Revisiting weak values through non-normality

Abstract

Quantum measurement is one of the most fascinating and discussed phenomena in quantum physics, due to the impact on the system of the measurement action and the resulting interpretation issues. Scholars proposed weak measurements to amplify measured signals by exploiting a quantity called a weak value, but also to overcome philosophical difficulties related to the system perturbation induced by the measurement process. The method finds many applications and raises many philosophical questions as well, especially about the proper interpretation of the observations. In this paper, we show that any weak value can be expressed as the expectation value of a suitable non-normal operator. We propose a preliminary explanation of their anomalous and amplification behavior based on the theory of non-normal matrices and their link with non-normality: the weak value is different from an eigenvalue when the operator involved in the expectation value is non-normal. Our study paves the way for a deeper understanding of the measurement phenomenon, helps the design of experiments, and it is a call for collaboration to researchers in both fields to unravel new quantum phenomena induced by non-normality.

Figures (18)

• Figure 1: Scheme of a weak measurement with post-selection. In the first step, pre-selection, the initial state of the system is set to ${|{\psi_i}\rangle}$ (blue). Then, the system and ancilla interact through a unitary operator $U=e^{-i\gamma\hat{O}\otimes\hat{P}}$ (green). This interaction should be weak, meaning that the interaction strength $\gamma$ should be small. After the weak interaction, a projective measurement is executed in the system and the final state is chosen to be ${|{\psi_f}\rangle}$ (yellow). If the post-selection is successful, the ancilla position is measured by applying a projective measurement in the ancilla (pink). The expectation value of the position is shifted by a quantity that is proportional to the weak value $O_w$ of the system's observable $\hat{O}$ for the pre- and post-selected system states.
• Figure 2: a) Levels set of the modulus of the weak value of $\hat{\sigma}_x$ in terms of the polar angles of the pre- and post-selected states $(\theta_i, \theta_f)$, imposing $\xi_i=0$, $\xi_f=0$. b) Levels set of the Henrici departure from normality of the non-normal matrix $\hat{A}$ associated to the weak value $\sigma_{x,w}$. a,b,d) The red curve corresponds to the border of the area in which the modulus of the weak value is larger than $1.0$. The black curve corresponds to the boundary of the area in which the modulus of the weak value is twice the maximum possible expectation weak value. c) Square modulus of the weak value as a function of the Henrici departure from normality of the operator $\hat{A}$ for anomalous weak values, $|\sigma_{x,w}|>1$, obtained by varying $\theta_f$ between $0$ and $\frac{\pi}{2}$, while $\theta_i$ is fixed (colored dots in the legend). d) Levels set of the Henrici departure from normality of the non-normal operator $\hat{A}'$ associated with the weak value $\sigma_{x,w}$.
• Figure 3: a) Levels set of the modulus of the weak value of $\hat{\sigma}_y$ in terms of the polar angles of the pre- and post-selected states $(\theta_i, \theta_f)$, imposing $\xi_i=0$, $\xi_f=0$. b) Levels set of the Henrici departure from normality of the non-normal matrix associated with the weak value, $\hat{A}$. a,b,d) The red curve corresponds to the boundary of the area in which weak value amplification occurs. The black curve corresponds to the frontier of the area in which the modulus of the weak value is twice the maximum possible expectation value. c) Square modulus of the amplified weak value as function of the Henrici departure from normality of the operator $\hat{A}$, for various $0<\theta_f<\frac{\pi}{2}$. d) Levels set of the Henrici departure from normality of the non-normal operator $\hat{A}'$ associated with the weak value $\sigma_{y,w}$.
• Figure 4: a) Levels set of the modulus of the weak value of $\hat{\sigma}_z$ in terms of the polar angles of the pre- and post-selected states $(\theta_i, \theta_f)$, imposing $\xi_i=0$, $\xi_f=0$. b) Levels set of the Henrici departure from normality of the non-normal matrix $\hat{A}$ associated with the weak value. c) Square modulus of the weak value as a function of the Henrici departure from normality, with $0<\theta_f<\frac{\pi}{2}$. d) Levels set of the Henrici departure from normality of the non-normal operator $\hat{A}'$ associated with the weak value $\sigma_{z,w}$.
• Figure 5: a) Levels set of the modulus of the weak value of the operator $\hat{O}=\frac{1}{\sqrt{3}}\left(\hat{\sigma}_y+\hat{\sigma}_x+\hat{\sigma}_z\right)$ in terms of the polar angles of the pre- and post-selected states $(\theta_i, \theta_f)$, imposing $\xi_i=0$, $\xi_f=0$. b) Levels set of the Henrici departure from normality of the non-normal matrix $\hat{A}$ associated with the weak value in terms of the polar angles of the pre- and post-selected states $(\theta_i, \theta_f)$. a,b,d) The red curve corresponds to the boundary of the area in which the weak value amplification occurs. The black curve corresponds to the frontier of the area in which the modulus of the weak value is twice the maximum possible expectation value. c) Square modulus of the weak value as a function of the Henrici departure from normality for amplified anomalous weak values, $|O_{w}|>1$, with $0<\theta_f<\frac{\pi}{2}$. d) Levels set of the Henrici departure from normality of the non-normal operator $\hat{A}'$ associated with the weak value $O_{w}$.