Sensitivity bounds for interferometry with Ising Hamiltonians

Sensitivity bounds for a generic interferometric phase estimation problem are the shot noise and the Heisenberg limits. The shot noise is the highest sensitivity that can be reached with separable states, while the Heisenberg limit is the ultimate bound in sensitivity which can be saturated with entangled states. The scaling of these bounds with the number of particles $N$ entering the interferometer depends on the specific Hamiltonian governing the same interferometer. In typical cases, the Hamiltonian is linear, and the shot-noise and Heisenberg limits scale with $N^{-1/2}$ and with $N^{-1}$, respectively. With interferometers described by generic, nonlinear Hamiltonian, the scalings with the number of particles can be rather different. Here we study the shot-noise and Heisenberg limits of Ising-like Hamiltonians in the presence of longitudinal and transverse fields, both in the nearest-neighbor and in the fully connected spin interaction cases. We provide the explicit forms of the states saturating the shot-noise and Heisenberg limits. These results can be relevant not only for precision measurement purposes but also to characterize quantum phase transitions and, more generally, the witnessing of multipartite entanglement.

Figure 1

Shot-noise limit and Heisenberg limit for nearest-neighbor coupling in longitudinal field: even case. (a) Values of $\langle {\widehat{\sigma }}_{\mathbf{n}}^{\left(i\right)}\rangle$ (with $N=20$) for the optimal separable state; see Eq. (23), as a function of $\gamma$. (b) Top view of panel (a) highlighting the position of the critical value of $\gamma$. (c) $F_{\mathrm{SN}}/N$ (dots), $F_{\mathrm{HL}}/N$ (solid black line) as a function of $\gamma$. The gray region corresponds to sensitivities beyond the shot-noise limit achieved with entanglement initial state. The solid blue line is Eq. (25), which is expected to hold for small values of $\gamma$. The dashed green line is Eq. (26), which is expected to hold for large $\gamma$. The dashed red line is the upper bound Eq. (27). The inset shows the crossing of the dashed green and solid blue lines at ${\gamma }_c=0.7302$ that determines the critical values of $\gamma$.

Figure 2

Shot-noise and Heisenberg limits for nearest-neighbor coupling in longitudinal field: odd case. (a) Values of $\langle {\widehat{\sigma }}_{\mathbf{n}}^{\left(i\right)}\rangle$ (with $N=21$) for the optimal separable state; see Eq. (23), as a function of $\gamma$. (b) Top view of panel (a). (c) $F_{\mathrm{SN}}/N$ (dots), $F_{\mathrm{HL}}/N$ (solid black line) as a function of $\gamma$. The solid blue line is the new Eq. (25), which is expected to hold for small values of $\gamma$. The dashed green line is Eq. (31) with $d=\sqrt{2}-1$, which is expected to hold for large $\gamma$. The dashed red line is the upper bound Eq. (27). The inset shows the crossing of the dashed green and solid blue lines at ${\gamma }_{\mathrm{co}}^{\prime }=0.766$ (see inset), which is close to ${\gamma }_{\mathrm{co}}\sim 0.743$ determined numerically.

Figure 3

The critical values ${\gamma }_{co}, {\gamma }_{co}^{\prime }$ with respect to odd $N$. Numerical critical ${\gamma }_{co}$ (red triangles) and analytical critical ${\gamma }_{co}^{\prime }$ (black points) are shown with respect to number of spins $N$. The solid red line denotes ${\gamma }_c$.

Figure 4

Shot-noise limit and Heisenberg limit for nearest-neighbor coupling in transverse field: even case. (a) Values of $|\langle {\widehat{\sigma }}_{\mathbf{m}}^{\left(i\right)}\rangle |$ (with $N=20$) for the optimal separable state as a function of $\gamma$. (b) Top view of panel (a), focusing on the region crossing the critical ${\gamma }_c$. (c) $F_{\mathrm{SN}}/N$ (dots), $F_{\mathrm{HL}}/N$ (solid black line) as a function of $\gamma$. Two limit cases are shown by solid blue line [Eq. (42), for small $\gamma$] and the dashed green line [Eq. (45) for large $\gamma$], and cross at the critical value ${\gamma }_c^{\prime }=3.771$ (see inset), which is larger than the numerical one ${\gamma }_c\sim 2.197$. The dashed red line is the upper bound (46).

Figure 5

Shot-noise limit and Heisenberg limit for nearest-neighbor coupling in transverse field: odd case. (a) Values of $|\langle {\widehat{\sigma }}_{\mathbf{m}}^{\left(i\right)}\rangle |$ (with $N=21$) for the optimal separable state as a function of $\gamma$. (b) Top view of panel (a), especially in the region crossing the critical ${\gamma }_{co}$. (c) $F_{\mathrm{SN}}/N$ (dots), $F_{\mathrm{HL}}/N$ (solid black line) as a function of $\gamma$. Two limit cases are shown by solid blue line [Eq. (42), for small $\gamma$] and the dashed green line [Eq. (48) with $d={(2\sqrt{2}-2)}^{1/2}$], and cross at the critical value ${\gamma }_{co}^{\prime }=4.022$ (see inset), which is larger than ${\gamma }_{co}\sim 2.238$. The dashed red line is the upper bound Eq. (50).

Figure 6

The critical values of ${\gamma }_{co}, {\gamma }_{co}^{\prime }$ with respect to odd $N$. Analytical critical ${\gamma }_{co}^{\prime }$ (black points) and numerical critical ${\gamma }_{co}$ (red points) with respect to odd spins $N$. With the increase of $N$, the critical value ${\gamma }_{co}$ approaches ${\gamma }_c=2.197$ (red line). As with the numerical result, ${\gamma }_{co}^{\prime }$ approaches ${\gamma }_c^{\prime }=3.771$ (black line).

Figure 7

Shot-noise limit and Heisenberg limit for fully connected coupling in longitudinal field. (a) Values of $\langle {\widehat{\sigma }}_{\mathbf{n}}^{\left(i\right)}\rangle$ (with $N=20$) for the optimal separable state [see Eq. (23)] as a function of $\tilde{\gamma }$. (b) $F_{\mathrm{SN}}/N$ (dots), $F_{\mathrm{HL}}/N$ (solid black line) as a function of $\tilde{\gamma }$. The solid blue lines are Eqs. (60) and (61). The dashed red line is the upper bound (62).

Figure 8

Shot-noise limit and Heisenberg limit for fully connected coupling in transverse field. (a) Values of $|\langle {\widehat{\sigma }}_{\mathbf{m}}^{\left(i\right)}\rangle |$ (with $N=10$) for the optimal separable state [see Eq. (23), replacing $\langle {\widehat{\sigma }}_{\mathbf{n}}^{\left(i\right)}\rangle$ by $\langle {\widehat{\sigma }}_{\mathbf{m}}^{\left(i\right)}\rangle$] as a function of $\tilde{\gamma }$. (b) $F_{\mathrm{SN}}/N$ (dots), $F_{\mathrm{HL}}/N$ (dash-dotted black line) as a function of $\tilde{\gamma }$. The solid blue lines are Eqs. (72) and (73). The dashed red line is the upper bound Eq. (62).