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Theoretical and Mathematical Physics

Optimal time evolution for pseudo-Hermitian Hamiltonians

Wenhua Wang, Z. L. Chen, Yun S. Song, Yajing Fan

Published Aug 1, 2020• 2 citations
Publisher

Abstract

If an initial state $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and a final state $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are given, then there exist many Hamiltonians under whose action $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ evolves into $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ . In this case, the problem of the transition of $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ to $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if $$|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle $$ and $$|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle $$ are orthogonal. But for a $$PT$$ -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.

DOI
10.1134/s0040577920080048

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If an initial state ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ and a final state ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ are given, then there exist many Hamiltonians under whose action ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ evolves into ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ . In this case, the problem of the transition of ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ to ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ and ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ are orthogonal. But for a PTPTPT -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.
We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric (inner product) dependence of the lower bound on the travel time and the universality (metric independence) of the upper bound on the speed of unitary evolutions.
Given an initial quantum state $|{\ensuremath{\psi}}_{I}⟩$ and a final quantum state $|{\ensuremath{\psi}}_{F}⟩$, there exist Hamiltonians HHH under which $|{\ensuremath{\psi}}_{I}⟩$ evolves into $|{\ensuremath{\psi}}_{F}⟩$. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of HHH is held fixed, which HHH achieves this transformation in the least time $\ensuremath{\tau}$? For Hermitian Hamiltonians $\ensuremath{\tau}$ has a nonzero lower bound. However, among non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonians satisfying the same energy constraint, $\ensuremath{\tau}$ can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from $|{\ensuremath{\psi}}_{I}⟩$ to $|{\ensuremath{\psi}}_{F}⟩$ can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.
For any pair of quantum states, an initial state |I and a final quantum state |F , in a Hilbert space, there are many Hamiltonians H under which |I evolves into |F .Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, E max and E min , is held fixed.We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time τ .For Hermitian Hamiltonians, τ has a nonzero lower bound.However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle.The minimum value of τ can be made arbitrarily small because for PT -symmetric Hamiltonians the path from the vector |I to the vector |F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short.The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole.This result may have applications in quantum computing.
Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state |ψ i into a specific final state |ψ f can be made arbitrarily small.Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by |ψ i and |ψ f .In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.
Abstract The Hamiltonian of a conventional quantum system is Hermitian, which ensures real spectra of the Hamiltonian and unitary evolution of the system. However, real spectra are just the necessary conditions for a Hamiltonian to be Hermitian. In this paper, we discuss the metric operators for pseudo-Hermitian Hamiltonian which is similar to its adjoint. We first present some properties of the metric operators for pseudo-Hermitian Hamiltonians and obtain a sufficient and necessary condition for an invertible operator to be a metric operator for a given pseudo-Hermitian Hamiltonian. When the pseudo-Hermitian Hamiltonian has real spectra, we provide a new method such that any given metric operator can be transformed into the same positive-definite one and the new inner product with respect to the positive-definite metric operator is well defined. Finally, we illustrate the results obtained with an example.
Recently the quantum brachistochrone problem has been considered for non-Hermitian PT\cal PTPT-symmetric quantum system, and it has been shown that the optimal evolution time required to transform a given initial state ∣ψi>|\psi_i>∣ψi​> into specific final state ∣ψf>|\psi_f>∣ψf​> can be made arbitrary small. Here we study the similar problem for generic non-Hermitian Hamiltonian and find that the same answer can be obtained. We show that at the exceptional point the distance between ∣ψi>|\psi_i>∣ψi​> and ∣ψf>|\psi_f>∣ψf​> becomes virtually vanishing and quantum tunneling occurring at the exceptional point yields the shortest timed required to evolve from the initial state ∣ψi>|\psi_i>∣ψi​> into the final state ∣ψf>|\psi_f>∣ψf​>. These results may have applications in holonomic quantum computation and quantum information processing.