A bipartite perfect quantum strategy (BPQS) allows two players who cannot communicate with each other to always win a nonlocal game. BPQSs are rare but fundamental in light of some recent results in quantum information, computation, and foundations. A more than 40-year-old open problem is how many inputs (measurement settings) a BPQS requires. A related problem is how many inputs are needed if, in addition, the quantum system has minimum dimension. A third, apparently unrelated, problem is what is the connection between BPQSs and state-independent contextuality, which inspired the first BPQSs. Here, we solve the third problem: we prove that every BPQS defines a Kochen-Specker set. We use this result to identify the BPQS with the smallest number of inputs, both in the general case and in the case of minimum dimension, and solve some related problems. We conjecture that the BPQSs presented here are the solutions to the first two problems.
Bipartite perfect quantum strategies (BPQSs) allow two players isolated from each other to win every trial of a nonlocal game. BPQSs have crucial roles in recent developments in quantum information and quantum computation. However, only few BPQSs with a small number of inputs are known and only one of them has been experimentally tested. It has recently been shown that every BPQS has an associated Kochen-Specker (KS) set. Here, we first prove that any BPQS of minimum input cardinality that can be obtained from a generalized KS set can also be obtained from a KS set of pure states. We address the problem of finding BPQSs of small input cardinality starting from KS sets. We introduce an algorithm that identifies the BPQS with the minimum number of settings for any given KS set. We apply it to many well-known KS sets of small cardinality in dimensions 3, 4, 5, 6, 7, and 8. In each dimension, the algorithm either recovers the best BPQS known or finds one with fewer inputs.
A perfect strategy is one that allows the mutually in-communicated players of a nonlocal game to win every trial of the game. Perfect strategies are basic tools for some fundamental results in quantum computation and crucial resources for some applications in quantum information. Here, we address the problem of producing qudit-qudit perfect quantum strategies with a small number of settings. For that, we exploit a recent result showing that any perfect quantum strategy induces a Kochen-Specker set. We identify a family of KS sets in even dimension
d≥6 that, for many dimensions, require the smallest number of orthogonal bases known:
d+1. This family was only defined for some
d. We first extend the family to infinitely many more dimensions. Then, we show the optimal way to use each of these sets to produce a bipartite perfect strategy with minimum input cardinality. As a result, we present a family of perfect quantum strategies in any
(2,d−1,d) Bell scenario, with
d=2kpm for
p prime,
m≥k≥0 (excluding
m=k=0),
d=8p for
p≥19,
d=kp for
p>((k−2)2k−2)2 whenever there exists a Hadamard matrix of order
k, other sporadic examples, as well as a recursive construction that produces perfect quantum strategies for infinitely many dimensions
d from any dimension
d′ with a perfect quantum strategy. We identify their associated Bell inequalities and prove that they are not tight, which provides a second counterexample to a conjecture of 2007.
Perfect prepare-and-measure games exhibit an all-or-nothing quantum advantage: a quantum system of dimension
d satisfies every prescribed winning constraint, whereas a classical
d-level message cannot. We establish three structural results for such forbidden-output support constraints. First, every binary-output support game reduces exactly to a conflict graph: perfect classical realization with a
d-level message is equivalent to
d-colorability, perfect
d-dimensional quantum realization is equivalent to a
d-dimensional orthogonal representation, and the minimum number of Bob inputs realizing a fixed conflict graph is its edge biclique-cover number. Second, for an arbitrary finite output alphabet, every perfect qubit strategy admits a perfect classical-bit realization. Third, with at most two Bob inputs every perfect qutrit strategy admits a perfect classical-trit realization. An explicit seven-preparation qutrit game with three Bob inputs satisfies
C3=20<Q3=S=21, so three Bob inputs are necessary and sufficient in dimension three. Conversely, a two-input, six-output game in dimension five satisfies
C5=39<Q5=S=40. Thus the smallest dimension admitting a two-input perfect same-dimensional separation is either four or five; the four-dimensional case remains open. As a flagship binary application, the
13-ray qutrit graph yields a compressed game
(X,Y,B)=(13,8,2) with
C3=39<Q3=S=40, and eight Bob inputs are minimal among all binary-output realizations of that graph. Graph extensions demonstrate the mechanism in every dimension, while Torpedo and antidistinguishability games illustrate the genuinely nonbinary regime. These results connect exact communication, graph coloring, contextuality, state exclusion, and zero-error information theory.
Kochen-Specker (KS) sets are fundamental in physics. Every time nature produces bipartite correlations attaining the nonsignaling limit, or two parties always win a nonlocal game impossible to always win classically, it is because the parties are measuring a KS set. The simplest quantum system in which all these phenomena occur is a pair of three-level systems. However, the simplest KS sets known in dimension three are asymmetrical and require a large number of bases (the current minimum is 16, set by Peres and Penrose). Here, we present a KS set that is much more symmetrical and easier to prove than any previous example. It sets a new record for minimum number of bases, 14, and enables us to refute Conjecture 2 in Cabello Simplest bipartite perfect quantum strategies, Phys. Rev. Lett. 134, 010201 (2025)PRLTAO0031-900710.1103/PhysRevLett.134.010201, setting a new record for qutrit-qutrit perfect strategies with a minimum number of inputs: 5-9. We establish the fundamental nature of this set in quantum theory.
One of the longstanding open problems in quantum nonlocality is to determine if maximally entangled states are complete for bipartite pseudo-telepathic games: namely, if every nonlocal game which admits a perfect entangled strategy admits such a strategy which uses a maximally entangled state. We exhibit a counterexample to this in the form of a bipartite nonlocal game with input sets of size 4 and 3 and output sets both of size 6. This game is part of a new class of nonlocal games, which we call inner product games, which could be of independent interest.
We study bipartite games that arise in the context of nonlocality with the help of graph theory. Our main results are alternate proofs that deciding whether a no-communication classical winning strategy exists for certain games (called forbidden-edge and covering games) is NP- complete, while the problem of deciding if these games admit a non-signalling winning strategy is in P. We discuss relations between quantum winning strategies and orthogonality graphs. We also show that every pseudo-telepathy game yields both a proof of the Bell-Kochen-Specker theorem and an instance of a two-prover interactive proof system that is classically sound, but that becomes unsound when provers use shared entanglement.