2,035 research outputs found
An equivariant isomorphism theorem for mod reductions of arboreal Galois representations
Let be a quadratic, monic polynomial with coefficients in , where is a localization of a number ring
. In this paper, we first prove that if is non-square and
non-isotrivial, then there exists an absolute, effective constant with
the following property: for all primes
such that the reduced polynomial is non-square and non-isotrivial, the squarefree
Zsigmondy set of is bounded by . Using this
result, we prove that if is non-isotrivial and geometrically stable then
outside a finite, effective set of primes of the geometric
part of the arboreal representation of is isomorphic to
that of . As an application of our results we prove R. Jones' conjecture
on the arboreal Galois representation attached to the polynomial .Comment: Comments are welcome
Detecting Gaussian entanglement via extractable work
We show how the presence of entanglement in a bipartite Gaussian state can be
detected by the amount of work extracted by a continuos variable Szilard-like
device, where the bipartite state serves as the working medium of the engine.
We provide an expression for the work extracted in such a process and
specialize it to the case of Gaussian states. The extractable work provides a
sufficient condition to witness entanglement in generic two-mode states,
becoming also necessary for squeezed thermal states. We extend the protocol to
tripartite Gaussian states, and show that the full structure of inseparability
classes cannot be discriminated based on the extractable work. This suggests
that bipartite entanglement is the fundamental resource underpinning work
extraction.Comment: 12 pages, 8 figure
Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
The convex set of quantum states of a composite system with
positive partial transpose is analysed. A version of the hit and run algorithm
is used to generate a sequence of random points covering this set uniformly and
an estimation for the convergence speed of the algorithm is derived. For this algorithm works faster than sampling over the entire set of states and
verifying whether the partial transpose is positive. The level density of the
PPT states is shown to differ from the Marchenko-Pastur distribution, supported
in [0,4] and corresponding asymptotically to the entire set of quantum states.
Based on the shifted semi--circle law, describing asymptotic level density of
partially transposed states, and on the level density for the Gaussian unitary
ensemble with constraints for the spectrum we find an explicit form of the
probability distribution supported in [0,3], which describes well the level
density obtained numerically for PPT states.Comment: 11 pages, 4 figure
On the volume of the set of mixed entangled states
A natural measure in the space of density matrices describing N-dimensional
quantum systems is proposed. We study the probability P that a quantum state
chosen randomly with respect to the natural measure is not entangled (is
separable). We find analytical lower and upper bounds for this quantity.
Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate
that P decreases exponentially with N. Analysis of a conditional measure of
separability under the condition of fixed purity shows a clear dualism between
purity and separability: entanglement is typical for pure states, while
separability is connected with quantum mixtures. In particular, states of
sufficiently low purity are necessarily separable.Comment: 10 pages in LaTex - RevTex + 4 figures in eps. submitted to Phys.
Rev.
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