It has recently been proposed classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the corresponding quantum system is in a Gaussian state. We generalized these results by providing classical analogs of the generalized purities, Bastiaans-Tsallis entropies, R\'enyi entropies, and logarithmic negativity for classical integrable systems. These classical analogs are entirely characterized by the classical covariance matrix. We compute these classical analogs exactly in the cases of linearly coupled harmonic oscillators, a generalized harmonic oscillator chain, and a one-dimensional circular lattice of oscillators. In all of these systems, the classical analogs reproduce the results of their quantum counterparts whenever the system is in a Gaussian state. In this context, our results show that quantum information of Gaussian states can be reproduced by classical information.
It has recently been proposed classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the corresponding quantum system is in a Gaussian state. We generalized these results by providing classical analogs of the generalized purities, Bastiaans-Tsallis entropies, R'enyi entropies, and logarithmic negativity for classical integrable systems. These classical analogs are entirely characterized by the classical covariance matrix. We compute these classical analogs exactly in the cases of linearly coupled harmonic oscillators, a generalized harmonic oscillator chain, and a one-dimensional circular lattice of oscillators. In all of these systems, the classical analogs reproduce the results of their quantum counterparts whenever the system is in a Gaussian state. In this context, our results show that quantum information of Gaussian states can be reproduced by classical information.
We obtain a classical analog of the quantum covariance matrix by performing its classical approximation for any continuous quantum state, and we illustrate this approach with the anharmonic oscillator. Using this classical covariance matrix, we propose classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the quantum counterpart of the system under consideration is in a Gaussian state. As is well known, this matrix completely characterizes the purity, linear quantum entropy, and von Neumann entropy for Gaussian states. These classical analogs can be interpreted as quantities that reveal how much information from the complete system remains in the considered subsystem. To illustrate our approach, we calculate these classical analogs for three coupled harmonic oscillators and two linearly coupled oscillators. We find that they exactly reproduce the results of their quantum counterparts. In this sense, it is remarkable that we can calculate these quantities from the classical viewpoint.
We obtain a classical analog of the quantum covariance matrix by performing its classical approximation for any continuous quantum state, and we illustrate this approach with the anharmonic oscillator. Using this classical covariance matrix, we propose classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the quantum counterpart of the system under consideration is in a Gaussian state. As is well known, this matrix completely characterizes the purity, linear quantum entropy, and von Neumann entropy for Gaussian states. These classical analogs can be interpreted as quantities that reveal how much information from the complete system remains in the considered subsystem. To illustrate our approach, we calculate these classical analogs for three coupled harmonic oscillators and two linearly coupled oscillators. We find that they exactly reproduce the results of their quantum counterparts. In this sense, it is remarkable that we can calculate these quantities from the classical viewpoint.
This thesis explores important concepts in the area of quantum information geometry and their relationships. We highlight the unique characteristics of these concepts that arise from their quantum mechanical foundations and emphasize the differences from their classical counterparts. However, we also demonstrate that for Gaussian states, classical analogs can be used to obtain the same mathematical results, providing a valuable tool for simplifying calculations. To establish the groundwork for the subsequent analysis, we introduce some fundamental ideas from quantum field theory. We then explore the structure of parameter space using the fidelity and the Quantum Geometric Tensor (QGT), which is composed of the Quantum Metric Tensor and the Berry curvature. We also introduce the Quantum Covariance Matrix (QCM) and show its relationship to the QGT. We present how the QCM can be used to study entanglement between quantum systems by obtaining the purity, linear entropy, and von Neumann entropy. To illustrate these concepts, we calculate all these quantities for several systems, including the Stern-Gerlach, a two qubits system, two symmetrically coupled harmonic oscillators, and N coupled harmonic oscillators. In the final section of this thesis, we examine how the aforementioned quantum concepts can be applied in a classical sense, following the approach taken by Hannay with the Berry phase. We examine classical analogs of the QGT and QCM and since for Gaussian states, all the necessary information to calculate purity, linear entropy, and von Neumann entropy is contained within the QCM, we also generate classical analogs for them. These results in turn can be used to derive measures of separability for classical systems.
The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space-a common meeting point to both classical and quantum density statistical descriptors. Here, this issue is tackled by investigating the behavior of classical analogs arising upon the removal of all interference traits displayed by the Wigner distribution functions associated with a given pure quantum state. Accordingly, the dynamical evolution of the linear and von Neumann entropies is numerically computed for a continuous-variable bipartite system, and compared with the corresponding classical counterparts, in the case of two quartic oscillators nonlinearly coupled under regular and chaos conditions. Three quantum states for the full system are considered: a Gaussian state, a cat state, and a Bell-type state. By comparing the quantum and classical entropy values, and particularly their trends, it is shown that, instead of entanglement production, such entropies rather provide us with information on the system (either quantum or classical) delocalization. This gradual loss of information translates into an increase in both the quantum and the classical realms, directly connected to the increase in the correlations between both parties' degrees of freedom which, in the quantum case, is commonly related to the production of entanglement.
In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated. Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded uncertainty relation by adding an additional parameter characterizing the state, namely its degree of non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation. The common points and differences between the two extensions of the uncertainty relation help us to draw more general conclusions concerning the bounds on the non-Gaussianity of mixed states.
The entropy of a quantum state measures the information or uncertainty contained in the quantum system and plays a crucial role in quantum information theory. Many entropic properties, such as the subadditivity, have been studied for the discrete system with a finite-dimensional state space, while much less is explored in the infinite-dimensional system. In this work, we study the entropic properties of the continuous system that models a collection of Gaussian modes or fields. In particular, we identify the parameter range for the Rényi, Tsallis, and Unified entropies of Gaussian states that they are additive, subadditive, and strong subadditive. We show that the Rényi entropies are subadditive, further giving rise to the subadditivity of the Tsallis and Unified entropies in certain parameter range. We also find that the Tsallis and Unified entropies are not strong subadditive, while the Rényi entropies with the order ranging from 1 to 2 are conjectured to admit the strong subadditivity. Our results are useful in studying quantum correlations and channel capacities in Gaussian systems.