#
Supervised Quantum State Discrimination^{ †}

^{*}

^{†}

## 1. Introduction

## 2. The Model

## 3. Results

- (i)
- ${\rho}_{1}$, ${\rho}_{2}$ have assigned purities—the moduli of their Bloch vectors being respectively ${r}_{1}$ and ${r}_{2}$; uniform prior on the Bloch vector’s directions,$$\begin{array}{c}{P}_{err,min}^{(n\gg 1)}=\frac{1}{2}-\frac{1}{24}\frac{{({r}_{1}+{r}_{2})}^{3}-{|{r}_{1}-{r}_{2}|}^{3}}{{r}_{1}{r}_{2}}\hfill \\ \hfill +\frac{5}{24\phantom{\rule{0.166667em}{0ex}}n}\frac{{({r}_{1}+{r}_{2})}^{3}+{|{r}_{1}-{r}_{2}|}^{3}}{{r}_{1}^{2}{r}_{2}^{2}}-\frac{1}{24\phantom{\rule{0.166667em}{0ex}}n}\frac{{({r}_{1}+{r}_{2})}^{5}-{|{r}_{1}-{r}_{2}|}^{5}}{{r}_{1}^{3}{r}_{2}^{3}}+o\left(\frac{1}{n}\right).\end{array}$$
- (ii)
- ${\rho}_{1}$ and ${\rho}_{2}$ are generically mixed qubit states, with a constant density Bloch sphere prior,$${P}_{err,min}^{(n\gg 1)}=\frac{17}{70}+\frac{18}{35n}+o\left(\frac{1}{n}\right)\phantom{\rule{0.277778em}{0ex}}.$$
- (iii)
- ${\rho}_{1}$ and ${\rho}_{2}$ are pure and they have a fixed overlap $\mathrm{Tr}\left[{\rho}_{1}{\rho}_{2}\right]={sin}^{2}\frac{\theta}{2}$; uniform prior on the global orientation,$${P}_{err,min}^{(n\gg 1)}=\frac{1}{2}\left(1-|cos{\textstyle \frac{\theta}{2}}|\right)+{\textstyle \frac{3+cos\theta}{8\sqrt{2}\sqrt{1+cos\theta}}}\frac{1}{n}+{\textstyle \frac{1-60cos\theta -5cos2\theta}{128\sqrt{2}{(1+cos\theta )}^{3/2}}}\frac{1}{{n}^{2}}+o\left(\frac{1}{{n}^{2}}\right)\phantom{\rule{0.277778em}{0ex}}.$$

## References

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**MDPI and ACS Style**

Fanizza, M.; Mari, A.; Giovannetti, V.
Supervised Quantum State Discrimination. *Proceedings* **2019**, *12*, 21.
https://doi.org/10.3390/proceedings2019012021

**AMA Style**

Fanizza M, Mari A, Giovannetti V.
Supervised Quantum State Discrimination. *Proceedings*. 2019; 12(1):21.
https://doi.org/10.3390/proceedings2019012021

**Chicago/Turabian Style**

Fanizza, Marco, Andrea Mari, and Vittorio Giovannetti.
2019. "Supervised Quantum State Discrimination" *Proceedings* 12, no. 1: 21.
https://doi.org/10.3390/proceedings2019012021