# State Entropy and Differentiation Phenomenon

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## Abstract

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## 1. Introduction

## 2. State Representation by Density Operator

## 3. Differentiation Phenomenon in Quantum Measurement Process

## 4. Characteristic Quantity of State Distribution

## 5. State Entropy

## 6. Model of Differentiation and Calculation of State Entropy

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. State Entropy and other Quantum Entropies

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**Figure 1.**Histograms of population rates of states with $\frac{l-1}{20}<{|\u2329{\psi}_{1}|{\Psi}_{\{{i}_{1},{i}_{2},\cdots ,{i}_{n}\}}\u232a|}^{2}\le \frac{l}{20}\phantom{\rule{4pt}{0ex}}(l=1,2,\cdots ,20)$ in the case of $n=0,10,100,500$ and 2000. The parameters are set by $M=L=2$, $\left|\Psi \right.\u232a=\sqrt{0.7}\left|{\psi}_{1}\right.\u232a+\sqrt{0.3}\left|{\psi}_{2}\right.\u232a$ (${P}_{1}=0.7,\phantom{\rule{4pt}{0ex}}{P}_{2}=0.3$), ${\nu}_{1|1}=\sqrt{0.5}\phantom{\rule{4pt}{0ex}}({\nu}_{2|1}=\sqrt{0.5})$ and ${\nu}_{1|2}=\sqrt{0.45}\phantom{\rule{4pt}{0ex}}({\nu}_{2|2}=\sqrt{0.55})$. If $\left|{\Psi}_{\{{i}_{1},{i}_{2},\cdots ,{i}_{n}\}}\right.\u232a\approx \left|{\psi}_{1}\right.\u232a(\left|{\psi}_{2}\right.\u232a)$, $|\u2329{\psi}_{1}|{\Psi}_{\{{i}_{1},{i}_{2},\cdots ,{i}_{n}\}}\u232a{|}^{2}$ takes a value nearby 1(0). With increasing n, the state distribution approaches to $\{\{{\psi}_{1},{\psi}_{2}\},\{0.7,0.3\}\}$.

**Figure 2.**Behaviors of state entropy, von Neumann entropy and $-\mathrm{log}(Tr({\rho}^{2}))$ at the parameters of $M=L=2$, $\left|\Psi \right.\u232a=\sqrt{0.7}\left|{\psi}_{1}\right.\u232a+\sqrt{0.3}\left|{\psi}_{2}\right.\u232a$ (${P}_{1}=0.7,\phantom{\rule{4pt}{0ex}}{P}_{2}=0.3$), ${\nu}_{1|1}=\sqrt{0.5}\phantom{\rule{4pt}{0ex}}({\nu}_{2|1}=\sqrt{0.5})$ and ${\nu}_{1|2}=\sqrt{0.45}\phantom{\rule{4pt}{0ex}}({\nu}_{2|2}=\sqrt{0.55})$.

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Asano, M.; Basieva, I.; Pothos, E.M.; Khrennikov, A. State Entropy and Differentiation Phenomenon. *Entropy* **2018**, *20*, 394.
https://doi.org/10.3390/e20060394

**AMA Style**

Asano M, Basieva I, Pothos EM, Khrennikov A. State Entropy and Differentiation Phenomenon. *Entropy*. 2018; 20(6):394.
https://doi.org/10.3390/e20060394

**Chicago/Turabian Style**

Asano, Masanari, Irina Basieva, Emmanuel M. Pothos, and Andrei Khrennikov. 2018. "State Entropy and Differentiation Phenomenon" *Entropy* 20, no. 6: 394.
https://doi.org/10.3390/e20060394