# 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

## References and Notes

- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Vidal, G. Entanglement monotones. J. Mod. Opt.
**2000**, 47, 355. [Google Scholar] [CrossRef] - Bengtsson, I.; Zyczkowski, K. Geometry of Quantum States; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Horodecki, R.; Horodecki, P.; Horodecki, M.; Horodecki, K. Quantum entanglement. Rev. Mod. Phys.
**2009**, 81, 865–942. [Google Scholar] [CrossRef] - Rockafellar, R.T. Convex Analysis; Princeton University Press: Princeton, NJ, USA, 1970. [Google Scholar]
- Bennett, C.; DiVincenzo, D.; Smolin, J.; Wootters, W. Mixed state entanglement and quantum error correction. Phys. Rev. A
**1996**, 54, 3824–3851. [Google Scholar] [CrossRef] [PubMed] - Hayden, P.; Leung, D.; Winter, A. Aspects of general entanglement. Comm. Math. Phys.
**2007**, 265, 95117. [Google Scholar] - Holevo, A.S. Quantum probability and quantum statistics (in Russian). Probl. Perdeachi Inf.
**1973**, 9, 3. [Google Scholar] - Connes, A.; Narnhofer, H.; Thirring, W. Dynamical entropy of C
^{*}-algebras and von Neumann algebras. Comm. Math. Phys.**1987**, 112, 681–719. [Google Scholar] [CrossRef] - Benatti, F.; Narnhofer, H.; Uhlmann, A. Decompositions of quantum states with respect to entropy. Rep. Math. Phys.
**1996**, 38, 123–141. [Google Scholar] [CrossRef] - Uhlmann, A. Entropy and optimal decompositions of states relative to a maximal commutative subalgebra. Open Sys. Inf. Dyn.,
**1998**, 5, 209–227. [Google Scholar] [CrossRef] - Hill, S.; Wootters, W.K. Entanglemant of a pair of quantum bits. Phys. Rev. Lett.
**1997**, 78, 5022–5025. [Google Scholar] [CrossRef] - Wootters, W.K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett.
**1998**, 80, 2243–2248. [Google Scholar] [CrossRef] - Uhlmann, A. Fidelity and concurrence between conjugated states. Phys. Rev. A
**2000**, 62, 032307. [Google Scholar] [CrossRef] - Wootters, W.K. Entanglement of formation and concurrence. Quantum Inf. Comput.
**2002**, 1, 27–47. [Google Scholar] - Hildebrand, R. Concurrence of Lorentz-positive maps. arXiv: quant-ph/0612064. 2006. [Google Scholar]
- Hildebrand, R. Concurrence revisted. J. Math. Phys.
**2008**, 48, 102108. [Google Scholar] [CrossRef] - Pòlik, I.; Terlaky, T. A Survey of the S-Lemma. SIAM Rev.
**2007**, 3, 371–418. [Google Scholar] [CrossRef] - Hellmund, M.; Uhlmann, A. Concurrence of stochastic 1-qubit maps. arXiv:0802.209 (quant-ph).
- Gorini, V.; Sudarshan, E.C.G. Extreme affine transformations. Commun. Math. Phys.
**1976**, 46, 43–52. [Google Scholar] [CrossRef] - Coffman, V.; Kundu, J.; Wootters, W.K. Distributet entanglement. Phys. Rev. A
**2000**, 61, 052306. [Google Scholar] [CrossRef] - Osborne, T.; Verstraete, F. General monogamy inequaltiy for bipartite qubit entanglement. Phys. Rev. Lett.
**2006**, 96, 22503. [Google Scholar] [CrossRef] - This does not affect the 1-tangle. Though (93) results from a subtraction procedure, it is not an application of the S-lemma: τ
_{T}becomes negatve for x_{00}→ ∞. - If the null space is 2-dimensional the convex leaves of C
_{T}become 2-dimensional too. A 3-dimensional null-space forces C_{T}to be constant. - Osterloh, A.; Siewert, J.; Uhlmann, A. Tangles of superpositions and the convex-roof extension. Phys. Rev. A
**2008**, 77, 032310. [Google Scholar] [CrossRef] - Vollbrecht, K.G.; Werner, R.F. Entanglement measures under symmetry. Phys. Rev. A
**2001**, 64, 0623307. [Google Scholar] [CrossRef] - Terhal, B.M.; Vollbrecht, K.G. The entanglement of formation for isotropic states. Phys. Rev. Lett.
**2000**, 85, 2625–2628. [Google Scholar] [CrossRef] [PubMed] - Ohya, M.; Petz, D. Quantum Entropy and Its Use; Springer: New York, NY, USA, 1993. [Google Scholar]
- A flat roof point can allow for several optimal decompositions, some of them flat and some of them not flat.
- Levitan, L.B. Entropy defect and information for two quantum states. Open Sys. Inf. Dyn.
**1994**, 2, 319–329. [Google Scholar] [CrossRef] - Thirring, W. 1994; unpublished.
- This is the very reason the present author introduced the notation “convex roof”.
- If g is not continuous, an optimal decomposition may not exist, even if we allow for infinite length.
- Mitchison, G.; Jozsa, R. Towards a geometrical interpretation of quantum information compression. Phys. Rev. A
**2004**, 69, 032304. [Google Scholar] [CrossRef] - Anti-linear Operators are also called a “conjugate linear” ones.
- Wigner, E.P. Über die Operation der Zeitumkehr in der Quantenmechanik. Nachr. Ges. Wiss. Göttingen, Math.-Physikal. Klasse
**1932**, 31, 546–559. [Google Scholar] - Wigner, E.P. Normal form of anitunitary operators. J. Math. Phys.
**1960**, 1, 409–413. [Google Scholar] [CrossRef] - Horn, R.A.; Johnson, C.R. Matrix Analysis; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- The polar decomposition for anti-linear operators is equivalent to the Takaki decomposing of symmetric matrices.
- Chen, K.; Albeverio, S.; Fei, Sh.-M. Entanglement of formation of bipartite quantum states. Phys. Rev. Lett.
**2005**, 95, 21051. [Google Scholar] [CrossRef] - Chen, K.; Albeverio, S.; Fei, Sh.-M. Concurrence of arbitrary dimensional bipartite quantum states. Phys. Rev. Lett.
**2005**, 95, 040504. [Google Scholar] [CrossRef] [PubMed] - Łoziński, A.; Buchleitner, A.; Życzkowski, K.; Wellens, T. Entanglement of 2×K quantum systems. Europhys. Lett.
**2003**, 62, 168. [Google Scholar] [CrossRef] - Mintert, F.; Kuś, M.; Buchleitner, A. Concurrence of mixed bipartite quantum states of arbitrary dimensions. Phys. Rev. Lett.
**2005**, 95, 260502. [Google Scholar] [CrossRef] [PubMed] - Osborne, T. Entanglement for rank-2 mixed states. Phys. Rev. A
**2005**, 72, 022399. [Google Scholar] [CrossRef] - Fei, S.; Li-Jost, X. R-function related to entanglement of formation. Phys. Rev. A
**2006**, 73, 024302. [Google Scholar] [CrossRef] - Rungta, R.; Buzek, V.; Caves, C.M.; Hillery, M.; Milburn, G.J.; Wootters, W.K. Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A
**2001**, 64, 042315. [Google Scholar] [CrossRef] - Fujiwara, A.; Algoet, P. One-to-one parametrization of quantum channels. Phys. Rev. A
**1999**, 59, 3290–3294. [Google Scholar] [CrossRef] - Ruskai, M.B.; Szarek, S.; Werner, E. An analysis of completely-positive trace-preserving maps on 2×2 matrices. Lin. Alg. Appl.
**2002**, 347, 159–187. [Google Scholar] [CrossRef] - King, C.; Ruskai, M.B. Minimal entropy of states rmerging from noisy quantum channels. IEEE Trans. Info. Theor.
**2002**, 47, 1–19. [Google Scholar] - Uhlmann, A. On concurrence and entanglement of rank two channels. Open Sys. & Inf. Dyn.
**2005**, 12, 1–14. [Google Scholar] - Uhlmann, A. Concurrence and foliations induced by some 1-qubit channels. Int. J. Theor. Phys.
**2003**, 42, 983–999. [Google Scholar] [CrossRef] - Hellmund, M.; Uhlmann, A. Concurrence and entanglement entropy for stochastic 1-qubit maps. Phys. Rev. A
**2009**, 70, 052319. [Google Scholar] [CrossRef] - Hellmund, M. personal communication, 2009.
- Benatti, F.; Narnhofer, H.; Uhlmann, A. Optimal decompositions with respect to entropy and symmetries. Lett. Math. Phys.
**1999**, 47, 237–253. [Google Scholar] [CrossRef] - Benatti, F.; Narnhofer, H.; Uhlmann, A. Broken symmetries in the entanglement of formation. Int. J. Theor. Phys.
**2003**, 42, 983–999. [Google Scholar] [CrossRef] - Fei, S.; Jost, J.; Li-Jost, X.; Wang, G. Entanglement of formation for a class of quantum states. Phys. Lett. A
**2003**, 310, 333. [Google Scholar] [CrossRef] - Θ
_{e}is the so-called modular conjugation with defining vector |e〉.

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