Tight qubit uncertainty relations studied through weak values in neutron interferometry

TL;DR

This work tests Ozawa's universally valid error-disturbance relation in a neutron Mach-Zehnder interferometer using a which-way observable $\hat{A}=\hat{\Pi}_1$ and an output observable $\hat{B}=\hat{\sigma}_x$, employing feedback compensation to access the real part of weak values $\omega_{1\pm}$. The experiment measures the error $\varepsilon(A)$, the standard deviation $\Delta(B)$, and the lower bound via the fringe gradient, demonstrating a tight fulfillment of the Ozawa-Hall relation across a range of initial phases $\chi$ for pure states; the error vanishes when the weak values are real. The results validate the operator-based framework, connect the error to the optimized weak-value estimates, and illustrate how the imaginary part of weak values would relate to the error in future measurements. This approach provides a concrete route to directly observe quantum measurement limits and informs precision control in interferometric quantum experiments.

Abstract

In its original formulation, Heisenberg's uncertainty principle describes a trade-off relation between the error of a quantum measurement and the thereby induced disturbance on the measured object. However, this relation is not valid in general. An alternative universally valid relation was derived by Ozawa in 2003, defining error and disturbance in a general concept, experimentally accessible via a tomographic method. Later, it was shown by Hall that these errors correspond to the statistical deviation between a physical property and its estimate. Recently, it was discovered that these errors can be observed experimentally when weak values are determined through a procedure named "feedback compensation". Here, we apply this procedure for the complete experimental characterization of the error-disturbance relation between a which-way observable in an interferometer and another observable associated with the output of the interferometer, confirming the theoretically predicted relation. As expected for pure states, the uncertainty is tightly fulfilled.

Figures (8)

• Figure 1: Scheme of feedback compensation (a) from Hofmann21 as applied to a Mach-Zehnder interferometer (b). After a coupling $\hat{U}^\alpha_{z}$ between object (interferometer paths) and probe system (spin), a compensation $\hat{U}_z^{\beta\pm}$ dependent on the output channel is applied, searching for the maximal value of $p_\pm^{\uparrow x}$ of the probe qubit in output $\ket{+}$ and $\ket{-}$, respectively. See Appendix D and E for details of the state evolution and calculation of observed intensities, respectively.
• Figure 2: Experimental setup: polarized monochromatic neutrons enter the interferometer and are split into paths $\ket{1}$ and $\ket{2}$ at the first interferometer plate at a ratio of 1:4. Before the interferometer, the probe qubit is prepared by a $\frac{\pi}{2}$ direct-current spin rotator (DC 1). In path $1$ the spin is rotated by an angle $\alpha$. The phase shifter adjusts the relative phase $\chi$ of the initial state $\ket{\psi(\chi)}$. Behind the interferometer (in the O-beam), the compensation is applied, that is a spin counterrotation by angle $\beta_\pm$, dependent of the respective measurement context. The spin is analyzed in $\pm x$ direction by the combination of a $\frac{\pi}{2}$ direct-current spin rotator (DC 2) and the magnetic supermirror. The neutrons are counted in a $^3$He detector.
• Figure 3: Measured probability $p_+$, vs. phase of initial state $\chi$, together with theoretical predictions.
• Figure 4: Spin $\ket{\uparrow_x}$ intensities vs. $\beta$ at $\chi=0.04\,\pi$ (top) and at $\chi=1.56\,\pi$ (bottom), measured at the $+$ port (blue) and $-$ port (orange). The optimal compensation angles $\beta^{\rm{opt}}_\pm$ are given by the positions of the maxima (indicated by vertical lines).
• Figure 5: Measured path presence $A^{\text{opt}}_+=\frac{\beta^{\text{opt}}_{+}}{\alpha}$ (red) and $A^{\text{opt}}_-=\frac{\beta^{\text{opt}}_{-}}{\alpha}$ (blue) vs. phase $\chi$ of the initial state, together with theoretical predictions $\Re(\omega_{1\pm})$.