66 research outputs found
A remark on the paper ``Randomizing quantum states: Constructions and applications''
The concept of \e-randomizing quantum channels has been introduced by
Hayden, Leung, Shor and Winter in connection with approximately encrypting
quantum states. They proved using a discretization argument that sets of
roughly random unitary operators provide examples of such channels
on \C^d. We show that a simple trick improves the efficiency of the argument
and reduces the number of unitary operators to roughly
Partial transposition of random states and non-centered semicircular distributions
Let W be a Wishart random matrix of size d^2 times d^2, considered as a block
matrix with d times d blocks. Let Y be the matrix obtained by transposing each
block of W. We prove that the empirical eigenvalue distribution of Y approaches
a non-centered semicircular distribution when d tends to infinity. We also show
the convergence of extreme eigenvalues towards the edge of the expected
spectrum. The proofs are based on the moments method.
This matrix model is relevant to Quantum Information Theory and corresponds
to the partial transposition of a random induced state. A natural question is:
"When does a random state have a positive partial transpose (PPT)?". We answer
this question and exhibit a strong threshold when the parameter from the
Wishart distribution equals 4. When d gets large, a random state on C^d tensor
C^d obtained after partial tracing a random pure state over some ancilla of
dimension alpha.d^2 is typically PPT when alpha>4 and typically non-PPT when
alpha<4.Comment: 24 pages. V2 : includes the convergence of extreme eigenvalues. V3 :
includes probability estimates showing that p=4d^2 is a sharp threshol
Stochastic domination for iterated convolutions and catalytic majorization
We study how iterated convolutions of probability measures compare under
stochastic domination. We give necessary and sufficient conditions for the
existence of an integer such that is stochastically dominated by
for two given probability measures and . As a consequence
we obtain a similar theorem on the majorization order for vectors in . In
particular we prove results about catalysis in quantum information theory
Catalytic majorization and norms
An important problem in quantum information theory is the mathematical
characterization of the phenomenon of quantum catalysis: when can the
surrounding entanglement be used to perform transformations of a jointly held
quantum state under LOCC (local operations and classical communication) ?
Mathematically, the question amounts to describe, for a fixed vector , the
set of vectors such that we have for
some , where denotes the standard majorization relation. Our main
result is that the closure of in the norm can be fully
described by inequalities on the norms: for all
. This is a first step towards a complete description of
itself. It can also be seen as a -norm analogue of Ky Fan dominance
theorem about unitarily invariant norms. The proofs exploits links with another
quantum phenomenon: the possibiliy of multiple-copy transformations
( for given ). The main new tool is a
variant of Cram\'er$ theorem on large deviations for sums of i.i.d. random
variables
Realigning random states
We study how the realignment criterion (also called computable cross-norm
criterion) succeeds asymptotically in detecting whether random states are
separable or entangled. We consider random states on \C^d \otimes \C^d
obtained by partial tracing a Haar-distributed random pure state on \C^d
\otimes \C^d \otimes \C^s over an ancilla space \C^s. We show that, for
large , the realignment criterion typically detects entanglement if and only
if . In this sense, the realignment criterion is
asymptotically weaker than the partial transposition criterion
Maximal inequality for high-dimensional cubes
We present lower estimates for the best constant appearing in the weak
maximal inequality in the space (\R^n,\|\cdot\|_{\iy}). We show that
this constant grows to infinity faster than when tends
to infinity. To this end, we follow and simplify the approach used by J.M.
Aldaz. The new part of the argument relies on Donsker's theorem identifying the
Brownian bridge as the limit object describing the statistical distribution of
the coordinates of a point randomly chosen in the unit cube (
large).Comment: The overall presentation has been changed. To appear in Confluentes
Mathematic
Catalysis in the trace class and weak trace class ideals
Given operators in some ideal in the algebra
of all bounded operators on a separable Hilbert space , can
we give conditions guaranteeing the existence of a trace-class operator
such that is submajorized (in the sense of Hardy--Littlewood) by
? In the case when , a necessary and
almost sufficient condition is that the inequalities hold for every . We show that the analogous
statement fails for by connecting it
with the study of Dixmier traces
Maximal exponent of the Lorentz cones
We show that the maximal exponent (i.e., the minimum number of iterations
required for a primitive map to become strictly positive) of the n-dimensional
Lorentz cone is equal to n. As a byproduct, we show that the optimal exponent
in the quantum Wielandt inequality for qubit channels is equal to 3.Comment: 1 figure. v2: simplified the proof, added a section about quantum
Wielandt inequalit
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