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arXiv:2407.21473

Perfect quantum strategies with small input cardinality

Stefan Trandafir, Junior R. Gonzales-Ureta, Adán Cabello

Published Jul 31, 2024• 0 citations
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Abstract

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 $d \ge 6$ 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 $d = 2^kp^m$ for $p$ prime, $m \geq k \geq 0$ (excluding $m=k=0$), $d = 8p$ for $p \geq 19$, $d=kp$ for $p > ((k-2)2^{k-2})^2$ 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.

DOI
10.48550/arxiv.2407.21473

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Physical Review A•Oct 26, 2021•12 citations
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≥6d \ge 6d≥6 that, for many dimensions, require the smallest number of orthogonal bases known: d+1d+1d+1. This family was only defined for some ddd. 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)(2,d-1,d)(2,d−1,d) Bell scenario, with d=2kpmd = 2^kp^md=2kpm for ppp prime, m≥k≥0m \geq k \geq 0m≥k≥0 (excluding m=k=0m=k=0m=k=0), d=8pd = 8pd=8p for p≥19p \geq 19p≥19, d=kpd=kpd=kp for p>((k−2)2k−2)2p > ((k-2)2^{k-2})^2p>((k−2)2k−2)2 whenever there exists a Hadamard matrix of order kkk, other sporadic examples, as well as a recursive construction that produces perfect quantum strategies for infinitely many dimensions ddd from any dimension d′d'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.
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 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.
Perfect prepare-and-measure games exhibit an all-or-nothing quantum advantage: a quantum system of dimension ddd satisfies every prescribed winning constraint, whereas a classical ddd-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 ddd-level message is equivalent to ddd-colorability, perfect ddd-dimensional quantum realization is equivalent to a ddd-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=21C_3=20<Q_3=S=21C3​=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=40C_5=39<Q_5=S=40C5​=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 131313-ray qutrit graph yields a compressed game (X,Y,B)=(13,8,2)(X,Y,B)=(13,8,2)(X,Y,B)=(13,8,2) with C3=39<Q3=S=40C_3=39<Q_3=S=40C3​=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.
Circuit depth is a central resource in complexity theory. While bounded-depth classical circuits admit well-understood hierarchy theorems, the internal structure of constant-depth quantum computation remains comparatively unexplored. We prove an explicit depth hierarchy theorem for QNC0\mathsf{QNC}^0QNC0. For each d≥12d\ge 12d≥12, we construct a family of two-round interactive problems on which no depth-(d−1)(d-1)(d−1) quantum circuit can achieve near-perfect success, regardless of gate set, circuit size, or ancillary qubits. In contrast, we prove that our construction admits realizations by simple bounded fan-in quantum circuits of depth larger than ddd by a small constant factor. Moreover, all bounded fan-in classical circuits of sublogarithmic depth (in the input size) fail to achieve perfect success on these tasks for every ddd, yielding a hierarchy of problems that show unconditional quantum advantage of QNC0\mathsf{QNC}^0QNC0 over NC0\mathsf{NC}^0NC0. A key obstacle is the scarcity of lower bound techniques for quantum circuits. To address this, we develop methods to analyze how depth affects a circuit's ability to realize nonlocal correlations amongst its output qubits in a fine-grained manner. Our approach exploits the correspondence between constraint systems and nonlocal games, translating group-theoretic constructions into rigid operator-valued constraint systems and then into non-local games. In particular, we construct constraint systems whose unique faithful operator-valued solutions require every perfect strategy, and every near-perfect strategy to a fixed precision, to implement multi-controlled phase operations. This reduces to a nonlocal unitary-synthesis problem, yielding depth lower bounds for both shallow quantum and classical circuits. These results show that increasing depth strictly increases computational power within QNC0\mathsf{QNC}^0QNC0, establishing a genuinely quantum hierarchy.
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.
We expand the quantum variant of the popular game Sudoku by introducing the notion of cardinality of a quantum Sudoku (SudoQ), equal to the number of distinct vectors appearing in the pattern. Our considerations are focused on the genuinely quantum solutions---the solutions of size N2{N}^{2}N2 that have cardinality greater than N2{N}^{2}N2, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parametrization of the genuinely quantum solutions of a $4\ifmmode\times\else\texttimes\fi{}4$ SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8, and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet [I. Nechita and J. Pillet, Quantum Inf. Comput. 21, 781 (2021)] for this special dimension. In general, we proved that for any NNN it is possible to find an ${N}^{2}\ifmmode\times\else\texttimes\fi{}{N}^{2}$ SudoQ solution of cardinality N4{N}^{4}N4, which for a prime NNN is related to a set of NNN mutually unbiased bases of size N2{N}^{2}N2. Such a construction of N4{N}^{4}N4 different vectors of size NNN yields a set of N3{N}^{3}N3 orthogonal measurements.