817 research outputs found
Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods
We consider the problem of approximating the von Neumann entropy of a large,
sparse, symmetric positive semidefinite matrix , defined as
where . After establishing some useful
properties of this matrix function, we consider the use of both polynomial and
rational Krylov subspace algorithms within two types of approximations methods,
namely, randomized trace estimators and probing techniques based on graph
colorings. We develop error bounds and heuristics which are employed in the
implementation of the algorithms. Numerical experiments on density matrices of
different types of networks illustrate the performance of the methods.Comment: 32 pages, 10 figure
Faster randomized partial trace estimation
We develop randomized matrix-free algorithms for estimating partial traces.
Our algorithm improves on the typicality-based approach used in [T. Chen and
Y-C. Cheng, Numerical computation of the equilibrium-reduced density matrix for
strongly coupled open quantum systems, J. Chem. Phys. 157, 064106 (2022)] by
deflating important subspaces (e.g. corresponding to the low-energy
eigenstates) explicitly. This results in a significant variance reduction for
matrices with quickly decaying singular values. We then apply our algorithm to
study the thermodynamics of several Heisenberg spin systems, particularly the
entanglement spectrum and ergotropy
Minimum Relative Entropy for Quantum Estimation: Feasibility and General Solution
We propose a general framework for solving quantum state estimation problems
using the minimum relative entropy criterion. A convex optimization approach
allows us to decide the feasibility of the problem given the data and, whenever
necessary, to relax the constraints in order to allow for a physically
admissible solution. Building on these results, the variational analysis can be
completed ensuring existence and uniqueness of the optimum. The latter can then
be computed by standard, efficient standard algorithms for convex optimization,
without resorting to approximate methods or restrictive assumptions on its
rank.Comment: 9 pages, no figure
There is entanglement in the primes
Large series of prime numbers can be superposed on a single quantum register
and then analyzed in full parallelism. The construction of this Prime state is
efficient, as it hinges on the use of a quantum version of any efficient
primality test. We show that the Prime state turns out to be very entangled as
shown by the scaling properties of purity, Renyi entropy and von Neumann
entropy. An analytical approximation to these measures of entanglement can be
obtained from the detailed analysis of the entanglement spectrum of the Prime
state, which in turn produces new insights in the Hardy-Littlewood conjecture
for the pairwise distribution of primes. The extension of these ideas to a Twin
Prime state shows that this new state is even more entangled than the Prime
state, obeying majorization relations. We further discuss the construction of
quantum states that encompass relevant series of numbers and opens the
possibility of applying quantum computation to Arithmetics in novel ways.Comment: 30 pages, 11 Figs. Addition of two references and correction of typo
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