Does the entropy of systems with larger internal entanglement grow stronger?

TL;DR

The paper addresses whether greater internal entanglement within a quantum system accelerates entropy growth when the system interacts with an environment. Using a simple qubit-plus-harmonic-oscillator model with $H_{SE}=A_S\otimes\widehat{n}_E$, the study shows that a universal relation does not hold; however, Haar-random states exhibit a positive average correlation between initial entanglement and final entropy due to measure concentration, and this correlation strengthens with system size. Special state constructions and constraints (e.g., fixed excitation, generalized Dicke states, and specific interaction types) can reverse or diminish the trend, indicating sensitivity to state structure and dynamics. The work underscores the important roles of sampling, entanglement depth, and interaction details in the entanglement–entropy interplay and suggests avenues for exploring how correlations influence quantum thermodynamics in more complex settings. Overall, the findings illuminate how internal entanglement contributes to entropy growth on average, while also highlighting scenarios where this contribution can be suppressed or flipped depending on the chosen model and state preparation.

Abstract

It is known that when a system interacts with its environment, the entanglement contained in the system is redistributed since parts of the system entangle with the environment. On the other hand, the entanglement of a system with its environment is closely related to the entropy of the system. However, does this imply that the entropy of systems with larger internal entanglement will grow stronger? We study the issue using the simplest model as an example: a system of qubits interacts with the environment described by the quantum harmonic oscillator. The answer to the posed question is ambiguous. However, the study of the situation on average (using the simulation of a set of random states) reveals certain patterns and we can say that the answer is affirmative. At the same time, the choice of states satisfying certain conditions in some cases can change the dependence to the opposite. Additionally, we show that the entanglement depth also makes a small contribution to entropy growth.

Figures (8)

• Figure 1: Final entropy vs initial entanglement entropy for states of type (\ref{['eq:ref20']}) for some choices of $A_S$ (lines of different colors correspond to different interactions). The values of the coefficients $\alpha_{x},\alpha_{y},\alpha_{z},\beta_{x},\beta_{y},\beta_{z}$ do not affect the final result; what matters is that condition (\ref{['eq:refne']}) is satisfied (if (\ref{['eq:refne']}) is not satisfied (\ref{['eq:ref36']}) implies three different eigenvalues and the picture will differ). The monotonicity of the curves is determined by the form of $|c_{jk}|^2$ (in the cases when one and two qubits interact these are coefficients in expansions (\ref{['eq:ref37']}) and (\ref{['exp2']}) correspondingly). The allowed area (for any possible initial states and interactions of type (\ref{['eq:ref31']})) is shaded in grey (see Appendix A). The area allowed in the case when only one qubit interacts is shaded in a darker grey.
• Figure 2: Final entropy vs initial entanglement for systems of different sizes. a) Results of generating $10^4$ random states (for each system size). b) Approximate average lines (solid) align with approximating straight lines (dashed). They indicate that, on average, the initial entanglement contributes to the growth of entropy. The clustering around mean values in a) and the increase in the approximating lines in b) are based on the phenomenon of measure concentration.
• Figure 3: The influence of the number of interacting qubits for systems of different sizes. a) Light gray dots correspond to $10^4$ random states for the 4-qubit system. The greater the number of interacting qubits, the higher the dots are arranged, and the greater their vertical blurring. The approximate average lines (solid) show dependencies for different system sizes and different numbers of interacting qubits. In regions where there was little data, we supplemented them with approximating straight lines (dashed). The lower group of lines corresponds to the case when only one qubit interacts with the environment, the second - when two, and so on. Systems of different sizes are represented by lines of different colors. Bold dots in the figure indicate mean points. b) The mean values of final entropy for different numbers of interacting qubits for systems of different sizes.
• Figure 4: Six qubit system consisting of clusters of entangled subsystems with different entanglement depth. a) Final entropy vs initial entanglement for $10^4$ generated states for different entanglement depths. The larger the entanglement depth, the stronger the clustering. b) The entanglement depth of subsystems affects the mean values (bold dots in the figure indicate mean points) and the slope of approximating straight lines.
• Figure 5: Imposing condition (\ref{['eq:ref37c']}) results in opposite dependencies of final entropy on initial entanglement for $\sigma_z$- and $\sigma_x$-interactions. Different colors correspond to different values of $E$. a),b) Regions occupied by the generated states. The dashed line shows the boundary of the region allowed for any pure states, in a) it is formed by states $\sqrt{1-E}\ket{00}+\sqrt{E}\ket{11}$, in b) by states (\ref{['border']}) with $E=0.5$. In a), b) approximate average lines are shown for $E=0.5, 0.2, 0.1$. c), d) States with real components having the same sign form segments of lines. In the case where both qubits interact with the environment, we draw them with solid lines; when only one qubit interacts, we use dashed lines.