# Perfect Commuting-Operator Strategies for Linear System Games

@article{Cleve2016PerfectCS, title={Perfect Commuting-Operator Strategies for Linear System Games}, author={Richard Cleve and Li Liu and William Slofstra}, journal={arXiv: Quantum Physics}, year={2016} }

Linear system games are a generalization of Mermin's magic square game introduced by Cleve and Mittal. They show that perfect strategies for linear system games in the tensor-product model of entanglement correspond to finite-dimensional operator solutions of a certain set of non-commutative equations. We investigate linear system games in the commuting-operator model of entanglement, where Alice and Bob's measurement operators act on a joint Hilbert space, and Alice's operators must commute…

## 40 Citations

### Tsirelson’s problem and an embedding theorem for groups arising from non-local games

- MathematicsJournal of the American Mathematical Society
- 2019

Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that…

### THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED

- MathematicsForum of Mathematics, Pi
- 2019

We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert…

### 3XOR Games with Perfect Commuting Operator Strategies Have Perfect Tensor Product Strategies and are Decidable in Polynomial Time

- Mathematics, Computer ScienceArXiv
- 2020

It is shown that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded, in contrast to the general3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage.

### Perfect Strategies for Non-Local Games

- Computer ScienceMathematical Physics, Analysis and Geometry
- 2020

A new class of non-local games, called imitation games, in which the players display linked behaviour, and which contain as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and unique games.

### Synchronous linear constraint system games

- Mathematics, Computer Science
- 2020

It is demonstrated that linear constraint system games are equivalent to graph isomorphism games on a pair of graphs parameterized by the linear system.

### Classification and Computability for Nonlocal Games

- Computer Science
- 2018

This work attempts to relate the two models, namely XOR games and linear system games, by studying the relationships between their strategies and refutations, and tries to understand when results for one model can be transferred to the other.

### Noncommutative Nullstellens\"atze and Perfect Games

- Mathematics
- 2021

The foundations of classical Algebraic Geometry and Real Algebraic Geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades the basic analogous theorems for matrix and…

### Robust self-testing for linear constraint system games

- Mathematics
- 2017

We study linear constraint system (LCS) games over the ring of arithmetic modulo $d$. We give a new proof that certain LCS games (the Mermin--Peres Magic Square and Magic Pentagram over binary…

### Entanglement in Non-local Games and the Hyperlinear Profile of Groups

- MathematicsAnnales Henri Poincaré
- 2018

We relate the amount of entanglement required to play linear system non-local games near-optimally to the hyperlinear profile of finitely presented groups. By calculating the hyperlinear profile of a…

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