Dynamical discontinuities in repeated weak measurements revealed by complex weak values

TL;DR

The paper addresses nonanalytic dynamical behavior in a minimal quantum protocol of repeated weak measurements with post-selection, controlled by the post-selection angle $\phi$. By analyzing the Kraus map and the complex weak value $O_{S,w}$, it shows that dynamical discontinuities in meter observables arise when $\mathrm{Im}(O_{S,w})$ vanishes, and that this is tied to an exchange of dominance among Kraus eigenvalues at a critical $\phi_c$. The relaxation time $\tau$ diverges as $\tau \propto |\phi - \phi_c|^{-1}$, giving a universal critical exponent $\nu = 1$ independent of system parameters, demonstrated both analytically and in a two-level (qubit-qubit) model with three such critical points. Although not a true thermodynamic phase transition, the results reveal a dynamical phase-transition–like structure and suggest routes to engineer non-analytic behavior in measurement-based quantum control and to probe criticality in post-selected quantum dynamics using weak values.

Abstract

Critical phenomena reveal universal behavior in complex systems, and uncovering analogous effects in quantum weak measurement protocols with post-selection provides new insight into how measurement backaction can shape quantum dynamics. This work investigates dynamical discontinuities that arise when the post-selected polar angle is used as a control parameter. The system evolves under repeated applications of a weak measurement protocol with post-selection, in which the meter state is retained after each iteration while the system is renewed. The emergence of these discontinuities is shown to be determined by the structure of the weak value: when the weak value has a nonzero imaginary component, a discontinuity appears in the expectation value of meter observables precisely at the point where the imaginary part of the weak value vanishes as a function of the post-selection polar angle. In contrast, no discontinuities occur when the weak value remains real for all post-selection angles. The phenomenon originates from the eigenstructure of the protocol's Kraus operator, with the stability of fixed points changing at the critical point where the discontinuity arises. Remarkably, the associated critical exponent is 1, independent of system parameters. These results open new perspectives for engineering non-analytic behavior in measurement-based quantum control and for probing criticality in post-selected quantum dynamics using weak measurements with weak values.

Figures (8)

• Figure 1: The protocol consists of the application of $n$ times the weak measurement protocol where the system is pre-selected $(\hat{\rho}_S)$, then it interacts with the meter (in the state $\hat{\rho}_A^{f,i}$) via a weak unitary operator and finally the system is post-selected ($\hat{\rho}_f$). After this protocol a new system is pre-selected ($\hat{\rho}_S$) and the protocol continues using the same meter.
• Figure 2: Three distinct discontinuities characterize the dependence of the expectation value of $\hat{\sigma}_x$ on the post-selected angle $\phi$, emphasizing the nontrivial structure that arises for different repetition numbers $n$. In this plot $\alpha=\frac{\pi}{7}$, $\theta=\frac{\pi}{4}$ and $\gamma=0.001$, while the initial meter state is along the $z$ axis.
• Figure 3: The imaginary part of the weak value vanishes when the absolute value of the two eigenvalues of the Kraus operator coincide. The plot shows the eigenvalues of the Kraus operator $\hat{K}$, $|\lambda_{+}|$ and $|\lambda_{-}|$, as functions of the post-selected angle $\phi$, together with their difference $|\lambda_{+}| - |\lambda_{-}|$ and the imaginary part of the weak value $\mathrm{Im}(O_{s,w})$. In this plot $\alpha=\frac{\pi}{7}$, $\theta=\frac{\pi}{4}$ and $\gamma=0.1$, while the initial meter state is along the $z$ axis.
• Figure 4: The Bloch sphere trajectory under unitary dynamics traces cycles at a constant polar angle, highlighting the periodic nature of the system. The path begins at the blue point and ends at the red point. System parameters are $\theta = \frac{\pi}{4}$, $\alpha = \frac{\pi}{7}$, $\gamma = 0.1$, $\phi = \frac{\pi}{2}$, with initial meter state $r = (\sqrt{0.5}, 0, \sqrt{0.5})^T$.
• Figure 5: Although initiated near the unstable fixed point, the trajectory on the Bloch sphere is repelled toward the other fixed point, starting at the blue point and ending at the red point. Parameters: $\theta = \frac{\pi}{4}$, $\alpha = \frac{\pi}{7}$, $\gamma = 0.1$, $\phi = 1$, initial state $r = (\sqrt{0.99}, 0, \sqrt{0.001})^T$.