Algebraic maximum violation of the Leggett-Garg inequality in a two-level system under PT symmetric dynamics

TL;DR

This paper demonstrates that a two-level quantum system under PT symmetric, non-Hermitian dynamics can violate the Leggett-Garg inequality beyond the Tsirelson bound, reaching the algebraic maximum during PT symmetry breaking.

Contribution

It reveals that PT symmetric dynamics enable maximum violation of the Leggett-Garg inequality, surpassing traditional quantum bounds in a two-level system.

Findings

LGI violation exceeds Tsirelson bound under PT symmetry breaking

Maximum violation approaches the algebraic limit

PT symmetric dynamics enhance quantum temporal correlations

Abstract

We investigate the three-term Leggett-Garg inequality (LGI) for a two-level quantum system undergoing parity-time (PT ) symmetric dynamics governed by a non-Hermitian Hamiltonian, when a sequence of dichotomic projective measurements are carried out at different time intervals. In contrast to the case of coherent unitary dynamics, violation of LGI is shown to increase beyond the temporal Tsirelson bound approaching the algebraic maximum in the limit of the spontaneous PT symmetry breaking

Equations

$$-3\le K_3$$

$$H=s\left(\begin{array}{cc}i\sin \alpha & 1\\ 1& -i\sin \alpha \end{array}\right)s,\alpha \in \mathbb{R};s\ne 0.$$

$$|E_{+}(\alpha )⟩=\frac{e^{i\alpha /2}}{\sqrt{2\cos \alpha }}\left(\begin{array}{c}1\\ e^{-i\alpha }\end{array}\right),|E_{-}(\alpha )⟩=\frac{i{e}^{-i\alpha /2}}{\sqrt{2\cos \alpha }}\left(\begin{array}{c}1\\ -e^{i\alpha }.\end{array}\right)$$

$$U_{\alpha }(t)$$

$$\rho (0)\stackrel{U_{\alpha }(t)}{⟶}\rho (t;\alpha )=\frac{U_{\alpha }(t)\rho (0)U_{\alpha }^{†}(t)}{\mathrm{Tr}[U_{\alpha }(t)\rho (0)U_{\alpha }^{†}(t)]}.$$

$${\rho }_{q_i}(t_i;\alpha )$$

$$p(q_j,t_j|q_i,t_i;\alpha )=\frac{\mathrm{Tr}[U_{\alpha }(t_j-t_i){\rho }_{q_i}(t_i;\alpha )U_{\alpha }^{†}(t_j-t_i){\Pi }_{q_j}]}{p(q_i,t_i;\alpha )}$$

$$C_{ji}(\alpha )=\sum _{q_i,q_j=\pm 1}{q}_i{q}_{j}p(q_j,t_j|q_i,t_i;\alpha )p(q_i,t_i;\alpha )$$

$$C_{ji}(\alpha )$$

$$R_{j\mu }(\alpha )$$

$$I_{j\mu }(\alpha )$$

$$K_{j\mu }(\alpha )$$

Full text

Introduction.– Quantum and classical descriptions of correlations between observables of spatially separated systems differ drastically. Bell-CHSH inequalities Bell (1964); Clauser et al. (1969) place bounds on spatial correlations based on the classical framework of local realism. However, quantum theory predicts the presence of non-local correlations, which cry out for explanations Bell (2004). Experimental violation of the Bell-CHSH inequalities by quantum entangled states project how quantum theory differs from our intuitive classical description of correlations in spatially separated systems Brunner et al. (2014). Moreover, it has been realized that the Tsirelson bound Cirel’son (1980), the maximum possible violation of Bell-CHSH inequality in quantum theory is less than the algebraic maximum value. Violation of the Bell-CHSH inequality beyond the Tsirelson bound, points towards the existence of post quantum theories Popescu and Rohrlich (1992, 1994), which attract attention from the perspective of quantum information theory.

On the other hand, Leggett and Garg probed the contrast between the conflicting worldviews underlying the quantum and classical theories of nature in terms of temporal correlations in a single system under the tenet of macrorealism Leggett and Garg (1985). Macrorealism is based on the twin assumptions: (i) Macrorealism per se: Physical properties of a macroscopic system exists independent of the act of observation. (ii) Non-invasive measurability: Measurements performed do not influence the subsequent system evolution. Leggett-Garg inequality (LGI) places macrorealistic bounds on the temporal correlations of a physical observable, measured sequentially at different time intervals during the evolution of a single system. It is well-known that LGI gets violated in the quantum domain serving as a test to validate the notion of macrorealism in the controversial twilight zone area separating the quantum and classical boundary.

LGI imposes bounds on the linear combination of two-time correlations between consecutive measurements of a dichotomic observable of a system respecting macrorealism Emary et al. (2014). As it exhibits a striking symmetry with the Bell-CHSH inequality Markiewicz et al. (2014); Das et al. (2013), it is also referred to as the temporal Bell inequality Brukner et al. . Numerous theoretical and experimental investigations have confirmed violations of LGI Kofler and Brukner (2007, 2008); Fritz (2010); Wilde et al. (2010); Palacios-Laloy et al. (2010); Kofler and Brukner (2013); Devi et al. (2013); Katiyar et al. (2013); Dressel et al. (2011); Goggin et al. (2011); Knee et al. (2012); Athalye et al. (2011); Waldherr et al. (2011).

Akin to the Tsirelson bound for the Bell-CHSH inequality Cirel’son (1980); Wehner (2006), investigations towards bounding the temporal correlations for the LGI have been explored Fritz (2010); Budroni et al. (2013). Moreover, Budroni and Emary Budroni and Emary (2014) have shown that the strength of violation of the LGIs for dichotomic observables can exceed the temporal Tsirelson bound (TTB) in $N$-level quantum systems approaching the algebraic maximum value (admitted by the inequality) in the limit of $N\rightarrow\infty$. In contrast to the strict adherence to the Tsirelson bound on the Bell-CHSH inequality, violation of the LGI beyond the TTB attracts special attention which sets the tone for the present letter.

The magnitude of the violation of LGI depends both on the size of the quantum system and the measurement schemes employed Budroni and Emary (2014). To this end, stronger violation of the three-term LGI, beyond the TTB, was reported in liquid state nuclear magnetic resonance (NMR) experiment on an ensemble of three level systems by implementing ideal negative result measurements on dichotomic observables Katiyar et al. (2017). More recently, experimental demonstration of significantly larger violations of three and four term LGI, with three measurement outcomes, have been reported in photonic three-level systems Wang et al. (2017). Possibility of robust violation exceeding the TTB and reaching up to the algebraic maximum value of the inequality has been theoretically investigated in multiqubit systems by different measurement schemes Lambert et al. (2016) and by constructing variants of LGI other than the standard LGI Pan et al. (2018).

With the interest in observing violations beyond the temporal Tsirelson bound growing, we explore if it is possible to witness violations of LGI higher than the TTB in a two-level system. To this end, we theoretically investigate the effect of non-unitary evolution generated by a parity-time  ($\mathcal{P}\mathcal{T}$) symmetric non-Hermitian Hamiltonian Bender and Boettcher (1998) on the possible violations of LGI in a two-level system. The $\mathcal{P}\mathcal{T}$ Hamiltonian commutes with the combined symmetry operations of parity ($\mathcal{P}$), and time reversal ($\mathcal{T}$); it has real eigenvalues and shares common eigenvectors with $\mathcal{P}\mathcal{T}$ operation up to the exceptional point or the $\mathcal{P}\mathcal{T}$ symmetry breaking point. Spontaneous $\mathcal{P}\mathcal{T}$ symmetry breaking occurs at the exceptional point, separating the broken and unbroken phases of $\mathcal{P}\mathcal{T}$ symmetry. The Hamiltonian and the $\mathcal{P}\mathcal{T}$ symmetry operator do not share common eigen states in the regime of broken $\mathcal{P}\mathcal{T}$ symmetry, even though they commute with each other Bender and Boettcher (1998); Bender et al. (1999, 2002); Bender (2005). Peculiar features exhibited by $\mathcal{P}\mathcal{T}$ symmetric systems (see Longhi (2017); Christodoulides and Yang (2018) for a recent review on theoretical and experimental investigations), close to the exceptional point, have attracted attention of researchers over the last two decades Bender and Boettcher (1998); Bender et al. (1999, 2002); Bender (2005); Ramezani et al. (2010); Schindler et al. (2011); Ramezani et al. (2012); Lee et al. (2014); Kawabata et al. (2017); Quijandría et al. (2018); Xiao et al. (2018); Wu et al. (2019). Simulating the ever gnawing energy exchange interaction of a quantum system with its environment in terms of $\mathcal{P}\mathcal{T}$ symmetric non-unitary dynamics offers a new perspective for the study of open quantum systems. There is a surge of activity in experimentally simulating $\mathcal{P}\mathcal{T}$ symmetric quantum dynamics Quijandría et al. (2018); Xiao et al. (2018); Wu et al. (2019). From the information theoretic perspective, in particular, it was shown that the information flow from the system to its environment can be completely retrieved in the unbroken regime of $\mathcal{P}\mathcal{T}$ symmetry and vice versa Kawabata et al. (2017). Such reversible-irreversible criticality of information flow under $\mathcal{P}\mathcal{T}$ symmetric dynamics has recently been demonstrated experimentally, where non-unitary quantum dynamics of a two-state system is simulated in a single-photon interferometric network Xiao et al. (2018). This unidirectional information flow behavior, in the unbroken phase of $\mathcal{P}\mathcal{T}$, suggests applications of $\mathcal{P}\mathcal{T}$ systems in quantum control. Furthermore, the paradigmatic features characterizing $\mathcal{P}\mathcal{T}$ symmetric systems open up avenues for their potential applicability in quantum information processing tasks.

In this letter, we report violation of the LGI higher than the temporal Tsirelson bound in a two-level system undergoing $\mathcal{P}\mathcal{T}$ symmetric non-unitary dynamics. We show that the magnitude of the violation attains the algebraic maximum value in the limit of the spontaneous breaking of $\mathcal{P}\mathcal{T}$ symmetry. While the excess violation of the three term LGI were reported for dimensions larger than 2, it was not known whether the same could be observed in a two-level system. Here we report for the first time that the violation of three term LGI beyond the TTB reaching up to the algebraic maximum value is indeed possible with $\mathcal{P}\mathcal{T}$ symmetry.

Three-term LGI.— We consider the three-term LGI:

[TABLE]

where $C_{ji}=\braket{Q(t_{j})\,Q(t_{i})}\equiv\braket{Q_{j}\,Q_{i}}$ denote temporal correlations of a dichotomic observable $Q$ with outcomes $q=\pm 1$, measured at two different time intervals $t_{j}>t_{i}$.

According to the notion of macrorealism, there exists definite values for the observable $Q$ implying that the outcome of measurement made at an instant of time $t_{j}$ is not influenced by that made at an earlier time instant $t_{i}$. This leads to the bound of Eq.(1). On the other hand, with appropriate choice of measurement time intervals, it is possible for the Leggett-Garg parameter $K_{3}$ to exceed the bound 1 (see Eq.(1)) in quantum systems. In particular, the maximum value of $K_{3}$ for a two-level quantum system undergoing unitary dynamics, when projective measurements $\Pi_{i}$ ($\Pi_{j}$) of an observable $Q$ with outcomes $q_{i}$ ($q_{j}$) at time instants $t_{i}$ ($t_{j}$) is given by $K_{3}^{\rm TTB}=3/2$ Leggett and Garg (1985).

Within the standard quantum theoretic framework, for a system evolving under coherent unitary dynamics, violation of the three-term LGI beyond the value $K_{3}^{\rm TTB}$ could be witnessed by extending the dimension of the Hilbert space (i.e., by considering quantum systems with number of levels $N>2$) and by employing a general measurement update rule Budroni and Emary (2014). Under this context, it is of interest to verify whether $\mathcal{P}\mathcal{T}$-symmetric dynamics leads to violation of LGI larger than $K_{3}^{\rm TTB}$ in two-level systems.

$\mathcal{P}\mathcal{T}$* symmetric dynamics of a two-level system.—* We consider evolution of a two-level quantum system generated by the $\mathcal{P}\mathcal{T}$ symmetric Hamiltonian:

[TABLE]

The Hamiltonian (2) commutes with the $\mathcal{P}\mathcal{T}$ operator i.e., $[H,\mathcal{P}\mathcal{T}]=0$. It is Hermitian for $\alpha=0$. Though the Hamiltonian is non-Hermitian for $\alpha\neq 0$, it possesses real eigenvalues (by virtue of $\mathcal{P}\mathcal{T}$ symmetry) given by $E_{\pm}=\pm s\cos{\alpha}$, with the corresponding non-orthogonal eigenvectors

[TABLE]

Both $H$ and $\mathcal{P}\mathcal{T}$ share common eigenvectors in the range $0\leq\alpha<\frac{\pi}{2}$. The eigenvalues and eigenvectors coalesce at the exceptional point $\alpha~{}=~{}\pi/2$. Here, we restrict ourselves to the parameter range $0\leq\alpha<\pi/2$, i.e., to the unbroken $\mathcal{P}\mathcal{T}$ symmetry regime.

The non-unitary time evolution generated by the $\mathcal{P}\mathcal{T}$ symmetric Hamiltonian (2) has the following explicit form:

[TABLE]

where $t^{\prime}=\frac{\Delta E}{2}\,t$ and $\Delta E=E_{+}-E_{-}=2\,s\,\cos\alpha$; we have set $\hbar=1$. Consider a two-level system prepared in a state $\rho(0)$ (at time $t=0$). The time evolved state $\rho(t;\alpha)$ under the $\mathcal{P}\mathcal{T}$ symmetric Hamiltonian (2) is given by

[TABLE]

The normalization term in the denominator of (4) arises due to the non-unitary dynamics of the system.

Temporal correlations in a two-level system undergoing $\mathcal{P}\mathcal{T}$ symmetric dynamics.— We consider two-time correlations of the observable $Q=\sigma_{y}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}$ when the system is undergoing $\mathcal{P}\mathcal{T}$ symmetric dynamics (see Eqns. (3) and (4)). One-dimensional projection operators employed for measuring dichotomic outcomes $q=\pm\,1$ of the observable $\sigma_{y}$ are denoted by $\Pi_{q}$.

When measurement of $\sigma_{y}$ results in the outcome $q_{i}$ at time $t_{i}$, the initial state transforms as $\rho(0)~{}\rightarrow~{}\rho_{q_{i}}(t_{i};\alpha)~{}=~{}\frac{\Pi_{q_{i}}\rho(t_{i};\alpha)\Pi_{q_{i}}}{p(q_{i},t_{i};\alpha)}$, with probability $p(q_{i},t_{i};\alpha)={\rm Tr}[\rho(t_{i};\alpha)\,\Pi_{q_{i}}]$ for different choices of the parameter $0\leq\alpha<\pi/2$. Conditioned on the measurement outcome $q_{i}$ at time $t_{i}$, a second measurement of $\sigma_{y}$ at a later time $t_{j}>t_{i}$ leads to (see Fig. (1))

[TABLE]

where

[TABLE]

denotes the probability of getting the measurement outcome $q_{j}$ at time $t_{j}$, when the first measurement outcome is registered as $q_{i}$ at an earlier time $t_{i}$.

We evaluate the two-time temporal correlations

[TABLE]

by considering a two level system initialized to a maximally mixed state $\rho(0)=\frac{I}{2}$, where $I$ denotes $2\times 2$ identity matrix, and obtain

[TABLE]

where

[TABLE]

with $j=1,2,3,$ and $\mu=0,1,2,\ j>\mu.$

Violation of LGI up to algebraic maximum.— Setting measurement   time intervals $t_{1}^{\prime}-t^{\prime}_{0}~{}=~{}\tau,\ t^{\prime}_{2}~{}-~{}t^{\prime}_{1}~{}=~{}\tau,t^{\prime}_{3}~{}-~{}t_{2}^{\prime}~{}=~{}\tau$ and $t^{\prime}_{3}-t_{1}^{\prime}=2\,\tau$, we compute the left hand side of the three-term LGI $K_{3}(\alpha)~{}=~{}C_{21}(\alpha)~{}+~{}C_{32}(\alpha)~{}-~{}C_{31}(\alpha)$. Standard three-term LGI parameter i.e., $K_{3}=2\,\cos(2\tau)-\,\cos(4\tau)$ is recovered when $\alpha=0$ i.e., when the dynamics of the system is unitary.

Having computed the two-time correlations, we have plotted $K_{3}(\alpha)$ as a function of dimensionless measurement time interval $\tau$ for different values of $\alpha$ in Fig. (2). It is clearly seen that the three-term LGI is violated all the way up to the algebraic maximum 3, at the $\mathcal{P}\mathcal{T}$ symmetry breaking point $\alpha\rightarrow\pi/2.$ Variation of $K^{\rm max}_{3}(\alpha)$ for different choices of the parameter $\alpha/\pi$ is illustrated in the Fig. (3), where it is clearly seen that $K^{\rm max}_{3}(\alpha)\rightarrow 3$ when $\alpha\rightarrow\pi/2$. We also observe that the shortest measurement time-step $\tau_{\rm min}$, at which $K^{\rm max}_{3}$ reaches the algebraic maximum 3, saturates to a constant value of $\pi/4$ as $\alpha\rightarrow\pi/2$ (see Figs. 2, 4(b)). In fact, when the measurement time-step is fixed to be $\tau=\pi/4$, the temporal correlations (see Eq. (5)) $C_{21}(\alpha)$, $C_{32}(\alpha)$ approach the value 1 as $\alpha\rightarrow\pi/2$, and $C_{31}(\alpha)=-1$, irrespective of the parameter $\alpha$ (see Fig. (4)(a)).

Discussion.— Negation of macrorealism in a two-level system is no surprise as it is genuinely quantum in nature. Nevertheless, the importance of tests of LGI on two-level systems elucidates the persistence and stability of the quantumness of the system in the presence of environmental interactions Athalye et al. (2011). This merits investigations on the violation of LGI in a two-level system under a more general framework of non-Hermitian, non-unital dynamics, thereby tapping the counter intuitive traits, which come underpinned for applications in future quantum technology. This is for the first instance, to the best of our knowledge, that investigation of macrorealism on a $\mathcal{P}\mathcal{T}$ symmetric system is carried out. We have shown here that violation of LGI beyond the TTB and up to the algebraic maximum is possible in a two-level $\mathcal{P}\mathcal{T}$ system. Our work reveals the non-trivial influence of $\mathcal{P}\mathcal{T}$ symmetric dynamics on the temporal correlations of an observable in a two level quantum system. Significantly, non-unitary dynamics of two-level quantum system under $\mathcal{P}\mathcal{T}$ symmetric Hamiltonians, both in the unbroken and broken regimes, has been simulated experimentally in single-photon interferometric networks Xiao et al. (2018) and also in a single nitrogen-vacancy (NV) center in diamond Wu et al. (2019). We believe that our theoretical result on the violation of LGI beyond the TTB, which is shown to be possible in $\mathcal{P}\mathcal{T}$ symmetric two-level systems, could be readily verified by the existing experimental platforms Xiao et al. (2018); Wu et al. (2019). Furthermore, understanding the basic principle resulting in the algebraic maximum violation of LGI at the $\mathcal{P}\mathcal{T}$ symmetry breaking point deserves a deeper analysis as this may shine light on the possible information content of quantum temporal correlations in two-level $\mathcal{P}\mathcal{T}$ symmetric systems. Finally, we believe that our work opens up avenues for further studies on $\mathcal{P}\mathcal{T}$ symmetric quantum systems, both from the foundational view point, as well as from the perspective of applicability in quantum information processing.

The authors gratefully acknowledge discussions with Professors A K Rajagopal, Sudha, Sibasish Ghosh. HSK and ASH thank Prof. R Srikanth for insightful comments on the manuscript. ASH is grateful to Prof. C M Chandrashekhar for hosting her visit at IMSc, Chennai during which this work was initiated and partly completed.