17,439 research outputs found
Trumping and Power Majorization
Majorization is a basic concept in matrix theory that has found applications
in numerous settings over the past century. Power majorization is a more
specialized notion that has been studied in the theory of inequalities. On the
other hand, the trumping relation has recently been considered in quantum
information, specifically in entanglement theory. We explore the connections
between trumping and power majorization. We prove an analogue of Rado's theorem
for power majorization and consider a number of examples.Comment: 8 page
Quantum Error Correction and One-Way LOCC State Distinguishability
We explore the intersection of studies in quantum error correction and
quantum local operations and classical communication (LOCC). We consider
one-way LOCC measurement protocols as quantum channels and investigate their
error correction properties, emphasizing an operator theory approach to the
subject, and we obtain new applications to one-way LOCC state
distinguishability as well as new derivations of some established results. We
also derive conditions on when states that arise through the stabilizer
formalism for quantum error correction are distinguishable under one-way LOCC.Comment: 20 page
Primes in arithmetic progressions and semidefinite programming
Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the
size of intervals that contain primes from a given arithmetic progression using
the approach developed by Carneiro, Milinovich and Soundararajan [Comment.
Math. Helv. 94, no. 3 (2019)]. For this we extend the Guinand-Weil explicit
formula over all Dirichlet characters modulo , and we reduce the
associated extremal problems to convex optimization problems that can be solved
numerically via semidefinite programming.Comment: 11 pages, 5 ancillary file
On Improving Local Search for Unsatisfiability
Stochastic local search (SLS) has been an active field of research in the
last few years, with new techniques and procedures being developed at an
astonishing rate. SLS has been traditionally associated with satisfiability
solving, that is, finding a solution for a given problem instance, as its
intrinsic nature does not address unsatisfiable problems. Unsatisfiable
instances were therefore commonly solved using backtrack search solvers. For
this reason, in the late 90s Selman, Kautz and McAllester proposed a challenge
to use local search instead to prove unsatisfiability. More recently, two SLS
solvers - Ranger and Gunsat - have been developed, which are able to prove
unsatisfiability albeit being SLS solvers. In this paper, we first compare
Ranger with Gunsat and then propose to improve Ranger performance using some of
Gunsat's techniques, namely unit propagation look-ahead and extended
resolution
Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory
We examine k-minimal and k-maximal operator spaces and operator systems, and
investigate their relationships with the separability problem in quantum
information theory. We show that the matrix norms that define the k-minimal
operator spaces are equal to a family of norms that have been studied
independently as a tool for detecting k-positive linear maps and bound
entanglement. Similarly, we investigate the k-super minimal and k-super maximal
operator systems that were recently introduced and show that their cones of
positive elements are exactly the cones of k-block positive operators and
(unnormalized) states with Schmidt number no greater than k, respectively. We
characterize a class of norms on the k-super minimal operator systems and show
that the completely bounded versions of these norms provide a criterion for
testing the Schmidt number of a quantum state that generalizes the
recently-developed separability criterion based on trace-contractive maps.Comment: 17 pages, to appear in JF
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