One from many

Models have free parameters

$$\mathbf{\theta}=(\theta^1,\ldots,\theta^N)$$

Estimates are always noisy

$$\text{Cov}\left(\hat{\mathbf{\theta}}\right)$$

Fisher information reigns

$$F_{jk}= \int dx\frac{\partial_j\Pr(x|\mathbf{\theta})\,\partial_k\Pr(x|\mathbf{\theta})} {\Pr(x|\mathbf{\theta})}$$

Fisher information reigns

$$\text{Cov}\left(\hat{\mathbf{\theta}}\right)\geq F^{-1}$$

How about a single number?

$q=\sum_jq_jB^j$

Optimal and secure measurement protocols for quantum sensor networks, Zachary Eldredge, Michael Foss-Feig, JAG, S. L. Rolston, and Alexey V. Gorshkov, Phys. Rev. A 97, 042337 (2018)

How about a single number?

Identify $\theta^k$ and calculate $F_{kk}$, right?

Optimal and secure measurement protocols for quantum sensor networks, Zachary Eldredge, Michael Foss-Feig, JAG, S. L. Rolston, and Alexey V. Gorshkov, Phys. Rev. A 97, 042337 (2018)

WRONG!

You will calculate unattainable sensitivity goals.

You will construct estimators with unnecessary errors.

Parameters define tangent vectors

$$F_{jk}= \int dx\frac{\partial_j\Pr(x|\mathbf{\theta})\,\partial_k\Pr(x|\mathbf{\theta})} {\Pr(x|\mathbf{\theta})}$$

Functions define differential forms

$q=\theta^1+\frac{1}{2}\theta^2$

Parameters define conditioning

$(F_{kk})^{-1}$ bounds conditional variance

Functions define marginalizing

$(F^{-1})^{kk}$ bounds marginal variance

Conditional isn’t marginal

$(F_{kk})^{-1}\neq(F^{-1})^{kk}$

Quantum models define many distributions

$$\begin{aligned} H(\mathbf{\theta})&=\sum_j\theta^jX_j \\ \rho(\mathbf{\theta})&=e^{-itH(\mathbf{\theta})}\rho_0e^{itH(\mathbf{\theta})} \\ \Pr(x|\mathbf{\theta})&=\operatorname{tr}[\rho(\mathbf{\theta})E_x] \end{aligned}$$

Optimize over choices

$$\Vert\theta^k\Vert^2=\max_{\rho_0,\{E_x\}}F_{jj}$$

Optimize over choices

$$\Vert\theta^k\Vert^2=\max_{\rho_0,\{E_x\}}F_{jj}$$

Optimize over choices

$$\Vert\theta^k\Vert^2=\max_{\rho_0,\{E_x\}}F_{jj}$$

We use norm to marginalize

Summary