Recent Advances in Mobile Robotics 312 quantum mechanics (Ballentine, 2006); for a complete review of quantum computation and information theory, see, e.g., (Nielsen & Chuang, 2000).
Basic notation and properties of quantum-mechanical systemsIn general, in the considerations of the finite quantum-mechanical systems, it is postulated (Ballentine, 2006) with the unit sum of the diagonal elements, ( ) tr 1 σ = ; and that the observed value of the quantum-mechanical system is specified by the eigenvalues of the matrix σ . Since matrix σ is Hermitial, its eigenvalues are real numbers. If matrix σ is diagonal, then such a representation of the state is equivalent to the representation of the state of stochastic classical system, in which diagonal elements jj σ form a probability vector (Holevo, 2001 From such a property of observation, it follows that in contrast to the classical systems, the actual state of the quantum-mechanical system obtains a value, which was measured by the observer, and further evolution of the system starts from this value. In the other words, the evolution the quantum-mechanical system depends on the fact of its observation.
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Navigation of Quantum-Controlled Mobile Robots 313An actual evolution of the quantum-mechanical system is governed by the evolution operators, which are applied to the state matrix σ or state vector s . Below, we consider the states and operators, which are used in quantum information theory.
Concepts of the quantum information theoryThe elementary state, which is considered in quantum information theory (Nielsen & Chuang, 2000), is called qubit (quantum bit) and is represented by a two-element complex vector , which correspond to the bit values "0" and "1" known in classical information theory (Cover & Thomas, 1991). In general, vectors 0 and 1 determine the states "spin up" and "spin down" of electron, i.e. 0" " ≡ ↑ and 1" " ≡ ↓ . 12" " =⋅ → −⋅ ← , the pairs of the vectors 0 a n d 1, a n d "" → and "" ← can be used interchangeably.In general, the evolution of the qubits is governed by the use of the following operators: The Pauli operators are the most known qubits operators that are in use in general quantum mechanics, while the other three operators are more specific for quantum information theory. According to the Kitaev-Solovey theorem (Nielsen & Chuang, 2000), an algebra, which consists of the qubits 0 and 1 , and CNOT, Hadamard and phase shift operators, forms a universal algebra that models any operation of the Boolean algebra :The other types of the qubit gates, e.g. phase shift operator S and its derivatives, cannot be represented by the classical operators and require quantum computing devices. In such computing, it is assumed that each matrix operation is conducted in one computation step providing a power of quantum computing. The indicated dependence of quantum states on the observation process allows an implementation of such operations by the use of adaptive computation schemes. Below, we demonstrate a relation between quantum operati...