Temporal correlation beyond quantum bounds in non-hermitian dynamics

TL;DR

This paper demonstrates that non-Hermitian Hamiltonians can violate the quantum bounds of the Leggett-Garg inequality, reaching the algebraic maximum and revealing new dynamics beyond standard quantum mechanics.

Contribution

It shows that non-Hermitian two-level systems can surpass quantum bounds on Leggett-Garg inequality violations, achieving the algebraic maximum of 3, unlike Hermitian systems.

Findings

Violates Leggett-Garg bound of 3/2 with non-Hermitian Hamiltonians.

Achieves algebraic maximum violation of 3 in two-level systems.

Exceeds quantum speed limits through non-linear Bloch dynamics.

Abstract

We study the dynamics of two level systems described by non-hermitian Hamiltonians with real eigenvalues. Within the framework of hermitian quantum mechanics, it is known that maximal violation of Leggett-Garg inequality is bounded by $3/2$ (Luder's bound). We show that this absolute bound can be evaded when dynamics is governed by non-hermitian Hamiltonians. Moreover, the extent of violation can be optimized to reach its algebraic maximum of $3$ which is otherwise only feasible when the Hilbert space is infinite dimensional in the hermitian case. The extreme violation of Leggett-Garg inequality is shown to be directly related to the two basic ingredients: (i) The Bloch equation for the two level system has non-linear terms which allow for accelerated dynamics of states on the Bloch sphere exceeding all known quantum speed limits of state evolution; and (ii) We need to ensure that the quantum trajectory of states always lies on a single great circle (geodesic path) on the Bloch sphere at all times.

Equations

$$\frac{d{\rho }_t}{dt}=-i\left[\vec{A}\cdot \vec{\sigma },{\rho }_t\right]-\left\{\vec{B}\cdot \vec{\sigma },{\rho }_t\right\}+2tr({\rho }_t\vec{B}\cdot \vec{\sigma }){\rho }_t$$

$${\rho }_t={\rho }_t=(1/2)\mathbb{I}+\vec{S}(t)\cdot \vec{\sigma }=\frac{e^{-iHt}{\rho }_0{e}^{i{H}^{†}t}}{tr(e^{-iHt}{\rho }_0{e}^{i{H}^{†}t})}$$

$$\frac{d\vec{S}(t)}{dt}=2\vec{A}\times \vec{S}(t)-\vec{B}+4\{\vec{B}\cdot \vec{S}(t)\}\vec{S}(t).$$

$$\frac{d{S}_A(t)}{dt}$$

$$\frac{d{S}_B(t)}{dt}$$

$$\frac{d{S}_n(t)}{dt}$$

$$S_B=\frac{1}{2}\sqrt{\frac{(A^2-B^2){\sin }^{2}2\sqrt{A^2-B^2}t}{(A-B\cos 2\sqrt{A^2-B^2}t{)}^2}},$$

$$S_n=\frac{1}{2}\frac{B-A\cos 2\sqrt{A^2-B^2}t}{A-B\cos 2\sqrt{A^2-B^2}t}.$$

$$|\psi (t)⟩=N(t)exp[-i{H}_{\theta }t]{|\uparrow ⟩}_y$$

$$\delta=\cos^{-1}\left[~{}{\Big{|}}~{}\sqrt{\frac{\sin^{2}(t)(1-\sin\theta)}{(1+\cos(2t)\sin\theta)}}~{}{\Big{|}}~{}\right].$$

$$H_T={\mathbb{I}}_2\otimes H_s+{\sigma }_y\otimes V$$

$$\begin{array}{rl}|{\Psi }_T(t)⟩& =N_T({|\uparrow ⟩}_z\otimes exp[-i{H}_{\theta }t]|\psi ⟩+\\ & {|\downarrow ⟩}_z\otimes \eta exp[-i{H}_{\theta }t]|\psi ⟩).\end{array}$$

$$\begin{array}{rl}\frac{d{\rho }_t}{dt}& =-i\left[\vec{A}\cdot \vec{\sigma },{\rho }_t\right]-\left\{\vec{B}\cdot \vec{\sigma },{\rho }_t\right\}+\\ & 2tr({\rho }_t\vec{B}\cdot \vec{\sigma }){\rho }_t+\kappa (\mathbb{I}-2{\rho }_t)\end{array}$$

$$|\psi (t)⟩$$

$$\delta$$

$$p(\delta t)$$

$$v=\frac{{\cos }^2\theta }{(1+\cos 2t\sin \theta {)}^2}.$$

$$P_{ij}(Q(t_i),Q(t_j))$$

$$P_{12}(+,+)$$

$$P_{12}(+,-)$$

$$P_{12}(-,+)$$

$$P_{23}(+,+)$$

$$P_{23}(+,-)$$

$$P_{23}(-,+)$$

$$P_{23}(-,-)$$

$$P_{13}(+,+)$$

$$P_{13}(+,-)$$

$$P_{13}(-,+)$$

$$C_{12}$$

$$C_{13}$$

$$C_{23}$$

$$P_{12}(+,+)$$

$$P_{12}(+,-)$$

$$P_{12}(-,+)$$

$$P_{23}(+,+)$$

$$P_{23}(+,-)$$

Full text

I Introduction

The subatomic world of particles follows the laws of quantum mechanics, while at macroscopic scales much of the phenomena are described by classical mechanics. There has been a persistent quest to understand the validity of quantum mechanics at the macroscopic scales. LGI has emerged as temporal analog of Bell’s inequality Bell1964 ; Peres1999 ; Mahler1993 ; Budroni2013 ; Kim2006 which provides test of quantum mechanics at macroscopic scales via violation of bounds on temporal correlations LG1985 ; Leggett2008 ; Legget2002 . Macroscopic realism (MR) and non-invasive measurability (NIM) underlies the basic construction of LGI LGIreview2014 , which are based on our intuition of the classical world and are not conformed by quantum mechanics. Thus, violation of the LGI surely indicates a breakdown of any one of the above assumptions or both. And, hence its violation can be considered as an indication of non-classical or quantum behaviour under appropriate experimental circumstances Mahler1993 ; Wilde2012 . Currently, there are theoretical studies Kunkun2018 ; Aditi2018 ; Zhan2017 ; MAL2016 ; Kofler2013 ; Suzuki2012 ; Pan2018 ; Naikoo2019 as well as experimental confirmation of violation of LGI in a large variety of macroscopic systems ranging from superconducting qubits Palacios2010 ; Xiang2013 to nuclear spins Athalye2011 ; Formaggio2016 ; Katiyar2013 .

Explicit construction of LGI for multi-level quantum system involves identification of a dichotomic observable $Q$ such that its eigenvalues are restricted to $\pm 1$. LGI denoted by $K_{3}$ with the assumption of MR and NIM is bounded as $-3\leq K_{3}=C_{12}+C_{23}-C_{13}\leq 1$. Here $C_{ij}=\sum_{Q(t_{i}),Q(t_{j})=\pm 1}Q(t_{i})Q(t_{j})P_{ij}(Q(t_{i}),Q(t_{j}))$, where $Q(t_{i}),Q(t_{j})$ represents the outcomes of the strong quantum measurements of the observable $Q$ at times $t_{i}$ and $t_{j}$ respectively and $P_{ij}(Q(t_{i}),Q(t_{j}))$ represents the joint probability for the outcome of quantum measurement performed at time $t_{i}$ and $t_{j}$ to be $Q(t_{i})$, $Q(t_{j})$ respectively. The maximum quantum bound of $K_{3}$ for any $N$ level system is $3/2$ which is known as the Luder’s bound  Budroni2013 . Violation of Luder’s bound for an $N$ level quantum system, where $N>2$ is possible provided further degeneracy breaking measurements are introduced Emary2014 but violation of Luder’s bound for $N=2$ i.e. TLS is impossible within the Hermitian dynamics.

In this article, we demonstrate the possibility of violation of the Luder’s bound of $3/2$ and asymptotically approaching the algebraic bound of $3$ for the LGI parameter ($K_{3}$) for a TLS. We show that the dynamics of a TLS governed by a non-Hermitian Hamiltonian with real eigenvalues  HFJones2002 ; Bender1998 leads to such extreme violation of LGI. The implications of our findings are non-trivial in two ways: (a) violation of the Luder’s bound provides a clear quantification for temporal correlation which are more correlated then what is allowed by Hermitian dynamics of TLS (b) approaching the maximum value of $K_{3}=3$ for LGI is a clear indicator of extreme temporal correlation within the Hilbert space of TLS, which has only been observed in the past for Hermitian quantum mechanics for Hilbert spaces dimensions tending to infinity Emary2014 .

I.1 Probabilistic considerations of $K_{3}=3$

We start by noting that the maximum violation of LGI corresponding to $K_{3}=3$ imposes stringent constraints on the joint probabilities $P_{ij}(Q(t_{i}),Q(t_{j}))$ which can not be satisfied within pure Hermitian dynamics of TLS. This can be understood as follows. Measurement of LGI involves quantum measurement at three different times (call it $t_{1},t_{2},t_{3}$ such that $t_{1}<t_{2}<t_{3}$) with successive measurement performed in pairs. Now, $K_{3}=3$ implies that $C_{12}=C_{23}=1$ while $C_{13}=-1$. $C_{ij}=1$ in turn implies that joint probabilities $P_{ij}(+,+)+P_{ij}(-,-)=1$ and $P_{ij}(+,-)=P_{ij}(-,+)=0$, owing to the normalization constraints satisfied by the probability given by $\sum_{Q(t_{i}),Q(t_{j})=\pm 1}P_{ij}(Q(t_{i}),Q(t_{j}))=1$. Similarly, $C_{ij}=-1$ implies that $P_{ij}(+,+)=P_{ij}(-,-)=0$ and $P_{ij}(+,-)+P_{ij}(-,+)=1$. Hence the condition of $C_{ij}=-1$ and $C_{ij}=1$ are mutually exclusive in the sense that the former condition needs $P_{ij}(+,-)=P_{ij}(-,+)=0$ and rest to be finite while the later condition demands just the opposite. And hence the condition for $K_{3}=3$ implies that the joint probability of observation of “spin flip state” between times $t_{1},t_{2}$ given by $P_{12}(-,+)/P_{12}(+,-)$ and between the times $t_{2},t_{3}$ given by $P_{23}(-,+)/P_{23}(+,-)$ have to be identically zero while the joint probability of observation of ”same spin state” between time $t_{1},t_{3}$ given by $P_{13}(+,+)/P_{13}(-,-)$ have to identically zero.

Now, let us consider an ideal situation where the three inputs, the initial state, the Hamiltonian and the times $t_{1},t_{2},t_{3}$ are chosen in such a way that the time evolution of the initial state, when not interrupted by any measurement, evolves into eigenstate of the dichotomic observable $Q$ at $t_{1},t_{2},t_{3}$. Even under this ideal situation where the measurement is non-invasive in the sense that measured state is ensured to be an eigenstate of measurement operator itself, satisfying the condition of $C_{12}=C_{23}=1$ and $C_{13}=-1$ simultaneously is impossible as no physically realizable dynamics can produce states which can simultaneously satisfy $P_{12}(-,+)/P_{12}(+,-)=P_{23}(-,+)/P_{23}(+,-)=0$ and $P_{13}(+,+)/P_{13}(-,-)=0$. So the interesting question that one can ask is if there is any possibility for approaching asymptotically closeness to this condition. Again, based on above argument it is straightforward to see that Hermitian dynamics of a TLS can not do this because the speed of evolution of non-eigenstate of TLS on the Bloch sphere is always uniform (constant in time). The crucial element which is essential for approaching $K_{3}=3$ is a nonuniform, accelerated (or decelerated) evolution of state between two successive quantum measurements.

In what follows, we first discuss the equation of motion (the Bloch equation) for pseudo spin corresponding to TLS evolving under the influence of non-Hermitian Hamiltonian with real eigenvalues. We identify the presence of non-linear term in Bloch equation which is responsible for such accelerated evolution and obtain analytic solution of the equation. For identifying the parameter space of $K_{3}=3$, using these solutions we first identify the subspace of parameters corresponding to $C_{13}=-1$ and then look for the point in the subspace which also satisfies $C_{12}=C_{23}=1$.

II Non-Hermitian Bloch equation

A general non-Hermitian Hamiltonian for a TLS can be written as $H=(\vec{A}-i\vec{B})\cdot\vec{\sigma}$, where $\vec{\sigma}$ represents a vector whose components are given by Pauli matrices and $\vec{A},\vec{B}$ are vectors which live in ${\mathbb{R}}^{3}$ and to ensure that the eigenvalues of $H$ stay real we impose that $\vec{A}\cdot\vec{B}=0$ and $|\vec{A}|>|\vec{B}|$  Jordan1969 . Such non-Hermitian systems have been shown to exhibit non trivial effects such as acceleration in the speed of the evolution of a quantum state Jones2007 . If any arbitrary initial state is evolved through such non-Hermitian Hamiltonian, then the governing equation is given by Graefe2012 :

[TABLE]

It is interesting to note that this type of time evolution as in eq. (1) has been extensively discussed in many contexts like constrained quantum motion Brody1 , approach to thermal equilibrium Korsch , non-Hermitian quantum motion and dissipation Mizrahi ; Sergi , and also has been realized experimentally Tang ; Li ; Peng . The formal solution of the eq.(1) is given by:

[TABLE]

An experimental implementation of such a evolution requires a careful balancing of loss and gain of energy of the quantum system such that eigenvalues of the Hamiltonian remain real at all times. It should be evident, however, that the physical realisation of such an evolution is difficult since it requires extreme fine-tuning. The extension of eq.(1) which takes account of the possible impact of uncontrollable ambient noise in energy has been studied in Graefe2012 , and we will also implement it while discussing the possible effects of such noise on the violation of Luder’s bound in later section.

The Bloch equation dictating the time evolution of the vector $\vec{S}$ corresponding to eq.(1) is given by:

[TABLE]

Time evolution in eq.(3) is such that it preserves the length of the vector $\vec{S}(t)$ in case of pure states only. Note the appearance of a term which is non-linear in $\vec{S}$ in eq.(3). This is an unusual addition to the standard Bloch equation which usually includes the precession term (first term in eq.(3)) and the decay type term (second term in eq.(3)). The third term in eq.(3) is the one which is responsible for an accelerated (or decelerated) evolution of pure states on the Bloch sphere which is a crucial ingredient for maximizing the $K_{3}$ as described above. Further we would like to emphasize that this equation has solution which forms periodic orbits on the Bloch sphere with periods determined by the difference in eigenvalues of $H$ given by $\pm\sqrt{\vec{A}^{2}-\vec{B}^{2}}$ just like the Hermitian case. To obtain analytical solutions of eq. 3, we work in a Cartesian coordinate system defined by the unit vectors $\hat{A}$, $\hat{B}$ and $\hat{n}=\hat{A}\times\hat{B}$. Rewriting eq.(3) in its component form, we get

[TABLE]

[TABLE]

[TABLE]

In absence of the non-Hermitian term ($\vec{B}=0$), $\vec{S}.\hat{A}=S_{A}$ is a constant of motion as expected however as $\vec{B}$ becomes finite this is no more true. But, if $S_{A}$ is set to zero initially then it stays zero as a function of time owing to the fact that solution to (4) takes the form $S_{A}(t)=S_{A}(t_{0})\exp[4\left|B\right|\int_{t_{0}}^{t}S_{B}(t)]$. Geometrically this fact has two implications: (i) all solutions with boundary condition $S_{A}=0$ lies in the plane spanned by ${\{}\hat{B},\hat{n}{\}}$, (ii) all such solutions corresponding to pure states will trace out geodesics paths on the Bloch sphere. Lastly, note that all these three equation are invariant under the transformation, $S_{A}\rightarrow-S_{A}$. This implies that all closed looped trajectories laying on the Bloch sphere which are solutions of this equation, trajectories generated by geometric reflection of these trajectories about the ${\{}\hat{B}$-$\hat{n}{\}}$-plane are also valid solutions to the same equation. Now it is straightforward to obtain analytic solution for $S_{B}(t)$ and $S_{n}(t)$ for the case of $S_{A}=0$ given by :

[TABLE]

[TABLE]

It is evident from the form of the solutions that they correspond to non-uniform speed of evolution of the state and being on geodesic path (the great circle) these non-uniformities are expected to take extreme values when compared with any other path on the Bloch sphere.

II.1 Speed of state evolution in $S_{A}=0$ subspace

Let us start by considering an explicit non-Hermitian Hamiltonian with real eigenvalue given by $H_{\theta}=\ \sec(\theta)\ \sigma_{x}+i\ \tan(\theta)\ \sigma_{z}$, such that $\hat{A}=\hat{x}$ and $\hat{B}=\hat{z}$, and we choose an initial state for evolution which respects $S_{A}=0$ given by $\ket{\uparrow}_{y}=\{i/\sqrt{2},1/\sqrt{2}\}^{t}$. Since the non-Hermiticity in our case is solely governed by the parameter $\theta$ we need to study optimization of speed of state evolution with respect to this parameter. To study the quantum speed limit (SQL) Mandel1945 we first start by studying the evolution of geodesic distance between two initially ($t=0$) orthogonal states (in the $S_{A}=0$ subspace) given by $\delta=\cos^{-1}(|\braket{\psi(t)}{\downarrow}_{y}|)$ Chru1999 where :

[TABLE]

and $N(t)=1/\sqrt{\bra{\uparrow}_{y}exp[iH^{\dagger}_{\theta}t]\ exp[-iH_{\theta}t]\ket{\uparrow}_{y}}$ is the time dependent normalization derived from eq.(1). The distance $\delta$ evaluated as a function of time for different values of $\theta$ given by (see Appendix A):

[TABLE]

As can be seen from fig. 1, the variation of distance, $\delta$ with time is at a constant rate for the Hermitian case ($\theta=0$) while for the case of extreme non-Hermiticity ($\theta\rightarrow\pi/2$) it mostly shows a vanishing slow variation deep inside the interval $t=[0,\pi/2)$ and $t=(\pi/2,\pi]$ while depicting a extremely fast variation corresponding to an almost instantaneous spin-flip (see variation of $S_{n}$ given by eq.(8) in the plot) process in neighbourhood of $t=\pi/2$. It is important to note that that physical range of $\theta$ is given by $[0,\pi/2)$ as both $S_{B}$ and $S_{n}$ are ill-defined at $\theta=\pi/2$. As shown by Anandan and Aharonov Anandan1990 , the speed of evolution can be determined by expanding $\left|\braket{\psi(t)}{\psi(t+\delta t)}\right|^{2}=1-v^{2}\delta t^{2}+O(\delta t^{3})$  , where $v=\sqrt{\ v_{1}+v_{2}+v_{3}}$ is identified as the speed of evolution. Here $v_{1}=(\Delta\vec{A}\cdot\vec{\sigma})^{2}$, $v_{2}=(\Delta\vec{B}\cdot\vec{\sigma})^{2}$ and $v_{3}=-i\left\langle[\vec{A}\cdot\vec{\sigma},\vec{B}\cdot\vec{\sigma}]_{C}\right\rangle$ (see Appendix B). Hence $v$ for our case is obtained as $v=({\cos^{2}\theta})/({(1+\cos 2t\ \sin\theta)^{2}})$.

Inset of fig. 1 shows the plot of $v$ as a function of time and it depicts the direction correlation of $v$ and $\delta$. It show unbounded acceleration of state evolution in the neighbourhood of $t=\pi/2$ as $\theta\rightarrow\pi/2$. It is important to notice that, unlike the Hermitian case Anandan1990 ; Graefe2012 , the speed of evolution $v$ can not be identified with the standard deviation of energy ($\Delta E$) for non-Hermitian dynamics. Also, note that the third term $v_{3}$ is responsible for extreme acceleration and deceleration of state evolution for $\theta\rightarrow\pi/2$. Hence we have established that $S_{A}=0$ subspace allows for unbounded acceleration of states and we will show now that this fact facilitates an access to the algebraic maxima of LGI.

II.2 LGI and its algebraic maxima

For evaluation of $K_{3}$ we consider dynamics of TLS with Hamiltonian $H_{\theta}$ and a dichotomic observable for performing the quantum measurement given by $Q=\sigma_{y}$ (anti-parallel to $\hat{n})$. We also assume $t_{1}=0$. We further consider a situation where the temporal spacing between successive measurement is equispaced, i.e., $t_{3}=2t_{2}=2t$. This leads to $C_{13}=({\cos 4t+\sin\theta})/({1+\cos 4t\ \sin\theta})$ (see Appendix C). Note that $C_{13}$ becomes $-1$ at $t=\pi/4$. The LGI parameter at $t=\pi/4$ takes a simple form given by $K_{3}=1+\sin\theta+\sin^{2}\theta$, with $C_{12}=\sin\theta$ and $C_{23}=\sin^{2}\theta$. It is easy to see that $C_{12}\rightarrow 1,~{}C_{23}\rightarrow 1$ and $K_{3}\rightarrow\ 3$ and as $\theta\rightarrow\pi/2$. Note that $\theta\rightarrow\pi/2$ is exactly the limit in which unbounded acceleration of states was observed in fig. 1. Hence we have established that it is possible to asymptotically approach the conditions, $C_{12}=C_{23}=1$ and $C_{13}=-1$ simultaneously, which is the key to extremizing $K_{3}$ as pointed out in the introduction section of this article. It is very interesting to note from fig. 2 that, as $\theta\rightarrow\pi/2$, $K_{3}$ stays at the boundary of classical and quantum dynamics (i.e. $K_{3}=1$) most of the time and then it momentarily shoots up to extreme values of value of $K_{3}=-3$ (deep in classical region) or $K_{3}=3$ ( much beyond the quantum limit).

Hence we have not only show that the non-Hermitian dynamics of TLS allows for violation of LGI up to the algebraic bound but we also established a directly connection of this violation with extreme speed of state evolution over geodesic paths.

Finally, we perform a numerical search over the full parameter space of $K_{3}$ which comprises of, (i) all possible initial state which lie on the Bloch sphere (amounting to two parameters), (ii) all possible dichotomic observable $Q$ (two parameters) and (iii) all possible values of times given by the variables $t_{1},t_{2}$ and $t_{3}$ (three parameters).Hence it amounts to a scan for finding maximum possible value of $K_{3}=K_{3}^{max}$ in the space of these seven parameters for a given $\theta$. Our findings are depicted in fig. 3 which shows $K_{3}^{max}$ in the main plot. We also perform an independent numerical scan for identifying the maximum velocity $v=v^{max}$ in this parameters space for a given $\theta$ and plot it in the inset. This numerical exercise shows a direct correlation between the maximum violation of LGI and the maximum velocity of state evolution hence reinforcing our analytic finding for the subspace of $S_{A}=0$ and establishing it as a generic fact. The study also demonstrates that the choice of non-Hermitian Hamiltonian (via choice of $\theta$) single-handedly limits the maximum allowed violation of LGI where the lowest value for the $K_{3}^{max}$ corresponds to the Luder’s bound of $3/2$ corresponding to the case of $\theta=0,\pi$, i.e., the case of Hermitian Hamiltonian and the highest value of $K_{3}^{max}$ corresponds to $\theta\rightarrow\pi/2$. Hence, an increasing $\theta$ starting from $\theta=0$ to $\theta=\pi/2$ defines Hamiltonian with an increasing degree of non-Hermiticity defined in the sense of increasing degree of violation of LGI.

III Embedding of TLS into larger Hilbert space

It is also worthwhile to remark that non-Hermitian Hamiltonian can be simulated using Hermitian systems. Such non-Hermitian Hamiltonian dynamics can be reinterpreted as a dynamics of a subsystem of a higher-dimensional Hermitian system as shown in Ref. Ueda, ; Gunther, (for real eigenvalues of non-Hermitian Hamiltonian) and Ref. Ray, (for imaginary eigenvalues). Following these references we consider embedding the TLS into a larger four dimensional Hilbert space $\mathcal{H}_{4}=\mathcal{H}_{2}\otimes\mathcal{H}_{TLS}$ by adding an ancilla which is represented by a Hermitian two level system with $\mathcal{H}_{2}$ as its space of states. In order to generate the effective non-Hermitian dynamics of the TLS given by $H_{\theta}=\ \sec(\theta)\ \sigma_{x}+i\ \tan(\theta)\ \sigma_{z}$, the corresponding embedding Hermitian Hamiltonian (ancilla $\otimes$ TLS) reads as :

[TABLE]

where $H_{s}=\cos\theta\ \sigma_{x}$ and $V=-\sin\theta\ \sigma_{z}$.

The idea is to study time evolution generated by $H_{T}$ for those states in $\mathcal{H}_{4}$ which can be expressed in a form given by $\ket{\Psi_{T}}=N_{T}\ (\ \ket{\uparrow}_{z}\otimes\ket{\psi}\,+\,\ket{\downarrow}_{z}\otimes\eta\ket{\psi}\ )$ where $N_{T}$ is the normalization constant and $\eta$ is such that $\eta\ H_{\theta}=H_{\theta}^{\dagger}\ \eta$. The remarkable fact about this construction is that the action of time evolution operator $exp[-iH_{T}t]$ on the state $\ket{\Psi_{T}}$ keeps it form invariant such that,

[TABLE]

Hence the state of the isolated TLS evolving under the influence of the non-Hermitian Hamiltonian $H_{\theta}$ at a time instant $t$ can be extracted by projecting $\ket{\Psi_{T}(t)}$ with operator $P_{\uparrow}=\ket{\uparrow}_{z}\bra{\uparrow}_{z}\otimes\mathbb{I}_{2}$ and post-selecting over $\ket{\uparrow}_{z}$.

Since we are interested in those states of TLS which belong to the $S_{A}=0$ subspace so that they could lead to maximal violation of LGI, we consider the initial state ($t=0$) of the TLS to be $\ket{\psi}=\ket{\uparrow}_{y}$ which is consistent with the discussion above in eq.(9) which implies that the normalization $N_{T}=1/\sqrt{\braket{\Psi_{T}(0)}{\Psi_{T}(0)}}=\cos\theta/(\sqrt{2(1-\sin\theta)})$ where $\eta=\sec\theta\ \mathbb{I}_{2}+\tan\theta\ \sigma_{y}$. As mentioned earlier, the state of the TLS at time instant $t$ can be extracted by projecting $\ket{\Psi_{T}(t)}$ with operator $P_{\uparrow}=\ket{\uparrow}_{z}\bra{\uparrow}_{z}\otimes\mathbb{I}_{2}$ at that instant of time and then post-selecting over $\ket{\uparrow}_{z}$.

The post selection results in a state which describes the sub-ensemble of out comes whose measurement over ancilla resulted in $\ket{\uparrow}_{z}$. For obtaining a correct description of the non-Hermitian evolution of the state we need to normalize again over this sub-ensemble. The state after the post-selection is given by $\ket{\psi_{T}^{\uparrow}(t)}=N_{T}\ (\ \ket{\uparrow}_{z}\otimes\ exp[-iH_{\theta}t]\ket{\uparrow}_{y}\ )/\sqrt{\bra{\Psi_{T}(t)}P_{\uparrow}\ket{\Psi_{T}(t)}}$ which can be rewritten as $\ket{\uparrow}_{z}\otimes\{N(t)\ exp[-iH_{\theta}t]\ket{\uparrow}_{y}\}$ where we have used that fact that $\sqrt{\bra{\Psi_{T}(t)}P_{\uparrow}\ket{\Psi_{T}(t)}}=N_{T}\sqrt{\bra{\uparrow}_{y}\ exp[iH_{\theta}^{\dagger}t]\ exp[-iH_{\theta}t]\ket{\uparrow}_{y}}=N_{T}/N(t)$. Note that $N(t)$ is the same time dependent normalization that appears in eq. (9) hence indicating that the fact that the embedding is reliable and the effective non-Hermitian evolution of the state is given by $\ket{\psi(t)}=N(t)\ exp[-iH_{\theta}t]\ \ket{\uparrow}_{y}$. Now one has to perform further measurement of the form $\mathbb{Q}=\mathbb{I}_{2}\otimes Q$ (where $Q$ represents dichotomic observable for the TLS) on the sub-ensemble to determine the $K_{3}$ corresponding to effective non-Hermitian dynamics.

For understanding the $\theta\rightarrow\pi/2$ limit (equivalently the $K_{3}\rightarrow 3$ limit) from the perspective of embedding, we rewrite the embedding Hamiltonian $H_{T}=\mathbb{I}_{2}\otimes\sin\delta\ \sigma_{x}-\sigma_{y}\otimes\cos\delta\ \sigma_{z}$ and total initial state $\ket{\Psi_{T}(0)}=(\sqrt{(1+\cos\delta)/2})\ket{\uparrow}_{z}\otimes\ket{\uparrow}_{y}+(\sqrt{(1-\cos\delta)/2})\ket{\downarrow}_{z}\otimes\ket{\uparrow}_{y}$ in terms of $\delta$ defined as $\theta=\pi/2-\delta$. Hence we note that the embedding Hamiltonian remains well defined and corresponding initial state $\ket{\Psi_{T}(0)}$ retains its desired form (representing finite entanglement between the ancilla and the TLS) in the limit of small $\delta$ hence assuring that the proposed embedding of the TLS into a four level system is good enough to asymptotically approach the algebraic maximum of $K_{3}^{max}=3$. We note that in the $\delta\rightarrow 0$ limit, the value of $K_{3}$ to leading order can be shown to be $K_{3}(\delta\rightarrow 0)=3(1-\delta^{2}/2)$. It should be further noted that at the $\delta=0$ point, $\ket{\Psi_{T}(0)}$ becomes a separable state $\ket{\Psi_{T}(0)}=\ket{\uparrow}_{z}\otimes\ket{\uparrow}_{y}$ and the strategy of generating non-Hermitian evolution by employing a higher embedding itself breaks down. Hence, only asymptotically close values to algebraic maximum of $K_{3}$ can be achieved using this strategy.

IV Possible influence of energy fluctuation on LGI

The dynamics considered in the paper corresponds to balanced loss-gain of energy owing to the reality of eigenvalues of the non-Hermitian Hamiltonian. On the other hand, any experimental implementation of temporally resolved balanced loss-gain of energy of a quantum state will be impossible to maintain and it will inevitably suffer from finite temporal fluctuation of energy. To take such effect into account we consider the approach outlined in Graefe2012 , where eq.(1) has been considered with Gaussian white noise. This leads to modification of evolution equation for the density matrix and it takes the form given by :

[TABLE]

where $\kappa$ is the strength of the noise. Using this equation we recalculate the LGI parameter $K_{3}$ for different values of strength parameter $\kappa$. We found that the point of extreme violation of Luder’s bound re-normalizes itself to smaller values and eventually settles at the $K_{3}=1$ line for larger values of $\kappa$ as can be seen from fig. 4. We also plot the variation of maximum value achievable for the LGI parameter $K_{3}^{max}$ with the scaled noise parameter $\bar{\kappa}=\kappa\times 10^{5}$ (fig. 5) which provides the value of $\kappa$ at which the $K_{3}^{max}$ attains the value of 1.5 and eventually saturates to unity.

V Conclusion & outlook

To conclude, we have demonstrated that the non-Hermitian dynamics of TLS leads to violation of the Luder’s bound of $K_{3}=3/2$ for LGI. We further show that the extent of violation can be optimized to asymptotically approach the algebraic maximum of $K_{3}^{max}=3$, which is otherwise only feasible when the Hilbert space is infinite dimensional in the Hermitian case. We established that this extreme violation of LGI is a consequence of unbounded growth of quantum speed of state evolution of TLS owing to the non-Hermiticity induced non-linear terms in the Bloch equation. Our finding uncovers a completely new view of extreme temporal correlations of two level systems in terms of speed of state evolution.

We have also shown that such dynamics can be simulated in a realistic quantum system by embedding the non-Hermitian TLS dynamics into higher dimensional Hilbert spaces where states follow unitary time evolution. We have established that a four dimensional embedding of the TLS is enough to host $K_{3}\rightarrow 3$ limit which is an encouraging result from the perspective of possible experimental realization of proposed non-Hermitian dynamics. It is worth noting that non-Hermitian Hamiltonians for quantum systems have already been realized experimentally Tang ; Li . More importantly the non-Hermitian dynamics dictated by eqn. (1) has also been implemented experimentally using a photon interferometric network in Ref. Peng in two different ways. Firstly, they have used the experimental setup for simulating a non-unitary dynamics of a single-qubit, i.e., a non-Hermitian two level system. In a different experimental setup, unitary dynamics of a two-qubit system is realized and an effective non-Hermitian dynamics of a single qubit is studied via the embedding approach following the proposal of Ref. [Ueda, ; Gunther, ]. In order to compare the experiment findings of Ref. Peng and our theoretical work, we compare the plot of the quantity called ”distinguishability” defined in their paper which is directly related to the distance $\delta$ defined in eq.(10) in an appropriate limit as discussed below. For a fair comparison with the Hamiltonian in Ref. [Peng, ] we re-scale our Hamiltonian $H_{\theta}\rightarrow\cos\theta\,H_{\theta}$ and then calculate the geodesic distance $\delta$ Chru1999 of two orthogonal states (namely $\ket{H}=\{1,0\}^{t}$ and $\ket{V}=\{0,1\}^{t}$) after evolving them for time ”t”. It is worth noting that the distance measure defined in Ref. Peng given by $D[\rho_{1},\rho_{2}]=(1/2)\,Tr[|\rho_{1}-\rho_{2}|]$ is different then the geodesic distance $\delta$ but in ultra non-Hermitian limit ($\theta\rightarrow\pi/2$) they turn out to be the same (see fig. 6). We extract the experimental data points from fig. 2(c) of Ref. Peng and plot it together with the corresponding value of $\delta$ evaluated from the experimental inputs of Ref. [Peng, ] in fig. 6. We observe an excellent fit of the data with both $\delta$ and $D$ in the $\theta\rightarrow\pi/2$. This is indeed very encouraging as it not only implies that the experimental realization of this kind of non-Hermitian dynamics is possible, but it also implies that the extreme non-Hermitian limit of $\theta\rightarrow\pi/2$ identified via the extreme limits of $K_{3}$ could be an experimentally achievable limit. For completeness we also plot $\delta$ and $D$ as a function of time in the same plot for value of $\theta$ different then the $\theta\rightarrow\pi/2$ limit which demonstrate that they are not the same quantity in general.

Finally we would like to point out that the implication of our findings are of importance for the quantum information and quantum computation community at large which can be understood as follows:

(i) A non-hermitian TLS demonstrates extreme speed up of time evolution of state with respect to its hermitian counterpart,

(ii) the non-hermitian dynamics of such a TLS can be simulated by appropriately embedding the TLS into a higher dimensional Hilbert space with Hermitian Hamiltonian.

Hence (i) and (ii) together implies, if we want to speed up the time evolution of quantum state of TLS beyond the limit defined by its dynamics in isolation, we can achieve this by appropriately coupling the TLS to an external degrees of freedom following the prescription given in Eq.11. This observation implies that the coupling of a qubit to an appropriately chosen external quantum degrees of freedom can be used as a resource for speeding up of gate operations on the qubit and hence can be of great relevance to fast quantum information processing.

Note added :- It should be noted that these results were first reported in an abstract for a talk submitted on June 5, 2019 by one of the authors (S. Das) in the second annual conference on Quantum Condensed Matter held at the Indian Institute of Sciences, Bengaluru (see http://www.qmatiisc2019.in and the abstracts therein conf ). During the finishing stages of this manuscript, we noted that another work appeared on the arXiv Usha2019 which deals with similar idea however our treatment provides a deeper insight into the phenomenon in terms of a renormalized Bloch equation and the quantum speed limit. Also our work presents a Hermitian embedding of the TLS which was not discussed in arXiv Usha2019 .

Acknowledgements

It is a pleasure to thank Poonam Mehta for several discussions and comments. AVV would like to thank the Council of Scientific and Industrial Research (CSIR), Govt. of India for financial support and S.D. acknowledges the kind hospitality of the Condensed Matter and Statistical Physics Division at the Abdus Salam ICTP under the Associateship Scheme during the finishing stages of this work. S.D. would like to acknowledge support from DST-SERB MATRICS grant (MTR/2019/001043) and ARF grant from IISER Kolkata.

Appendix A The geodesic distance

Geodesic distance between two arbitrary pure states is defined as: $\delta=\cos^{-1}(|\braket{\psi}{\phi}|)$ Chru1999 . The two states here are: $\ket{\psi(t)}=N(t)\exp[iH_{\theta}t]\ket{\uparrow}_{y}$ and $\ket{\downarrow}_{y}=\{-i/\sqrt{2},1/\sqrt{2}\}^{T}$. The state $\ket{\psi(t)}$ is given by

[TABLE]

where $N(t)=\sqrt{\frac{1+\sin\theta}{2+2\cos 2t\ \sin\theta}}$. The time evolution of this state is shown in fig. 4. Also, the state $\ket{\psi}=\{-i/\sqrt{2},1/\sqrt{2}\}^{T}$ for $t=\pi/2$, irrespective of the values of $\theta$. Hence the distance $\delta$ reads as

[TABLE]

Appendix B The speed of evolution

For evaluating the speed of evolution we need to calculate the overlap between state a time $t$ give by $\ket{\psi_{t}}$ and state at time $t+\delta t$ given by $\ket{\psi_{\delta t}}$ which reads as

[TABLE]

where $U=e^{-iH_{\theta}(t+\delta t)}=\cos(t+\delta t)\ \mathbb{I}-i\ H_{\theta}\ \sin(t+\delta t)$ since $H_{\theta}^{2}=H_{\theta}^{\dagger\ 2}=\mathbb{I}$ for our case. Here we have considered normalized state, hence $\braket{\Psi_{t}}{\Psi_{t}}=1$. Now, we can expand the denominator as well as numerator in $\delta t$. Then writing the whole expression upto second order in $\delta t$ gives us three terms. Now, comparing with the expansion discussed above and $\left|\bra{\psi(t)}\ket{\psi(t+\delta t)}\right|^{2}=1-v^{2}\delta t^{2}+O(\delta t^{3})$, one identifies $v$ as the speed of the evolution of the state. For the given Hamiltonian and initial state (discussed in the main text), we arrive the following expression for the speed of evolution given by

[TABLE]

Appendix C Correlation optimization

The two time correlation $C_{ij}$ is given by $\sum_{Q(t_{i}),Q(t_{j})=\pm 1}P_{ij}(Q(t_{i}),Q(t_{j}))$, where the joint probability is defined as

[TABLE]

where $\ket{\psi_{I}}$ is the initial state. Now $C_{ij}$ are evaluated for the case of equal time spacing defined by $t_{3}-t_{2}=t_{2}-t_{1}=t$, with $t_{1}=0$ with inital state taken as the $\ket{\psi_{I}}=\ket{\uparrow}_{y}=\{i/\sqrt{2},1/\sqrt{2}\}^{T}$ for the Hamiltonian $H_{\theta}$ and measurement operator $Q=\sigma_{y}$ given in the main text. The joint probabilities corresponding to the correlation $C_{12}$ are given by

[TABLE]

Similarly, joint probabilities for the correlations $C_{23}$ and $C_{13}$ reads as follows

[TABLE]

Then the corresponding correlations take the form

[TABLE]

and

[TABLE]

Now putting $t=\pi/4$, the joint probabilities reduce to

[TABLE]

Similarly, joint probabilities for the correlations $C_{23}$ and $C_{13}$ reduce to

[TABLE]

and the correlations are given by $C_{12}=\sin\theta,C_{23}=\sin^{2}\theta$ and $C_{13}=-1$.