# Beating Standard Quantum Limit with Weak Measurement

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## Abstract

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## 1. Introduction

## 2. Standard-Quantum-Limited WM Protocols

## 3. Heisenberg Scaling WM Protocols

#### 3.1. Heisenberg Scaling WM Protocol Probing with Mixed States

#### 3.2. Heisenberg Scaling WM Protocol by Measuring the Post-Selection Probability

## 4. The Implementation of the Scheme in a Ramsey-Type Model with Qubits

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Scheme for a standard weak measurement (WM) protocol. The quantum system (QS) is pre-selected into a superposition of $|+1\rangle $ and $|-1\rangle $, between which an extra phase is introduced after the coupling with the measurement apparatus (MA). After the post-selection, the two observables of the MA, namely p and q, are shifted in proportion to the real and imaginary parts of the weak value, as shown in (

**a**,

**b**), respectively. (Adapted from [12]).

**Figure 2.**The precision to estimate coupling strength g with different probing states. (

**a**) Probing with coherent state $\left|\alpha \right.\u232a$: the precision of measuring a phase $Arg\left(\alpha \right)$ is limited by the inherent uncertainty $\Delta \mathrm{Re}\left[\alpha \right]=\Delta \mathrm{Im}\left[\alpha \right]=1/2$, hence it is standard-quantum-limited. (

**b**) Probing with mixed state: the initial state is a statistical ensemble of $\left|\alpha \right.\u232a$, each of which acquires a phase decided by $\left|\alpha \right|$, and interferes (with itself) with varying probability due to the post-selection. A shift in the radical direction is generated and its level is proportional to the length squared. For a mixed state, the length can be increased to $\sim \left|\alpha \right|$, which leads to Heisenberg scaling precision. (Adapted from [21]).

**Figure 3.**Setup of estimating single photon Kerr effect with mixed probe state. Single photons of 785 nm interact with strong pulses of 800 nm in the photonic crystal fiber (PCF). The amplitude of the strong pulses is modulated to generate a mixture with different coherent states. Post-selecting the single photons and detecting the intensity of corresponding strong pulses with a full HD oscilloscope, the coupling strength g can be estimated from the mean photon number shift of the strong pulses. The achieved precision $\Delta g$ is inversely proportional to the mean photon number N, which is the so-called Heisenberg scaling. (Adapted from [21]).

**Figure 4.**A practical Heisenberg scaling in experiment.The estimation precision of g against the mean photon number N of the mixed state is plotted. For $N<{10}^{5}$, the precision follows a Heisenberg scaling of $\Delta g\simeq 6.3\times {10}^{-4}{N}^{-1}$ rad, shown as a green line, obtained by fitting these points. The red line is a bound on the precision for mixed states, taking account of the QFI for each member in the ensemble, given by $\Delta {g}_{\mathrm{min}}\simeq 0.95\times {10}^{-4}{N}^{-1}$ rad. (Adapted from [21]).

**Figure 5.**The classical information ${F}_{p}$ contained in the post-selection process. ${F}_{p}$ is calculated for varying interaction strength g and post-selection parameter $\epsilon $, when the mean photon number of strong pulses is $n=5\times {10}^{4}$. As $g\to 0$, ${F}_{p}$ becomes dominant in ${F}_{tot}$ and scales with ${n}^{2}$, which means a practical Heisenberg scaling precision is attainable by measuring the successful post-selection probability. (Adapted from [22]).

**Figure 6.**Setup for estimating single photon Kerr effect by measuring the post-selection probability. The 815 nm photons serve as triggers and the heralded 785 nm photons interact with strong pulses (800 nm) in an 8 m long photonic crystal fiber (PCF), centering in a polarization Sagnac interferometer (PSI). The interaction strength g is estimated from the distribution of successful and failed post-selection probabilities. (Adapted from [22]).

**Figure 7.**A practical Heisenberg scaling approaching the Heisenberg limit.(

**a**) Experimental verification of Heisenberg-limited precision. The achieved precision shows good agreement with the $1/n$ fitting line up to $n={10}^{6}$ photons, and an ultimate precision of $\sim {10}^{-10}$ rad is obtained. (

**b**) The amount of extracted Fisher information (FI) for different n. The ${n}^{2}$ scaling also indicates that the Heisenberg limit is approached in this measurement. (Adapted from [22]).

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**MDPI and ACS Style**

Chen, G.; Yin, P.; Zhang, W.-H.; Li, G.-C.; Li, C.-F.; Guo, G.-C.
Beating Standard Quantum Limit with Weak Measurement. *Entropy* **2021**, *23*, 354.
https://doi.org/10.3390/e23030354

**AMA Style**

Chen G, Yin P, Zhang W-H, Li G-C, Li C-F, Guo G-C.
Beating Standard Quantum Limit with Weak Measurement. *Entropy*. 2021; 23(3):354.
https://doi.org/10.3390/e23030354

**Chicago/Turabian Style**

Chen, Geng, Peng Yin, Wen-Hao Zhang, Gong-Chu Li, Chuan-Feng Li, and Guang-Can Guo.
2021. "Beating Standard Quantum Limit with Weak Measurement" *Entropy* 23, no. 3: 354.
https://doi.org/10.3390/e23030354