Quantum metrology with partially accessible chaotic sensors

Abstract

Most quantum metrology protocols harness highly entangled probe states and globally accessible measurements to surpass the standard quantum limit. However, it is challenging to satisfy these requirements in realistic many-body sensors. We demonstrate that both of these constraints can be overcome in quantum chaotic sensors. Crucially, we establish that even in the presence of partial measurement accessibility, chaotic dynamics enables initial unentangled states to exhibit Heisenberg scaling of the quantum Fisher information, $I_α$ with time. In the weakly chaotic regime, we identify spin-coherent states placed at the edge of the regular islands in the mixed classical phase space as optimal initial states for enhanced sensitivity. On the other hand, in the strongly chaotic regime, $I_α$ is insensitive to the choice of the initial state. Notably, quantum-enhanced sensitivity is achieved even when a very low fraction ($\sim 5\%$) of the qubits are accessible. These results establish quantum chaos as a robust resource for quantum-enhanced sensing under realistic accessibility constraints on accessibility.

Figures (8)

• Figure 1: (a) A schematic illustration of the "all-to-all" interacting quantum kicked top model under partial access. (b) The phase space averaged Lyapunov exponent that captures the sensitive dependence on initial conditions. The average is performed over 40,000 trajectories evolved over 50,000 drive periods. (c) The phase space map for (a) $\kappa = 3.0$ and (b) $\kappa = 30.0$. Different initial coherent states $\ket{\theta, \phi}$ are marked: non-equatorial island $\ket{2.20, 2.44}$ (blue circle), chaotic sea $\ket{2.46, 0.32}$ (red circle), and an edge state $\ket{2.56, 2.31}$ (green circle). Details regarding map generation are discussed in Appendix \ref{['Map']}. In the case of $\kappa = 30.0$, the system is strongly chaotic, and hence there are no distinguishable regular islands.
• Figure 2: Scaling of the QFI, $I_\alpha$, is presented as a function of time $t$, for $\kappa = 3.0$ and different subsystem access $Q$, out of a total of $N =1000$ qubits. Each panel shows the QFI for three different initial coherent states: a non-equatorial island state $\ket{2.20, 2.44}$ (blue), a state at the chaotic sea $\ket{2.46, 0.32}$ (red), and an edge state $\ket{2.56, 2.31}$ (green). At sufficiently later times, the edge state exhibits the highest QFI, followed closely by the chaotic sea state. Importantly, even in only $\mathbf{5\%}$ access, the QFI for the edge state nearly reaches the same order of magnitude as the full access case. The QFI vs time plots for more initial states are shown in Appendix \ref{['MR']}.
• Figure 3: Scaling of the QFI is shown as a function of the subsystem size $Q$ for $\kappa = 3.0$ and at three different times : (a) $t = t_E = 35$, the Ehrenfest time for the edge state. (b) $t = t_H = 167$, the Heisenberg time of the system. (c) $t = 2^{15}$, a much later time $t\gg t_H$. With time evolution, the separation between the QFI of the initial chaotic and edge states reduces. Eventually, the QFI of the edge state surpasses that of the chaotic state and reaches its highest QFI value. A transition in QFI scaling is observed starting from $Q=100$. However, at short times, the chaotic initial states provide the larger QFI values and hence the best candidates for sensing. QFI vs $Q$ plots for more initial states are shown in Appendix \ref{['MR']}.
• Figure 4: Scaling of the QFI as a function of time $t$ is shown at $\kappa = 30.0$ and for three different subsystem sizes: (a) $Q = 50$ qubits. (b) $Q = 100$ qubits. (c) $Q = 150$ qubits. (d) Fully accessed case with a total of $1000$ qubits. The dotted line represents the Ehrenfest time $t_E = 3$ and the solid line represents the Heisenberg time $t_H = 167$. The trends of the QFI with time for the partially accessible case behave the same as the fully accessible QKT for the large value of the chaoticity parameter $\kappa$.
• Figure 5: Fractional QFI versus fractional access $Q/N$ are shown for $\kappa = 3.0$ and $30.0$ at different times. Panels (a)-(c) are for $\kappa = 3.0$ case and (d)-(f) are for $\kappa = 30.0$ case. A transition in the slope is observed around $Q/N = 0.1$ for both the cases.