Generalized measurements and non-Hermitian quantum mechanics


Generalized measurements and non-Hermitian quantum mechanics

Samsonov, B.; Günther, U.

The basics of the theory of generalized quantum measurements is briefly discussed. Starting from positive operator valued measures (POVMs) (as generalization of decompositions of unity in terms of orthogonal projectors) it is demonstrated how to associate the POVMs in a natural way with the nonorthogonal eigenvectors of non-Hermitian operators with non-degenerate real spectrum. This approach allows for a self-consistent treatment of the non-Hermitian operators connected with a state discrimination technique for their non-orthogonal eigenvectors and the definition of a probabilistic content of corresponding generalized measurement procedures. The technique is applied to the brachistochrone problem of PT-symmetric quantum mechanics (PTSQM) in a measurement frame with Hermitian Hamiltonian and non-Hermitian generalized spin-type observables. The Naimark dilation (extension) technique for POVMs is briefly sketched.

Keywords: quantum mechanics; non-Hermitian operators; positive operator valued measure; POVM; PT-symmetric quantum mechanics; brachistochrone; Naimark dilation; Naimark extension

  • Invited lecture (Conferences)
    Quantum Physics with Non-Hermitian Operators (PHHQP VII), 29.06.-11.07.2008, Benasque, Spain

Permalink: https://www.hzdr.de/publications/Publ-11524
Publ.-Id: 11524