# Concepts and Criteria for Blind Quantum Source Separation and Blind Quantum Process Tomography

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## Abstract

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## 1. Introduction

## 2. An Unentanglement Criterion for a Qubit Pair

## 3. Random Quantum Sources and Their Independence

## 4. BQSS and Probabilities in Spin Component Measurements

#### 4.1. Some General Considerations

#### 4.2. Probabilities in Measurements, Classical versus Quantum World

#### 4.3. An Unentanglement Criterion Using Probabilities

#### 4.4. Knowing 2-Qubit Pure States from ${s}_{ij}$ Measurements

## 5. Disentanglement and Cylindrical-Symmetry Heisenberg Coupling

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. About Applications of Blind Conventional and Quantum Source Separation

#### Appendix A.1. Conventional BSS

#### Appendix A.2. Blind Quantum Source Separation

## References

- Laloë, F. Comprenons-Nous Vraiment la MéCanique Quantique; EDP Sciences Les Ulis: Les Ulis, France, 2011; English version: Do We Really Understand Quantum Mechanics? Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Cohen-Tannoudji, C.; Diu, B.; Laloë, F. Mécanique Quantique; Hermann: Paris, France, 1973; English version: Quantum Mechanics; John Wiley: New York, NY, USA, 1977. [Google Scholar]
- Dirac, P. Quantum Mechanics of Many-Electron Systems. Proc. R. Soc. A
**1929**, 123, 714–733. [Google Scholar] [CrossRef] - Timpson, C.G. Quantum Information Theory and the Foundations of Quantum Mechanics. Ph.D. Thesis, University of Oxford, Oxford, UK, 2004. [Google Scholar]
- Comon, P.; Jutten, C. (Eds.) Handbook of Blind Source Separation: Independent Component Analysis and Applications; Academic Press: Oxford, UK, 2010. [Google Scholar]
- Deville, Y. Blind Source Separation and Blind Mixture Identification Methods. In Wiley Encyclopedia of Electrical and Electronics Engineering; Webster, J., Ed.; Wiley: Hoboken, NJ, USA, 2016; pp. 1–33. [Google Scholar]
- Deville, Y.; Deville, A. Blind separation of quantum states: Estimating two qubits from an isotropic Heisenberg spin coupling model. In Proceedings of the 7th International Conference on Independent Component Analysis and Signal Separation, London, UK, 9–12 September 2007; Davies, M.E., James, C.J., Abdallah, S.A., Plumbley, M.D., Eds.; Springer: Berlin, Germany, 2007; pp. 706–713. [Google Scholar]
- Deville, Y.; Deville, A. Classical-processing and quantum-processing signal separation methods for qubit uncoupling. Quantum Inf. Process.
**2012**, 11, 1311–1347. [Google Scholar] [CrossRef] - Deville, Y.; Deville, A. A quantum-feedforward and classical-feedback separating structure adapted with monodirectional measurements; blind qubit uncoupling capability and links with ICA. In Proceedings of the 23rd IEEE International Workshop on Machine Learning for Signal Processing, Southampton, UK, 22–25 September 2013. [Google Scholar]
- Deville, Y.; Deville, A. Blind qubit state disentanglement with quantum processing: Principle, criterion and algorithm using measurements along two directions. In Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing, Florence, Italy, 4–9 May 2014; pp. 6262–6266. [Google Scholar]
- Deville, Y.; Deville, A. Quantum-Source Independent Component Analysis and Related Statistical Blind Qubit Uncoupling Methods. In Blind Source Separation: Advances in Theory, Algorithms and Applications; Naik, G.R., Wang, W., Eds.; Springer: Berlin, Germany, 2014; pp. 3–37. [Google Scholar]
- Deville, Y.; Deville, A. From blind quantum source separation to blind quantum process tomography. In Proceedings of the 12th International Conference on Latent Variable Analysis and Signal Separation, Liberec, Czech Republic, 25–28 August 2015; Vincent, E., Yeredor, A., Koldovský, Z., Tichavský, P., Eds.; Springer: Berlin, Germany, 2015; pp. 184–192. [Google Scholar]
- Deville, Y.; Deville, A. Blind quantum computation: Blind quantum source separation and blind quantum process tomography. In Proceedings of the 19th Conference on Quantum Information Processing, Banff, AB, Canada, 10–15 January 2016. [Google Scholar]
- Deville, Y.; Deville, A. Blind quantum source separation: Quantum-processing qubit uncoupling systems based on disentanglement. Digit. Signal Process.
**2017**, 67, 30–51. [Google Scholar] [CrossRef] - Deville, Y. Traitement du Signal: Signaux Temporels et Spatiotemporels—Analyse des Signaux, Théorie de L’information, Traitement D’antenne, Séparation Aveugle de Sources; Ellipses Editions Marketing: Paris, France, 2011. (In French) [Google Scholar]
- Feynman, R.P. Quantum Mechanical Computers. Opt. News
**1985**, 11, 11–20. [Google Scholar] [CrossRef] - Feynman, R.P. Feynman Lectures on Computation; Perseus Publishing: Cambridge, MA, USA, 1996. [Google Scholar]
- Peres, A. Separability Criterion for Density Matrices. Phys. Rev. Lett.
**1996**, 77, 1413–1415. [Google Scholar] [CrossRef] [PubMed] - Horodecki, M.; Horodecki, P.; Horodecki, R. Separability of mixed states: Necessary and sufficient conditions. Phys. Lett. A
**1996**, 223, 1–8. [Google Scholar] [CrossRef] - Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Buchleitner, A.; Viviescas, C.; Tiersch, M. (Eds.) Entanglement and Decoherence (Lectures Notes in Physics); Springer: Berlin, Germany, 2009. [Google Scholar]
- Köhler, J.; Disselhorst, J.A.J.M.; Donckers, M.C.J.M.; Groenen, E.J.J.; Schmidt, J.; Moerner, W.E. Magnetic resonance of a single molecular spin. Nature
**1993**, 363, 242–244. [Google Scholar] [CrossRef] - Gruber, A.; Dräbenstedt, A.; Tietz, C.; Fleury, L.; Wrachtrup, J.; von Borczyskowski, C. Scanning Confocal Optical Microscopy and Magnetic Resonance on Single Defect Centers. Science
**1997**, 276, 2012–2014. [Google Scholar] [CrossRef] - Rugar, D.; Budakian, R.; Mamin, H.J.; Chui, B.W. Single spin detection by magnetic resonance force microscopy. Nature
**2004**, 430, 329–332. [Google Scholar] [CrossRef] [PubMed] - Otte, A.F. Can data be stored in a single magnetic atom? Europhys. News
**2008**, 38, 31–34. [Google Scholar] [CrossRef] - Bienfait, A.; Pla, J.J.; Kubo, Y.; Stern, M.; Zhou, X.; Lo, C.C.; Weis, C.D.; Schenkel, T.; Thewalt, M.L.W.; Vion, D.; et al. Reaching the quantum limit of sensitivity in electron spin resonance. arXiv
**2015**. [Google Scholar] - Hyvärinen, A.; Karhunen, J.; Oja, E. Independent Component Analysis; Wiley: New York, NY, USA, 2001. [Google Scholar]
- Abragam, A. The Principles of Nuclear Magnetism; Oxford University Press: Oxford, UK, 1961. [Google Scholar]
- Tolman, R.C. The Principles of Statistical Mechanics; Oxford University Press: Oxford, UK, 1938; p. 327. [Google Scholar]
- Von Neumann, J. Les Fondements Mathématiques de la Mécanique Quantique; Alcan: Paris, France, 1946. Editions Jacques Gabay: Paris, France, 1988. (In French) [Google Scholar]
- Barnett, S.M. Quantum Information; Oxford University Press: Oxford, UK, 2009. [Google Scholar]
- Fuchs, C.A.; Peres, A. Quantum theory needs no “interpretation”. Phys. Today
**2000**, 53, 70–71. [Google Scholar] [CrossRef] - Margenau, H. Quantum-Mechanical description. Phys. Rev.
**1936**, 49, 240–242. [Google Scholar] [CrossRef] - Margenau, H. Critical Points in Modern Physical Theory. Philos. Sci.
**1937**, 4, 337–370. [Google Scholar] [CrossRef] - Feynman, R.P. Statistical Mechanics; Basic Books: New York, NY, USA, 1972. [Google Scholar]
- Abragam, A.; Bleaney, B. Electron Paramagnetic Resonance of Transition Ions; Oxford University Press: Oxford, UK, 1970. [Google Scholar]
- DiVincenzo, D.P. Quantum Computation. Science
**1995**, 270, 255–261. [Google Scholar] [CrossRef] - Fazekas, P. Electron Correlation and Magnetism; World Scientific: Hackensack, NJ, USA, 1999. [Google Scholar]

**Figure 2.**Block diagram of a system using BQSS, with quantum processing in the forward path (no cloning [14], with permision from Elsevier).

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Deville, A.; Deville, Y.
Concepts and Criteria for Blind Quantum Source Separation and Blind Quantum Process Tomography. *Entropy* **2017**, *19*, 311.
https://doi.org/10.3390/e19070311

**AMA Style**

Deville A, Deville Y.
Concepts and Criteria for Blind Quantum Source Separation and Blind Quantum Process Tomography. *Entropy*. 2017; 19(7):311.
https://doi.org/10.3390/e19070311

**Chicago/Turabian Style**

Deville, Alain, and Yannick Deville.
2017. "Concepts and Criteria for Blind Quantum Source Separation and Blind Quantum Process Tomography" *Entropy* 19, no. 7: 311.
https://doi.org/10.3390/e19070311