438 research outputs found
On the Grothendieck Theorem for jointly completely bounded bilinear forms
We show how the proof of the Grothendieck Theorem for jointly completely
bounded bilinear forms on C*-algebras by Haagerup and Musat can be modified in
such a way that the method of proof is essentially C*-algebraic. To this
purpose, we use Cuntz algebras rather than type III factors. Furthermore, we
show that the best constant in Blecher's inequality is strictly greater than
one.Comment: 9 pages, minor change
Quantum Steering and Space-Like Separation
In non-relativistic quantum mechanics, measurements performed by separate
observers are modeled via tensor products. In Algebraic Quantum Field Theory,
though, local observables corresponding to space-like separated parties are
just required to commute. The problem of determining whether these two
definitions of "separation" lead to the same set of bipartite correlations is
known in non-locality as Tsirelson's problem. In this article, we prove that
the analog of Tsirelson's problem in steering scenarios is false. That is,
there exists a steering inequality that can be violated or not depending on how
we define space-like separation at the operator level.Comment: Some typos corrected. Short discussion about Algebraic Quantum Field
Theory. Modified introduction and conclusio
The Kt-Functional for the Interpolation Couple L1(A0), Lâ(A1)
AbstractLet (A0, A1) be a compatible couple of Banach spaces in the interpolation theory sense. We give a formula for the Kt-functional of the interpolation couples (l1(A0), c0(A1)) or (l1(A0), lâ(A1)) and (L1(A0), Lâ(A1))
Matrix Product State and mean field solutions for one-dimensional systems can be found efficiently
We consider the problem of approximating ground states of one-dimensional
quantum systems within the two most common variational ansatzes, namely the
mean field ansatz and Matrix Product States. We show that both for mean field
and for Matrix Product States of fixed bond dimension, the optimal solutions
can be found in a way which is provably efficient (i.e., scales polynomially).
This implies that the corresponding variational methods can be in principle
recast in a way which scales provably polynomially. Moreover, our findings
imply that ground states of one-dimensional commuting Hamiltonians can be found
efficiently.Comment: 5 pages; v2: accepted version, Journal-ref adde
Tsirelson's problem and Kirchberg's conjecture
Tsirelson's problem asks whether the set of nonlocal quantum correlations
with a tensor product structure for the Hilbert space coincides with the one
where only commutativity between observables located at different sites is
assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products
of C*-algebras would imply a positive answer to this question for all bipartite
scenarios. This remains true also if one considers not only spatial
correlations, but also spatiotemporal correlations, where each party is allowed
to apply their measurements in temporal succession; we provide an example of a
state together with observables such that ordinary spatial correlations are
local, while the spatiotemporal correlations reveal nonlocality. Moreover, we
find an extended version of Tsirelson's problem which, for each nontrivial Bell
scenario, is equivalent to the QWEP conjecture. This extended version can be
conveniently formulated in terms of steering the system of a third party.
Finally, a comprehensive mathematical appendix offers background material on
complete positivity, tensor products of C*-algebras, group C*-algebras, and
some simple reformulations of the QWEP conjecture.Comment: 57 pages, to appear in Rev. Math. Phy
Phase transitions for random states and a semi-circle law for the partial transpose
For a system of N identical particles in a random pure state, there is a
threshold k_0 = k_0(N) ~ N/5 such that two subsystems of k particles each
typically share entanglement if k > k_0, and typically do not share
entanglement if k < k_0. By "random" we mean here "uniformly distributed on the
sphere of the corresponding Hilbert space." The analogous phase transition for
the positive partial transpose (PPT) property can be described even more
precisely. For example, for N qubits the two subsystems of size k are typically
in a PPT state if k
k_1. Since, for a given state of the entire system, the induced state of a
subsystem is given by the partial trace, the above facts can be rephrased as
properties of random induced states. An important step in the analysis depends
on identifying the asymptotic spectral density of the partial transposes of
such random induced states, a result which is interesting in its own right.Comment: 5 pages, 2 figures. This short note contains a high-level overview of
two long and technical papers, arXiv:1011.0275 and arXiv:1106.2264. Version
2: unchanged results, editorial changes, added reference, close to the
published articl
More efficient Bell inequalities for Werner states
In this paper we study the nonlocal properties of two-qubit Werner states
parameterized by the visibility parameter 0<p<1. New family of Bell
inequalities are constructed which prove the two-qubit Werner states to be
nonlocal for the parameter range 0.7056<p<1. This is slightly wider than the
range 0.7071<p<1, corresponding to the violation of the
Clauser-Horne-Shimony-Holt (CHSH) inequality. This answers a question posed by
Gisin in the positive, i.e., there exist Bell inequalities which are more
efficient than the CHSH inequality in the sense that they are violated by a
wider range of two-qubit Werner states.Comment: 7 pages, 1 figur
Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality
In noncommutative probability theory independence can be based on free
products instead of tensor products. This yields a highly noncommutative
theory: free probability . Here we show that the classical Shannon's entropy
power inequality has a counterpart for the free analogue of entropy .
The free entropy (introduced recently by the second named author),
consistently with Boltzmann's formula , was defined via volumes of
matricial microstates. Proving the free entropy power inequality naturally
becomes a geometric question.
Restricting the Minkowski sum of two sets means to specify the set of pairs
of points which will be added. The relevant inequality, which holds when the
set of "addable" points is sufficiently large, differs from the Brunn-Minkowski
inequality by having the exponent replaced by . Its proof uses the
rearrangement inequality of Brascamp-Lieb-L\"uttinger
Amenability of algebras of approximable operators
We give a necessary and sufficient condition for amenability of the Banach
algebra of approximable operators on a Banach space. We further investigate the
relationship between amenability of this algebra and factorization of
operators, strengthening known results and developing new techniques to
determine whether or not a given Banach space carries an amenable algebra of
approximable operators. Using these techniques, we are able to show, among
other things, the non-amenability of the algebra of approximable operators on
Tsirelson's space.Comment: 20 pages, to appear in Israel Journal of Mathematic
Noncommutative Figa-Talamanca-Herz algebras for Schur multipliers
We introduce a noncommutative analogue of the Fig\'a-Talamanca-Herz algebra
on the natural predual of the operator space of
completely bounded Schur multipliers on Schatten space . We determine the
isometric Schur multipliers and prove that the space of bounded
Schur multipliers on Schatten space is the closure in the weak operator
topology of the span of isometric multipliers.Comment: 24 pages; corrected typo
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