248 research outputs found
Large bipartite Bell violations with dichotomic measurements
In this paper we introduce a simple and natural bipartite Bell scenario, by
considering the correlations between two parties defined by general
measurements in one party and dichotomic ones in the other. We show that
unbounded Bell violations can be obtained in this context. Since such
violations cannot occur when both parties use dichotomic measurements, our
setting can be considered as the simplest one where this phenomenon can be
observed. Our example is essentially optimal in terms of the outputs and the
Hilbert space dimension
Unbounded violations of bipartite Bell Inequalities via Operator Space theory
In this work we show that bipartite quantum states with local Hilbert space
dimension n can violate a Bell inequality by a factor of order (up
to a logarithmic factor) when observables with n possible outcomes are used. A
central tool in the analysis is a close relation between this problem and
operator space theory and, in particular, the very recent noncommutative
embedding theory. As a consequence of this result, we obtain better Hilbert
space dimension witnesses and quantum violations of Bell inequalities with
better resistance to noise
Quantum query algorithms are completely bounded forms
We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials
Quantum Query Algorithms are Completely Bounded Forms
We prove a characterization of t-query quantum algorithms in terms of the unit ball of a
space of degree-2t polynomials. Based on this, we obtain a refined notion of approximate polynomial
degree that equals the quantum query complexity, answering a question of Aaronson et
al. (CCC’16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct.
Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels
to multilinear forms. Using our characterization, we show that many polynomials of degree
four are far from those coming from two-query quantum algorithms. We also give a simple and
short proof of one of the results of Aaronson et al. showing an equivalence between one-query
quantum algorithms and bounded quadratic polynomials
Connes' embedding problem and Tsirelson's problem
We show that Tsirelson's problem concerning the set of quantum correlations
and Connes' embedding problem on finite approximations in von Neumann algebras
(known to be equivalent to Kirchberg's QWEP conjecture) are essentially
equivalent. Specifically, Tsirelson's problem asks whether the set of bipartite
quantum correlations generated between tensor product separated systems is the
same as the set of correlations between commuting C*-algebras. Connes'
embedding problem asks whether any separable II factor is a subfactor of
the ultrapower of the hyperfinite II factor. We show that an affirmative
answer to Connes' question implies a positive answer to Tsirelson's.
Conversely, a positve answer to a matrix valued version of Tsirelson's problem
implies a positive one to Connes' problem
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