Parity-time symmetric holographic principle

Originating from the Hamiltonian of a single qubit system, the phenomenon of the avoided level crossing is ubiquitous in multiple branches of physics, including the Landau-Zener transition in atomic, molecular and optical physics, the band structure of condensed matter physics and the dispersion relation of relativistic quantum physics. We revisit this fundamental phenomenon in the simple example of a spinless relativistic quantum particle traveling in (1+1)-dimensional space-time and establish its relation to a spin-1/2 system evolving under a $\mathcal{PT}$-symmetric Hamiltonian. This relation allows us to simulate 1-dimensional eigenvalue problems with a single qubit. Generalizing this relation to the eigenenergy problem of a bulk system with $N$ spatial dimensions reveals that its eigenvalue problem can be mapped onto the time evolution of the edge state with $(N-1)$ spatial dimensions governed by a non-Hermitian Hamiltonian. In other words, the bulk eigenenergy state is encoded in the edge state as a hologram, which can be decoded by the propagation of the edge state in the temporal dimension. We argue that the evolution will be $\mathcal{PT}$-symmetric as long as the bulk system admits parity symmetry. Our work finds the application of $\mathcal{PT}$-symmetric and non-Hermitian physics in quantum simulation and provides insights into the fundamental symmetries.