Information Retrieval and Criticality in Parity-Time-Symmetric Systems

By investigating information flow between a general parity-time ($PT$-)symmetric non-Hermitian system and an environment, we find that the complete information retrieval from the environment can be achieved in the $PT$-unbroken phase, whereas no information can be retrieved in the $PT$-broken phase. The $PT$-transition point thus marks the reversible-irreversible criticality of information flow, around which many physical quantities such as the recurrence time and the distinguishability between quantum states exhibit power-law behavior. Moreover, by embedding a $PT$-symmetric system into a larger Hilbert space so that the entire system obeys unitary dynamics, we reveal that behind the information retrieval lies a hidden entangled partner protected by $PT$ symmetry. Possible experimental situations are also discussed.

Figure 1

Information flow in the $PT$-symmetric two-level system. (a) Time evolution of the distinguishability $D(t)$ in the $PT$-unbroken phase ($0<a<1$). The distinguishability oscillates with period $T=\pi /\sqrt{1-a^2}$. The more closely we approach the exceptional point ($a=1$), the longer the period becomes and the larger the degree of the information retrieval grows. (b) Time evolution of the distinguishability in the $PT$-broken phase ($a>1$). The distinguishability decays with a relaxation time $\tau =1/(2\sqrt{a^2-1})$. Near the exceptional point, $D(t)$ asymptotically behaves as $D\sim 1/t^2$. (c) Time evolution of the entanglement entropy between the system and the ancilla with $a=0.75$. The blue (green) solid curve represents the entanglement entropy for the $PT$ dynamics whose initial state is $|\uparrow ⟩$ ($|\downarrow ⟩$), while the dotted curve represents the distinguishability between them. The entanglement clearly oscillates with period $T_E=\pi /(2\sqrt{1-a^2})$, which is one-half of the period of $D(t)$.