# zbMATH — the first resource for mathematics

Note on generalized quantum gates and quantum Operations. (English) Zbl 1160.81350
Summary: 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, No. 1, 268–279 (2008; Zbl 1145.81020)]. 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 $$\mathcal H$$. Meanwhile, we proved that the quantum operation deduced by an isometry is an extreme point of the set of all quantum operations on $$\mathcal H$$.

##### MSC:
 81P68 Quantum computation 81P15 Quantum measurement theory, state operations, state preparations
Full Text:
##### References:
 [1] Busch, P., Singh, J.: Lüders theorem for unsharp quantum measurements. Phys. Lett. A 249, 10–12 (1998) · doi:10.1016/S0375-9601(98)00704-X [2] Du, H.K., Wang, Y.Q., Xu, J.L.: Applications of the generalized Lüders theorem. J. Math. Phys., in press · Zbl 1153.81352 [3] Gudder, S.: Mathematical theory of duality quantum computers. Quantum Inf. Process. 6(1), 37–48 (2007) · Zbl 1119.81027 · doi:10.1007/s11128-006-0040-3 [4] Gudder, S.: Duality quantum computers and quantum operations. Int. J. Math. Phys. 47(1), 268–279 (2008) · Zbl 1145.81020 · doi:10.1007/s10773-007-9512-1 [5] Long, G.L.: Mathematical theory of the duality computer in the density matrix formalism. Quantum Inf. Process. 6, 49–54 (2007) · Zbl 1119.81032 · doi:10.1007/s11128-006-0042-1 [6] Long, G.L.: The general quantum interference principle and the duality computer. arxiv:quant-ph/0512120 (2005) [7] Nielsen, M., Chuang, J.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) · Zbl 1049.81015 [8] Pedersen, G.K.: Analysis Now. Springer/World Publishing, Beijing (1990) · Zbl 0668.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.