# Generic Entanglement Entropy for Quantum States with Symmetry

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Haar Measure, Haar Random Unitaries, and Haar Random States

#### 2.2. Entanglement Entropy, and Entanglement Spectrum

## 3. Generic Entanglement without Symmetry

**Theorem**

**1**

## 4. Concentration of Entanglement Entropy of a Random State in a Subspace

**Theorem**

**2.**

**Proof of Theorem**

**2.**

## 5. Generic Entanglement of States with an Axial Symmetry

## 6. Generic Entanglement of States with Permutation Symmetry

## 7. Generic Entanglement of States with Translation Symmetry

## 8. Entanglement Phases and Symmetries

## 9. Is Generic Entanglement with Symmetry Physical?

## 10. Summary and Discussions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The function $\overline{S}\left({\mathcal{H}}_{m}\right)/{n}_{A}$ is plotted by × as a function of $m/n$ for a $n=100$ and ${n}_{A}=1$ (

**A**), ${n}_{A}=25$ (

**B**), and ${n}_{A}=50$ (

**C**). We also provide a function $f(m/n):=4m/n(1-m/n)$ by a red dashed line in each figure. It is clear that $\overline{S}\left({\mathcal{H}}_{m}\right)/{n}_{A}\approx f(m/n)$ for any ${n}_{A}$ and m.

**Figure 2.**The distributions of entanglement over the random states without/with symmetry, which are numerically obtained for $d=2$, $n=10$, and ${n}_{A}=5$. The number of samples is ${10}^{5}$, binned in intervals of $0.02$ for Panels (

**A**,

**B**,

**D-I**,

**D-II**), $0.2$ for Panel (

**C**). Panel (

**A**) shows the distribution of the entanglement entropy ${E}_{A}$ over a Haar random state without symmetry (red), that over a random symmetric state (purple), and that over a random translation invariant state for $\theta =0$ (blue). We observe that only a random symmetric state has significantly less entanglement entropy, which is consistent with our analytical investigations. Panels (

**B**,

**C**,

**D-I**,

**D-II**) show the rescaled purity $R\left(\right|\varphi \rangle )$ of a random state without symmetry, a random symmetric state, a random translation invariant state for $\theta =0$, and that for $\theta =\pi $, respectively. The rescaled purity is more suitable to see the entanglement phases. The insets numerically provide $-ln\left[p(R\left(\right|\varphi \rangle =s)\right]/{2}^{2{n}_{A}+1}$ as a function of s, where $p\left(R\right(|\varphi \rangle =s)$ is the probability density function. In the insets, we also plotted quadratic functions (brown dotted lines) fitted to the numerical data as a reference, which may be useful to detect the phase transition. See the main text for the detail.

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**MDPI and ACS Style**

Nakata, Y.; Murao, M.
Generic Entanglement Entropy for Quantum States with Symmetry. *Entropy* **2020**, *22*, 684.
https://doi.org/10.3390/e22060684

**AMA Style**

Nakata Y, Murao M.
Generic Entanglement Entropy for Quantum States with Symmetry. *Entropy*. 2020; 22(6):684.
https://doi.org/10.3390/e22060684

**Chicago/Turabian Style**

Nakata, Yoshifumi, and Mio Murao.
2020. "Generic Entanglement Entropy for Quantum States with Symmetry" *Entropy* 22, no. 6: 684.
https://doi.org/10.3390/e22060684