. The maximum value of K 3 for an N level system is 3/2 and is termed as Lüders bound 5. Violation of this bound for an N level quantum system, where N > 2 is possible provided further degeneracy breaking measurements are performed 12 but violation of Lüders bound for N = 2 i.e. a two level system (TLS) is impossible within the unitary dynamics.
However, any quantum system in a natural setting is expected to have finite coupling with its environment de-grees of freedom leading to non-unitary dynamics. All dynamics which are completely positive (CP) and trace preserving are considered as valid dynamics in quantum mechanics [13][14][15][16][17][18]. Unitary dynamics is the most common example of such CP dynamics. Rest of the set of CP dynamics is comprised of unital and non-unital dynamics. It is known by now, in case of both the unitary and unital dynamics that upper bound on the LG parameter K 3 is Lüders bound 9,19. However, set of dynamics outside the sub-set of CP dynamics, which is only positive and trace preserving has not been explored much earlier. In this article we focus on on such set of dynamics popularly referred as non-Hermitian dynamics i.e. C N H (see FIG. 1 (Left)). Recently it has been shown that K 3 can take values upto its algebraic bound in case of a TLS undergoing non-Hermitian dynamics (PT-symmetric with fixed eigenspectrum) 20,21 and close to the exceptional point in the PT-symmetric phase 22. Whereas we explore the non-Hermitian dynamics of a TLS across the exceptional point separating the PT-symmetric and PTbroken phases. We find that when the dynamics is in PT-symmetric phase it is impossible to find values of K 3 approaching its algebraic maximum i.e. 3, except close to the exceptional point only. More interestingly we find that it is always possible to approach the maxima K 3 → 3 in the PT-broken phase. We therefore explore and present the understanding of how the initial state, measurement operator and SOE of the state conspire together leading to such extreme temporal correlations for PT-broken phase as well as close to the exceptional point in the PT-symmetric phase. We find that change in the SOE and the time interval dependence of the evolution, which are impossible in unitary dynamics are responsible for such extreme correlations when measurement is chosen appropriately. The maximum SOE must reach its maximum value in the optimized parameter space, which is comprised of parameters of initial state, and time interval between measurements. Moreover, the minimum value of SOE must become zero during the evolution of the state between the two successive measurements for algebraic maximum of K 3 → 3. Interestingly, we find that this minimum SOE can be treated as order parameter to characterize the PT-transition as its value is finite on the PT-symmetric side and identically zero on the PT-broken side.. The remainder of this article is organized as follows. In section II we revisit solution of the dynamical equation representing non-Hermitian dynamics in terms of spin components of the Bloch vector and find their relation with the SOE of the state. In section III we find maximum values of LG parameter K 3 in both PT-symmetric and broken regime optimized in the full parameter space. We then present the dynamical process and its relation with the SOE, resulting in the algebraic maximum i.e. K 3 = 3. Section IV is devoted to the realization of the non-Hermitian dynamics of TLS by exploiting a three level atom coupled to an environment undergoing Lindbladian dynamics. Finally, we dedicate section V on discussion and conclude in the section VI.

II. DYNAMICAL SOLUTION AND SOE
If any arbitrary initial state ρ 0 of a TLS is evolved through a generic non-Hermitian Hamiltonian H P T = ( A − i B). σ, then the governing dynamical equation is given by 23 : The formal solution of the eq.(1) is given by:. ( It should be noticed that the two eigenstates of the Hamiltonian H P T are the fixed points of this dynamics. Assuming | ± are the right eigenstates of t implying the fixed point behaviour of the two eigenstates. The corresponding Bloch like equation controlling the time evolution of the Bloch vector S corresponding to eq.(1) is given by: To obtain analytical solutions of eq. (3), we transform the coordinate system and work in the Cartesian coordinate system defined by the unit vectorsÂ,B and n =Â ×B. Rewriting eq. (3) in its component form, we obtain where We can now define the SOE of the Bloch vector S. Using the definition of SOE by Anandan and Aharonov 24, we can write by expanding the term 23. Numerical scan shows that the maximum values of K 3 lies in theB −n plane. Therefore, we now work in the S A = 0 subspace. In this subspace i.e. S a (t) = 0. We find that all the we can write down the solutions of eq. (5-6) in a compact form as: where constant C is determined by initial values of S B and S n at time t = 0. Since these solutions represent the evolution on the geodesic in the planeB −n, and there are two fixed points on the the geodesic. Therefore, it is important to ask which of the two paths on the geodesic separated by two fixed points the state follows.
In order to specify that we Taylor expand the solution S B (t) in time upto first order. We find that the sign of the coefficient of first order term which is proportional to S n (0) ( A + 2BS n (0) ) decides the path followed by the initial state.
It should be noted that √ A 2 − B 2 > 0 i.e. γ/J < 2 then we are in PT-symmetric regime and when √ A 2 − B 2 < 0 i.e. γ/J > 2 then we are in PT-broken regime and the sinusoidal functions in eq. (7) and (8) change to hyperbolic functions, and periodicity of S B and S n is lost.
We now derive explicit expressions for the correlations for a typical TLS. For simplification of analytic expressions we choose A = Jx, and B = γ 2ẑ. Then the non-Hermitian Hamiltonian H P T = Jσ x − i γ 2 σ z has two exceptional points at γ = ±2J. It should be noted that work along similar line has been done in 20. However there is a subtle difference that should be pointed out. Since a PT-symmetric Hamiltonian H P T is connected with a Hermitian Hamiltonianĥ via similarity transformation P as: H P T = Pĥ P −1. For the Hermitian Hamiltonianĥ = σ x the form of the similarity transformation used in 20 is given by P = e θσy, where θ = (1/2) tanh −1 (sin α). This transformation is such that it keeps the eigenspectrum fixed. However, here the similarity transformation connecting the H P T with the We can now write each components of the SOE (v(t)) which are connected with the components of S(t) with the following relations: where n denotes the directionn =x ×ẑ = −ŷ. Utilizing the analytic expressions in the eqs. (9-11) above we can write the maximized SOE, v max = 1 + γ/2 given J = 1. On the other hand minimum value of SOE, v min = 1 − γ/2 in the PT-symmetric phase and identically zero in the PT-broken phase.

III. LGI AND PT-SYMMETRY
In this section we analyse the LG parameter K 3 across the exceptional point. We consider initial state as | ψ = [ cos θ 2 e iφ, sin θ 2 ] T such that ρ 0 = | ψ ψ | and measurement operator of the form Q = n · σ with n = ( sin θ m cos φ m, sin θ m sin φ m, cos θ m ). We also assume that t 1 = 0. Therefore, we have two time intervals namely, t 2 − t 1 and t 3 − t 2 when a state evolves, and measurements are made at three time instants t 1, t 2 and t 3. We then numerically optimize the expression of K 3 over the complete parameter space comprising of six parameters { θ, φ, θ m, φ m, t 2, t 3 }, for a fixed value of ratio γ/J as illustrated in FIG. 1 (Middle).
We find that for any non-zero value of γ/J, the maximum accessible value of K 3 is always greater than Lüders bound of 3/2. Moreover, as we move across the exceptional point i.e. γ/J > 2, it is always possible to access the algebraic maximum of K 3. We now focus on the initial state and measurement operator dictating the optimization of the temporal correlations leading to this maximization of the LG parameter K 3. We note that the parameters φ and φ m are 3π/2 and π/2 respectively, for all the maximized values of K 3 for different γ values. This corresponds to initial state | ψ and measurement direction n lying in theŷ −ẑ plane, in the optimized parameter space. This fact implies that all the trajectories followed by the state upon evolution that maximizes K 3 lie on the geodesic in theŷ −ẑ plane on the Bloch sphere.
PT-symmetric regime:-In this regime γ/J < 2, implying all the trajectories on the Bloch sphere are periodic in nature.
The measurement operator Q represented