# Roofs and Convexity

## Abstract

**:**

## 1. Introduction

_{b}, as a function of a pure bipartite states π.

_{b}, we call the resulting quantity the entanglement of T and denote it by ${E}_{T}$. For a channel, ${E}_{T}$ is equivalent (in many ways) to the restriction of E onto a face of a bipartite state space of sufficiently high dimension, [7]. In this way, T is seen as a sub-channel of the partial trace: The tensor product is partitioned into subspaces on which the sub-channels are defined. A 1-qubit channel, for example, can be represented by a $2\times m$ bipartite quantum systems as the restriction of the partial trace over the larger dimensional part onto the density operators supported by a suitable 2-dimensional subspace.

## 2. Roofs, Roof Extensions

**Definition 2.1a: Roof points**

**Definition 2.1b: Flat roof points**

**Definition 2.2: Roofs, flat roofs**

**Definition 2.3: Roof extensions**

**Proposition 2.1**

#### 2.1. Examples

#### Example 2.1: A Bloch ball construction

#### Example 2.2

#### Example 2.3: Affine functions on the state space

**Proposition 2.2**

**Proposition 2.3**

#### Example 2.4: An application to the diagonal map

#### Example 3

## 3. Roofs and Convexity

#### 3.1. Convex and Concave Extensions

**Definition 3.1: Convex (concave) extensions**

**Proposition 3.1**

**Proposition 3.2**

**Definition 3.2: ${g}^{\cap}$**and ${g}^{\cup}$

**Proposition 3.3**

**Proposition 3.4**

#### 3.2. Convex and Concave Roofs

**Proposition 3.5**

**Definition 3.2: convex leaves**

**Proposition 3.6**

**Proposition 3.6.a: (36) is a complete ${g}^{\cup}$-leaf.**

**Proposition 3.7**

#### 3.3. Illustrating Examples

#### Example 3.1: Minimum and maximum of g

#### Example 3.2: Again the diagonal map

**Proposition 3.8**

## 4. Wootters’ Method

#### 4.1. Anti-Linearity in Short

#### 4.2. Building Roofs with an Anti-Linear Hermitian ϑ

**Proposition 4.1**

**Proposition 4.2**

#### 4.3. Cases of Application

**Proposition 4.3**

**Proposition 4.4**

#### 4.4. How to Find ϑ

#### 4.5. Applications

## 5. A Subtraction Procedure

#### 5.1. Concurrence of Stochastic 1-Qubit Maps

**Proposition 5.1**

**S-lemma**

**Proposition 5.2**

#### 5.2. Axial Symmetric Maps, Concurrence

#### 5.3. Axial Symmetric Maps, Tangle

## 6. Symmetries

**Proposition 6.1:**

**Proposition 6.2:**

#### 6.1. Entanglement of the Diagonal Channel

**Proposition 6.3:**

**Proposition 6.4:**

#### 6.2. An Embedding

**Proposition 6.5:**

#### 6.3. A Further Embedding

## 7. Summary and Outlook

## Acknowledgements

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Uhlmann, A.
Roofs and Convexity. *Entropy* **2010**, *12*, 1799-1832.
https://doi.org/10.3390/e12071799

**AMA Style**

Uhlmann A.
Roofs and Convexity. *Entropy*. 2010; 12(7):1799-1832.
https://doi.org/10.3390/e12071799

**Chicago/Turabian Style**

Uhlmann, Armin.
2010. "Roofs and Convexity" *Entropy* 12, no. 7: 1799-1832.
https://doi.org/10.3390/e12071799