Automatic hermiticity for mixed states
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Automatic hermiticity for mixed states. / Nagao, Keiichi; Nielsen, Holger Bech.
I: Progress of Theoretical and Experimental Physics, Bind 2023, Nr. 3, 031B01, 09.03.2023.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Automatic hermiticity for mixed states
AU - Nagao, Keiichi
AU - Nielsen, Holger Bech
PY - 2023/3/9
Y1 - 2023/3/9
N2 - We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product I-Q that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator Q. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time T-A but also a future state at the final time T-B is given, we study a couple of candidates for it, and introduce a "skew density matrix" composed of both ensembles of the future and past states such that the trace of the product of it and an operator O matches a normalized matrix element of O. We argue that the skew density matrix defined with I-Q at the present time t for large T-B - t and large t - T-A approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product I-QJ for large t - T-A.
AB - We previously proposed a mechanism to effectively obtain, after a long time development, a Hamiltonian being Hermitian with regard to a modified inner product I-Q that makes a given non-normal Hamiltonian normal by using an appropriately chosen Hermitian operator Q. We studied it for pure states. In this letter we show that a similar mechanism also works for mixed states by introducing density matrices to describe them and investigating their properties explicitly both in the future-not-included and future-included theories. In particular, in the latter, where not only a past state at the initial time T-A but also a future state at the final time T-B is given, we study a couple of candidates for it, and introduce a "skew density matrix" composed of both ensembles of the future and past states such that the trace of the product of it and an operator O matches a normalized matrix element of O. We argue that the skew density matrix defined with I-Q at the present time t for large T-B - t and large t - T-A approximately corresponds to another density matrix composed of only an ensemble of past states and defined with another inner product I-QJ for large t - T-A.
KW - PSEUDO-HERMITICITY
KW - PT-SYMMETRY
KW - FUTURE
U2 - 10.1093/ptep/ptad025
DO - 10.1093/ptep/ptad025
M3 - Journal article
VL - 2023
JO - Progress of Theoretical Physics Supplement
JF - Progress of Theoretical Physics Supplement
SN - 2050-3911
IS - 3
M1 - 031B01
ER -
ID: 341918991