Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1107.4647 · Physical Review A

Approximate simulation of entanglement with a linear cost of communication

A. Montina

Published Oct 5, 2011• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.84.042307

Related papers

Classical simulation of quantum entanglement without local hidden variables

Serge Massar, Dave Bacon, Nicolas J. Cerf +1 more

Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when the communication is merely required to be finite on average, Bell states can be simulated without any local hidden variables. More generally, we show that arbitrary positive operator valued measurements on systems of n Bell states can be simulated with ${O(n2}^{n})$ bits of communication on average (again, without local hidden variables). On the other hand, when the communication is required to be absolutely bounded, we show that a finite number of bits of local hidden variables is insufficient to simulate a Bell state. This latter result is based on an analysis of the nondeterministic communication complexity of the NOT-EQUAL function, which is constant in the quantum model and logarithmic in the classical model.

arXiv:quant-ph/0009088•Physical Review A•Apr 16, 2001•82 citations

Classical Cost of Transmitting a Qubit

Martin J. Renner, Armin Tavakoli, Marco Túlio Quintino

We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state.

arXiv:2207.02244•Physical Review Letters•Mar 22, 2023•16 citations

The communication cost of simulating POVMs over maximally entangled qubits

Ricard Ravell Rodríguez

In [Toner and Bacon, Phys. Rev. Lett. 91, 187904 (2003)], $1$ bit of communication was proven to be enough to simulate the statistics of local projective measurements over the maximally entangled state. Ever since then, the question of whether $1$ bit is also enough for the case of generalized measurements has been open. In this thesis, we retort to inefficiency-resistant Bell functionals, a powerful technique to prove lower bounds communication complexity, to numerically study this question. The results obtained suggest that, indeed, as is the case with projective measurements, $1$ bit of communication suffices to simulate POVMs over maximally entangled qubits.

•Sep 9, 2019•0 citations

The minimal communication cost for simulating entangled qubits

Martin J. Renner, Marco Túlio Quintino

We analyze the amount of classical communication required to reproduce the statistics of local projective measurements on a general pair of entangled qubits, |ΨAB⟩=p |00⟩+1−p |11⟩ (with 1/2≤p≤1 ). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when 2p(1−p)2p−1log⁡(p1−p)+2(1−p)≤1 , approximately 0.835≤p≤1 , we present a classical protocol that requires only a single bit of communication. The latter model even allows a perfect classical simulation with an average communication cost that approaches zero in the limit where the degree of entanglement approaches zero ( p→1 ). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.

arXiv:2207.12457•Quantum•Oct 24, 2023•5 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.

quant-pharXiv:2609.04182•Sep 4, 2026•0 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.

arXiv:0904.1390•Physical Review A•Dec 8, 2009•15 citations
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 nnn Bell states, but it grows exponentially in nnn. 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 11%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.
Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when the communication is merely required to be finite on average, Bell states can be simulated without any local hidden variables. More generally, we show that arbitrary positive operator valued measurements on systems of n Bell states can be simulated with O(n2n){O(n2}^{n})O(n2n) bits of communication on average (again, without local hidden variables). On the other hand, when the communication is required to be absolutely bounded, we show that a finite number of bits of local hidden variables is insufficient to simulate a Bell state. This latter result is based on an analysis of the nondeterministic communication complexity of the NOT-EQUAL function, which is constant in the quantum model and logarithmic in the classical model.
We consider general prepare-and-measure scenarios in which Alice can transmit qubit states to Bob, who can perform general measurements in the form of positive operator-valued measures (POVMs). We show that the statistics obtained in any such quantum protocol can be simulated by the purely classical means of shared randomness and two bits of communication. Furthermore, we prove that two bits of communication is the minimal cost of a perfect classical simulation. In addition, we apply our methods to Bell scenarios, which extends the well-known Toner and Bacon protocol. In particular, two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state.
In [Toner and Bacon, Phys. Rev. Lett. 91, 187904 (2003)], 111 bit of communication was proven to be enough to simulate the statistics of local projective measurements over the maximally entangled state. Ever since then, the question of whether 111 bit is also enough for the case of generalized measurements has been open. In this thesis, we retort to inefficiency-resistant Bell functionals, a powerful technique to prove lower bounds communication complexity, to numerically study this question. The results obtained suggest that, indeed, as is the case with projective measurements, 111 bit of communication suffices to simulate POVMs over maximally entangled qubits.
We analyze the amount of classical communication required to reproduce the statistics of local projective measurements on a general pair of entangled qubits, |ΨAB⟩=p |00⟩+1−p |11⟩ (with 1/2≤p≤1 ). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when 2p(1−p)2p−1log⁡(p1−p)+2(1−p)≤1 , approximately 0.835≤p≤1 , we present a classical protocol that requires only a single bit of communication. The latter model even allows a perfect classical simulation with an average communication cost that approaches zero in the limit where the degree of entanglement approaches zero ( p→1 ). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.
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 357357357 classical bits, and consequently, all correlations of two entangled qutrits.
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.