538 research outputs found
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 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
Explicit lower and upper bounds on the entangled value of multiplayer XOR games
XOR games are the simplest model in which the nonlocal properties of
entanglement manifest themselves. When there are two players, it is well known
that the bias --- the maximum advantage over random play --- of entangled
players can be at most a constant times greater than that of classical players.
Recently, P\'{e}rez-Garc\'{i}a et al. [Comm. Math. Phys. 279 (2), 2008] showed
that no such bound holds when there are three or more players: the advantage of
entangled players over classical players can become unbounded, and scale with
the number of questions in the game. Their proof relies on non-trivial results
from operator space theory, and gives a non-explicit existence proof, leading
to a game with a very large number of questions and only a loose control over
the local dimension of the players' shared entanglement.
We give a new, simple and explicit (though still probabilistic) construction
of a family of three-player XOR games which achieve a large quantum-classical
gap (QC-gap). This QC-gap is exponentially larger than the one given by
P\'{e}rez-Garc\'{i}a et. al. in terms of the size of the game, achieving a
QC-gap of order with questions per player. In terms of the
dimension of the entangled state required, we achieve the same (optimal) QC-gap
of for a state of local dimension per player. Moreover, the
optimal entangled strategy is very simple, involving observables defined by
tensor products of the Pauli matrices.
Additionally, we give the first upper bound on the maximal QC-gap in terms of
the number of questions per player, showing that our construction is only
quadratically off in that respect. Our results rely on probabilistic estimates
on the norm of random matrices and higher-order tensors which may be of
independent interest.Comment: Major improvements in presentation; results identica
Algorithms, Bounds, and Strategies for Entangled XOR Games
We study the complexity of computing the commuting-operator value
of entangled XOR games with any number of players. We introduce necessary and
sufficient criteria for an XOR game to have , and use these
criteria to derive the following results:
1. An algorithm for symmetric games that decides in polynomial time whether
or , a task that was not previously known to be
decidable, together with a simple tensor-product strategy that achieves value 1
in the former case. The only previous candidate algorithm for this problem was
the Navascu\'{e}s-Pironio-Ac\'{i}n (also known as noncommutative Sum of Squares
or ncSoS) hierarchy, but no convergence bounds were known.
2. A family of games with three players and with , where it
takes doubly exponential time for the ncSoS algorithm to witness this (in
contrast with our algorithm which runs in polynomial time).
3. A family of games achieving a bias difference
arbitrarily close to the maximum possible value of (and as a consequence,
achieving an unbounded bias ratio), answering an open question of Bri\"{e}t and
Vidick.
4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games:
that is, we show that there exists a constant depending
only on the number of players, such that a random -XOR game over an
alphabet of size has with high probability when the number
of clauses is above .
5. A lower bound of on the number of levels
in the ncSoS hierarchy required to detect unsatisfiability for most random
3-XOR games. This is in contrast with the classical case where the -th level
of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all
possible solutions.Comment: 55 page
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