862 research outputs found
A priori probability that a qubit-qutrit pair is separable
We extend to arbitrarily coupled pairs of qubits (two-state quantum systems)
and qutrits (three-state quantum systems) our earlier study (quant-ph/0207181),
which was concerned with the simplest instance of entangled quantum systems,
pairs of qubits. As in that analysis -- again on the basis of numerical
(quasi-Monte Carlo) integration results, but now in a still higher-dimensional
space (35-d vs. 15-d) -- we examine a conjecture that the Bures/SD (statistical
distinguishability) probability that arbitrarily paired qubits and qutrits are
separable (unentangled) has a simple exact value, u/(v Pi^3)= >.00124706, where
u = 2^20 3^3 5 7 and v = 19 23 29 31 37 41 43 (the product of consecutive
primes). This is considerably less than the conjectured value of the Bures/SD
probability, 8/(11 Pi^2) = 0736881, in the qubit-qubit case. Both of these
conjectures, in turn, rely upon ones to the effect that the SD volumes of
separable states assume certain remarkable forms, involving "primorial"
numbers. We also estimate the SD area of the boundary of separable qubit-qutrit
states, and provide preliminary calculations of the Bures/SD probability of
separability in the general qubit-qubit-qubit and qutrit-qutrit cases.Comment: 9 pages, 3 figures, 2 tables, LaTeX, we utilize recent exact
computations of Sommers and Zyczkowski (quant-ph/0304041) of "the Bures
volume of mixed quantum states" to refine our conjecture
How many zeros of a random polynomial are real?
We provide an elementary geometric derivation of the Kac integral formula for
the expected number of real zeros of a random polynomial with independent
standard normally distributed coefficients. We show that the expected number of
real zeros is simply the length of the moment curve
projected onto the surface of the unit sphere, divided by . The
probability density of the real zeros is proportional to how fast this curve is
traced out. We then relax Kac's assumptions by considering a variety of random
sums, series, and distributions, and we also illustrate such ideas as integral
geometry and the Fubini-Study metric.Comment: 37 page
Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration
Several fundamental problems that arise in optimization and computer science
can be cast as follows: Given vectors and a
constraint family , find a set that
maximizes the squared volume of the simplex spanned by the vectors in . A
motivating example is the data-summarization problem in machine learning where
one is given a collection of vectors that represent data such as documents or
images. The volume of a set of vectors is used as a measure of their diversity,
and partition or matroid constraints over are imposed in order to ensure
resource or fairness constraints. Recently, Nikolov and Singh presented a
convex program and showed how it can be used to estimate the value of the most
diverse set when corresponds to a partition matroid. This result was
recently extended to regular matroids in works of Straszak and Vishnoi, and
Anari and Oveis Gharan. The question of whether these estimation algorithms can
be converted into the more useful approximation algorithms -- that also output
a set -- remained open.
The main contribution of this paper is to give the first approximation
algorithms for both partition and regular matroids. We present novel
formulations for the subdeterminant maximization problem for these matroids;
this reduces them to the problem of finding a point that maximizes the absolute
value of a nonconvex function over a Cartesian product of probability
simplices. The technical core of our results is a new anti-concentration
inequality for dependent random variables that allows us to relate the optimal
value of these nonconvex functions to their value at a random point. Unlike
prior work on the constrained subdeterminant maximization problem, our proofs
do not rely on real-stability or convexity and could be of independent interest
both in algorithms and complexity.Comment: in FOCS 201
Constrained Triangulations, Volumes of Polytopes, and Unit Equations
Given a polytope P in R^d and a subset U of its vertices, is there a triangulation of P using d-simplices that all contain U? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of P. Our proof relates triangulations of P to triangulations of its "shadow", a projection to a lower-dimensional space determined by U. In particular, we obtain a formula relating the volume of P with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations
Asymptotic properties of entanglement polytopes for large number of qubits
Entanglement polytopes have been recently proposed as the way of witnessing
the SLOCC multipartite entanglement classes using single particle information.
We present first asymptotic results concerning feasibility of this approach for
large number of qubits. In particular we show that entanglement polytopes of
-qubit system accumulate in the distance from the
point corresponding to the maximally mixed reduced one-qubit density matrices.
This implies existence of a possibly large region where many entanglement
polytopes overlap, i.e where the witnessing power of entanglement polytopes is
weak. Moreover, the witnessing power cannot be strengthened by any entanglement
distillation protocol as for large the required purity is above current
capability.Comment: 5 pages, 4 figure
The Aemulus Project I: Numerical Simulations for Precision Cosmology
The rapidly growing statistical precision of galaxy surveys has lead to a
need for ever-more precise predictions of the observables used to constrain
cosmological and galaxy formation models. The primary avenue through which such
predictions will be obtained is suites of numerical simulations. These
simulations must span the relevant model parameter spaces, be large enough to
obtain the precision demanded by upcoming data, and be thoroughly validated in
order to ensure accuracy. In this paper we present one such suite of
simulations, forming the basis for the AEMULUS Project, a collaboration devoted
to precision emulation of galaxy survey observables. We have run a set of 75
(1.05 h^-1 Gpc)^3 simulations with mass resolution and force softening of
3.51\times 10^10 (Omega_m / 0.3) ~ h^-1 M_sun and 20 ~ h^-1 kpc respectively in
47 different wCDM cosmologies spanning the range of parameter space allowed by
the combination of recent Cosmic Microwave Background, Baryon Acoustic
Oscillation and Type Ia Supernovae results. We present convergence tests of
several observables including spherical overdensity halo mass functions, galaxy
projected correlation functions, galaxy clustering in redshift space, and
matter and halo correlation functions and power spectra. We show that these
statistics are converged to 1% (2%) for halos with more than 500 (200)
particles respectively and scales of r>200 ~ h^-1 kpc in real space or k ~ 3 h
Mpc^-1 in harmonic space for z\le 1. We find that the dominant source of
uncertainty comes from varying the particle loading of the simulations. This
leads to large systematic errors for statistics using halos with fewer than 200
particles and scales smaller than k ~ 4 h^-1 Mpc. We provide the halo catalogs
and snapshots detailed in this work to the community at
https://AemulusProject.github.io.Comment: 16 pages, 12 figures, 3 Tables Project website:
https://aemulusproject.github.io
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