Nanowires: Exponential speedup in quantum computing

Markovian dynamics for a quantum/classical system and quantum trajectories

“…The transition instruments (29) are uniquely determined by their Fourier transform, the characteristic operator:…”

“…In the construction of the hybrid dynamics, we have used a two-step procedure: firstly, the SDEs ( 2) and ( 10) under the reference probability Q have been introduced, and, then, the SDEs ( 20) and ( 22) under the physical probability P. By following this path, already developed in the field of time-continuous measurements [23], we were able to give rise to the instruments (29) with the right properties, given in proposition 5, and with the connection (33) with the physical probability P. In this way, the connection with the general formulation of a quantum theory is preserved.…”

“…As we shall see T t is a contraction; so, it is enough to give T t on the product elements and then to extend it by linearity and continuity. By using the instruments (29), we can also write…”

“…On the other hand, given T t , we can get the instruments (29) and their characteristic operator (32) by…”

“…One of the main motivations of the study of hybrid systems is that the output of a monitored system is classical; then, implicitly or explicitly, the dynamics of quantum/classical systems is involved in the theory of quantum measurements in continuous time and quantum filtering [1][2][3][4][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Moreover, hybrid systems could be used as an approximation to complicate quantum systems, as an effective theory [1,4,[9][10][11][12]29]. Another motivation is in the connections between a theory of gravity and quantum theories; perhaps gravity is 'classical', and a dynamical theory of gravity and matter needs a hybrid dynamics [1,4,5,28,[30][31][32][33].…”