Abstract
In quantum mechanics, physical states are represented by rays in Hilbert space $\mathscr H$, which is a vector space imbued by an inner product $\langle\,|\,\rangle$, whose physical meaning arises as the overlap $\langle\phi|\psi\rangle$ for $|\psi\rangle$ a pure state (description of preparation) and $\langle\phi|$ a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum channel, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.
Highlights
Hilbert-space inner product is fundamental to quantum mechanics (QM), and its physicality relates to norm through the Born interpretation and to fidelity and distinguishability through its complex angle [1]
A C∗-algebraic approach shows that a set of non-Hermitian operators comprises the observables of a quantum mechanical system if and only if the operators are Hermitian with respect to a new Hilbertspace inner product [26]
These experiments simulate PT-symmetric dynamics on classical [6,7,9,10] or quantum [8,11,39,40] systems by balancing loss and gain. Another way to simulate PT-symmetric Hamiltonians with real spectra is by dilating the nonunitary propagator to a nonlocal unitary operator over multiple subsystems, which has been demonstrated on qubit systems [13,41,42,43,44,45,46,47,48]
Summary
For a possibly infinite dimensional Hilbert space H , we denote by L(H ) and B(H ) the algebra of linear and bounded linear operators on H respectively. For any Hilbert space H = (V , | ) and a self-adjoint, positive-definite operator η ∈ B(H ), 1. We construct a ∗-representation of the algebra A on the new Hilbert space Hη constructed in Theorem 1. H and Hη are two Hilbert spaces with their inner product related by the metric operator η as in Theorem 1, A is a C∗ algebra of operators and π : A → B(H ) is a ∗representation of A. The following lemma, which establishes the inverse of the metric operator η, is adapted from the Appendix A of Ref. Any self-adjoint and positive-definite operator η ∈ B(H ) is invertible. The lemma relates † to ‡, with the latter denoting the adjoint with respect to the inner product | of Hη. Πη is a ∗-representation of A on Hη
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
