Universality in fidelity-based quantum metrology

Abstract

We consider the problem of identifying the quantum spin states that are the optimal sensors of a given transformation averaged over all possible orientations of the spin system. Our geometric approach to the problem is based on a fidelity criterion and is entirely general, encompassing both unitary transformations (such as rotations and squeezing) and non-unitary transformations (such as Lorentz boosts). This formalism leads to a universality result: There exists a zero-measure subset of states that will be optimal sensors for certain transformations and the worst sensors for others, and this set does not depend on the transformation under consideration. In other words, some spin states are simply the best (or worst) sensors, regardless of what they detect.

Figures (9)

• Figure 1: Optimal squeezing sensors for $s = 3$. Top: Average fidelity as a function of $\eta$ for coherent states, the $W$ state, the $2$-AC triangular prism state and optimal states. Bottom: SU(2) invariants of the optimal states $r_1$ (blue), $r_2$ (green) and $r_3$ (orange) as a function of $\eta\in[0,\pi]$.
• Figure 2: Locus $\mathcal{R}^{(1)}$ (horizontal mustard yellow segment) and curves corresponding to rotations $R_\eta$ (black) and squeezing $Q_\mu$ (blue) in the $\tilde{r}$-plane. The cyan disc on the left end of $\mathcal{R}^{(1)}$ stands for the coherent state while the magenta square on the right end denotes the anticoherent one. For any transformation $V$ such that the $\phi$ coordinate of $\tilde{r}_{V}$ is less than $\pi/2$, the optimal $V$-sensor is the coherent state, while for $\phi>\pi/2$ the anticoherent one. The little white disc on the rotation curve denotes the critical rotation angle $\eta_1=\arccos(-2/3)$ for which the optimal sensor changes discontinuously. Little red dots on both curves correspond to $30^\circ$ parameter increments. The black dot at the top denotes the identity transformation. Note that both curves are traversed twice, from the identity at the top to the red dot at the "corner" on the right for $0 \leq \eta,\mu \leq \pi$, and retracing the same locus back to the identity for $\pi \leq \eta,\mu \leq 2\pi$. Special to the $s=1$ case, the preimage of any point of $\mathcal{R}^{(1)}$ in projective space is a single SU$(2)$ orbit, i.e., only states with same-shape Majorana constellations project to the same point of $\mathcal{R}^{(1)}$.
• Figure 3: Locus $\mathcal{R}^{(3/2)}$ (horizontal mustard yellow segment) and curves corresponding to rotations $R_\eta$ (black) and squeezing $Q_\mu$ (blue) in the $\tilde{r}$-plane (see the Fig. \ref{['spin1locus']} caption for the visual conventions used). Unlike the $s=1$ case, states with Majorana constellations of different shapes can project to the same point in the $\tilde{r}$-plane.
• Figure 4: Locus $\mathcal{R}^{(2)}$ of $r_\rho$ for $s=2$. $\mathcal{R}^{(2)}$ is the triangle in the figure. The vertices correspond to the (classes of) coherent (little cyan sphere), GHZ (magenta square) and tetrahedral (yellow tetrahedron) states. The frame has been rescaled (with respect to the original $r_\rho$ referred to) and rotated so that the triangle has constant $\tilde{r}_3$, equal to $1/\sqrt{15}$, and the coherent state vector $\tilde{r}_\circ$ has $\phi=0$. Also shown are generic vectors $\tilde{r}_\rho$ (in red) and $\tilde{r}_{V}$ (in green) for a state $\rho$ and a unitary operator $V$ ($\tilde{r}_{V}$ is normalized to $|\tilde{r}_{V}|=0.4$ as the optimal state depends only on its direction). The color the vector $\tilde{r}_{V}$ points to on the sphere indicates the optimal state for detecting the transformation implemented by $V$ (note that the top half of the yellow sector has been removed to provide visual access to the interior). For a given $\tilde{r}_{V}$, the optimal state is the one minimizing the Euclidean inner product $\tilde{r}_\rho \cdot \tilde{r}_{V}$.
• Figure 5: Optimal roto- and squeeze-sensors for $s=2$. The locus $\mathcal{R}^{(2)}$ of Fig. \ref{['locuss2Plot1:fig']} is seen here from above. The color transitions are shown in Tab. \ref{['tab_phi_s2']}. The black curve corresponds to the rotation operators $R_\eta=e^{-i \eta S_z}$, starting, at $\eta=0$, at the identity (at the position of $\tilde{r}_3$, in the center of the figure), looping in the yellow (tetrahedral) sector and crossing to magenta (GHZ) at $\eta_1=96.47^\circ$, and later on, into cyan (coherent), at $\eta_2=139.95^\circ$. The end point of the curve corresponds to $\eta=\pi$ --- the curve turns back there and retraces itself symmetrically in the $[\pi,2\pi]$-interval. Similarly, the blue curve corresponds to the squeezing operators $Q_\mu=e^{-i \mu S_z^2}$ --- there are three critical points in this case, $\mu_1=84.82^\circ$, $\mu_2=120^\circ$, and $\mu_3=157.45^\circ$. Like in the case of rotations, the curve retraces itself in the interval $[\pi,2\pi]$. Little white spheres on both curves mark the critical angles $\eta_i$, $\mu_j$, while even smaller red spheres mark $30^\circ$ parameter increments.