# 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.;
et al. 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,
et al. 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,
and et al. 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