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.
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.

1. Number theory allows definition of entanglement negentropy

p-Adic approach leads to the generalization of the notion of Shannon entropy and thus of entanglement entropy defined by the same formula. The idea is to replace logarithms of probabilities appearing in the formula with the logarithms of their p-adic norms. This certainly makes sense if probabilities are rational numbers. It makes also sense for algebraic extensions of rationals provided an appropriate extension of p-adic numbers is introduced and norm generalized appropriately. What is fascinating is that the resulting entanglement entropy can be negative so that entropy becomes negentropy! One can always find the prime for which it is maximally negative and identify it as a prime characterizing the system in question. This assignment can be generalized so that it applies at space-time level too. In this case maximal non-deterministic regions of space-time surface define an ensemble, and the prime p assignable to space-time sheet is some prime factor of the number N of these regions. One can say that rational and even algebraic entanglement carries genuine information. When the number of entangled state is prime or power of prime, entanglement negentropy increases dramatically. This would predict that physical systems for which state space is p^n-dimensional, p prime, should be of special importance. In p-adic physics this kind of systems appear naturally. Also this idea generalizes to space-time level. In TGD inspired theory of consciousness rational entanglement can be identified as a correlate for an experience of understanding. Quite generally, rational, and more generally algebraic, entanglement suggests strongly the identification as bound state entanglement. Thus in quantum jump this kind of entanglement would be stable whereas transcendental entanglement would be unstable. For ideal quantum computer entanglement probabilities are indeed simple rational numbers. That bound state entanglement is in question might imply stability and allow to circumvent the fundamental problem caused by the fact that entanglement is highly unstable in the world governed by the standard quantum theory. In quantum computing p^n-state systems should be of special importance since qupits are expected to be exceptionally stable. A general mechanism of macroscopic and macrotemporal quantum coherence based on spin glass property of TGD Universe, and possible applications to quantum biology are discussed in Macro-Temporal Quantum Coherence and Spin Glass Degeneracy .

2. Rationals and algebraic numbers are special also physically

The exceptional role of rationals has a nice correlate in what I take a liberty to call arithmetic dynamics. Rationals can be regarded as islands of order in the chaotic sea of transcendentals in the sense that the expansion of rational in terms of any integer n> 1 is periodic and thus analog to a periodic orbit of a dynamical system. For transcendentals this is not the case. Algebraic numbers are islands of order in weaker sense since they become rational by a finite number of algebraic operations. Rationals have a special role also in classical mechanics. Resonances occur in many particle systems (such as planetary systems) when the frequencies of periodic motions are rational multiples of basic frequency so that the entire system can behave strictly periodically and oscillate coherently. This is in accordance with the interpretation as formation of bound states. One of the basic characteristics of brain is entrainment with external frequencies. There is evidence that radiation with cyclotron frequencies of important biological ions in Earth's magnetic field has special effects on brain. Schumann frequencies appear in EEG also. TGD interpretation would be that the magnetic body associated with the biological body is of Earth size at least (entire scale hierarchy of them is predicted), and that there is also a strong resonant coupling with the magnetic body of Earth, even to the extent that bound states are formed. Libet's findings about strange time delays of conscious experience find a nice explanation if EEG mediates sensory input from brain to magnetic body and intentional motor control from magnetic body to brain.

3. Primes for tensor product are prime-dimensional vector spaces

For physicist it might become as a surprise how general the notion of prime is. When you have some algebraic product you can immediate ask whether there exist elements of algebra serving as "elementary particles" of the algebra in the sense that you can express any algebra element as product of these elements. In case of number fields these building bricks are highly unique. For instance, all classical division fields allow the notion of primeness and the notions of ordinary, complex, hyper-quaternionic and hyper-octonionic primes are central for TGD. "Hyper" means that one considers primes in the complexification of number field and having the property that imaginary part of integer is multiplied by commuting sqrt(-1): "hyper" is needed in order to obtain number theoretic norm with Minkowskian signature. Also the tensor product of vector spaces defines an algebraic multiplication and now primes are vector spaces with prime-valued dimension. Hence p-adic hierarchy emerges already at the level of basic mathematics of quantum theory. Obviously this observation should have deep relevance if Universe is topological quantum computer as TGD indeed suggests. Matti Pitkänen

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