On pins and needles

Geometry is important

Quantum inference requires optimization w.r.t. geometric constraints.

$$\hat{\rho}=\underset{\rho\geq0}{\mathrm{arg\,max}}\operatorname{Pr} (\mathrm{data}|\rho)$$

Qubits are trivial

$$\rho=\frac{1}{2}(\mathbb{1}+\mathbf{r}\cdot\boldsymbol{\sigma})$$

$$\rho\geq0\,\Leftrightarrow\,\Vert\mathbf{r}\Vert\leq1\,\Leftrightarrow\, \operatorname{tr}(\rho^2)\leq1$$

Higher dimensions are complicated

$\operatorname{tr}(\rho^2)\leq1$, $\quad3\operatorname{tr}(\rho^2)-2\operatorname{tr}(\rho^3)\leq1$

Bloch sphere becomes pointed!

Solid angle measures pointedness:

$$\Omega=\frac{\mathrm{vol}(\text{ball}\cap\text{state-space})} {\mathrm{vol}(\text{ball})}$$

Pointedness grows rapidly

JAG, Carlton M. Caves, Rank deficiency and the Euclidean geometry of quantum states, APS March Meeting A52.00002 (2017).

Pointedness is important

Gives valuable information about

• Estimator bias
  Silva et al., Investigating bias in maximum-likelihood quantum-state tomography, Phys. Rev. A 95, 022107 (2016).
• Optimization success probability
  Amelunxen et al., Living on the edge: Phase transitions in convex programs with random data, arXiv:1303.6672.

Fisher information dictates geometry

$$F_{jk}(\rho)=\sum_{E\in\text{POVM}}\frac{\operatorname{tr}(E\partial_j\rho) \operatorname{tr}(E\partial_k\rho)}{\operatorname{tr}(E\rho)}$$

Fisher information dictates geometry

$$F_{jk}(\rho)=\sum_{E\in\text{POVM}}\frac{\operatorname{tr}(E\partial_j\rho) \operatorname{tr}(E\partial_k\rho)}{\operatorname{tr}(E\rho)}$$

Quantify difference with condition number

$$\kappa=\sigma_{\mathrm{max}}^2/\sigma_{\mathrm{min}}^2$$

Have proved $\kappa>1$, but how much bigger?

Li et al., Fisher-symmetric informationally complete measurements for pure states, Phys. Rev. Lett. 116, 180402 (2016).

Case study: Qutrit

Measure a SIC POVM.

Case study: Qutrit

Evaluate Fisher information at $2^{14}$ uniformly sampled pure states.

Condition numbers are large!

$\operatorname{min}\kappa=3$

Condition numbers are large!

Ellipses at 25th and 75th percentiles

Ignoring stretching causes errors

Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise, Smolin et al., Phys Rev. Lett. 108, 070502 (2017).

Need to further quantify anisotropy

unm.edu/~jagross

• How does the excess MSE scale with dimension?
• How does anisotropy affect pointedness?
• How does anisotropy change with different measurements?