# New Equilibrium Ensembles for Isolated Quantum Systems

## Abstract

**:**

## 1. Introduction

## 2. Isolated Quantum Systems

## 3. Results

## 4. Meaning of the Approximation

- First, since we are maximizing the entropy, adding constraints can not increase the optimal value:$${S}_{\mathrm{vN}}({\gamma}_{n+1})\le {S}_{\mathrm{vN}}({\gamma}_{n}).$$$$logD={S}_{\mathrm{vN}}({\gamma}_{1})\ge {S}_{\mathrm{vN}}({\gamma}_{2})\ge \dots \ge {S}_{\mathrm{vN}}({\gamma}_{D-1})\ge {S}_{\mathrm{vN}}({\gamma}_{D})={S}_{\mathrm{vN}}({\rho}_{\mathrm{DE}})$$
- Second, given that we are including progressively higher moments of the energy probability distribution ${P}_{E}$, the moment generating function ${M}_{n}(t)$ of the n-th level ensemble provides increasingly better approximations to the moment generating functions ${M}_{\mathrm{DE}}(t)$ of the diagonal ensemble. These are defined as:$${M}_{\mathrm{DE}}(t):={\langle {e}^{iHt}\rangle}_{\mathrm{DE}}=Tr{\rho}_{\mathrm{DE}}{e}^{iHt}=\sum _{n=1}^{D}{|{c}_{n}|}^{2}{e}^{i{E}_{n}t}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}{M}_{n}(t):={\langle {e}^{iHt}\rangle}_{{\gamma}_{n}}=Tr{\gamma}_{n}{e}^{iHt}$$$$\begin{array}{c}{M}_{n}(t)=\sum _{l=1}^{\infty}{\left.\frac{{\partial}^{l}{M}_{\mathrm{DE}}(t)}{\partial {t}^{l}}\right|}_{t=0}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{t}^{l}=1+{m}_{1}^{E}t+{m}_{2}^{E}{t}^{2}+\dots +{m}_{n}^{E}{t}^{n}+\frac{{\partial}^{k}{M}_{n}(t)}{\partial {t}^{k}}+\dots \hfill \end{array}$$$$\begin{array}{c}{M}_{n}(t)-{M}_{\mathrm{DE}}(t)=\sum _{l=n+1}^{\infty}\left.{\left(\frac{{\partial}^{l}{M}_{n}(t)}{\partial {t}^{l}}\right|}_{t=0}-{m}_{l}^{E}\right){t}^{l}\hfill \end{array}$$
- Third, we now prove that the ${\gamma}_{n}$ provide progressively better approximation to the diagonal ensemble ${\rho}_{\mathrm{DE}}={\gamma}_{D}$. This is relevant to make predictions about the equilibrium physics, which go beyond the thermal ansatz. We note that, thanks to the exponential form of the ${\gamma}_{n}$, we have:$${S}_{\mathrm{vN}}({\gamma}_{n})-{S}_{\mathrm{vN}}({\gamma}_{D})={D}_{KL}\left({\gamma}_{D}|\phantom{\rule{-0.166667em}{0ex}}|{\gamma}_{n}\right)$$$${D}_{KL}\left({\gamma}_{D}|\phantom{\rule{-0.166667em}{0ex}}|{\gamma}_{n+1}\right)\le {D}_{KL}\left({\gamma}_{D}|\phantom{\rule{-0.166667em}{0ex}}|{\gamma}_{n}\right).$$$$\begin{array}{c}T(\rho ,\sigma )\le \sqrt{\frac{1}{2}{D}_{KL}(\rho |\phantom{\rule{-0.166667em}{0ex}}|\sigma )}\hfill \end{array}$$$$0\le {t}_{k}\le \sqrt{\frac{1}{2}{d}_{k}}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\underset{k\to D}{lim}{d}_{k}=0\phantom{\rule{2.em}{0ex}}\u27f9\phantom{\rule{2.em}{0ex}}\underset{k\to D}{lim}{t}_{k}=0$$

## 5. Examples

#### 5.1. First Example: L = 4

#### 5.2. Second Example: L = 10

## 6. Summary and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Constrained Entropy Maximization

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**Figure 1.**Relative entropy ${D}_{KL}({\rho}_{\mathrm{DE}}|\phantom{\rule{-0.166667em}{0ex}}|{\gamma}_{n})$ between ${\gamma}_{n}$ and ${\rho}_{\mathrm{DE}}={\gamma}_{D}$. The diagonal ensemble ${\rho}_{\mathrm{DE}}$ is built from $|{\psi}_{0}\rangle =|\uparrow ,\downarrow ,\dots \rangle $ and the eigenstates of the Hamiltonian in Equation (25). As n increases, we can see that ${\gamma}_{n}$ provides increasingly better approximations of ${\rho}_{\mathrm{DE}}$.

**Figure 2.**Here we compare the shape of the true energy probability distribution (blue dots) with the maximum entropy distribution obtained with different numbers of constrained moments: 1 (top left); 5 (top right); 10 (bottom left) and 15 (bottom right).

**Figure 3.**Relative entropy ${D}_{KL}({\rho}_{\mathrm{DE}}|\phantom{\rule{-0.166667em}{0ex}}|{\gamma}_{n})$ between ${\gamma}_{n}$ and ${\rho}_{\mathrm{DE}}={\gamma}_{D}$. The diagonal ensemble ${\rho}_{\mathrm{DE}}$ is built from $|{\psi}_{0}{\rangle =|\uparrow}_{x},{\downarrow}_{x},\dots \rangle $ and the eigenstates of the Hamiltonian in Equation (25) for system size $L=10$. As n increases, we can see that ${\gamma}_{n}$ provides increasingly better approximations of ${\rho}_{\mathrm{DE}}$. However, we notice that only the first two moments provide a significant decrease in the relative entropy.

**Figure 4.**Here we compare the shape of the true energy probability distribution (blue dots) with the maximum entropy distribution obtained with different numbers of constrained moments: 2 (top left); 10 (top right); 20 (bottom left) and 30 (bottom right).

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Anza, F.
New Equilibrium Ensembles for Isolated Quantum Systems. *Entropy* **2018**, *20*, 744.
https://doi.org/10.3390/e20100744

**AMA Style**

Anza F.
New Equilibrium Ensembles for Isolated Quantum Systems. *Entropy*. 2018; 20(10):744.
https://doi.org/10.3390/e20100744

**Chicago/Turabian Style**

Anza, Fabio.
2018. "New Equilibrium Ensembles for Isolated Quantum Systems" *Entropy* 20, no. 10: 744.
https://doi.org/10.3390/e20100744