# Protective Measurement—A New Quantum Measurement Paradigm: Detailed Description of the First Realization

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## Abstract

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## 1. Introduction

#### Theoretical Framework

## 2. Experimental Implementation

#### Experimental Setup

## 3. Results

- An acquisition with only the crystals in the optical path and $|\psi \rangle =|H\rangle $ or $|\psi \rangle =|V\rangle $, which allows us to calibrate the system;
- An acquisition without protection (only crystals in the optical path), corresponding to the traditional PJ scenario;
- An acquisition with both weak interaction and active Zeno-like protection (both birefringent crystals and polarizers in the optical path), realizing the PM;
- Two acquisitions, one with only the polarizing plates and one with a free optical path, allowing us to complete the system calibration by evaluating and properly subtracting unwanted position biases introduced by crystals and polarizing plates.

#### 3.1. Output State Verification

#### 3.2. Expectation Values

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Expectation Value Analysis

#### Appendix A.1. Projective Measurements

#### Appendix A.2. Protective Measurements

## References

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**Figure 1.**Theoretical framework (ideal pictorial representation). (

**a**) Projective measurement: the two polarization components are completely separated, with the single photons impinging on two regions of the detector corresponding to the polarization operator eigenvalues $A=\pm 1$. The expectation value $\langle A\rangle $ is evaluated as the weighted average of the events, following Equation (8). (

**b**) Protective measurements: all the photons fall in the same region, centered in a position proportional to the polarization expectation value (see Equation (9)).

**Figure 2.**Experimental setups for projective measurement (

**a**), extracting the expectation value of an observable by measuring an ensemble of identical particles, and protective measurement (

**b**), able to reliably estimate such an expectation value with just a single detection event. Ti:Sa ML laser: titanium–sapphire mode-locked laser; SHG: second harmonic generator; SMF: single-mode fiber; SPAD: single-photon avalanche diode; PBS: polarizing beam splitter; HWP: half-wave plate.

**Figure 3.**Density matrix reconstructions for the state outgoing the measurement process, considering an initial polarization state $|\frac{17}{60}\pi \rangle $: (

**a**) theoretical real part $\mathrm{Re}\left[{\rho}_{in}\right]$ of the initial state density matrix ($\mathrm{Im}\left[{\rho}_{in}\right]=0$); (

**b**,

**c**), respectively: the reconstructed real ($\mathrm{Re}\left[{\rho}_{PM}^{rec}\right]$) and imaginary ($\mathrm{Im}\left[{\rho}_{PM}^{rec}\right]$) parts of the density matrix of the single-photon state after the protective measurement; (

**d**) theoretically expected real part $\mathrm{Re}\left[{\rho}_{dec}\right]$ of the density matrix of our state at the end of the projective measurement ($\mathrm{Im}\left[{\rho}_{dec}\right]=0$); and (

**e**,

**f**), respectively: reconstructed real ($\mathrm{Re}\left[{\rho}_{PJ}^{rec}\right]$) and imaginary ($\mathrm{Im}\left[{\rho}_{PJ}^{rec}\right]$) parts of the density matrix of the single-photon state after undergoing the projective measurement.

**Figure 4.**Plots of the photon counts distributions obtained with projective (

**a**,

**c**,

**d**) and protective (

**b**,

**d**,

**f**) measurements, for three different linearly polarized initial states $|{\psi}_{\theta}\rangle $. (

**a**,

**b**) $|{\psi}_{\frac{17}{60}\pi}\rangle \approx 0.629|H\rangle +0.777|V\rangle $; (

**c**,

**d**) $|{\psi}_{\frac{\pi}{4}}\rangle \approx 0.707|H\rangle +0.707|V\rangle $; and (

**e**,

**f**) $|{\psi}_{\frac{\pi}{8}}\rangle \approx 0.924|H\rangle +0.383|V\rangle $.

**Figure 5.**Few-event photon counts distributions for the input state $|\frac{17}{60}\pi \rangle $. In both figures, the first detection event is marked in yellow. While PJ (

**a**) requires a measurement on an ensemble of identically prepared particles, PM (

**b**) allows extracting the expectation value of our observable with just a single click. Yellow dashed line: x position corresponding to the theoretical expectation value of the polarization $\langle {A\rangle}^{\mathrm{th}}$ = −0.208; red circles: FWHM of the corresponding distributions for the state $|\frac{17}{60}\pi \rangle $ reported in Figure 4.

**Table 1.**Comparison between theoretical and reconstructed density matrices. $F({\rho}_{PM}^{rec},{\rho}_{in})$ and $F({\rho}_{PJ}^{rec},{\rho}_{dec})$: fidelities between reconstructed density matrices ${\rho}_{PM\left(PJ\right)}^{rec}$ and their theoretical counterparts ${\rho}_{in\left(dec\right)}$ in the PM (PJ) case; $F({\rho}_{PM}^{rec},{\rho}_{PJ}^{rec})$: fidelities between reconstructed protected and unprotected states; and $\mathcal{P}\left({\rho}_{PM}^{rec}\right)$ and $\mathcal{P}\left({\rho}_{PJ}^{rec}\right)$: purities of the reconstructed states in the PM and PJ case, respectively.

State | $\mathit{F}({\mathit{\rho}}_{\mathit{PM}}^{\mathit{rec}},{\mathit{\rho}}_{\mathit{in}})$ | $\mathit{F}({\mathit{\rho}}_{\mathit{PJ}}^{\mathit{rec}},{\mathit{\rho}}_{\mathit{dec}})$ | $\mathit{F}({\mathit{\rho}}_{\mathit{PM}}^{\mathit{rec}},{\mathit{\rho}}_{\mathit{PJ}}^{\mathit{rec}})$ | $\mathcal{P}\left({\mathit{\rho}}_{\mathit{PM}}^{\mathit{rec}}\right)$ | $\mathcal{P}\left({\mathit{\rho}}_{\mathit{PJ}}^{\mathit{rec}}\right)$ |
---|---|---|---|---|---|

$|+\rangle $ | 0.999 | 0.998 | 0.720 | 0.998 | 0.540 |

$|\frac{17}{60}\pi \rangle $ | 0.996 | 0.999 | 0.751 | 0.992 | 0.520 |

$|\frac{\pi}{8}\rangle $ | 0.992 | 0.999 | 0.894 | 0.992 | 0.789 |

**Table 2.**Comparison between the experimental and theoretical expectation values. Experimental expectation values—$\langle {A\rangle}^{\mathrm{th}}$: theoretical expectation values; $\langle {A\rangle}^{\mathrm{PJ}}$: experimental expectation value with projective measurements; and $\langle {A\rangle}^{\mathrm{PM}}$: experimental expectation value with protective measurements.

State | $\langle {\mathit{A}\rangle}^{\mathbf{th}}$ | $\langle {\mathit{A}\rangle}^{\mathbf{PJ}}$ | $\langle {\mathit{A}\rangle}^{\mathbf{PM}}$ |
---|---|---|---|

$|+\rangle $ | 0 | $-0.03\pm 0.04$ | $0.012\pm 0.014$ |

$|\frac{17}{60}\pi \rangle $ | −0.208 | $-0.21\pm 0.02$ | $-0.19\pm 0.02$ |

$|\frac{\pi}{8}\rangle $ | 0.707 | $0.72\pm 0.02$ | $0.72\pm 0.02$ |

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Rebufello, E.; Piacentini, F.; Avella, A.; Lussana, R.; Villa, F.; Tosi, A.; Gramegna, M.; Brida, G.; Cohen, E.; Vaidman, L.; Degiovanni, I.P.; Genovese, M. Protective Measurement—A New Quantum Measurement Paradigm: Detailed Description of the First Realization. *Appl. Sci.* **2021**, *11*, 4260.
https://doi.org/10.3390/app11094260

**AMA Style**

Rebufello E, Piacentini F, Avella A, Lussana R, Villa F, Tosi A, Gramegna M, Brida G, Cohen E, Vaidman L, Degiovanni IP, Genovese M. Protective Measurement—A New Quantum Measurement Paradigm: Detailed Description of the First Realization. *Applied Sciences*. 2021; 11(9):4260.
https://doi.org/10.3390/app11094260

**Chicago/Turabian Style**

Rebufello, Enrico, Fabrizio Piacentini, Alessio Avella, Rudi Lussana, Federica Villa, Alberto Tosi, Marco Gramegna, Giorgio Brida, Eliahu Cohen, Lev Vaidman, Ivo Pietro Degiovanni, and Marco Genovese. 2021. "Protective Measurement—A New Quantum Measurement Paradigm: Detailed Description of the First Realization" *Applied Sciences* 11, no. 9: 4260.
https://doi.org/10.3390/app11094260