397 research outputs found
The structure of optimal and nearly-optimal quantum strategies for non-local XOR games
We study optimal and nearly-optimal quantum strategies for non-local XOR
games. First, we prove the following general result: for every non-local XOR
game, there exists a set of relations with the properties: (1) a quantum
strategy is optimal for the game if and only if it satisfies the relations, and
(2) a quantum strategy is nearly optimal for the game if and only if it
approximately satisfies the relations. Next, we focus attention on a specific
infinite family of XOR games: the CHSH(n) games. This family generalizes the
well-known CHSH game. We describe the general form of CHSH(n) optimal
strategies. Then, we adapt the concept of intertwining operator from
representation theory and use that to characterize nearly-optimal CHSH(n)
strategies
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
Robust Randomness Amplifiers: Upper and Lower Bounds
A recent sequence of works, initially motivated by the study of the nonlocal
properties of entanglement, demonstrate that a source of
information-theoretically certified randomness can be constructed based only on
two simple assumptions: the prior existence of a short random seed and the
ability to ensure that two black-box devices do not communicate (i.e. are
non-signaling). We call protocols achieving such certified amplification of a
short random seed randomness amplifiers.
We introduce a simple framework in which we initiate the systematic study of
the possibilities and limitations of randomness amplifiers. Our main results
include a new, improved analysis of a robust randomness amplifier with
exponential expansion, as well as the first upper bounds on the maximum
expansion achievable by a broad class of randomness amplifiers. In particular,
we show that non-adaptive randomness amplifiers that are robust to noise cannot
achieve more than doubly exponential expansion. Finally, we show that a wide
class of protocols based on the use of the CHSH game can only lead to (singly)
exponential expansion if adversarial devices are allowed the full power of
non-signaling strategies. Our upper bound results apply to all known
non-adaptive randomness amplifier constructions to date.Comment: 28 pages. Comments welcom
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Bell Violations through Independent Bases Games
In a recent paper, Junge and Palazuelos presented two two-player games
exhibiting interesting properties. In their first game, entangled players can
perform notably better than classical players. The quantitative gap between the
two cases is remarkably large, especially as a function of the number of inputs
to the players. In their second game, entangled players can perform notably
better than players that are restricted to using a maximally entangled state
(of arbitrary dimension). This was the first game exhibiting such a behavior.
The analysis of both games is heavily based on non-trivial results from Banach
space theory and operator space theory. Here we present two games exhibiting a
similar behavior, but with proofs that are arguably simpler, using elementary
probabilistic techniques and standard quantum information arguments. Our games
also give better quantitative bounds.Comment: Minor update
Test for a large amount of entanglement, using few measurements
Bell-inequality violations establish that two systems share some quantum
entanglement. We give a simple test to certify that two systems share an
asymptotically large amount of entanglement, n EPR states. The test is
efficient: unlike earlier tests that play many games, in sequence or in
parallel, our test requires only one or two CHSH games. One system is directed
to play a CHSH game on a random specified qubit i, and the other is told to
play games on qubits {i,j}, without knowing which index is i.
The test is robust: a success probability within delta of optimal guarantees
distance O(n^{5/2} sqrt{delta}) from n EPR states. However, the test does not
tolerate constant delta; it breaks down for delta = Omega~(1/sqrt{n}). We give
an adversarial strategy that succeeds within delta of the optimum probability
using only O~(delta^{-2}) EPR states.Comment: 17 pages, 2 figures. Journal versio
Self-testing of binary observables based on commutation
We consider the problem of certifying binary observables based on a Bell
inequality violation alone, a task known as self-testing of measurements. We
introduce a family of commutation-based measures, which encode all the distinct
arrangements of two projective observables on a qubit. These quantities by
construction take into account the usual limitations of self-testing and since
they are "weighted" by the (reduced) state, they automatically deal with
rank-deficient reduced density matrices. We show that these measures can be
estimated from the observed Bell violation in several scenarios and the proofs
rely only on standard linear algebra. The trade-offs turn out to be tight and,
in particular, they give non-trivial statements for arbitrarily small
violations. On the other extreme, observing the maximal violation allows us to
deduce precisely the form of the observables, which immediately leads to a
complete rigidity statement. In particular, we show that for all the
-partite Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the
-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubit
measurements on every party. Our results imply that any pair of projective
observables on a qubit can be certified in a truly robust manner. Finally, we
show that commutation-based measures give a convenient way of expressing
relations among more than two observables.Comment: 5 + 4 pages. v2: published version; v3: formatting errors fixe
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