Vaibhav Madhok

The authors regret an oversight in acknowledging funding. This work was supported in part by grant No. SRG/2019/001094/PMS from SERB and MHRD/DST grants SB20210807PHMHRD008128, SB20210854EEMHRD008074 and DST/ICPS/QusT/Theme-3/2019/Q69. The authors would like to apologise for any inconvenience caused.

The central goal of a dynamical theory of evolution is to abstract the mean evolutionary trajectory in the trait space by considering ecological processes at the level of the individual. In this work we develop such a theory for a class of deterministic individual-based models describing individual births and deaths, which captures the essential features of standard stochastic individual-based models and becomes identical to the latter under maximal competition. The key motivation is to derive the canonical equation of adaptive dynamics from this microscopic ecological model, which can be regarded as a paradigm to study evolution in a simple way and give it an intuitive geometric interpretation. Another goal is to study evolution and sympatric speciation under maximal competition. We show that these models, in the deterministic limit of adaptive dynamics, lead to the same equations that describe the unraveling of the mean evolutionary trajectory as those obtained from the standard stochastic models. We further study conditions under which these models lead to evolutionary branching and find them to be similar to those obtained from the standard stochastic models. We find that, although deterministic models result in a strong competition that leads to a speedup in the temporal dynamics of a population cloud in the phenotypic space as well as an increase in the rate of generation of biodiversity, they do not seem to result in an absolute increase in biodiversity as far as the total number of species is concerned. Hence, they essentially capture all the features of the standard stochastic model. Interestingly, the notion of a fitness function does not explicitly enter in our derivation of the canonical equation, thereby advocating a mechanistic view of evolution based on fundamental birth-death events where fitness is a derived quantity rather than a fundamental ingredient. We illustrate our work with the help of several examples and qualitatively compare the rate of unraveling of evolutionary trajectory and generation of biodiversity for the deterministic and standard individual-based models by showing the motion of population clouds in the trait space.

The out-of-time-ordered correlators (OTOCs) and the Loschmidt echo are two measures that are now widely being explored to characterize sensitivity to perturbations and information scrambling in complex quantum systems. Studying few qubits systems collectively modeled as a kicked top, we solve exactly the three- and four-qubit cases, giving analytical results for the OTOC and the Loschmidt echo. While we may not expect such few-body systems to display semiclassical features, we find that there are clear signatures of the exponential growth of OTOC even in systems with as low as four qubits in appropriate regimes, paving way for possible experimental measurements. We explain qualitatively how classical phase space structures like fixed points and periodic orbits have an influence on these quantities and how our results compare to the large-spin kicked top model. Finally we point to a peculiar case at the border of quantum-classical correspondence which is solvable for any number of qubits and yet has signatures of exponential sensitivity in a rudimentary form.

We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step, and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in a fixed basis even although the measurement record is not informationally complete. We then compare this with the information generated and fidelities obtained by application of a different Haar random unitary at each time step. We give an upper bound on the maximal information that can be obtained in tomography and show that a covariance matrix taken from the Wishart-Laguerre ensemble of random matrices and the associated Marchenko-Pastur distribution saturates this bound. We find that physically, this corresponds to an application of a different Haar random unitary at each time step. We show that repeated application of random diagonal unitaries gives a covariance matrix in tomographic estimation that corresponds to a new ensemble of random matrices. We analytically and numerically estimate eigenvalues of this ensemble and show the information gain to be bounded from below by the Porter-Thomas distribution.

Does chaos in the dynamics enable or impede information gain in quantum tomography? We address this question by considering continuous measurement tomography in which the measurement record is obtained as a sequence of expectation values of a Hermitian observable evolving under the repeated application of the Floquet map of the quantum kicked top. For a given dynamics and Hermitian observables, we observe completely opposite behavior in the tomography of well-localized spin coherent states compared to random states. As the chaos in the dynamics increases, the reconstruction fidelity of spin coherent states decreases. This contrasts with the previous results connecting information gain in tomography of random states with the degree of chaos in the dynamics that drives the system. The rate of information gain and hence the fidelity obtained in tomography depends not only on the degree of chaos in the dynamics and to what extent it causes the initial observable to spread in various directions of the operator space, but, more importantly, how well these directions are aligned with the density matrix to be estimated. Our study also gives an operational interpretation for operator spreading in terms of fidelity gain in an actual quantum information tomography protocol.

Out-of-time-ordered correlators (OTOC) are a quantifier of quantum information scrambling and are useful in characterizing quantum chaos. We propose an efficient quantum algorithm to measure OTOCs that provides an exponential speed-up over the best known classical algorithm provided the OTOC operator to be estimated admits an efficient gate decomposition. We also discuss a scheme to obtain information about the eigenvalue spectrum and the spectral density of OTOCs as well as an efficient algorithm to estimate gate fidelities.

We study operator growth in a bipartite kicked coupled tops (KCTs) system using out-of-time ordered correlators (OTOCs), which quantify “information scrambling” due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system’s dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed-phase space, we use Percival’s conjecture to partition the eigenstates of the Floquet map into “regular” and “chaotic.” Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.