# The Volume of Two-Qubit States by Information Geometry

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## Abstract

**:**

## 1. Introduction

## 2. Structure of a Set of Two-Qubit States

- (i)
- For any ρ, the matrix T belongs to the tetrahedron $\mathcal{T}$ with vertices $(-1,-1,-1)$, $(-1,1,1)$, $(1,-1,1)$, $(1,1,-1)$.
- (ii)
- For any separable state ρ, the matrix T belongs to the octahedron $\mathcal{O}$ with vertices $(0,0,\pm 1)$, $(0,\pm 1,0)$, $(0,0,\pm 1)$.

- (i)
- Any operator (1) with $\mathit{r},\mathit{s}=0$ and diagonal T is a state (density operator) iff T belongs to the tetrahedron $\mathcal{T}$.
- (ii)
- Any state ρ with maximally disordered subsystems and diagonal T is separable iff T belongs to the octahedron $\mathcal{O}$.

## 3. Fisher Metrics

#### 3.1. Classical Fisher Metric in Phase Space

**Proposition**

**1.**

**Proof.**

#### 3.2. Quantum Fisher Metrics

**Proposition**

**2.**

**Proof.**

## 4. Volume of States with Maximally Disordered Subsystems

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Ratio of volumes (43) vs purity P. Solid line refers to the classical Fisher metric; dashed line refers to the quantum Fisher metric.

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Rexiti, M.; Felice, D.; Mancini, S.
The Volume of Two-Qubit States by Information Geometry. *Entropy* **2018**, *20*, 146.
https://doi.org/10.3390/e20020146

**AMA Style**

Rexiti M, Felice D, Mancini S.
The Volume of Two-Qubit States by Information Geometry. *Entropy*. 2018; 20(2):146.
https://doi.org/10.3390/e20020146

**Chicago/Turabian Style**

Rexiti, Milajiguli, Domenico Felice, and Stefano Mancini.
2018. "The Volume of Two-Qubit States by Information Geometry" *Entropy* 20, no. 2: 146.
https://doi.org/10.3390/e20020146