Note on Generalized Quantum Gates and Quantum Operations

Abstract

Recently, Gudder proved that the set of all generalized quantum gates coincides the set of all contractions in a finite-dimensional Hilbert space (S. Gudder, Int. J. Theor. Phys. 47:268-279, 2008). In this note, we proved that the set of all generalized quantum gates is a proper subset of the set of all contractions on an infinite dimensional separable Hilbert space H. Meanwhile, we proved that the quantum operation deduced by an isometry is an extreme point of the set of all quantum operations on H.Keywords Generalized quantum gate · Duality quantum computer · Extreme point of quantum operations

Duality Quantum Computing and Duality Quantum Information Processing

“…is the combiner structure in (5). Physically, this corresponds to a multiple asymmetric slits with some unitary lenses at each slit so that they modify the phases of the wave function at each slit.…”

“…The mathematical descriptions can be found in [1][2][3][4][5][6][7][8][9][10]. Concise mathematical expressions together with the studies of the properties of divider and combiner, and duality gates are given.…”

“…This theorem tells us that the duality computer is able to perform any operator in a Hilbert spaces H. It will be interesting to study the computability of duality computer in terms of this theorem. However, the theorem by Wang, Du and Dou [5] has given explicitly what can not be a generalized quantum gate when the dimension is infinite. That will be reviewed shortly.…”

“…In a wider context, it is an important issue to study the computing capabilities of duality computing. The recent theorem of Wang, Du and Dou [5] which limits what can not be a duality gate in a Hilbert space with infinite degrees of freedom, is an important step toward this direction.…”

“…There have been intensive interests in the theory of duality computer in recent years [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Here in this review article we will briefly describe the main results.…”