152 research outputs found
Robust self-testing of many-qubit states
We introduce a simple two-player test which certifies that the players apply
tensor products of Pauli and observables on the tensor
product of EPR pairs. The test has constant robustness: any strategy
achieving success probability within an additive of the optimal
must be -close, in the appropriate distance
measure, to the honest -qubit strategy. The test involves -bit questions
and -bit answers. The key technical ingredient is a quantum version of the
classical linearity test of Blum, Luby, and Rubinfeld.
As applications of our result we give (i) the first robust self-test for
EPR pairs; (ii) a quantum multiprover interactive proof system for the local
Hamiltonian problem with a constant number of provers and classical questions
and answers, and a constant completeness-soundness gap independent of system
size; (iii) a robust protocol for delegated quantum computation.Comment: 36 pages. Improves upon and supersedes our earlier submission
arXiv:1512.0209
Entanglement in non-local games and the hyperlinear profile of groups
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 certain group, we give an example
of a finite non-local game for which the amount of entanglement required to
play -optimally is at least , for some
. Since this function approaches infinity as approaches
zero, this provides a quantitative version of a theorem of the first author.Comment: 27 pages. v2: improved results based on a suggestion by N. Ozaw
Unbounded entanglement in nonlocal games
Quantum entanglement is known to provide a strong advantage in many two-party
distributed tasks. We investigate the question of how much entanglement is
needed to reach optimal performance. For the first time we show that there
exists a purely classical scenario for which no finite amount of entanglement
suffices. To this end we introduce a simple two-party nonlocal game ,
inspired by Lucien Hardy's paradox. In our game each player has only two
possible questions and can provide bit strings of any finite length as answer.
We exhibit a sequence of strategies which use entangled states in increasing
dimension and succeed with probability for some .
On the other hand, we show that any strategy using an entangled state of local
dimension has success probability at most . In addition,
we show that any strategy restricted to producing answers in a set of
cardinality at most has success probability at most .
Finally, we generalize our construction to derive similar results starting from
any game with two questions per player and finite answers sets in which
quantum strategies have an advantage.Comment: We have removed the inaccurate discussion of infinite-dimensional
strategies in Section 5. Other minor correction
Elementary Proofs of Grothendieck Theorems for Completely Bounded Norms
We provide alternative proofs of two recent Grothendieck theorems for jointly
completely bounded bilinear forms, originally due to Pisier and Shlyakhtenko
(Invent. Math. 2002) and Haagerup and Musat (Invent. Math. 2008). Our proofs
are elementary and are inspired by the so-called embezzlement states in quantum
information theory. Moreover, our proofs lead to quantitative estimates.Comment: 14 page
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
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