Observables and Geometric Quantities

An accurate state or spectrum is an intermediate result. The final physical quantity may require derivatives, traces, statistical reconstruction, or a limit that is harder to converge than the state itself. This chapter organizes those final calculations around their own normalization, resolution, and uncertainty checks.

Enter this chapter

Thermal states and partition functions uses stable Boltzmann weights and bounds missing spectral weight. Correlation functions follows operator-applied states, Fourier conventions, and sum rules. Entanglement from exact states extracts Schmidt spectra while respecting subsystem ordering and small singular values.

For synthetic experiments, tomography simulation distinguishes data generation from constrained state estimation. Randomized benchmarking distinguishes sequence averaging from shot noise and checks the assumptions behind decay fits. These pages implement protocols; their operational interpretation remains in Quantum Information.

For lattice models, start with band-structure numerics, then Berry curvature, Chern numbers, and Wilson loops. Transport adds open leads; Green functions develops resolvents and controlled broadening.

Choose the representation before the estimator. A state vector and a declared subsystem partition suffice for pure-state entanglement. Thermal traces require an ensemble or a controlled approximation to one. A rank-$r$ occupied frame over the Brillouin zone supports geometric calculations only while that subspace remains isolated. Transport additionally requires a lead model and matching convention; diagonalizing a closed sample does not provide these boundary data.

Converge the quantity being reported

An eigenvalue residual controls proximity to an energy, not the stability of a derivative or eigenvector at degeneracy. A smooth band plot does not certify a global gap. A normalized density matrix does not certify a complete thermal spectrum. An integer Chern number does not certify a sufficiently fine momentum mesh.

A useful hierarchy is to verify algebraic identities first, then numerical refinement, then the physical limiting procedure. For example, a spectral function should satisfy

$$\int\frac{d\omega}{2\pi}S_{B^\dagger B}(\omega) =\langle B^\dagger B\rangle.$$

After the sum rule passes, vary time window, sampling interval, and broadening independently. Finally examine whether the finite system represents the continuum or thermodynamic quantity of interest.

For stochastic characterization, distinguish finite-shot error from reconstruction bias and model mismatch. Increasing shot count narrows sampling uncertainty but can make a biased model’s incorrect answer look more precise.

The random unit of an experiment also determines the uncertainty estimate. In randomized benchmarking, different sequences generally have different survival probabilities, and shots sharing a sequence share that source of variation. Record both sequence count and shots per sequence; increasing the latter cannot eliminate the uncertainty from a fixed collection of sampled sequences. Tomography instead requires the likelihood for its actual measurement settings and shot allocations. Projecting an unphysical estimate onto the positive cone does not by itself maximize that likelihood.

Geometry requires conventions as data

Use $\mathcal A=i\langle u|\nabla u\rangle$ and a stated orientation. With forward overlap links, positively oriented plaquette phases give minus the Berry flux. Reversing the connection sign or momentum orientation changes curvature and Chern signs.

Eigenvectors also depend on orbital basis, unit-cell embedding, and reciprocal-lattice sewing. Preserve these alongside the Hamiltonian and band selection. Random-phase and random-frame transformations provide powerful tests because physical geometric output should remain invariant.

For the geometric definitions, see the Berry connection. For band applications and material interpretation, see Berry curvature and band dynamics. Here we develop the numerical algorithms and explain how to estimate their errors.

A compact benchmark route

Begin with a two-level thermal system, a single-spin dynamic correlation, and a Bell-state entropy. Then test a dimerized chain in both real and momentum space. Add a two-dimensional two-band model with known gapped Chern sectors, and finally a single resonant level coupled to leads.

These models exercise different invariants: Boltzmann normalization, spectral weight, Schmidt probabilities, Fourier consistency, gauge covariance, and current conservation. Agreement with one cannot substitute for the others.

Include a known failure along with each successful benchmark. The Chern-number example has three successively finer meshes that agree on the wrong integer because they miss a narrow curvature region. The band example has a positive direct gap but overlapping valence and conduction energy ranges. These failures test whether diagnostics detect the relevant physical problem rather than merely confirm algebraic consistency.

For a reported result, retain the model and units, representation and basis ordering, estimator definition, and parameters that control each error. Examples include the discarded thermal weight, time window and Fourier sign, subsystem axes, sequence and shot counts, momentum mesh and seam map, or lead self-energy and resolvent broadening. Report convergence of the observable under changes in those parameters, together with the analytic identity or independently solvable model used to check it.

References

• Fukui, T., Hatsugai, Y., and Suzuki, H., “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” Journal of the Physical Society of Japan 74, 1674–1677 (2005), arXiv:cond-mat/0503172.
• Magesan, E., Gambetta, J. M., and Emerson, J., “Characterizing quantum gates via randomized benchmarking,” Physical Review A 85, 042311 (2012), arXiv:1109.6887.
• Weiße, A., Wellein, G., Alvermann, A., and Fehske, H., “The kernel polynomial method,” Reviews of Modern Physics 78, 275–306 (2006), arXiv:cond-mat/0504627.