Download Mathematical Foundations of Quantum Information and by Masanori Ohya, I. Volovich PDF

By Masanori Ohya, I. Volovich

This monograph offers a mathematical origin to the speculation of quantum info and computation, with functions to numerous open structures together with nano and bio systems.

It contains introductory fabric on set of rules, sensible research, chance thought, details conception, quantum mechanics and quantum box thought. except common fabric on quantum details like quantum set of rules and teleportation, the authors talk about findings at the concept of entropy in C*-dynamical platforms, space-time dependence of quantum entangled states, entangling operators, adaptive dynamics, relativistic quantum info, and a brand new paradigm for quantum computation past the standard quantum Turing computing device. additionally, a few very important purposes of data concept to genetics and lifestyles sciences, in addition to fresh experimental and theoretical discoveries in quantum photosynthesis are described.

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Additional info for Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems

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And An ∩ Am = ∅, m = n ⇒ μ( n An ) = n μ(An ). For example, the ordinary (Lebesgue) measure dx on the unite segment [0, 1] on the real line is a probability measure. Let us consider an example of a probability space. M. Ohya, I. V. 3 (Binomial distribution. Coin tossing) Let us discuss a model for the experiment in which a coin is tossed N times independently with probability p (where 0 ≤ p ≤ 1) of heads (for a symmetrical coin p = 1/2). Record the results as an ordered set (a1 , . . , aN ) where ai = 1 for heads (“success”) and ai = 0 for tails (“failure”).

Next let us assume f (x1 , . . , xn ) = 0. Note that x1σ1 x2σ2 · · · xnσn = 1 only if x1 = σ1 , . . , xn = σn , which means that the right-hand side x1x1 x2x2 · · · xnxn f (x1 , . . , xn ) = 0 form the assumption f (x1 , . . , xn ) = 0. The basic logical operations AND, OR and NOT are not independent. For example, the conjunction function gAND can be represented as the superposition of the functions gOR and gNOT because one has gAND (x, y) = gNOT ◦ gOR gNOT (x), gNOT (y) or xy(≡ x ∧ y) = x ∨ y.

A P-problem and an NP-problem are defined as follows: Let n be the size of input. 17 (1) A P-problem is a problem with a time needed for solving it at worst a polynomial in n. Equivalently, it is a problem which can be recognized in a time polynomial in n by a deterministic Turing machine. 4 Computational Complexity, P-problem and NP-problem 27 (2) An NP-problem is a problem that can be solved in polynomial time by a nondeterministic Turing machine (there must be a verifier that runs in polynomial time).

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