The Complexity of Divisibility
Abstract
We address two sets of longstanding open questions in probability theory, from a computational complexity perspective: divisibility of stochastic maps, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic maps is an NPcomplete problem, and extend this result to nonnegative matrices, and completelypositive tracepreserving maps, i.e. the quantum analogue of stochastic maps. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is in P, but decomposability is NPhard. For the former, we give an explicit polynomialtime algorithm. All results on distributions extend to weakmembership formulations, proving that the complexity of these problems is robust to perturbations.
 Publication:

arXiv eprints
 Pub Date:
 November 2014
 DOI:
 10.48550/arXiv.1411.7380
 arXiv:
 arXiv:1411.7380
 Bibcode:
 2014arXiv1411.7380B
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Quantum Physics;
 6008 (Primary);
 81Q08;
 68Q30 (Secondary);
 F.2.1;
 G.3;
 J.2
 EPrint:
 50 pages, 11 figures. Journalaccepted version