# Maximum Geometric Quantum Entropy

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Motivation

## 2. Existing Results

#### 2.1. HJW Theorem

#### 2.2. Physically Realizable Ensembles

#### 2.3. Gaussian Adjusted Projected Measure

#### 2.4. Geometric Approach

#### 2.5. Summary

## 3. Geometric Quantum States

#### 3.1. Quantum State Space

#### 3.2. Observables

#### 3.3. Geometric Quantum States

#### 3.4. GQS as Conditional Probability Measures

#### 3.5. Quantifying Quantum Entropy

#### 3.6. Quantum Information Dimension and Geometric Entropy

## 4. Principle of Maximum Geometric Quantum Entropy

**Proposition**

**1**

#### 4.1. Finite Environments: $\mathfrak{D}=0$

#### 4.2. Full Support: $\mathfrak{D}=2({d}_{S}-1)$

#### 4.3. Integer, but Otherwise Arbitrary, $\mathfrak{D}$

#### 4.4. Noninteger $\mathfrak{D}$: Fractal Ensembles

## 5. How Does ${\mathit{\mu}}_{\mathit{ME}}$ Emerge?

#### 5.1. Emergence of ${\mu}_{ME}^{0}$

#### 5.2. Emergence of ${\mu}_{ME}^{2({d}_{S}-1)}$

#### 5.3. Stationary Distribution of Some Dynamic

#### 5.4. Emergence of ${\mu}_{ME}^{{d}_{S}-1}$

#### 5.5. Comment on the Generic ${\mu}_{ME}^{\mathfrak{D}}$

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Calculating the Partition Function

## Appendix B. Calculating Lagrange Multipliers

## References

- Pathria, R.K.; Beale, P.D. Statistical Mechanics; Elsevier B.V.: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Greiner, W.; Neise, L.; Stöcker, H. Thermodynamics and Statistical Mechanics; Springer: New York, NY, USA, 1995. [Google Scholar] [CrossRef]
- Jaynes, E.T. Information Theory and Statistical Mechanics. Phys. Rev.
**1957**, 106, 620. [Google Scholar] [CrossRef] - Jaynes, E.T. Information Theory and Statistical Mechanics. II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; Wiley-Interscience: New York, NY, USA, 2006. [Google Scholar]
- Anza, F.; Crutchfield, J.P. Beyond Density Matrices: Geometric Quantum States. Phys. Rev. A
**2020**, 103, 062218. [Google Scholar] [CrossRef] - Anza, F.; Crutchfield, J.P. Geometric Quantum Thermodynamics. Phys. Rev. E
**2022**, 106, 054102. [Google Scholar] [CrossRef] - Strocchi, F. Complex Coordinates and Quantum Mechanics. Rev. Mod. Phys.
**1966**, 38, 36–40. [Google Scholar] [CrossRef] - Kibble, T.W.B. Geometrization of quantum mechanics. Comm. Math. Phys.
**1979**, 65, 189–201. [Google Scholar] [CrossRef] - Heslot, A. Quantum mechanics as a classical theory. Phys. Rev. D
**1985**, 31, 1341–1348. [Google Scholar] [CrossRef] [PubMed] - Gibbons, G.W. Typical states and density matrices. J. Geom. Phys.
**1992**, 8, 147–162. [Google Scholar] [CrossRef] - Ashtekar, A.; Schilling, T.A. Geometry of quantum mechanics. In AIP Conference Proceedings; AIP: Melville, NY, USA, 1995; Volume 342, pp. 471–478. [Google Scholar] [CrossRef]
- Ashtekar, A.; Schilling, T.A. Geometrical formulation of quantum mechanics. In On Einstein’s Path; Springer: New York, NY, USA, 1999; pp. 23–65. [Google Scholar] [CrossRef]
- Brody, D.C.; Hughston, L.P. Geometric quantum mechanics. J. Geom. Phys.
**2001**, 38, 19–53. [Google Scholar] [CrossRef] - Bengtsson, I.; Zyczkowski, K. Geometry of Quantum States; Cambridge University Press: Cambridge, UK, 2017; p. 419. [Google Scholar] [CrossRef]
- Cariñena, J.F.; Clemente-Gallardo, J.; Marmo, G. Geometrization of quantum mechanics. Theoret. Math. Phys.
**2007**, 152, 894–903. [Google Scholar] [CrossRef] - Chruściński, D. Geometric aspects of quantum mechanics and quantum entanglement. J. Phys. Conf. Ser.
**2006**, 30, 9–16. [Google Scholar] [CrossRef] - Marmo, G.; Volkert, G.F. Geometrical description of quantum mechanics—transformations and dynamics. Phys. Scr.
**2010**, 82, 038117. [Google Scholar] [CrossRef] - Avron, J.; Kenneth, O. An elementary introduction to the geometry of quantum states with pictures. Rev. Math. Phys.
**2020**, 32, 2030001. [Google Scholar] [CrossRef] - Pastorello, D. A geometric Hamiltonian description of composite quantum systems and quantum entanglement. Intl. J. Geom. Meth. Mod. Phys.
**2015**, 12, 1550069. [Google Scholar] [CrossRef] - Pastorello, D. Geometric Hamiltonian formulation of quantum mechanics in complex projective spaces. Intl. J. Geom. Meth. Mod. Phys.
**2015**, 12, 1560015. [Google Scholar] [CrossRef] - Pastorello, D. Geometric Hamiltonian quantum mechanics and applications. Int. J. Geom. Methods Mod. Phys.
**2016**, 13, 1630017. [Google Scholar] [CrossRef] - Clemente-Gallardo, J.; Marmo, G. The Ehrenfest picture and the geometry of quantum mechanics. Il Nuovo C. C
**2013**, 3, 35–52. [Google Scholar] [CrossRef] - Hughston, L.P.; Jozsa, R.; Wootters, W.K. A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A
**1993**, 183, 14–18. [Google Scholar] [CrossRef] - Wiseman, H.M.; Vaccaro, J.A. Inequivalence of pure state ensembles for Open Quantum Systems: The preferred ensembles are those that are physically realizable. Phys. Rev. Lett.
**2001**, 87, 240402. [Google Scholar] [CrossRef] [PubMed] - Goldstein, S.; Lebowitz, J.L.; Tumulka, R.; Zanghi, N. On the Distribution of the Wave Function for Systems in Thermal Equilibrium. J. Stat. Phys.
**2006**, 125, 1193–1221. [Google Scholar] [CrossRef] - Brody, D.C.; Hughston, L.P. Information content for quantum states. J. Math. Phys.
**2000**, 41, 2586–2592. [Google Scholar] [CrossRef] - Karasik, R.I.; Wiseman, H.M. How Many Bits Does It Take to Track an Open Quantum System? Phys. Rev. Lett.
**2011**, 106, 020406. [Google Scholar] [CrossRef] [PubMed] - Schroedinger. The exhcange of energy in wave mechanics. Ann. Der Phys.
**1927**, 387, 257–264. [Google Scholar] - Schrödinger, E. Statistical Thermodynamics; Dover Publications: New York, NY, USA, 1989; p. 95. [Google Scholar]
- Walecka. Fundamentals of Statistical Mechanics. Manuscript and Notes by Felix Bloch; Stanford University Press: Stanford, CA, USA, 1989. [Google Scholar]
- Goldstein, S.; Lebowitz, J.L.; Mastrodonato, C.; Tumulka, R.; Zanghi, N. Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment. Commun. Math. Phys.
**2016**, 342, 965–988. [Google Scholar] [CrossRef] - Reimann, P. Typicality of Pure States Randomly Sampled According to the Gaussian Adjusted Projected Measure. J. Stat. Phys.
**2008**, 132, 921–935. [Google Scholar] [CrossRef] - Anza, F.; Crutchfield, J.P. Quantum Information Dimension and Geometric Entropy. Phys. Rev. X Quatum
**2022**, 3, 020355. [Google Scholar] [CrossRef] - Pollard, D. A User’s Guide to Measure Theoretic Probability; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Kolmogorov, A. Foundations of the Theory of Probability; Chelsea Publishing Company: Chelsea, MI, USA, 1933. [Google Scholar]
- Vourdas. Quantum systems with finite Hilbert spaces. Rep. Prog. Phys.
**2004**, 67, 267. [Google Scholar] [CrossRef] - Bengtsson. Three ways to look at Mutually Unbiased Bases. AIP Conf. Proc.
**2007**, 889, 40–51. [Google Scholar] - Lawrence, J.; Brukner, Č.; Zeilinger, A. Mutually unbiased binary observable sets on N qubits. Phys. Rev. A
**2002**, 65, 032320. [Google Scholar] [CrossRef] - Gordon-Rodriguez, E.; Loaiza-Ganem, G.; Cunningham, J. The continuous categorical: A novel simplex-valued exponential family. In Proceedings of the 37th International Conference on Machine Learning, Vienna, Austria, 8 June 2020; pp. 3637–3647. [Google Scholar]
- Anza, F.; Gogolin, C.; Huber, M. Eigenstate Thermalization for Degenerate Observables. Phys. Rev. Lett.
**2018**, 120, 150603. [Google Scholar] [CrossRef] - Neumann, J.v. Beweis des Ergodensatzes und desH-Theorems in der neuen Mechanik. Z. Phys.
**1929**, 57, 30–70. [Google Scholar] [CrossRef]

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Anza, F.; Crutchfield, J.P.
Maximum Geometric Quantum Entropy. *Entropy* **2024**, *26*, 225.
https://doi.org/10.3390/e26030225

**AMA Style**

Anza F, Crutchfield JP.
Maximum Geometric Quantum Entropy. *Entropy*. 2024; 26(3):225.
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**Chicago/Turabian Style**

Anza, Fabio, and James P. Crutchfield.
2024. "Maximum Geometric Quantum Entropy" *Entropy* 26, no. 3: 225.
https://doi.org/10.3390/e26030225