Berlin 2024 – scientific programme

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QI: Fachverband Quanteninformation

QI 29: Quantum Information: Concept and Methods II

QI 29.4: Talk

Thursday, March 21, 2024, 15:45–16:00, HFT-TA 441

Virtual subsystems under pseudo-Hermitian evolution — •Himanshu Badhani — The Institute of Mathematical Sciences, Chennai

Given a bipartite system undergoing evolution under a pseudo-Hermitian Hamiltonian, its evolution is unitary in the appropriately chosen metric Hilbert space. If the metric operator does not have a tensor product decomposition (even if the underlying vector space has a tensor product structure), then the Hilbert space does not have a tensor product structure. It is therefore not clear how one should define a subsystem in such Hilbert spaces. In our work, we establish a general prescription to define the "subsystems" in such scenarios. We propose two different methods of partially tracing out the degrees of freedom: one of the methods exploits the equivalence of the pseudo-Hermiticity and Hermiticity. The other method considers a purely geometric approach wherein the Hilbert space is given a vector bundle structure over a base space. Partial tracing is then defined as the sum of the parallel transported states to any one of the fibers of the vector bundle. We show that these two methods of ``partial trace" are equivalent. While the choice of the metric has no bearing on the system's properties, these subsystems can depend on the chosen metric since they correspond to the virtual subsystems of the original system. We prove this statement by establishing the C *-algebra in the non-trivial Hilbert space and its *-isomorphism to the algebra of the virtual bi-partition.

arXiv:2309.03042v1 and arXiv:2109.10682v2

Keywords: PT-symmetry; pseudo Hermiticity; Virtual subsystems; Partial trace; Quantum walks