301 research outputs found
Entanglement of approximate quantum strategies in XOR games
We show that for any Īµ > 0 there is an XOR game G = G(Īµ) with Ī(Īµ^(ā1/5)) inputs for one player and Ī(Īµ^(ā2/5)) inputs for the other player such that Ī©(Īµ^(ā1/5)) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1āĪµ) from optimal. This gives an exponential improvement in both the number of inputs or outputs and the noise tolerance of any previously-known self-test for highly entangled states. Up to the exponent ā1/5 the scaling of our bound with Īµ is tight: for any XOR game there is an Īµ-optimal strategy using āĪµ^(ā1)ā ebits, irrespective of the number of questions in the game
Entanglement of Approximate Quantum Strategies in XOR Games
We characterize the amount of entanglement that is sufficient to play any XOR game near-optimally. We show that for any XOR game and \eps>0 there is an \eps-optimal strategy for using \lceil \eps^{-1} \rceil ebits of entanglement, irrespective of the number of questions in the game. By considering the family of XOR games CHSH() introduced by Slofstra (Jour. Math. Phys. 2011), we show that this bound is nearly tight: for any \eps>0 there is an n = \Theta(\eps^{-1/5}) such that \Omega(\eps^{-1/5}) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1-\eps) from optimal in CHSH()
Lower bounds on the entanglement needed to play XOR non-local games
We give an explicit family of XOR games with O(n)-bit questions requiring 2^n
ebits to play near-optimally. More generally we introduce a new technique for
proving lower bounds on the amount of entanglement required by an XOR game: we
show that near-optimal strategies for an XOR game G correspond to approximate
representations of a certain C^*-algebra associated to G. Our results extend an
earlier theorem of Tsirelson characterising the set of quantum strategies which
implement extremal quantum correlations.Comment: 20 pages, no figures. Corrected abstract, body of paper unchange
Quantum XOR Games
We introduce quantum XOR games, a model of two-player one-round games that
extends the model of XOR games by allowing the referee's questions to the
players to be quantum states. We give examples showing that quantum XOR games
exhibit a wide range of behaviors that are known not to exist for standard XOR
games, such as cases in which the use of entanglement leads to an arbitrarily
large advantage over the use of no entanglement. By invoking two deep
extensions of Grothendieck's inequality, we present an efficient algorithm that
gives a constant-factor approximation to the best performance players can
obtain in a given game, both in case they have no shared entanglement and in
case they share unlimited entanglement. As a byproduct of the algorithm we
prove some additional interesting properties of quantum XOR games, such as the
fact that sharing a maximally entangled state of arbitrary dimension gives only
a small advantage over having no entanglement at all.Comment: 43 page
Entangled Games Are Hard to Approximate
We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1-1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be āroundedā to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games
Quantum bounds on multiplayer linear games and device-independent witness of genuine tripartite entanglement
Here we study multiplayer linear games, a natural generalization of XOR games
to multiple outcomes. We generalize a recently proposed efficiently computable
bound, in terms of the norm of a game matrix, on the quantum value of 2-player
games to linear games with players. As an example, we bound the quantum
value of a generalization of the well-known CHSH game to players and
outcomes. We also apply the bound to show in a simple manner that any
nontrivial functional box, that could lead to trivialization of communication
complexity in a multiparty scenario, cannot be realized in quantum mechanics.
We then present a systematic method to derive device-independent witnesses of
genuine tripartite entanglement.Comment: 7+8 page
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