No-signaling principle and quantum brachistochrone problem in $PT$-symmetric fermionic two- and four-dimensional models

Abstract

Fermionic systems differ from bosonic ones in several ways, in particular that the time-reversal operator $T$ is odd, $T^2=-1$. For $PT$-symmetric bosonic systems, the no-signaling principle and the quantum brachistochrone problem have been studied to some degree, both of them controversially. In this paper, we apply the basic methods proposed for bosonic systems to {\it fermionic} two- and four-dimensional $PT$-symmetric Hamiltonians, and obtain several surprising results: We find - in contrast to the bosonic case - that the no-signaling principle is upheld for two-dimensional fermionic Hamiltonians, however, the $PT$ symmetry is broken. In addition, we find that the time required for the evolution from a given initial state, the spin-up, to a given final state, the spin-down, is a constant, independent of the parameters of the Hamiltonian, under the eigenvalue constraint. That is, it cannot - as in the bosonic case - be optimized. We do, however, also find a dimensional dependence: four-dimensional $PT$-symmetric fermionic Hamiltonians considered here again uphold the no-signaling principle, but it is not essential that the $PT$ symmetry be broken. The symmetry is, however, broken if the measure of entanglement is conserved. In the four-dimensional systems, the evolution time between orthogonal states is dependent on the parameters of the Hamiltonian, with the conclusion that it again can be optimized, and approach zero under certain circumstances. However, if we require the conservation of entanglement, the transformation time between these two states becomes the same constant as found in the two-dimensional case, which coincides with the minimum time for such a transformation to take place in the Hermitian case.

arXiv comment8 pages, no figures

References in corpus (17)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Unknown Quantum States and Operations, a Bayesian View
• Faster than Hermitian Quantum Mechanics
• Quantum Brachistochrone
• Local PT symmetry violates the no-signaling principle
• Elementary Derivation for Passage Times
• On optimum Hamiltonians for state transformations
• A New Class of non-Hermitian Quantum Hamiltonians with PT Symmetry
• Consistency of PT-symmetric quantum mechanics
• PT-symmetric quantum state discrimination
• The quantum brachistochrone problem for non-Hermitian Hamiltonians
• Statistical Origin of Pseudo-Hermitian Supersymmetry and Pseudo-Hermitian Fermions
• No-signaling principle and Bell inequality in PT-symmetric quantum mechanics
• Two- and four-dimensional representations of the PT- and CPT-symmetric fermionic algebras

Cited by in corpus (6)

• Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions
• Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions
• $\mathcal{P}\mathcal{T}$-Symmetry in Hartree-Fock Theory
• Local preservation of no-signaling in multiparty PT-symmetric evolutions
• Remarks on the preservation of no-signaling principle in parity-time-symmetric quantum mechanics
• The Analytic Eigenvalue Structure of the 1+1 Dirac Oscillator