# Measuring the Tangle of Three-Qubit States

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## Abstract

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## 1. Introduction

## 2. Tangle in Three-Qubit States

## 3. Quantum Algorithm for Measuring the Tangle

## 4. Simulations

`Scipy`[24] for the optimization procedure. In particular, we employed the Powell method as it was found to provide accurate results [25]. The mean number of optimization steps is of the order of a few hundred.

#### 4.1. Error Model

#### 4.2. Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Probability density of three-qubit random states as a function of the tangle. (

**b**) Probability density of two-qubit random states as a function of the concurrence. Three-qubit random states tend to populate values around $\sim 0.3$, while two-qubit random states are mostly distributed at high values.

**Figure 2.**Quantum circuit required for driving an unknown state $|\psi {\rangle}_{ABC}$ into its up-to-phases canonical form $|\tilde{\phi}\rangle $. The optimal parameters ${({\overrightarrow{\theta}}_{A},{\overrightarrow{\theta}}_{B},{\overrightarrow{\theta}}_{C})}_{\mathrm{opt}}$ are chosen variationally.

**Figure 3.**Error model for the simulations. Single-qubit and measurements errors can occurr following the scheme of the figure, and may happen with probabilities $0.1\phantom{\rule{0.166667em}{0ex}}t\%$ and $1\phantom{\rule{0.166667em}{0ex}}t\%$, respectively, for $t=\{0,1,2,3,4,5\}$. All errors are uncorrelated. This circuit is to be applied after that in Figure 2.

**Figure 4.**Tangle of the $|\mathrm{GHZ}\rangle $ state vs. parameter t quantifying gate and measurement errors. Solid lines represent averaged results for the tangle obtained without optimization, while the shadowed regions span all results (again without optimization). The dots are the results for the full optimization method applied to the $|\mathrm{GHZ}\rangle $ state as if it were an unknown input state. Colors indicate whether post-selection was applied or not. The results indicate that the optimization procedure does not degrade the quality of the estimation of the tangle.

**Figure 5.**Measured tangle vs. exact tangle, for three-qubit random states. Results in green were obtained without applying post-selection, in contrast to those in red. (

**a**) Results with no gate errors. (

**b**) Results considering the maximum gate error allowed in this paper, t = 5. In all figures, the solid black line represents ideal measurement of the tangle. As the errors decrease, we observe convergence towards the exact tangle.

**Figure 6.**Relative error of the tangle of 1000 random states, with a $2\phantom{\rule{0.166667em}{0ex}}t\%$ threshold in the cost function value, as a function of the error parameter t. Dots correspond to average values, and error bars span $70\%$ of the measurements. Colors indicate whether post-selection has been applied or not. Note that the algorithm measures the correct tangle in the absence of noise, but tends to underestimate the tangle under its presence.

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**MDPI and ACS Style**

Pérez-Salinas, A.; García-Martín, D.; Bravo-Prieto, C.; Latorre, J.I.
Measuring the Tangle of Three-Qubit States. *Entropy* **2020**, *22*, 436.
https://doi.org/10.3390/e22040436

**AMA Style**

Pérez-Salinas A, García-Martín D, Bravo-Prieto C, Latorre JI.
Measuring the Tangle of Three-Qubit States. *Entropy*. 2020; 22(4):436.
https://doi.org/10.3390/e22040436

**Chicago/Turabian Style**

Pérez-Salinas, Adrián, Diego García-Martín, Carlos Bravo-Prieto, and José I. Latorre.
2020. "Measuring the Tangle of Three-Qubit States" *Entropy* 22, no. 4: 436.
https://doi.org/10.3390/e22040436