arXiv:2008.02370 · Physical Review Research

Quantum magic rectangles: Characterization and application to certified randomness expansion

Sean A. Adamson, Petros Wallden

Published Dec 4, 2020• 7 citations

Abstract

The authors generalize the magic square game of Mermin and Peres to arbitrary rectangular dimensions, characterize the optimal quantum strategies, and apply these games in the context of quantum-certified randomness expansion.

DOI: 10.1103/physrevresearch.2.043317

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arXiv:2008.02370 · Physical Review Research

Quantum magic rectangles: Characterization and application to certified randomness expansion

Sean A. Adamson, Petros Wallden

Published Dec 4, 2020• 7 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevresearch.2.043317

Related papers

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The $2\times3$ magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. Its quantum value was suggested by numerical bounds and known quantum random access codes. We prove that this value is $(1+\sqrt{2/3})/2$ and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements: up to local isometries and ancillary systems, each contains a maximally entangled pair of four-dimensional systems with fixed measurements on the local state supports. The same value holds for commuting-operator strategies and for the almost-quantum relaxation, confirming the Adamson--Wallden conjecture for this rectangle. For finite-dimensional magic-square strategies, the largest score compatible with perfect prediction of Alice's complete first row by an adversary with quantum side information is one third plus two thirds of the rectangle value. We classify the strategies attaining this threshold with perfect prediction and give linear-order upper and lower bounds on one minus the best compatible guessing probability above it. The rectangle value is also the optimal average success probability of finite-dimensional three-bit random access codes that hide the full input parity, and we classify all optimal codes.The $2\times3$ magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. Its quantum value was suggested by numerical bounds and known quantum random access codes. We prove that this value is $(1+\sqrt{2/3})/2$ and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements: up to local isometries and ancillary systems, each contains a maximally entangled pair of four-dimensional systems with fixed measurements on the local state supports. The same value holds for commuting-operator strategies and for the almost-quantum relaxation, confirming the Adamson--Wallden conjecture for this rectangle. For finite-dimensional magic-square strategies, the largest score compatible with perfect prediction of Alice's complete first row by an adversary with quantum side information is one third plus two thirds of the rectangle value. We classify the strategies attaining this threshold with perfect prediction and give linear-order upper and lower bounds on one minus the best compatible guessing probability above it. The rectangle value is also the optimal average success probability of finite-dimensional three-bit random access codes that hide the full input parity, and we classify all optimal codes.

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We present a generalization of the Mermin-Peres magic square game to

N

players, in which the square is replaced by an

N

-dimensional hypercube. We show that the winning probability of the optimal classical strategy remains

8/9

for all

N

, while for

N \ge 3

, a generalization of the quantum strategy for

N = 2

achieves a winning probability between

1/3

and

3/4

. This loss of quantum advantage for

N \ge 3

is a consequence of monogamy.

2 \times n

quantum magic rectangle games with odd

n

, and introduce a corresponding quantum magic rectangle inequality to characterize optimal non-perfect strategies. While our results are structural, they may provide a foundation for future developments in quantum information and quantum cryptography based on perfect nonlocal correlations.

Exact value and rigidity of the

2\times3

magic rectangle

The

2\times3

magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. Its quantum value was suggested by numerical bounds and known quantum random access codes. We prove that this value is

(1+\sqrt{2/3})/2

and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements: up to local isometries and ancillary systems, each contains a maximally entangled pair of four-dimensional systems with fixed measurements on the local state supports. The same value holds for commuting-operator strategies and for the almost-quantum relaxation, confirming the Adamson--Wallden conjecture for this rectangle. For finite-dimensional magic-square strategies, the largest score compatible with perfect prediction of Alice's complete first row by an adversary with quantum side information is one third plus two thirds of the rectangle value. We classify the strategies attaining this threshold with perfect prediction and give linear-order upper and lower bounds on one minus the best compatible guessing probability above it. The rectangle value is also the optimal average success probability of finite-dimensional three-bit random access codes that hide the full input parity, and we classify all optimal codes.