# A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics

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## Abstract

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## 1. Introduction

## 2. Methods: Entanglement Coarse-Graining and the Surfaces of Ignorance

#### 2.1. Macro and Microstates

#### 2.2. Surfaces of Ignorance: Metric Components and Volume

## 3. Results: Volume Examples

#### 3.1. Arbitrary N-Dimensional Unitary Transformations

#### 3.2. Example: $SU\left(2\right)$

#### 3.3. Example: $SO\left(3\right)$

#### 3.3.1. Computing Volume

#### 3.3.2. Analyzing the Entanglement Entropy of Macrostates

#### 3.4. Example: $SO\left(N\right)$

## 4. Generalizing the Entanglement Coarse-Graining

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CG | Coarse-graining |

ECG | Entanglement coarse-graining |

SOI | Surfaces of ignorance |

S | System |

E | Environment |

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**Figure 1.**Illustration of Boltzmann’s original approach to coarse-graining inspired by Figure 2 in [14]. On the left are examples of distributions on the single particle phase space, the $\mu $-space, while the right depicts the coarse-graining of the $6N$-dimensional phase space, the $\gamma $-space. By dividing the $\mu $-space into equal cells, macrostates are defined by simply counting the number of particles in each cell. Since each particle is indistinguishable, interchanging which particle occupies each cell does not change the macrostate; thus, there are many equivalent microstates for each macrostate. The size of each macrostate depends on the number of microstates it has. Boltzmann showed that distributions on the $\mu $-space that are more uniform have more microstates, and the largest macrostate, ${\Gamma}_{eq}$, is associated with a gas in equilibrium.

**Figure 2.**A conceptual example of an entangling process between ${\rho}_{S}$ and ${\rho}_{E}$. From the perspective of ${\rho}_{S}$, $|{\psi}_{ES}\rangle $ evolves from macrostates ${F}^{{\rho}_{S}}$ with a smaller volume to ${F}^{{\rho}_{S}}$ with a larger volume. If an observer only has access to the information in ${\rho}_{S}$, they cannot resolve the actual state of $|{\psi}_{ES}\rangle $ beyond the SOI depicted by the blue, orange, and red macrostates. For a global observer with access to $|{\psi}_{ES}\rangle $, the entangling process is a continuous curve of pure states from $|{\psi}_{ES}\left({t}_{0}\right)\rangle $ to $|{\psi}_{ES}\left({t}_{f}\right)\rangle $. This is the black curve in ${\mathcal{H}}_{ES}$. Each ${\rho}_{S}\in \mathcal{S}\subset \mathcal{P}\left({\mathcal{H}}_{S}\right)$ has one unique ${F}^{{\rho}_{S}}\subset {\mathcal{H}}_{ES}$. This implies a unique coarse-graining of $\mathcal{S}$ in ${\mathcal{H}}_{ES}$.

**Figure 3.**Plot of the normalized volume, von Neumann, and linear entropies for 2-level systems whose purifications are generated using $SU\left(2\right)$.

**Figure 4.**Comparison between the normalizations of ${V}_{SO\left(3\right)}$, von Neumann entropy, and linear entropy. This demonstrates that ${V}_{SO\left(3\right)}$ satisfies feature (1) of Boltzmann’s original CG for the example considered.

**Figure 5.**Discretization of the probability simplex $\mathcal{S}$ into a discrete ${\rho}_{l}$ of equal area, and the interval $L=[0,1]$ into segments of equal length for $\ell =5$ and $k=10$. In (

**a**), we have the division of $\mathcal{S}$ in the $\overrightarrow{\eta}$ basis while (

**b**) is that in the $\overrightarrow{\lambda}$ basis; the transformation is given by Equations (37)–(39). In (

**c**), we have the sorting of ${\rho}_{l}$ into volume-equivalent classes ${L}_{a}$.

**Figure 6.**Results of coarse-graining ${\mathcal{H}}_{ES}={\mathbb{R}}^{3}\otimes {\mathbb{R}}^{3}$. Row one is the discretization of $\mathcal{S}$ where each ${\rho}_{l}$ is colored using the volume or entropy of each column. Row two is the result of discretizing the interval $L=[0,1]$ and sorting equivalent ${\rho}_{l}$ into segments ${L}_{a}$. Row three is the fraction of ${\rho}_{l}$ belonging to each ${L}_{a}$. Finally, row four is the average von Neumann entropy of each ${L}_{a}$. It should be noted that the data from the graphs do not include the triangular distortions caused by the discretization of $\mathcal{S}$. We only used data from Weyl chambers that do not include triangles.

**Figure 7.**Plot comparing volumes given by Equation (42) with a direct numerical integration of $d{V}_{SO\left(4\right)}$. Both are normalized on their maximum values. To generate the plots, one thousand $\overrightarrow{\lambda}$’s were selected uniformly by generalizing Equations (37)–(39) to four dimensions and computing the corresponding volumes. The list of volumes and eigenvalues are sorted, $\mathrm{k}\in [1,1000]$, from largest to smallest. The red plot was computed from Equation (42), and the blue plot is a direct integration of $d{V}_{SO\left(4\right)}$ using a Monte Carlo integration. The inset is given to show that the plots are not exact but very close.

**Figure 8.**Plot of ${V}_{SO\left(N\right)}^{\mathrm{norm}}$ for $N=3,5,7,11,30$. The dashed vertical lines are located at the minimal value of ${\lambda}^{1}$ for each plot, which is $1/N$, the maximally mixed state. Notice how the centroids tend toward maximally mixed states as pure states subsume less volume as N increases.

**Figure 9.**Plot of the average von Neumann entropy (normalized to the maximally mixed state) with ${\lambda}^{1}\in [1/N,{\lambda}^{1*}]$ as a function of N. This quantifies the results of Figure 8 by showing that the average von Neumann entropy of states whose volumes take over $99.99\%$ of ${\mathcal{H}}_{ES}$ tends toward 1 where 1 corresponds to the maximum entanglement entropy.

**Figure 10.**Depiction of generalized entanglement coarse-graining procedure to allow unitary transformations of $\mathcal{S}$ in $\mathcal{P}\left({\mathcal{H}}_{S}\right)$. The green simplex on the left associated with $\rho $ is ${\mathcal{S}}^{\rho}$, and the orange simplex on the right associated with $\sigma $ is ${\mathcal{S}}^{\sigma}$. The orthonormal basis of ${\mathcal{S}}^{\sigma}$ is generated from unitary transformations ${U}_{S}$ applied to the orthonormal basis of ${\mathcal{S}}^{\rho}$. Each simplex has a coarse-graining of ${\mathcal{H}}_{ES}$ associated with it which is identical.

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**MDPI and ACS Style**

Ray, S.; Alsing, P.M.; Cafaro, C.; Jacinto, H.S.
A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics. *Entropy* **2023**, *25*, 788.
https://doi.org/10.3390/e25050788

**AMA Style**

Ray S, Alsing PM, Cafaro C, Jacinto HS.
A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics. *Entropy*. 2023; 25(5):788.
https://doi.org/10.3390/e25050788

**Chicago/Turabian Style**

Ray, Shannon, Paul M. Alsing, Carlo Cafaro, and H S. Jacinto.
2023. "A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics" *Entropy* 25, no. 5: 788.
https://doi.org/10.3390/e25050788