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How Useful are Computational Models to Traders in Their Decision Making?

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Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: References

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix D: QMPE in region (e): 3rd order EP The region (e) in the (d,˜ Γ) plane of Hatano’s model is composed of a single point It is a third order EP where all the non-zero eigenvalues become equal and their eigenvectors coalesce. Since this is a single point with fixed values of ˜d and Γ, the right eigenvectors are explicitly given by Of course |r1i and hℓ1| have same expressions both in Eqs. (3) and (D2). The left eigenvectors are ) The non-zero eigenvalue of the Lindbladian [Eq. (2)] in region (e) is −iλ = −i4√ 3. Importantly, as a consequence of the 3rd order EP, we can only achieve an approximately diagonal form i.e. Jordan normal form LbJn leading to the equation Due to the off-diagonal nonzero defects in Eq. (D5), the time evolution of density matrix elements and other observables pick up additional algebraic terms in time. 1. Density matrix elements In region (e), the density matrix elements have the expressions below In Fig. 15(a), we present existence of t˜+ and t˜− as function of the control parameters ˜dI and ˜dII using the fixed dissipation protocol. The solutions’ surfaces in Fig. 15(a) reveal that there are regions where only one of the solutions t˜+ or t˜− exist leading to single QMPE, and in other regions both the solutions exist resulting in double QMPE. Particularly, lower values of ˜dI and ˜dII correspond to single intersection time solution and therefore single QMPE. To understand the behaviors of t˜+,− more clearly, we fix ˜dI and display t˜+,− as functions of ˜dII in Fig. 15(b). The Fig. 15(b) shows that the two branches t˜+ and t˜− interchange as ˜dI < ˜dII. More specifically, for ˜dI > ˜dII we have t˜+ > t˜− and for ˜dI < ˜dII we have t˜+ < t˜−. The underlying reason is that the discriminant p γ 2 1 − 4γ2γ0 in Eq. (D8) is proportional to sign of ( ˜dI − ˜dII). Thus the two intersection times interchange their roles by changing the sign accompanying the discriminant as we go from ˜dI > ˜dII to ˜dI < ˜dII. The Fig. 16(a) demonstrates an explicit example of double QMPE in ground state probability in region (e). Like other regions, here also the second intersection (shown in the inset of Fig. 16(a)) is much feeble in comparison to the first intersection. 2. Average energy For the same parameters used to obtain QMPE for ρgg(t) in Fig. 16(a), we would like to examine the possibility of QMPE in average energy E(t). Indeed, we observe in Fig. 16(b) that energy also exhibits double QMPE, marked by the twice intersection of zero by the energy difference ∆E(t) between the two copies. The insets of Fig. 16(b) shows separately the two intersections in the time evolution of the energies for the two copies. The second intersection is much weaker in magnitude compared to the first intersection just like the case of ground state probability. Note that the first intersection between the energies of I and II occur after they overshoot their steady state value which they reach during some early time of the relaxation process. 3. von Neumann entropy Here we investigate the von Neumann entropy SvN(t), starting from two copies I and II. We use same parameters that have produced double QMPE both in ground state probability [Fig. 16(a)] and average energy [Fig. 16(b)]. In fact, the Fig. 17(a) shows that the entropy also exhibits double QMPE indicated by twice zero intersections (the second one shown in inset of Fig. 17(a)) of the entropy difference ∆SvN(t) between the two copies. 4. Temperature In this section we study the thermal QMPE in region (e). We observe thermal QMPE in region (e) by using variable dissipation protocol in Fig. 17(b). There we observe single intersection and reversal of identities (i.e. hotter becoming colder and vice versa) occur after the negative temperatures have happened. So, such temperatures can be considered as well-defined and therefore this intersection in Fig. 17(b) corresponds to single QMPE in temperature in region (e).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix D: QMPE in Region (e)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix C: QMPE in region (b): purely exponential relaxation In this Appendix we discuss QMPE in region (b) of Hatano’s model that contain eigenvalues 0, −iλ2, −iλ3, −iλ4 obeying 0 < λ2 < λ3. 1. Density matrix elements where In Fig. 12(a), we present the intersection time solutions denoting QMPE in ρgg(t) as functions of the control parameters ˜dI and ˜dII. Indeed, we observe that there are parameter regions where both t˜+ and t˜− are positive and finite, indicating double QMPE. Other regions have only one intersection time, either t˜+ and t˜−, implying single QMPE. To demonstrate double QMPE, the Fig. 12(b) exhibits a specific example of double intersection in ρgg(t). However, the two copies become close enough so that the second intersection (shown in inset of Fig. 12(b)) is weak in magnitude and hard to detect. 2. Energy We look for solutions to ∆Es(t) = 0, which can give to single or multiple QMPE depending on the number of positive finite intersection time solutions. The Fig. 13(a) shows that the average energy exhibits double QMPE indicated by the twice zero intersections of ∆Es(t), for the same set of parameter values producing double QMPE in ρgg(t) in Fig. 12(b). 3. von-Neumann entropy Here we ask if the entropy, like density matrix and energy, also exhibit double QMPE for the same set of parameters used in Figs. 12(b) and 13(a). However, in Fig. 13(b), we observe that the entropy difference ∆SvN(t) between two copies intersect zero only once, giving rise to single QMPE instead of double QMPE. 4. Temperature In Fig. 14(a), we present the time evolution of temperatures for the two copies I and II, one initially starting as hotter with respect to the other. We observe that the temperatures of the two copies intersect each other i.e. the hotter becomes colder and vice versa. Notably, unlike ground state probability [Fig. 12(b)], energy [Fig. 13(a)] that can exhibit double QMPE, we have observed only single QMPE in temperature. If observation of multiple QMPE is possible or not with purely exponential relaxations, remains an open question. 5. Kullback-Leibler divergence Here we study the possibility of QMPE in KL divergence for the same parameters used to explore thermal QMPE in Figs. 14(a). The Fig. 14(b) shows QMPE in KL divergence, however the intersection time corresponds to negative temperatures in Fig. 14(a).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix C: QMPE in Region (b)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix B: Eigenvectors of L in region (a1) (a1) In this Appendix, we give the explicit expressions for the right eigenvectors |rki and left eigenvectors hℓk| of the Lindbladian [Eq. (2)] in region (a1) [Sec. V], where k = 2, 3, 4. The right and left eigenvectors corresponding to the zero eigenvalue i.e. steady state. have already been provided in Eq. (3). Below are the expressions for the right eigenvectors The left eigenvectors are given by where

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix B: Eigenvectors

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References VI. SUMMARY We have studied QMPE in a two-level quantum system subjected to an oscillatory electric field and dissipative coupling with the environment. The Lindbladian of this two-level driven dissipative system, termed as Hatano’s model, has rich variety of eigen-spectrum, especially EPs and complex eigenvalues which are generic features of non-Hermitian quantum systems. We have investigated the roles of complex eigenvalues and EPs on the QMPE. The exact analytical formulae obtained for time evolved density matrix elements enable us to propose analytical criteria for the QMPE in observables. Interestingly, at the EPs, the density matrix elements and consequently observables (e.g. energy) attain additional algebraic time dependent relaxations apart from the usual exponential relaxations. Such additional algebraic time dependence at the EPs leads to double QMPE (twice intersections between trajectories starting at different initial conditions and relaxing towards the same steady state). We have procured the exact expressions for intersection times of trajectories, that mark the occurrence of QMPE and we provide analytical criteria that demarcate control parameters’ regimes with single or double or no QMPE. Whereas the multiple QMPE at EPs in the present model is restricted to maximum double QMPE i.e. two intersections, the multiple QMPE at the complex eigenvalues can result in even more number of intersections in a oscillatory manner. Notably, we have established the existence of thermal QMPE (single QMPE at EPs and multiple QMPE at complex eigenvalues), in presence of coherence in Hatano’s model. It is fascinating that, similar to energy and entropy, temperature also exhibits multiple QMPE where the initially hotter and colder copies intersect each other multiple times. Such multiple reversals of identities (i.e. hotter becoming colder and vice versa) result in multiple thermal QMPE. In spite of achieving multiple QMPE in several observables (ground state probability, energy, entropy, temperature) using either additional algebraic time dependence at EPs or oscillations at complex eigenvalues, surprisingly the KL divergence (a distance measure from steady state) shows single QMPE only. Overall, our theoretical analysis exhibits that EPs and complex eigenvalues can be used as systematic routes to generate multiple QMPE. However, we should mention that the presence of EPs or oscillations are not necessary for multiple QMPE. In this connection, we have shown that multiple QMPE can be achieved even with purely exponential relaxations, by putting suitable constraints on the gap statistics of the eigen-spectrum (Appendix C). An important and generic future goal in QMPE research would be to understand its utility in speeding up quantum processes. In fact, a recent work on accelerating relaxation in open quantum systems by engineering EPs [84] hints towards the connection of such acceleration to QMPE. Our present work analyzes in detail how EPs lead to QMPE and provide criteria on control parameters to observe the QMPE. It is important to find out the deeper connections between the QMPE and faster relaxations at EPs and for generic open quantum systems. In this connection, an intriguing question is to find how the quantum speed limit may affect the degree of faster relaxation that can be achieved through the QMPE. Our work on QMPE by controlling the amplitudes of oscillatory electric fields hints towards possible experimental realizations of the QMPE in qubit, keeping in mind the recent advancement in qubit related experiment [119]. The analysis of QMPE in the present study paves path for future investigations of QMPE in quantum many-body systems. - We thank Naomichi Hatano, Sosuke Ito, Ryo Hanai, Arnab Pal, Naruo Ohga and Kohei Yoshimura for useful discussions. This work is partially supported by the Grants-in-Aid for Scientific Research (Grant No. 21H01006 and No. 20K14428). A.K.C. gratefully acknowledges postdoctoral fellowship from the YITP. The numerical calculations have been done on Yukawa-21 at the YITP. Acknowledgements.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Summary

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References V. QMPE IN REGION (a1): OSCILLATIONS In this section, we explore QMPE in region (a1) of Hatano’s model which contains complex eigenvalues with both non-zero real and imaginary parts. The eigenvalues are non-degenerate and they are denoted by 0, λ2, λ3 = λre+iλim, λ4 = λre−iλim, where the subscripts re and im correspond to real and imaginary parts, respectively. In addition, the eigenvalues satisfy the relation 0 < λ2 < λre meaning λ2 is the slowest relaxation mode. In contrast to the nontrivial approximate diagonalization [Eqs. (10)- (11)] for second-order EP in Sec. IV, here the complete diagonalization can be obtained through the usual method A. Density matrix elements The the individual density matrix elements ρj (t) (j = 1, 2, 3, 4) are given by To discuss QMPE in the density matrix, let us focus on the ground state probability i.e. ρgg(t). Of course, there is no algebraic time dependence in the relaxation observed in Sec. IV. Rather, the interesting part of the relaxation in region (a1) originates from the complex nature of the eigenvalues and the corresponding coefficients ak in Eq. (27). Specifically, we have found that a3 and a4 are complex conjugates of each other, they take the forms a3 = are + iaim and a4 = are − iaim. Using such expressions, we obtain the following formula for ρgg(t): In Fig. 7 we present an example of multiple QMPE in the ground state density matrix element ρgg(t), where we have chosen parameters that satisfy the QMPE criteria Eq. (30) for region (a1). We observe that the two copies I and II relax in oscillatory manner and intersect each other multiple times. The inset explicitly shows that the difference between the two copies ∆ρgg(t) intersects zero multiple times, confirming the multiple QMPE. Comparing Fig. 3(a) [region (d)] with the inset of Fig. 7 [region (a1)], we understand that the number of intersections in region (a1) is not restricted to two as in region (d). Therefore, in region (a1) one can have many more intersections and higher order QMPE compared to the maximum double QMPE in the region (d). Since the amplitudes of the oscillations can be controlled by tuning the initial driving strengths ˜dI and ˜dII, complex eigenvalues seem to be useful to obtain multiple intersections and thereby multiple QMPE. It would be important in the future to find the exact number of solutions to ∆ρgg(t) = 0 either analytically or numerically. B. Energy The formal expression for average energy E(t) is given by Eq. (18). In the previous Sec. IV, we have already shown that QMPE in ground state probability does not necessarily imply QMPE in energy, in general. However, it is possible to find control parameters that can give rise to QMPE in both observables, although the intersection time(s) are generally different. Following the same path, we would like to see if E(t) shows multiple QMPE for the same set of parameters used to achieve multiple QMPE for ρgg(t) in Fig. 7. Indeed, Fig. 8 exhibits multiple intersections between time evolved copies of energies starting from I and II, both relaxing towards the same final energy. The inset of Fig. 8 alternatively demonstrates the multiple QMPE through multiple intersections of the energy difference ∆E(t) = EI(t)−EII(t) with zero, remarkably different from Fig. 3(b) with a maximum of two such intersections in region (d). C. von Neumann entropy Next we consider the von Neumann entropy SvN(t) [Eq. (22)]. To observe QMPE in SvN(t), we consider the intersection times for the entropy difference between two copies ∆SvN(t) = S I vN − S II vN. In Fig. 9, we present the behavior of ∆SvN(t) for the same set of parameter values used in Figs. 7 and 8. Indeed, we observe that ∆SvN(t) intersects zero multiple times showing the existence of multiple QMPE in the von Neumann entropy. D. Temperature Here, we study the thermal QMPE in region (a1) using the system temperature defined in Eq. (23). In Fig. 10, we present the time evolution of the temperatures of two copies I and II, one initially hotter than the other. It is fascinating that the temperatures of both the copies relax in oscillatory manner towards the same steady state temperature and during the relaxations, they intersect each other multiple times, leading to multiple thermal QMPE. This, in turn, implies that the hotter and colder copies reverse their identities multiple times (after each intersection the previous hotter copy turns into colder copy and remains so until the next intersection). Such oscillatory multiple thermal QMPE is observed here for the first time in quantum systems (as far as our knowledge is concerned), although multiple thermal MPE in classical inertial suspensions has already been reported [42]. The physical interpretation and utility of multiple thermal QMPE have to be explored in future in more details. Note that we have used the same parameter set for multiple thermal QMPE as in Figs. 7-9. Similar to Ref. [65], here also we observe negative temperature [114– 118] in the early stage of relaxation. This is due to nonmonotonic natures of energy and entropy as functions of time. However, it is satisfying that even if we neglect such initial negative temperatures, we observe multiple QMPE at later times of the relaxation process where the temperatures of both the copies remain positive. E. Kullback-Leibler divergence In this subsection we explore the QMPE in KL divergence defined in Eq. (24). Since we have been able to obtain multiple QMPE in region (a1) in all other observables considered here, it is natural to ask about the possibility of multiple QMPE in DKL(t) as well. However, we have found that the KL divergence does not show any intersection between two copies I and II for the same parameter values used in Figs. 7-10. Therefore, we have to use different values of the initial drives (still within the fixed dissipation protocol) to achieve QMPE in DKL(t), presented in Fig. 11(a). Intriguingly, we have only observed single intersection between the two time evolved copies, resulting in single QMPE in KL divergence. This means that once the initially higher distant copy intersects the initially lower distant copy producing QMPE, thereafter the new lower distant copy (that was initially higher distant) remains closer to the steady state for the rest of the relaxation process. In fact, for all other parameter sets that we have checked, we could not find multiple QMPE in DKL(t). The proof of the possibility or impossibility of multiple QMPE in KL divergence in presence of complex eigenvalues remains unsolved and could be an important question to investigate later. In this scenario, it would be interesting to examine the corresponding behaviors of the speeds vKL(t) [Eq. (25)] of the two copies I and II. In Fig. 11(b), we present the time evolution of speeds for the same set of parameters that gave rise to single QMPE in DKL(t). It is fascinating to observe that the speeds, unlike the distances, intersect each other multiple times producing multiple QMPE. The first intersection between the speeds of the two copies [Fig. 11(b)] shows a time lag in comparison to the intersection between distances of the same copies [Fig. 11(a)]. Such a time lag between the QMPE in KL divergence and the speed is in agreement with the observation in region (d) [Fig. 6]. However, contrary to single QMPE in vKL(t) in region (d), we have obtained multiple QMPE in the speed in region (a1). The contrasting behavior of single QMPE in DKL(t) and multiple QMPE in vKL(t) imply that the initially higher distant copy that becomes lower distant after the intersection in distances, can remain at lower distance even if its speed sometimes become smaller than the speed of the current higher distant copy. This is interesting because the system almost behaves like an intelligent and efficient runner in a race who knows how to adjust the speed (i.e. sometimes running faster and sometimes running slower than the people behind) and still remains in the first position to finish the race.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: QMPE in Region (a1)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix A: Eigenvectors of L in region (d) In this Appendix, we give the explicit expressions for the right eigenvectors |rki and left eigenvectors hℓk| of the Lindbladian [Eq. (2)] in region (d) [Sec. IV], where k = 2, 3, 4. The right and left eigenvectors corresponding to the zero eigenvalue i.e. the steady state. have already been provided in Eq. (3). Below are the expressions for the right eigenvectors where the superscript T denotes transpose. The left eigenvectors are obtained to be where the elements lkn are explicitly given by where the eigenvalues λ2 and λ4 are given by Eq. (12) and the parameters η, δ1, δ2, δ3 used in Eq. (A3) have the following forms

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