Assume Alice and Bob share some bipartite d-dimensional quantum state. As is well known, by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained classically. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.Assume Alice and Bob share some bipartite d-dimensional quantum state. As is well known, by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained classically. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.

Annual Symposium on Foundations of Computer ScienceOct 1, 20072 citations

Simulating Quantum Correlations with Finite Communication

Oded Regev, Ben Toner

Assume Alice and Bob share some bipartite d-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.Assume Alice and Bob share some bipartite d-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.

SIAM Journal on ComputingDec 4, 200936 citations

Quantum communication and Bell nonlocality require infinite classical communication to simulate

Carlos de Gois, Thyago S. R. Santos, Carlos Vieira

A quantum system of any fixed dimension can be prepared in a continuum of states, yet it cannot be used to transmit an unlimited amount of classical information. Similarly, the correlations observed between measurement outcomes on separate parts of a shared quantum system can be stronger than classical correlations, but they cannot transmit information. These fundamental limitations suggest that the statistics observed from quantum communication and quantum correlations may admit a simulation using a finite amount of classical communication. This expectation is confirmed in the smallest nontrivial quantum dimension, with two classical bits being necessary and sufficient to exactly simulate qubit communication and all correlations between qubits. Despite significant efforts during the previous decades, this remained the only solved case. Here we resolve both problems for every quantum dimension. The solution reveals an unexpected qualitative transition starting at dimension four: no finite amount of classical communication can exactly simulate ququart communication nor all quantum correlations of two entangled ququarts, even with unlimited shared randomness. One might have expected this transition, if it existed, to appear already for qutrits. Instead, we construct an explicit protocol that exactly simulates qutrit communication using $357$ classical bits, and consequently, all correlations of two entangled qutrits.A quantum system of any fixed dimension can be prepared in a continuum of states, yet it cannot be used to transmit an unlimited amount of classical information. Similarly, the correlations observed between measurement outcomes on separate parts of a shared quantum system can be stronger than classical correlations, but they cannot transmit information. These fundamental limitations suggest that the statistics observed from quantum communication and quantum correlations may admit a simulation using a finite amount of classical communication. This expectation is confirmed in the smallest nontrivial quantum dimension, with two classical bits being necessary and sufficient to exactly simulate qubit communication and all correlations between qubits. Despite significant efforts during the previous decades, this remained the only solved case. Here we resolve both problems for every quantum dimension. The solution reveals an unexpected qualitative transition starting at dimension four: no finite amount of classical communication can exactly simulate ququart communication nor all quantum correlations of two entangled ququarts, even with unlimited shared randomness. One might have expected this transition, if it existed, to appear already for qutrits. Instead, we construct an explicit protocol that exactly simulates qutrit communication using $357$ classical bits, and consequently, all correlations of two entangled qutrits.

quant-pharXiv:2609.04182Sep 4, 20260 citations

Simulating Quantum Correlations with Finite Communication

Oded Regev, Ben Toner

Assume Alice and Bob share some bipartite $d$ -dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension $d$ . Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.Assume Alice and Bob share some bipartite $d$-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension $d$. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.

arXiv:0708.0827Aug 7, 200712 citations

Lower bound on the communication cost of simulating bipartite quantum correlations

Tamás Vértesi, Erika Bene

Suppose Alice and Bob share a maximally entangled state of any finite dimension and each perform two-outcome measurements on the respective part of the state. It is known, due to the recent result of Regev and Toner, that if a classical model is augmented with two bits of communication, then all the quantum correlations arising from these measurements can be reproduced. Here, we show that two bits of communication are in fact necessary for the perfect simulation. In particular, we prove that a pair of maximally entangled four-dimensional quantum systems cannot be simulated by a classical model augmented by only one bit of communication.Suppose Alice and Bob share a maximally entangled state of any finite dimension and each perform two-outcome measurements on the respective part of the state. It is known, due to the recent result of Regev and Toner, that if a classical model is augmented with two bits of communication, then all the quantum correlations arising from these measurements can be reproduced. Here, we show that two bits of communication are in fact necessary for the perfect simulation. In particular, we prove that a pair of maximally entangled four-dimensional quantum systems cannot be simulated by a classical model augmented by only one bit of communication.

arXiv:0904.1390Physical Review ADec 8, 200915 citations

Correlations in Entanglement-Assisted Prepare-and-Measure Scenarios

Armin Tavakoli, Jef Pauwels, Erik Woodhead +1 more

We investigate the correlations that can arise between Alice and Bob in prepare-and-measure communication scenarios where the source (Alice) and the measurement device (Bob) can share prior entanglement.The paradigmatic example of such a scenario is the quantum dense-coding protocol, where the communication capacity of a qudit can be doubled if a two-qudit entangled state is shared between Alice and Bob.We provide examples of correlations that actually require more general protocols based on higher-dimensional entangled states.This motivates us to investigate the set of correlations that can be obtained from communicating either a classical or a quantum d-dimensional system in the presence of an unlimited amount of entanglement.We show how such correlations can be characterized by a hierarchy of semidefinite programming relaxations by reducing the problem to a noncommutative polynomial optimization problem.We also introduce an alternative relaxation hierarchy based on the notion of informationally restricted quantum correlations, which, though it represents a strict (nonconverging) relaxation scheme, is less computationally demanding.As an application, we introduce device-independent tests of the dimension of classical and quantum systems that, in contrast to previous results, do not make the implicit assumption that Alice and Bob share no entanglement.We also establish several relations between communication with and without entanglement as resources for creating correlations.We investigate the correlations that can arise between Alice and Bob in prepare-and-measure communication scenarios where the source (Alice) and the measurement device (Bob) can share prior entanglement.The paradigmatic example of such a scenario is the quantum dense-coding protocol, where the communication capacity of a qudit can be doubled if a two-qudit entangled state is shared between Alice and Bob.We provide examples of correlations that actually require more general protocols based on higher-dimensional entangled states.This motivates us to investigate the set of correlations that can be obtained from communicating either a classical or a quantum d-dimensional system in the presence of an unlimited amount of entanglement.We show how such correlations can be characterized by a hierarchy of semidefinite programming relaxations by reducing the problem to a noncommutative polynomial optimization problem.We also introduce an alternative relaxation hierarchy based on the notion of informationally restricted quantum correlations, which, though it represents a strict (nonconverging) relaxation scheme, is less computationally demanding.As an application, we introduce device-independent tests of the dimension of classical and quantum systems that, in contrast to previous results, do not make the implicit assumption that Alice and Bob share no entanglement.We also establish several relations between communication with and without entanglement as resources for creating correlations.

arXiv:2103.10748PRX QuantumDec 22, 202141 citations

357

classical bits, and consequently, all correlations of two entangled qutrits.

Assume Alice and Bob share some bipartite

d

-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension

d

. Our protocol and analysis are based on a power series method, resembling Krivine's bound on Grothendieck's constant, and on the computation of volumes of spherical tetrahedra.