Leggett-Garg inequalities and temporal correlations for a qubit under $\mathcal{PT}$-symmetric dynamics

Macrorealistic description governing the classical realm is known to get uprooted in the quantum domain. The Leggett-Garg inequality [A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)] proposes to probe the macrorealistic limit emerging from the quantum scenario. It places a bound on the linear combinations of temporal correlations of observables measured sequentially at different time intervals. Violation of the Leggett-Garg inequality, up to the so-called temporal Tsirelson bound (TTB), has been realized in the quantum domain. It is important to understand whether quantum theory permits larger violation beyond the TTB and up to the algebraic maximum value of the Leggett-Garg inequality. To this end, it has been shown [C. Budroni and C. Emary, Phys. Rev. Lett. 113, 050401 (2014)] that a three-term Leggett-Garg inequality can be violated beyond the TTB $3/2$ in the time evolution of $N>2$ level quantum systems by choosing suitable measurement schemes and state-update rules. Furthermore, it is revealed that the algebraic maximum value 3 can be realized in the limit $N\to \infty$. Their measurement scheme rules out violation of the Leggett-Garg inequality beyond the TTB by a qubit. Being genuinely quantum in nature, we ask whether such enhanced violation of the Leggett-Garg inequality could be witnessed in a qubit itself. Here, we show that a qubit undergoing time evolution generated by a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian indeed violates the three-term Leggett-Garg inequality beyond the TTB and up to its algebraic maximum value 3.

Figure 1

Leggett-Garg inequality two-time measurement scheme.

Figure 2

Leggett-Garg parameter $K_3(\alpha ;\tau )$ for a qubit evolving under a $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian [Eq. (2)] for $\alpha /\pi =0.1,0.2,0.3,0.4$. The (dimensionless) measurement times for measuring the dichotomic observable ${\sigma }_y$ are set as $t_1^{\prime }-t_0^{\prime }=t_2^{\prime }-t_1^{\prime }=t_3^{\prime }-t_2^{\prime }=\tau$. The standard three-term LGI corresponding to unitary dynamics is recovered for $\alpha =0$ (solid line); the maximum value of LGI parameter $K_3^{\max }(\alpha ;\tau )$ grows beyond the TTB $\frac{3}{2}$ (for $\alpha =0$) as $\alpha$ increases and approaches the algebraic maximum 3 in the $\mathcal{P}\mathcal{T}$-symmetry-breaking limit, i.e., $\alpha \to \pi /2$.

Figure 3

Two-time temporal correlations $C_{21}(\alpha ;\tau ),C_{32}(\alpha ;\tau )$ and $C_{31}(\alpha ;\tau )$ as a function of $\alpha /\pi$ and $\tau /\pi$. It is seen that $C_{21},C_{32}\to 1$ and $C_{32}\to -1$ as $\alpha \to \pi /2, \tau \to \pi /4$, thus displaying perfect correlations and anticorrelations during the $\mathcal{P}\mathcal{T}$-symmetric dynamics of a qubit.

Figure 4

Plot of the Leggett-Garg quantity $K_3(\alpha ;\tau )$ as a function of the non-Hermiticity parameter $\alpha /\pi$ and the measurement time step $\tau /\pi$. This establishes that $K_3^{\max }$ approaches the algebraic maximum value 3 when the $\mathcal{P}\mathcal{T}$-symmetry-breaking point $\alpha \to \pi /2$ is approached.

Figure 5

Plot showing saturation of the shortest value of the measurement time step ${\tau }_{\min }$ for which $K_3(\alpha ;{\tau }_{\min })$ attains its maximum value $K_3^{\max }$. It may be seen that ${\tau }_{\min }\to \pi /4$ as $K_3^{\max }$ attains the algebraic maximum value 3 in the limit $\alpha \to \pi /2$ of the spontaneous $\mathcal{P}\mathcal{T}$-symmetry breaking.

Figure 6

Maximum value $K_3^{\max }(\alpha ;\tau )$ for different choices of $\alpha /\pi$. Here, the measurement time step is restricted to $\tau =[0,\pi /4]$ during which the first instance of reaching the maximum value is observed. It may be noted that $K_3^{\max }$ approaches the algebraic maximum value 3 of LGI in the limit $\alpha \to \pi /2$.

Figure 7

Temporal correlations $C_{21}(\alpha ;\tau ),C_{32}(\alpha ;\tau )$, and $C_{31}(\alpha ;\tau )$ for different choices of $\alpha /\pi$ at a fixed measurement time step $\tau =\pi /4$. Note that $C_{21}(\alpha ;\pi /4), C_{32}(\alpha ;\pi /4)\to 1$ as $\alpha \to \pi /2$ and $C_{31}(\alpha ;\pi /4)=-1$.

Figure 8

Plot of $K_3\left(\tau \right)$ as a function of $\tau$ in the limit $\alpha \to \pi /2$. It is clearly seen that $K_3\left(\tau \right)$ reaches the algebraic maximum value 3 at $\tau =\pi /4$ momentarily.