Quantum Edge in Nonlocal Games Survives When the Inputs Are No Longer Fair

In the strange world of quantum mechanics, two players who can never communicate can still win a game far more often than any classical strategy allows. This is the essence of a nonlocal game, and the most famous of them all is the CHSH game, named after Clauser, Horne, Shimony and Holt, whose 1969 experiment turned a philosophical debate into testable physics. For decades, however, most analyses of these games have rested on a convenient assumption: that the inputs handed to the players are perfectly unbiased, each possibility equally likely. A new study published in Quantum Information Processing by Jyotirmoy Basak of the Okinawa Institute of Science and Technology Graduate University and Anurag Ghosh of the Indian Institute of Engineering Science and Technology, Shibpur, asks what happens when that assumption is relaxed, and the answer is reassuring for quantum enthusiasts.

The motivation is thoroughly practical. In real laboratories and real cryptographic devices, generating perfectly random, unbiased bits is notoriously difficult. Random number generators drift, hardware is imperfect, and any protocol that silently collapses when the inputs lean even slightly toward one outcome would be fragile in deployment. Device-independent quantum key distribution, quantum private query, and certified randomness generation all lean on nonlocal games as their engine, so knowing how quantum advantage behaves under biased inputs is not an academic luxury but an engineering necessity. Basak and Ghosh set out to map exactly how much imperfection these games can tolerate.

Their starting point was a recent discovery by Basak and collaborators, published in Cryptography and Communications in 2023, which showed that the CHSH game is not alone. Among all two-party nonlocal games with binary inputs and binary outputs, a small family of additional games also exhibits quantum advantage when inputs are uniformly distributed. These games can be described by Boolean functions, algebraic expressions over the bits that the players feed in and produce, and each can be characterized by how its winning condition partitions the sixteen possible input-output combinations. The new work takes each of these games and subjects it to a symmetrically biased input distribution, where the probability of a zero input is some value p and the probability of a one input is one minus p, applied identically to both players.

The technical machinery is elegant. Each game’s winning condition can be written in algebraic normal form, a polynomial over the Boolean algebra of the inputs x and y and the outputs a and b. For example, one of the games corresponds to the expression x plus a plus xy plus xa plus xb plus yb plus ab plus xya plus xyb plus yab, all combined with exclusive-or operations. Once the input distribution is biased, the success probability of each of the sixteen possible deterministic classical strategies becomes a function of p, and the authors determine, for every value of the bias, which classical strategy is optimal and what success probability it achieves. This exhaustive classical analysis forms the benchmark against which quantum performance is measured.

On the quantum side, the authors deliberately restrict themselves to a specific family of strategies, one in which each player measures their share of an entangled state along one of two possible directions, with the measurement angles theta-zero and theta-one for Alice and psi-zero and psi-one for Bob. Within this family, the quantum success probability of each game can be written as a closed-form trigonometric expression involving the cosines of the differences between the measurement angles, weighted by powers of p and one minus p. For each game and each input bias, the optimal angles are found by exhaustive grid search over the four-dimensional parameter space, with the code and data made publicly available in a GitHub repository accompanying the paper.

The central finding is striking: quantum advantage is remarkably robust. For every game considered, there is a suitable range of input bias within which the restricted quantum strategy still outperforms the best classical strategy. The size of that range and the magnitude of the advantage both depend on the bias, and the optimal measurement angles shift as the bias changes, meaning the quantum players must adapt their strategy to the statistical properties of the inputs. This adaptivity is a genuinely new feature that does not appear in the uniform-input analysis, where a single fixed set of angles, the celebrated CHSH angles, suffices.

Perhaps the most elegant result is a symmetry statement: for all the games studied, the quantum advantage is maximized precisely at the uniform input distribution, where p equals one half, even within the restricted strategy family. In other words, bias never helps the quantum players relative to their classical competitors; it only erodes their edge. This gives the uniform case a privileged status that goes beyond mathematical convenience. It suggests that the standard textbook analyses, which assume fair coins, capture the games at their most quantum-friendly point, and that any deviation from fairness moves the system along a predictable path toward classical behavior.

The implications ripple outward into several areas of quantum information science. Nonlocal games are the workhorses of device-independent protocols, where the security or correctness of a task is certified purely by observed winning statistics, without trusting the internal workings of the devices. The landmark results of Reichardt, Unger and Vazirani on classical command of quantum systems, and of Vazirani and Vidick on fully device-independent quantum key distribution, rely on rigidity properties of games like CHSH. Knowing how the advantage degrades under biased inputs tells protocol designers how much randomness their input sources must guarantee, and it also informs recent work on benchmarking quantum computers using nonlocal game strategies, where input distributions may not be perfectly controlled.

The study also connects to a broader theoretical program. Recent work has bounded the quantum value of compiled nonlocal games, shown that quantum advantage can be extracted from any nonlocal game with a gap, and explored no-signalling correlations in operator-algebraic terms. By grounding the analysis in Boolean functions, Basak and Ghosh provide a combinatorial handle on the problem: the algebraic normal form of each game’s predicate directly shapes the trigonometric structure of its quantum success probability. The seven games analyzed, corresponding to partitions of their truth tables with labels like three-plus-two-plus-two-plus-one and two-plus-one-plus-one-plus-one, each yield a distinct expression, yet the qualitative conclusions hold uniformly across the family, hinting at a deeper structural principle waiting to be formalized.

There remain open questions that the restricted strategy family leaves untouched. The true optimal quantum success probability under bias may be achievable by more general entangled states and measurements, and extending the analysis beyond this family is a natural next step. Asymmetric biases, where Alice and Bob face different input distributions, and games with more inputs or outputs, are further directions. But the message of the current work is clear and encouraging: the quantum edge in binary nonlocal games is not a delicate flower that wilts the moment inputs become imperfect. It bends with the bias, adapts its angles, and holds on over meaningful ranges, with fairness remaining the point of maximum quantum power. For a field racing to turn Bell-inequality magic into working technology, that robustness is very good news.

Subject of Research: Robustness of quantum advantage in two-party nonlocal games under biased input distributions

Article Title: Analysis of Boolean functions related to two-party nonlocal games for biased inputs

Article References: Basak, J., & Ghosh, A. (2026). Analysis of Boolean functions related to two-party nonlocal games for biased inputs. Quantum Information Processing, 25(10), Article 326. https://doi.org/10.1007/s11128-026-05346-3

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05346-3

Keywords: nonlocal games, CHSH game, Boolean functions, quantum advantage, biased inputs, Bell inequalities, device independence, quantum information, entanglement, input bias, quantum strategies, Quantum Information Processing