45 research outputs found
A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs
The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier
analysis of real-valued functions on the Boolean cube. In this paper we present
a version of this inequality for matrix-valued functions on the Boolean cube.
Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also
present a number of applications. First, we analyze maps that encode
classical bits into qubits, in such a way that each set of bits can be
recovered with some probability by an appropriate measurement on the quantum
encoding; we show that if , then the success probability is
exponentially small in . This result may be viewed as a direct product
version of Nayak's quantum random access code bound. It in turn implies strong
direct product theorems for the one-way quantum communication complexity of
Disjointness and other problems. Second, we prove that error-correcting codes
that are locally decodable with 2 queries require length exponential in the
length of the encoded string. This gives what is arguably the first
``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf
using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the
proceedings of the IEEE FOCS 08 conferenc
A doubly exponential upper bound on noisy EPR states for binary games
This paper initiates the study of a class of entangled games, mono-state
games, denoted by , where is a two-player one-round game and
is a bipartite state independent of the game . In the mono-state game
, the players are only allowed to share arbitrary copies of .
This paper provides a doubly exponential upper bound on the copies of
for the players to approximate the value of the game to an arbitrarily small
constant precision for any mono-state binary game , if is a
noisy EPR state, which is a two-qubit state with completely mixed states as
marginals and maximal correlation less than . In particular, it includes
,
an EPR state with an arbitrary depolarizing noise .The structure of
the proofs is built the recent framework about the decidability of the
non-interactive simulation of joint distributions, which is completely
different from all previous optimization-based approaches or "Tsirelson's
problem"-based approaches. This paper develops a series of new techniques about
the Fourier analysis on matrix spaces and proves a quantum invariance principle
and a hypercontractive inequality of random operators. This novel approach
provides a new angle to study the decidability of the complexity class MIP,
a longstanding open problem in quantum complexity theory.Comment: The proof of Lemma C.9 is corrected. The presentation is improved.
Some typos are correcte
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
Quantum Random Access Codes for Boolean Functions
An random access code (RAC) is an encoding of
bits into bits such that any initial bit can be recovered with probability
at least , while in a quantum RAC (QRAC), the bits are encoded into
qubits. Since its proposal, the idea of RACs was generalized in many different
ways, e.g. allowing the use of shared entanglement (called
entanglement-assisted random access code, or simply EARAC) or recovering
multiple bits instead of one. In this paper we generalize the idea of RACs to
recovering the value of a given Boolean function on any subset of fixed
size of the initial bits, which we call -random access codes. We study and
give protocols for -random access codes with classical (-RAC) and quantum
(-QRAC) encoding, together with many different resources, e.g. private or
shared randomness, shared entanglement (-EARAC) and Popescu-Rohrlich boxes
(-PRRAC). The success probability of our protocols is characterized by the
\emph{noise stability} of the Boolean function . Moreover, we give an
\emph{upper bound} on the success probability of any -QRAC with shared
randomness that matches its success probability up to a multiplicative constant
(and -RACs by extension), meaning that quantum protocols can only achieve a
limited advantage over their classical counterparts.Comment: Final version to appear in Quantum. Small improvements to Theorem 2