# Communication/resource cost of classically simulating quantum correlations

Thread report for Argus. Date: 2026-09-15.

## Bottom line

For a single singlet/Bell pair with arbitrary local projective measurements, Toner and Bacon give an exact one-bit worst-case classical communication protocol with shared randomness. It reproduces the full joint distribution, not merely the correlator, because it gives zero marginals and the singlet correlator <alpha beta> = -a . b for binary outcomes.

The one bit is optimal in the weak worst-case sense: zero bits are impossible by Bell/CHSH, and any nonzero bounded one-shot bit protocol costs at least one bit. It is not an average-communication lower bound. Average/asymptotic costs can be fractional; Pironio gives sqrt(2)-1 ~= 0.4142 bits necessary and sufficient on average for the correlations that maximally violate CHSH.

The constant-cost miracle does not simply persist for richer tasks. For n Bell pairs with coherent measurements, Brassard-Cleve-Tapp prove an exact worst-case lower bound Omega(2^n). For arbitrary finite-dimensional bipartite states with two-outcome measurements, Regev-Toner give a two-bit exact simulation of the correlator only, and Vertesi-Bene prove one bit is not enough already in local dimension 4. Full joint-distribution simulation beyond qubits remains much less resolved.

## 1. Toner & Bacon central item

### Citation

VERIFIED: B. F. Toner and D. Bacon, "Communication cost of simulating Bell correlations," Phys. Rev. Lett. 91, 187904 (2003), arXiv:quant-ph/0304076, DOI 10.1103/PhysRevLett.91.187904.

### Exact claim

VERIFIED quote from abstract:

> "For the simplest and most important case of local projective measurements on an entangled Bell pair state, we show that exact simulation is possible using local hidden variables augmented by just one bit of classical communication."

VERIFIED quote from the body:

> "Our protocol exactly simulates the quantum mechanical probability distribution for projective measurements on the singlet Bell pair state."

What it covers:

- State: singlet Bell pair, and by local basis changes any maximally entangled two-qubit state.
- Measurements: all local projective spin measurements, parameterized by arbitrary unit Bloch vectors a and b. Not POVMs.
- Communication: exactly one bit from Alice to Bob in each run, i.e. bounded worst-case one-way communication. They also note block compression to about 0.85 bits when many simulations are run in parallel and Alice's randomized measurement vector is uniform; that is not the base theorem.
- Randomness: shared continuous random variables: two independent random unit vectors lambda_1, lambda_2 uniformly distributed over the sphere.
- Exactness: exact, not approximate.
- Distribution: full joint distribution for binary projective singlet measurements. They explicitly prove <alpha> = <beta> = 0 and <alpha beta> = -a . b; for binary +/-1 outcomes this fixes p(alpha,beta) = (1 - alpha beta a . b)/4.

Protocol structure, VERIFIED:

Alice and Bob share independent uniform unit vectors lambda_1, lambda_2 in R^3.

1. Alice outputs alpha = -sgn(a . lambda_1).
2. Alice sends one bit c in {-1,+1}, where c = sgn(a . lambda_1) sgn(a . lambda_2).
3. Bob outputs beta = sgn[b . (lambda_1 + c lambda_2)].

The proof uses the symmetries of the shared vectors to get zero marginals and evaluates the integral to get <alpha beta> = -a . b.

Important Toner-Bacon context, VERIFIED quote:

> "Brassard et al. take the cost to be the number of bits sent in the worst case; Steiner, the average."

and

> "Surprisingly, the only lower bound for the amount of communication is given by Bell's theorem: at least some communication is needed. Here we present a simple protocol that uses just one bit of communication."

## 2. Lower bounds and optimality

### Bell / CHSH: communication is necessary

VERIFIED via Toner-Bacon and Pironio. Bell's theorem/CHSH imply zero communication plus shared randomness cannot reproduce singlet correlations. For the one-shot worst-case integer-bit resource used by Toner-Bacon, this makes one bit optimal: zero fails, one suffices.

### Brassard-Cleve-Tapp

VERIFIED: Gilles Brassard, Richard Cleve, Alain Tapp, "Cost of exactly simulating quantum entanglement with classical communication," Phys. Rev. Lett. 83, 1874-1877 (1999), arXiv:quant-ph/9901035, DOI 10.1103/PhysRevLett.83.1874.

Their own abstract states:

> "We show that, in the case of a single pair of qubits in a Bell state, a constant number of bits of communication is always sufficient--regardless of the number of measurements under consideration. We also show that, in the case of a system of n Bell states, a constant times 2^n bits of communication are necessary."

Exact theorems opened:

- Theorem 2: for a single Bell state with real-plane von Neumann measurements, four bits from Alice to Bob exactly simulate it.
- Theorem 3: for all von Neumann measurements on one Bell state, eight bits from Alice to Bob exactly simulate it.
- Theorem 4: for n Bell states, there are measurement sets M_A and M_B, each of size 2^(2^n), such that exact simulation of |Phi+>^{tensor n} requires c 2^n bits for some c > 0. The proof reduces restricted equality/EQ' to the simulation; if f(n) bits simulated the scenario, restricted equality would be solved with f(n)+n bits, hence f(n) >= c 2^n - n >= c' 2^n.

This is the clean exponential exact lower bound.

### Pironio: Bell violation gives average-communication lower bounds

VERIFIED: Stefano Pironio, "Violations of Bell inequalities as lower bounds on the communication cost of non-local correlations," Phys. Rev. A 68, 062102 (2003), arXiv:quant-ph/0304176, DOI 10.1103/PhysRevA.68.062102.

VERIFIED quote from abstract:

> "To reproduce in a local hidden variables theory correlations that violate Bell inequalities, communication must occur between the parties. We show that the amount of violation of a Bell inequality imposes a lower bound on the average communication needed to produce these correlations."

Exact CHSH number, VERIFIED:

> "to produce using classical resources the correlations that maximally violate the CHSH inequality, sqrt(2)-1 ~= 0.4142 bits of communication are necessary and sufficient."

This is average communication, not Toner-Bacon worst-case communication.

### Degorre-Laplante-Roland framing

VERIFIED: Julien Degorre, Sophie Laplante, Jeremie Roland, "Simulating quantum correlations as a distributed sampling problem," Phys. Rev. A 72, 062314 (2005), arXiv:quant-ph/0507120, DOI 10.1103/PhysRevA.72.062314.

They recast singlet simulation as distributed sampling. Useful verified facts:

- Bell theorem stated as: "No local hidden variable model may simulate the quantum correlations exhibited by the EPR experiment" (their Theorem 3).
- Sampling theorem: if Alice and Bob share lambda_s with density rho_a(lambda_s)=|a . lambda_s|/(2 pi), they can simulate the EPR experiment without further resource.
- Communication theorem: "there exists a protocol using exactly one bit of communication" to share such a sample; corollary: one-way communication simulates EPR with exactly one bit.
- They summarize prior average protocols: Maudlin 1.17 bits on average for real-plane measurements; Steiner 1.48 bits on average for real-plane measurements; Cerf-Gisin-Massar 1.19 bits on average for arbitrary projective measurements.

### Other lower-bound items

VERIFIED: Tamas Vertesi and Erika Bene, "Lower bound on the communication cost of simulating bipartite quantum correlations," Phys. Rev. A 80, 062316 (2009), arXiv:0904.1390, DOI 10.1103/PhysRevA.80.062316.

Their abstract:

> "we show that two bits of communication is 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."

Scope: maximally entangled state of arbitrary finite dimension, two-outcome measurements; this is about reproducing the quantum correlations in the Regev-Toner sense, not full arbitrary POVM joint distributions.

## 3. Beyond the singlet

### Higher-dimensional bipartite systems

VERIFIED: Oded Regev and Ben Toner, "Simulating Quantum Correlations with Finite Communication," SIAM J. Comput. 39, 1562-1580 (2010), arXiv:0708.0827, DOI 10.1137/080723909.

VERIFIED theorem:

> "There is a public-coin protocol for exactly simulating quantum correlations using two bits of one-way communication."

Crucial limitation, VERIFIED from their problem statement and open-problems section:

- They simulate only the correlator/parity E[alpha beta] = Tr(A tensor B rho), equivalently E[alpha beta] = <a,b> in Tsirelson's vector formulation.
- They explicitly say the general local-measurement simulation problem asks for "the entire output (as opposed to just the correlation)" and that their protocol does not resolve even m=2 full simulation because it gives uniform marginals, not necessarily the quantum marginals.
- Quote: "although our protocol gives the correct correlations, it generates uniform marginal distributions, and not those predicted by quantum theory."

VERIFIED: Degorre, Laplante, Roland, "Simulation of bipartite qudit correlations," Phys. Rev. A 75, 012309 (2007), arXiv:quant-ph/0608064, DOI 10.1103/PhysRevA.75.012309.

Abstract:

> "We present a protocol to simulate the quantum correlations of an arbitrary bipartite state, when the parties perform a measurement according to two traceless binary observables. We show that log(d) bits of classical communication is enough on average, where d is the dimension of both systems."

Opened detail: for arbitrary states they reproduce joint correlations E(AB), not necessarily marginals; for maximally entangled qudit pairs, the marginals are uniform so the full distribution for traceless binary observables is reproduced.

VERIFIED: BCT lower bound above gives Omega(d) when d=2^n for d-outcome coherent measurements on a maximally entangled d-dimensional state, stated in Regev-Toner as "Omega(d) bits of communication are necessary for exact simulation of d-outcome measurements on the maximally-entangled state in C^d tensor C^d."

### n entangled pairs / exponential scaling

VERIFIED: BCT, Phys. Rev. Lett. 83, 1874 (1999), theorem 4, exact worst-case lower bound c 2^n for n Bell states and specific coherent Deutsch-Jozsa measurement sets.

VERIFIED: Serge Massar, Dave Bacon, Nicolas J. Cerf, Richard Cleve, "Classical simulation of quantum entanglement without local hidden variables," Phys. Rev. A 63, 052305 (2001), arXiv:quant-ph/0009088, DOI 10.1103/PhysRevA.63.052305.

Abstract:

> "arbitrary positive operator valued measurements on systems of n Bell states can be simulated with O(n 2^n) bits of communication on average"

Opened text also says the BCT Omega(2^n) lower bound "with minor modifications ... also carries over to the average communication model," and therefore the O(n 2^n) average protocol is almost optimal.

VERIFIED: Alberto Montina, "Approximate simulation of entanglement with a linear cost of communication," Phys. Rev. A 84, 042307 (2011), arXiv:1107.4647, DOI 10.1103/PhysRevA.84.042307.

Abstract:

> "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."

This is the explicit "cost grows exponentially" framing; it contrasts exact simulation with approximate low-dimensional protocols.

### POVMs

VERIFIED: Andre Allan Methot, "Simulatings POVMs on EPR pairs with six bits of expected communication," Eur. Phys. J. D 29, 445-446 (2004), arXiv:quant-ph/0304122, DOI 10.1140/epjd/e2004-00045-y.

Abstract:

> "We present a classical protocol for simulating correlations obtained by bipartite POVMs on an EPR pair. The protocol uses shared random variables ... augmented by six bits of expected communication."

Opened protocol: Alice sends two bits, Bob sends one accept/reject bit; success probability 1/2 per round, so expected two rounds and 2(2+1)=6 expected bits. With block coding, 5.7 expected bits. Methot explicitly says he does not know a worst-case bounded-communication POVM protocol.

VERIFIED: Degorre-Laplante-Roland 2005 Theorem 19 gives for EPR/singlet POVMs: postselection with p(abort_A)=p(abort_B)=2/3; 6 bits expected communication; or 2 nonlocal-box bits plus 4 communication bits on average.

VERIFIED: Martin J. Renner, Armin Tavakoli, Marco Tulio Quintino, "Classical Cost of Transmitting a Qubit," Phys. Rev. Lett. 130, 120801 (2023), arXiv:2207.02244, DOI 10.1103/PhysRevLett.130.120801.

Abstract:

> "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."

and for Bell scenarios:

> "two bits of communication are enough to simulate all quantum correlations associated to arbitrary local POVMs applied to any entangled two-qubit state."

This is a modern worst-case two-bit qubit/POVM result, stronger than the old expected-communication POVM simulations for qubits.

### Multipartite / GHZ

VERIFIED: Tracey E. Tessier, Ivan H. Deutsch, Carlton M. Caves, "Efficient classical-communication-assisted local simulation of n-qubit GHZ correlations," Phys. Rev. A 72, 032305 (2005), arXiv:quant-ph/0407133.

Abstract:

> "measurements of all products of Pauli operators on an n-qubit GHZ state" can be simulated with communication that "scales linearly with the number of qubits"; specifically the structure uses n-2 bits.

VERIFIED: Cyril Branciard and Nicolas Gisin, "Quantifying the nonlocality of GHZ quantum correlations by a bounded communication simulation protocol," Phys. Rev. Lett. 107, 020401 (2011), arXiv:1102.0330, DOI 10.1103/PhysRevLett.107.020401.

Abstract:

> "3 bits in total turn out to be sufficient to simulate all equatorial Von Neumann measurements on the 3-partite GHZ state."

VERIFIED: Gilles Brassard, Luc Devroye, Claude Gravel, "Exact Classical Simulation of the GHZ Distribution," IEEE Trans. Inf. Theory 62, 876-890 (2016), arXiv:1303.5942, DOI 10.1109/TIT.2015.2504525.

Abstract:

> "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 for equatorial measurements:

> "a protocol that needs only O(n log n) bits of communication"

Scope: expected communication, exact sampling, no prior shared random variables in their random-bit model.

## 4. No-communication alternatives

### Post-selection / detection loophole

VERIFIED: Degorre-Laplante-Roland 2005, Theorem 8 and Corollary 9:

- One-sided postselection: Alice aborts with probability 1/2, Bob never aborts; conditioned on no abort, singlet projective correlations are simulated.
- Symmetrized Gisin-Gisin version: p(abort_A)=p(abort_B)=1/3.

Interpretation: detector efficiency in the symmetric construction is 2/3 per party. This is a construction, not stated there as the optimal detection-loophole threshold.

VERIFIED: For POVMs, their Theorem 19 gives postselection with p(abort_A)=p(abort_B)=2/3, i.e. much lower efficiency 1/3, in that protocol.

Citation for the underlying postselection protocol from their text: Nicolas Gisin and Bernard Gisin, "A local hidden variable model of quantum correlation exploiting the detection loophole," Phys. Lett. A 260, 323-327 (1999). I did not open this paper directly in this thread; the Gisin-Gisin details here are VERIFIED only through Degorre-Laplante-Roland's opened discussion.

### Measurement dependence / superdeterminism

VERIFIED: M. J. W. Hall, "Local deterministic model of singlet state correlations based on relaxing measurement independence," Phys. Rev. Lett. 105, 250404 (2010), arXiv:1007.5518, DOI 10.1103/PhysRevLett.105.250404.

Hall defines M = sup integral |rho_XY(lambda)-rho_X'Y'(lambda)| and F = 1 - M/2.

Exact singlet constants, VERIFIED:

- M_singlet = 2(sqrt(2)-1)/3 ~= 0.276.
- F_singlet = (4 - sqrt(2))/3 ~= 86%.
- Meaning: give up about 14% of measurement independence to model all singlet spin correlations in a deterministic no-signalling model.

For maximum algebraic CHSH violation E=4, Hall gives:

- E = min{2+3M, 4}.
- Maximum CHSH violation E=4 is achievable with F=2/3, "via giving up just 1/3 of measurement independence."

Important distinction: Hall 2010's 1/3 is not "bits"; it is fraction of measurement independence in his variational-distance measure.

Bits of setting-source correlation, VERIFIED through later Hall-Branciard paper:

VERIFIED: Michael J. W. Hall and Cyril Branciard, "Measurement-dependence cost for Bell nonlocality: causal vs retrocausal models," Phys. Rev. A 102, 052228 (2020), arXiv:2007.11903, DOI 10.1103/PhysRevA.102.052228.

They measure cost as mutual information I(X,Y:Lambda) in bits.

For unbiased CHSH settings and the maximal quantum violation S_Q = 2 sqrt(2):

- General/retrocausal optimal separable model: I_R(S_Q) ~= 0.046 bits.
- Fully causal measurement-dependence model: I_C(S_Q) ~= 0.080 bits.
- Their abstract: "the maximal quantum violation requires a mutual information of just ~0.080 bits" for the most efficient causal simulation; the previous ~0.046-bit model is the optimal retrocausal simulation.
- Fully superdeterministic models in their strict sense, where settings and outcomes are fully determined by Lambda, require I_SD(S)=2 bits for unbiased CHSH settings, independent of S.

Thus the answer to "how many bits of correlation between source and settings are needed to fake a maximal CHSH violation?" depends on the causal model:

- ~0.080 bits for a fully causal measurement-dependent separable model faking Tsirelson/maximal quantum CHSH violation 2 sqrt(2).
- ~0.046 bits if retrocausal measurement dependence is allowed.
- 2 bits for their strict superdeterministic model with unbiased CHSH settings because Lambda fully determines one of four setting pairs.

## 5. Simulator-of-the-universe framing

I found no paper in this thread that explicitly frames these communication costs as "costs to a simulator of our universe" in the simulation-hypothesis sense. The opened literature frames the results as:

- quantifying Bell nonlocality as communication/nonlocal resources;
- communication complexity of entanglement simulation;
- classical hidden-variable augmentation costs;
- device-independent security loopholes via measurement dependence.

The closest language is Methot's quote of Maudlin: "The key to understanding violations of Bell's inequality is not operator algebras but information transmission," and Methot's comment that the approach gauges "the amount of information hidden in the EPR pair itself or, in some sense, the amount of information that must be space-like transmitted, in a local hidden variable model." This is resource accounting for classical simulation, not universe-simulator speculation.

## WHERE I DID NOT LOOK

- I did not open Maudlin 1992 or Steiner 2000 directly; constants 1.17 and 1.48 are taken from opened Degorre-Laplante-Roland / Massar-Bacon-Cerf-Cleve summaries.
- I did not open Gisin-Gisin 1999 directly; the symmetric 1/3 abort construction is from opened Degorre-Laplante-Roland.
- I did not exhaust recent 2024-2026 preprints on infinite communication for continuous-variable/quantum-channel simulation; the requested core Bell/singlet/n-pair literature is covered.
- I did not try to prove optimal constants beyond the papers' statements; where a result is a protocol upper bound rather than a lower bound, I marked it as such.
- I did not search philosophy/simulation-hypothesis venues beyond targeted web searches for "simulator of our universe," "simulation hypothesis," and "computational cost" with Bell-correlation terms; those returned no relevant source hits.