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Bounding Two-Way Average Communication Cost of Simulating Quantum Correlations

This paper establishes lower bounds on the input-average two-way communication cost required to exactly simulate quantum correlations, demonstrating that certain parallel nonlocal games necessitate unbounded communication while providing specific asymptotic rates for Magic-Square and CHSH scenarios and a hierarchical framework to compute exact costs.

Original authors: Kai-Siang Chen, Gelo Noel M. Tabia, Bo-An Tsai, Swati Kumari, Yeong-Cherng Liang

Published 2026-10-06

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CC BY 4.0

Original authors: Kai-Siang Chen, Gelo Noel M. Tabia, Bo-An Tsai, Swati Kumari, Yeong-Cherng Liang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, particles can become linked in a way that defies our everyday experience of cause and effect. When two such particles are measured, their results are correlated with a precision that cannot be explained by any pre-existing agreement or hidden plan they might have carried with them. This phenomenon, known as quantum nonlocality, suggests that the universe is far more interconnected than classical physics allows. However, if we strip away the mystery of quantum mechanics and try to mimic these strange connections using only ordinary, classical tools, we hit a wall. To reproduce the results of quantum experiments without the particles actually being entangled, two people sharing the data would need to exchange information. The question that has long puzzled scientists is not just whether this information exchange is possible, but exactly how much of it is required. Is a single whisper enough, or does the task demand a flood of data? Understanding this cost is crucial because it quantifies the true "distance" between the quantum world and the classical world we live in.

A team of researchers has now mapped out the precise limits of this communication cost, revealing that for certain quantum scenarios, the amount of information needed to fake the results grows steadily and inevitably as the experiment gets larger. They focused on a specific type of challenge where two parties, traditionally called Alice and Bob, receive questions and must provide answers that satisfy a winning condition. In the quantum realm, they can win these games with perfect certainty using entangled particles. In the classical world, without any communication, they are doomed to lose some of the time. The researchers asked: if Alice and Bob are allowed to talk to each other to coordinate their answers, how much talking is necessary to win every single time, exactly as the quantum version would?

The team developed a new method to calculate the minimum amount of information that must be exchanged on average to simulate these quantum correlations. They applied this method to two famous types of games: the "Magic Square" game and the "CHSH" game. In the Magic Square game, the researchers proved that to perfectly simulate the quantum results for a series of $n$ parallel games, the communication cost grows linearly with the number of games. Specifically, they showed that the cost is at least $n$ times the logarithm of 1.5 bits. This means that for every additional game added to the series, a fixed, non-zero amount of extra communication is strictly required. They also constructed a specific strategy that achieves this limit, proving that their lower bound is tight and that the cost cannot be reduced further, even if Alice and Bob are allowed to talk back and forth as much as they like.

For the CHSH game, which is slightly different because quantum players cannot win with absolute certainty, the researchers found a similar linear growth in communication cost. They calculated that to simulate the quantum results for $n$ parallel copies of this game, the average communication required is approximately $0.04627$ bits per copy. While this number is small, it is significant because it proves that no matter how much communication budget is set in advance, there is a finite number of game copies that will exceed it. In other words, you cannot simulate a large enough quantum experiment with a fixed, finite amount of classical chatter. The researchers demonstrated that even with the most sophisticated, interactive two-way communication strategies, the cost inevitably outpaces any fixed limit.

To ensure these findings were robust, the team also built a step-by-step computational framework, or hierarchy, to check the communication cost for specific, smaller scenarios. This method works by breaking down the complex quantum behavior into simpler, deterministic pieces and calculating the cost of each. They found that for small, specific setups, their calculated lower bounds matched their upper bounds perfectly, confirming the exact cost of simulation for those cases. This dual approach—using broad mathematical proofs for large series and precise calculations for small instances—gives a complete picture of the problem. The results confirm that quantum correlations are not just slightly harder to mimic than classical ones; they require a fundamentally different scale of communication that scales up with the size of the system.

The implications of this work extend beyond abstract theory. By establishing that the communication cost grows linearly, the researchers have provided a concrete way to distinguish between quantum and classical behaviors in a rigorous, quantitative manner. They showed that for the Magic Square game, the most efficient way to simulate the quantum results does not actually require two-way conversation; a one-way message is sufficient to achieve the same asymptotic rate. This finding is surprising because it suggests that the complexity of the quantum correlation itself, rather than the back-and-forth nature of the communication, is the primary driver of the cost. For the CHSH game, the cost is lower per copy, but the principle remains the same: the classical world cannot replicate the quantum world without paying a price that grows with the size of the experiment.

Ultimately, this research settles a long-standing question about the resources needed to bridge the gap between quantum and classical physics. It proves that there is no "free lunch" where a small, fixed amount of communication can simulate arbitrarily large quantum systems. The researchers have provided explicit formulas and bounds that tell us exactly how much information is needed to fake quantum entanglement for any given number of parallel games. Their work transforms a vague intuition about quantum strangeness into a precise, measurable quantity, showing that the more we try to replicate the quantum world with classical tools, the more we have to talk.

Technical Summary: Bounding Two-Way Average Communication Cost of Simulating Quantum Correlations

Problem Statement
Bell nonlocal correlations, arising from entanglement, cannot be reproduced by local hidden-variable models without classical communication. While it is established that finite communication suffices to simulate any correlation in a fixed finite Bell scenario, determining the minimum amount of communication required for exact simulation remains a challenging problem. This is particularly true in the input-average communication cost setting, where the cost is averaged over the input distribution and random variables, and where protocols are allowed to be variable-length and fully interactive (two-way). The paper addresses the question: Given an arbitrary finite communication budget, can one exhibit a finite quantum correlation whose exact simulation necessarily exceeds that budget, even under these relaxed two-way, variable-length conditions?

Methodology
The authors employ two complementary methodological approaches to derive lower bounds on communication costs:

Nonlocal Game-Based Bounds:
- Winning Rectangles (Perfect Winning): For games where a correlation wins with certainty (pseudo-telepathy), the authors adapt the standard "rectangle method." They define a "winning rectangle" as a subset of input pairs $S \times T$ where local deterministic strategies can win. They prove that the largest winning-rectangle probability, $\alpha$ , is multiplicative under parallel repetition ( $\alpha_n = \alpha^n$ ). By relating the average communication cost to the entropy of the leaf distribution in a communication tree, they derive a lower bound of $\log_2(1/\alpha)$ per copy.
- Winning Probabilities (Imperfect Winning): For correlations that do not win with certainty, the authors utilize information-theoretic arguments based on the relative entropy between the achieved winning probability and the classical limit. They define a function $\Phi_{\omega_c}(w)$ based on the binary relative entropy $D_{bin}(w \| \omega_c)$ . They show that for parallel games with product input distributions, the total communication cost is bounded by the sum of these functions across copies.
Linear Programming (LP) Hierarchy:
- Recognizing that exact simulation costs can be formulated as an optimization over convex decompositions into deterministic correlations, the authors introduce a computationally tractable hierarchy.
- They group deterministic correlations by their minimum input-average communication costs ( $\kappa_k$ ).
- Lower Bounds: At level $L$ , the hierarchy minimizes the cost by including only deterministic correlations with costs $\le \kappa_L$ , while charging the remaining probability weight at the cost of the next level ( $\kappa_{L+1}$ ).
- Upper Bounds: Complementary upper bounds are obtained by restricting the exact decomposition to deterministic correlations with costs $\le \kappa_k$ .
- The hierarchy converges to the exact input-average cost at its final level. If lower and upper bounds coincide at a lower level, the exact cost is determined without enumerating all strategies.

Key Contributions and Results

Magic Square (MS) Game:
- The authors prove a lower bound of $n \log_2(3/2)$ bits for the exact simulation of $n$ parallel copies of the MS game under uniform inputs. This is derived from the fact that the largest winning rectangle covers at most $2/3$ of the input space.
- They construct a one-way communication protocol that achieves an asymptotic rate of $\log_2(3/2)$ bits per copy. This demonstrates that allowing two-way interaction does not reduce the asymptotic simulation rate for MS.
- Consequently, for any prescribed finite budget $B$ , there exists a finite number of copies $n \approx B / \log_2(1.5)$ such that exact simulation exceeds $B$ bits.
Parallel CHSH Game:
- For the $n$ -copy CHSH game, where the quantum correlation wins with probability $w_T = (2+\sqrt{2})/4$ (exceeding the classical $3/4$ ), the authors derive a lower bound of $n \times D_{bin}(w_T \| 3/4) \approx 0.04627n$ bits.
- This result establishes that even for correlations that do not win perfectly, the exact simulation cost grows linearly with $n$ , exceeding any finite budget for sufficiently large $n$ .
- The authors note that while their coefficient is smaller than previous worst-case one-way bounds, their result applies to the more general setting of variable-length, fully interactive two-way protocols under input-average cost.
Hierarchy of Bounds:
- The proposed LP hierarchy recovers exact costs when bounds agree.
- Numerical results for small Bell scenarios (e.g., $2 \times 2$ inputs/outputs) recover known exact costs (e.g., $\sqrt{2}-1 \approx 0.4142$ for CHSH).
- For larger scenarios, the first level of the hierarchy provides non-trivial lower bounds, often matching the EPR2 nonlocal content.

Significance
The paper claims to answer affirmatively the question of whether finite quantum correlations exist that require more communication than any prescribed finite budget, even when variable-length, fully interactive two-way protocols are permitted.

Explicit Finite Separations: Unlike previous asymptotic results that only guarantee growing costs for large $n$ , this work provides explicit finite-copy correlations (specifically for MS and CHSH) where the exact simulation cost strictly exceeds any given budget.
Tightness of Bounds: For the Magic Square game, the derived lower bound is shown to be asymptotically tight, as it is matched by a constructive one-way protocol. This determines the exact asymptotic simulation rate as $\log_2(3/2)$ bits per copy.
Methodological Framework: The introduction of the LP hierarchy offers a systematic way to compute exact input-average costs for specific correlations without needing to consider the infinite space of all possible communication strategies, provided the bounds converge at computationally accessible levels.
Robustness: The bounds hold for input-average costs, which are generally lower than worst-case costs, and apply to fully interactive protocols, making the results robust against protocol optimizations.

The authors conclude that these results provide a framework for quantifying the classical communication required to simulate observed quantum correlations in finite Bell scenarios, with implications for understanding the strength of nonlocality and the limitations of classical models with limited cross-talk.

Technical Summary: Bounding Two-Way Average Communication Cost of Simulating Quantum Correlations

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