Appendix F — Functional-analysis foundation: Riesz–Fréchet
This appendix is conceptual and adds no results. It is Appendix A of the research notes, which Chapter 4 cites: it explains why \(qg_Z\) is the canonical linear readout of a state and why \(qg_S\) cannot be one. The three numerical checks are done by tools/make_appendices.py.
F.1 The setting
For \(n\) qubits, the operators on \(\mathbb{C}^{2^n}\) form a finite-dimensional Hilbert space with the Hilbert–Schmidt inner product
\[ \langle A,B\rangle=\mathrm{Tr}(A^\dagger B), \]
and density matrices live in it.
F.2 The theorem
Riesz–Fréchet representation (Riesz 1907; Fréchet 1907). Every continuous linear functional \(f\) on a Hilbert space \(H\) has the form \(f(x)=\langle y_f,x\rangle\) for a unique \(y_f\in H\). In finite dimension continuity is automatic and the theorem reduces to linear algebra: with an orthonormal basis \(\{E_i\}\), \(y_f=\sum_i\overline{f(E_i)}\,E_i\). That is why it is a foundation and not a result.
F.3 Application: \(qg_Z\) has a representative
The readout \(\rho\mapsto qg_Z(\rho)=\mathrm{Tr}(\rho\,\sigma_z)\) is linear in \(\rho\), so it has a unique Hilbert–Schmidt representative: \(\sigma_z\) on one qubit, and \(\sigma_z^{(i)}\otimes I\) on qubit \(i\) of a register. Every linear readout of a state, be it an expectation value, a population or a per-shot average such as linear XEB, has this form. \(qg_Z\) is the one whose representative is the \(Z\) observable.
The script checks it on two qubits with the normalized Pauli basis, \(E_P=P/2\). It rebuilds the representative of \(\rho\mapsto\mathrm{Tr}(\rho\,Z\otimes I)\) and gets \(Z\otimes I\) with error 0. For a population readout with random weights, \(\rho\mapsto\sum_x w_x\langle x|\rho|x\rangle\), it gets \(\mathrm{diag}(w)\) with error \(8.7\cdot10^{-18}\).
F.4 What has no representative: \(qg_S\)
\(qg_S(\rho)=H(\langle0|\rho|0\rangle)\) is strictly concave in \(\rho\), not linear, so no operator represents it. The shortest example: \(|0\rangle\langle0|\) and \(|1\rangle\langle1|\) both have \(qg_S=0\), and their equal mixture has \(qg_S=1\) bit; a linear readout would give the average, 0. With \(qg_Z\) the average does hold: \((+1-1)/2=0\).
The same structural fact is behind two results of the notes:
- §10.3 B: linear readouts have unbiased shot-average estimators, and entropy does not (Jensen’s inequality). The table gives the exact bias, by binomial sum, of the two direct estimators.
- §7.1: the identity \(qg_S=H\big((1+qg_Z)/2\big)\) writes the non-linear metric as a fixed scalar function of the linear one. \(qg_S\) carries no information about the state beyond what the representative of \(qg_Z\) already extracts.
| \(p_0\) | shots \(N\) | bias of the \(qg_Z\) average | bias of \(H(k/N)\) (bits) |
|---|---|---|---|
| 0.5 | 10 | 0 | -0.07650 |
| 0.5 | 100 | 0 | -0.00725 |
| 0.5 | 1000 | 0 | -0.00072 |
| 0.9 | 10 | 0 | -0.08440 |
| 0.9 | 100 | 0 | -0.00735 |
| 0.9 | 1000 | 0 | -0.00072 |
F.5 On Riesz’s lemma (1918)
The almost-orthogonal-vector lemma is used to show that the closed unit ball of an infinite-dimensional normed space is not compact. In the finite dimensions of this book it is trivial, because an exactly orthogonal vector always exists. It plays no role in qang and should not be cited as if it did.
Bibliographic notes
The representation theorem is Riesz (1907) and Fréchet (1907). The binary entropy is that of Shannon (1948). The numbered notes (§) and the script are in the repository (Monteverde 2026).