Classical analogs of generalized purities, entropies, and logarithmic negativity

TL;DR

The results show that quantum information of Gaussian states can be reproduced by classical information.

Abstract

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, Rényi 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.

Figures (9)

• Figure 1: Plots of the classical analogs, as a function of $C$ for different values $\alpha$, of (a) generalized purities, (b) Bastiaans-Tsallis, and (c) Rényi entropies.
• Figure 2: Plots of the classical analogs, as a function of $Y_2$ for different values of $\alpha$, (a) generalized purities, (b) Bastiaans-Tsallis and (c) Rényi entropies.
• Figure 3: Illustration of the setup used in the first case. Two disjoint groups of $n_{1}=50$ and $n_{2}=50$ oscillators of a circular lattice of $N=200$ oscillators. The groups are separated by $d$ oscillators.
• Figure 4: Classical analog of the logarithmic negativity for two groups of $n_1=n_2=50$ oscillators embedded in a harmonic chain of $N=200$ oscillators with $k=0.1$. $d$ is the number of oscillators between the two groups, as shown in Fig. \ref{['Fig:LNconf1']}.
• Figure 5: Illustration of the setup considered in the second case. Two adjacent groups of $n_{1}$ and $n_{2}=100-n_1$ oscillators of a circular lattice of $N=200$ oscillators.