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WORLD SCIENTIFIC eBooks

Linear Quantum Mechanics: Its Successes and Problems

Published Apr 1, 2005• 0 citations
Publisher
DOI
10.1142/9789812567789_0001

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We formulate a (sub-optimal) quantum Linear Quadratic Gaussian (LQG) problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Our problem may be viewed as a polynomial matrix programming problem and we show that by utilizing a non-linear change of variables, it can be systematically converted to a Linear Matrix Inequality (LMI) rank constrained problem. As an initial demonstration of the feasibility of this approach, we provide a fully quantum controller design example in which a numerical solution to the problem was successfully obtained.
Time flow has been embodied in time-dependent Schroedinger equation representing one of the foundations of quantum mechanics. Pauli's criticism (1933) has, however, indicated that the assumptions concerning representation Hilbert space have led to some contradictions. Many authors have tried to solve this discrepancy practically without any actual success. The reason may be seen in that two different problems have been mixed: the problem of Pauli (being more general and more important) and non-unitarity of exponential phase operator of linear harmonic oscillator introduced by Dirac, as demonstrated in 1964. The problem will be discussed in a broad historical context and a solution based on extension of representation Hilbert space will be shown.