Essential role of quantum speed limit in violation of Leggett-Garg inequality across a PT-transition

Abstract

We study Leggett-Garg inequality (LGI) of a two level system (TLS) undergoing non-Hermitian dynamics governed by a non-linear Bloch equation (derived in J. Phys. A: Math. Theor. 54, 115301 (2021)) across a PT-transition. We present an algebraic identification of the parameter space for the maximum violation of LGI (in particular K3). In the PT-symmetric regime the maximum allowed value for K3 is always found to be greater than the quantum bound (Lüders bound) of 3/2 but it does not reach the algebraic maximum of K3 = 3 in general. However, in the limit where PTsymmetry breaking parameter approaches the exceptional point from the PT-symmetric side, K3 is found to asymptotically approach its algebraic maximum of 3. In contrast, the maximum value of K3 always reaches its algebraic maximum in the PT-broken phase i.e. K3 → 3. We find that (i) the speed of evolution (SOE) must reach its maximum value (in the parameter space of initial state and the time interval between successive measurements) to facilitate the value of K3 → 3, (ii) together with the constraint that its minimum value must run into SOE equals to zero during the evolution of the state. In fact we show that the minimum speed of evolution can serve as an order parameter which is finite on the PT-symmetric side and identically zero on the PT-broken side. Finally, we discuss a possible experimental realization of this dynamics by quantum measurement followed by post-selection procedure in a three level atom coupled to cavity mode undergoing a Lindbladian dynamics.