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 O ( m 2 ) bits in order to simulate exactly the outcomes that these measurements would have produced on those systems.
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 O ( m 2 ) bits in order to simulate exactly the outcomes that these measurements would have produced on those systems.
John Bell has shown that the correlations entailed by quantum mechanics cannot be reproduced by a classical process involving non-communicating parties. But can they be simulated with the help of bounded communication? This problem has been studied for more than two decades and it is now well understood in the case of bipartite entanglement. However, the issue was still widely open for multipartite entanglement, even for the simplest case, which is the tripartite Greenberger-Horne-Zeilinger (GHZ) state. We give an exact simulation of arbitrary independent von Neumann measurements on general n-partite GHZ states. Our protocol requires O(n^2) bits of expected communication between the parties, and O(n log n) expected time is sufficient to carry it out in parallel. Furthermore, we need only an expectation of O(n) independent unbiased random bits, with no need for the generation of continuous real random variables nor prior shared random variables. In the case of equatorial measurements, we improve on the prior art with a protocol that needs only O(n log n) bits of communication and O(log^2 n) parallel time. At the cost of a slight increase in the number of bits communicated, these tasks can be accomplished with a constant expected number of rounds.
Let H be a finite dimensional Hilbert space and ρ, σ ∈ D(H) be quantum states in H \nsuch that S(ρ||σ) is finite. In this thesis, we consider the following communication task involving two parties Alice and Bob. Suppose that Alice is given a classical description of the states ρ and σ. Given an unlimited number of copies of an entangled state whose marginal on Bob’s side is σ, Alice’s goal is to help Bob output a single copy of the state ρ by sending a single message to Bob in a one-way LOCC (short for local operation and classical communication) protocol. We propose a class of one-way LOCC protocols for this task which we refer to as quantum rejection sampling protocols. Inspired by the classical rejection sampling protocol of Harsha, Jain, McAllester, and Radhakrishnan [25] for a similar classical communication task, we introduce the Greedy Quantum Rejection Sampler. We characterize the expected communication cost of the protocol in terms of max-relative entropy of ρ with respect to σ, in the case where the state ρ is a pure state and prove that the Greedy Quantum Rejection Sampler is an optimal quantum rejection sampling protocol in this case. We describe an optimal quantum rejection sampling protocol in terms of a semidefinite program and we find general lower bounds and upper bounds on the expected communication cost of the optimal protocol. We propose an LOCC compression protocol based on the Greedy Quantum Rejection Sampler protocol, for lossless compression of an arbitrary pure state quantum source in the visible compression model and we show an upper bound on the average length of this encoding. The upper bound is always less than or equal to the Shannon entropy of the quantum source and the gap between the two quantities can be arbitrary large. Finally, we introduce a high-entanglement deterministic exact protocol for remote preparation of an arbitrary ensemble of quantum states. Our protocol is based on a quantum rejection sampling protocol which uses a prefix-free encoding for communication of the messages. We establish an upper bound on the expected communication cost of this protocol for the worst case choice of the target state in terms of the max-information in Bob’s output register at the end of the protocol about Alice’s classical input register. Furthermore, this protocol can be used as a non-oblivious universal protocol for exact remote preparation of an arbitrary d-dimensional state at an expected communication cost of at most lg(d) +O (lg(lg(d))).
John Bell has shown that the correlations entailed by quantum mechanics cannot be reproduced by a classical process involving non-communicating parties. But can they be simulated with the help of bounded communication? This problem has been studied for more than twenty years and it is now well understood in the case of bipartite entanglement. However, the issue was still widely open for multipartite entanglement, even for the simplest case, which is the tripartite Greenberger-Horne-Zeilinger (GHZ) state. We give an exact simulation of arbitrary independent von Neumann measurements on general n-partite GHZ states. Our protocol requires O(n^2) bits of expected communication between the parties, and O(n*log(n)) expected time is sufficient to carry it out in parallel. Furthermore, we need only an expectation of O(n) independent unbiased random bits, with no need for the generation of continuous real random variables nor prior shared random variables. In the case of equatorial measurements, we improve earlier results with a protocol that needs only O(n*log(n)) bits of communication and O(log^2(n)) parallel time. At the cost of a slight increase in the number of bits communicated, these tasks can be accomplished with a constant expected number of rounds.
John Bell has shown that the correlations entailed by quantum mechanics cannot be reproduced by a classical process involving non-communicating parties. But can they be simulated with the help of bounded communication? This problem has been studied for more than two decades, and it is now well understood in the case of bipartite entanglement. However, the issue was still widely open for multipartite entanglement, even for the simplest case, which is the tripartite Greenberger- Horne-Zeilinger (GHZ) state. We give an exact simulation of arbitrary independent von Neumann measurements on general n-partite GHZ states. Our protocol requires O(n2) bits of expected communication between the parties, and O(n log n) expected time is sufficient to carry it out in parallel. Furthermore, we need only an expectation of O(n) independent unbiased random bits, with no need for the generation of continuous real random variables nor prior shared random variables. In the case of equatorial measurements, we improve on the prior art with a protocol that needs only O(n log n) bits of communication and O(log2n) parallel time. At the cost of a slight increase in the number of bits communicated, these tasks can be accomplished with a constant expected number of rounds.
For two non-communicating parties, quantum theory can give rise to probability distributions of outcomes that no local classical model can reproduce without communication. However, in the case of two-dimensional systems ([Formula: see text]), it is known that allowing a finite amount of classical communication in addition to shared classical resources makes it possible to simulate these quantum correlations. Whether such a simulation remains possible in higher dimensions is still an open question. In this work, we identify the key features of the exact classical protocol in [Formula: see text], and use them to construct robust approximate protocols in higher dimensions. We assess their performance through a randomized numerical study based on the Total Variation Distance. Our approach exactly reproduces the quantum probability distributions for [Formula: see text], and performs very well compared to existing protocols for higher dimensions, being the most robust protocol in all cases studied. These results offer new insights into the analytical structure of classical protocols in higher dimensions.
Bell's theorem implies that the outcomes of local measurements on two maximally entangled systems cannot be simulated without classical communication between the parties. The communication cost is finite for n Bell states, but it grows exponentially in n. Three simple protocols are presented that provide approximate simulations for low-dimensional entangled systems and require a linearly growing amount of communication. We have tested them by performing some simulations for a family of measurements. The maximal error is less than 1 in three dimensions and grows sublinearly with the number of entangled bits in the range numerically tested. One protocol is the multidimensional generalization of the exact Toner-Bacon [Phys. Rev. Lett. 91, 187904 (2003)] model for a single Bell state. The other two protocols are generalizations of an alternative exact model, which we derive from the Kochen-Specker [J. Math. Mech. 17, 59 (1967)] scheme for simulating single-qubit measurements. These protocols can give some indication for finding optimal one-way communication protocols that classically simulate entanglement and quantum channels. Furthermore they can be useful for deciding if a quantum communication protocol provides an advantage on classical protocols.