We show that the analysis of entanglement distillation protocols for qudits of arbitrary dimension D benefits from applying basic concepts from number theory, since the set \zdn associated to Bell diagonal states is a module rather than a vector space. We find that a partition of \zdn into divisor classes characterizes the invariant properties of mixed Bell diagonal states under local permutations. We construct a very general class of recursion protocols by means of unitary operations implementing these local permutations. We study these distillation protocols depending on whether we use twirling operations in the intermediate steps or not, and we study them both analytically and numerically with Monte Carlo methods. In the absence of twirling operations, we construct extensions of the quantum privacy algorithms valid for secure communications with qudits of any dimension D. When D is a prime number, we show that distillation protocols are optimal both qualitatively and quantitatively.The result means that qupits represented by quantum systems with p states, p prime, are in a very special position as far quantum computation is considered. I have been talking about the importance of qupits for quantum computation and information processing in TGD Universe for about half decade now, see for instance the chapters Negentropy Maximization Principle, Intentionality, Cognition, and Physics as Number Theory or Space-Time Point as Platonia , Topological Quantum Computation in TGD Universe . As one might guess, colleagues who read only respected journals have not shown a slightest interest. There are however notable exceptions. In particular, Roger Penrose who mentions my work with p-adic numbers in his newest book "Road to Reality". My sincere hope is that this could help these revolutionary ideas to find young brains as templates for growth.
Wednesday, March 02, 2005
Primes and Quantum Computation
The morning visit to hep-th is nowadays an irritating experience since most of the stuff is wastes produced by the dying M-theory. In this morning cross-listings however contained a highly interesting looking article quant-ph/0503013 with title "Entanglement Distillation Protocols and Number Theory" by H. Bombin, M.A. Martin-Delgado. I attach here the abstract.