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Weak Value Advantage in Overcoming Noise

Zohar Schwartzman-Nowik, Dorit Aharonov, Eliahu Cohen

TL;DR

This paper tackles whether the weak value measurement protocol (WVMP) can offer a metrological advantage under noise acting on the primary system. It formalizes the task of learning an unknown Hermitian operator $A$ under amplitude and phase damping and defines the noisy weak value $A_{w,\\mathcal E}$. The main result proves that WVMP cancels the linear-in-$\\gamma$ bias for these channels, yielding an $O(\\gamma^2)$ bias and thus a qualitative advantage over strong measurements, even when postselection is allowed; however, for Pauli or unital channels the advantage can be diminished, underscoring a nuanced landscape. The work highlights the role of weak entanglement plus postselection in robustness, and suggests future extension to more general noise models, higher-dimensional systems, and a bias-variance tradeoff.

Abstract

The weak value exhibits numerous intriguing characteristics, such as values outside the operator spectrum, leading to unexpected phenomena. Nevertheless, the measurement protocol used for measuring the weak value has been the subject of an on-going controversy. In particular, the possibility of gaining a metrological advantage using weak measurements was questioned. A rigorous characterization of this advantage when the primary system is noisy is still missing. We thus consider here the challenge of learning an unknown operator under the influence of noise on the primary system which could lead to bias in the results. For amplitude and phase damping noise channels, we prove that the weak value measurement protocol (WVMP) eliminates the bias to linear order, and this cannot be done with strong measurements. Since the WVMP makes use both of weak entanglement as well as postselection, one might suspect that the advantage is solely due to the postselection aspect of the WVMP. We prove that this is not the case, and that the same advantage of the WVMP is kept even over strong measurement protocols that are allowed to apply postselection. By this we rigorously prove for the first time the existence of settings in which the WVMP possesses a strict advantage in robustness to noise, even over strong measurements augemented with postselection. However, for some noise channels, we show that no advantage is exhibited once both measurement regimes are equipped with postselection.

Weak Value Advantage in Overcoming Noise

TL;DR

This paper tackles whether the weak value measurement protocol (WVMP) can offer a metrological advantage under noise acting on the primary system. It formalizes the task of learning an unknown Hermitian operator under amplitude and phase damping and defines the noisy weak value . The main result proves that WVMP cancels the linear-in- bias for these channels, yielding an bias and thus a qualitative advantage over strong measurements, even when postselection is allowed; however, for Pauli or unital channels the advantage can be diminished, underscoring a nuanced landscape. The work highlights the role of weak entanglement plus postselection in robustness, and suggests future extension to more general noise models, higher-dimensional systems, and a bias-variance tradeoff.

Abstract

The weak value exhibits numerous intriguing characteristics, such as values outside the operator spectrum, leading to unexpected phenomena. Nevertheless, the measurement protocol used for measuring the weak value has been the subject of an on-going controversy. In particular, the possibility of gaining a metrological advantage using weak measurements was questioned. A rigorous characterization of this advantage when the primary system is noisy is still missing. We thus consider here the challenge of learning an unknown operator under the influence of noise on the primary system which could lead to bias in the results. For amplitude and phase damping noise channels, we prove that the weak value measurement protocol (WVMP) eliminates the bias to linear order, and this cannot be done with strong measurements. Since the WVMP makes use both of weak entanglement as well as postselection, one might suspect that the advantage is solely due to the postselection aspect of the WVMP. We prove that this is not the case, and that the same advantage of the WVMP is kept even over strong measurement protocols that are allowed to apply postselection. By this we rigorously prove for the first time the existence of settings in which the WVMP possesses a strict advantage in robustness to noise, even over strong measurements augemented with postselection. However, for some noise channels, we show that no advantage is exhibited once both measurement regimes are equipped with postselection.
Paper Structure (44 sections, 160 equations, 1 figure)

This paper contains 44 sections, 160 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and definitions
  3. Results
  4. Conclusion and open questions
  5. The weak measurement protocol of the weak value
  6. Weak value for pure states
  7. Weak value for mixed initial state
  8. Weak value for mixed initial and final states
  9. The limit of strong measurement
  10. Strong limit for pure state
  11. Strong limit for mixed state
  12. Strong measurement with postselection
  13. Coinciding with PVM with postselection
  14. Lindblad Noise Approximated by Noise Before Ideal WMP
  15. Proof of Theorem - Weak Value advantage at learning $A$ under Amplitude and Phase damping
  16. ...and 29 more sections

Figures (1)

  • Figure 1: a. The WVMP of the WV, consisting of a pre-selected state $|\psi_s\rangle$, noise $\mathcal{E}$, weak entanglement $\exp\left({igA\otimes \mathcal{P}}\right)$ and postselection $|\psi_f\rangle$. b. The strong measurement protocol without postselection, c. The strong measurement protocol with postselection. The entanglement is weak (strong) if the standard deviation of the probe state is large (small) compared to the interaction strength $g$.