Non-stabilizerness in quantum-enhanced metrological protocols

Abstract

Non-stabilizerness (colloquially "magic") characterizes genuinely quantum (beyond-Clifford) operations necessary for preparation of quantum states, and can be measured by stabilizer Rényi entropy (SRE). For permutationally symmetric states, we show that the SRE depends, for sufficiently large systems, only on a constant number of expectation values of collective spin operators. This compact description is leveraged for analysis of spin-squeezing protocols, which inherently generate non-stabilizerness. Under one-axis twisting (OAT), the generation of optimal squeezing is accompanied by a logarithmic divergence of SRE with system system size. Continued time evolution under OAT produces metrologically useful "kitten" states-superpositions of rotated GHZ states-that feature many-body Bell correlations but exhibit a smaller, system-size-independent SRE that decreases with increasing Bell-correlation strength. Our results reveal connections between non-stabilizerness, multipartite correlations, and quantum metrology, and provide a practical route to quantify non-stabilizerness in experiments for precision sensing.

Figures (7)

• Figure 1: Dynamics of SRE, $\mathcal{M}_2$ (upper panel), and many-body Bell correlator, $\mathcal{E}$ (lower panel), during OAT dynamics in the time scale $t \le \pi/2$ for $N=100$. Markers indicate exact numerical SRE calculations, while solid lines show the value of $\mathcal{M}_2$ calculated with our compact formula \ref{['eq:SRE_approx']}. Insets show the corresponding Husimi functions at marked times. Shaded areas correspond to the spin-squeezing time scale while dashed lines correspond to times at which kitten states are generated.
• Figure 2: Left panel shows scaling of $\mathcal{M}_2$ with $N$ for spin-squeezed states at fixed $\xi^2$. Markers indicate exact SRE calculations, while solid lines show results for \ref{['eq:SRE_approx']}. The label $\xi^2_\mathrm{best}$ corresponds to the best squeezed state with $\xi^2_{\rm best} \propto N^{-2/3}$. Right panel shows scaling of $\mathcal{M}_2$ with $\xi^2$ for $N = 100,500, 1000$.
• Figure 3: Left panel shows SRE scaling with $N$ for kitten states $\ket{\psi(\chi t = \pi / n)}$, for $n \in \{2, 4,6,8,10\}$. Black line and markers show the numerically obtained maximal SRE for $\ket{\psi(\chi t \le \pi / 2)}$. Right panels shows SRE of particular kitten-states $n$ for $N=1000$ spins.
• Figure 4: SRE scaling with $N$ for the best squeezed state generated with OAT (orange), generated with TACT (purple), and the Dicke state with zero magnetization $\ket{J = N/2, m = 0}$ (black). The dashed black line represents \ref{['eq:Dicke']}. Markers indicate exact SRE calculations, while solid lines show results for \ref{['eq:SRE_approx']}.
• Figure 5: Time evolution of SRE, ${\cal M}_2(\ket{\psi(t)})$ (top row), and spin-squeezing parameter $\xi^2$ (bottom row), under OAT (left, orange) and TACT (right, purple) dynamics for $N=100$ spins, on their respectives spin squeezing time scales. Markers indicate exact SRE calculations, while solid lines show results for \ref{['eq:SRE_approx_sup']}. Insets show the Husimi function $Q(\theta,\phi) = |\bra{\theta, \phi}\ket{\psi(t)}|^2$ at different marked times.