Explicit definition ofPTsymmetry for nonunitary quantum walks with gain and loss

Abstract

PT symmetry, that is, a combined parity and time-reversal symmetry, is a key milestone for non-Hermitian systems exhibiting entirely real eigenenergy. In the present work, motivated by a recent experiment, we study PT symmetry of the time-evolution operator of nonunitary quantum walks. We present the explicit definition of PT symmetry by employing a concept of symmetry time frames. We provide a necessary and sufficient condition so that the time-evolution operator of the nonunitary quantum walk retains PT symmetry even when parameters of the model depend on position. It is also shown that there exist extra symmetries embedded in the time-evolution operator. Applying these results, we clarify that the nonunitary quantum walk in the experiment does have PT symmetry.

Controlling probability transfer in the discrete-time quantum walk by modulating the symmetries

“…Because the rotation angle takes 0 or 2 p , the eigenstates of the a is taken, the chiral term e i d a does not affect the chiral symmetry of the effective Hamiltonian. The existence of chiral symmetry guarantees that the energy spectrum is symmetric with respect to zero energy [53,54]. Considering that the chiral symmetry, PHS and TRS have a direct connection, we can obtain that the system does not possess the TRS with d a of the chiral term satisfying…”

“…is satisfied. The emergence of chiral symmetry in the DTQW makes the eigenvalue spectrum of effective Hamiltonian H distribute symmetrically with respect to zero energy [52][53][54].…”

Generalized bulk-edge correspondence for non-Hermitian topological systems

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The Witten index for 1D supersymmetric quantum walks with anisotropic coins

“…which is known to have two linearly independent algebraic solutions. Here, we need not only to algebraically solve Equation (5), but also to ensure the solutions Ψ to be square summable. This is precisely why the difference on the right-hand side of (4) can still be non-zero.…”