of local operations only (no communication is allowed) and after completion of the protocol, Bob holds the state σsU | in one of his registers, and Alice and Bob have classical information α and β (measurement outcomes) that together determine s = s(α, β) ∈ {0, 1, 2, 3}2n. At the end of this protocol, Alice and Bob share U | in A : B2
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