Abstract
We show how to sample exactly discrete probability distributions whose defining parameters are distributed among remote parties. For this purpose, von Neumann’s rejection algorithm is turned into a distributed sampling communication protocol. We study the expected number of bits communicated among the parties and also exhibit a trade-off between the number of rounds of the rejection algorithm and the number of bits transmitted in the initial phase. Finally, we apply remote sampling to the simulation of quantum entanglement in its essentially most general form possible, when an arbitrary finite number m of parties share systems of arbitrary finite dimensions on which they apply arbitrary measurements (not restricted to being projective measurements, but restricted to finitely many possible outcomes). In case the dimension of the systems and the number of possible outcomes per party are bounded by a constant, it suffices to communicate an expected bits in order to simulate exactly the outcomes that these measurements would have produced on those systems.
Highlights
Let X be a nonempty finite set containing n elements and p = ( p x ) x∈X be a probability vector parameterized by some vector θ = (θ1, . . . , θm ) ∈ Θm for an integer m ≥ 2
We show how to generate a random vector X = ( X1, . . . , Xm ) ∈ X = X1 × . . . × Xm sampled from the exact joint probability distribution that would be obtained if each custodian i had the ith share of ρ and measured it according to the ith POVM, producing outcome xi ∈ Xi
We have introduced and studied the general problem of sampling a discrete probability distribution characterized by parameters that are scattered in remote locations
Summary
It may seem paradoxical that the leader can sample exactly the probability vector p with a finite expected number of bits sent by the custodians, who may hold continuous parameters that define p This counterintuitive possibility has been known to be achievable for more than a quarter-century in earlier work on the simulation of quantum entanglement by classical communication, starting with Refs. × Xm sampled from the exact joint probability distribution that would be obtained if each custodian i had the ith share of ρ (of dimension di ) and measured it according to the ith POVM, producing outcome xi ∈ Xi. We show how to generate a random vector X = This result subsumes that of Ref. [1] since all di ’s and ni ’s are equal to 2 for projective measurements on individual qubits of the m-partite GHZ state
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