diff --git a/docs/CLAIM_SUPPORT.md b/docs/CLAIM_SUPPORT.md index 2274789..c199b55 100644 --- a/docs/CLAIM_SUPPORT.md +++ b/docs/CLAIM_SUPPORT.md @@ -47,12 +47,32 @@ analytic deductions. | Global coordinatewise execution scaling is `O((1 + log(n/delta))/epsilon^2)` | Paper-level analytic deduction | Record definitions and norm tests | Norm and decoder tests above | Absolute raw-coordinate error. | | The concatenated magnitude record has norm `2*sqrt(n)` and fixed `l_2` accuracy costs `O(n/epsilon_2^2)` up to confidence | Appendix-B analytic result with executable support | `supporting_analysis.py`, `STATISTICAL_ACCURACY.md` | `test_supporting_analysis.py` | Raw magnitude vector; the complex phase stream is separate. | | Relative/directional control and magnitude-block natural-gradient conditioning follow from the `l_2` bound and metric factors | Appendix-B conditional result | `supporting_analysis.py`, `STATISTICAL_ACCURACY.md` | `test_supporting_analysis.py` | Requires a nonzero-gradient margin; damping is an optimizer choice. | -| Raw-coordinate accuracy does not imply normalized-frame accuracy without metric conditioning | Exact supporting separation | `swap_reflection`, `STATISTICAL_ACCURACY.md` | `test_coordinate_frame_separation_with_hopf_split` | Existence result; no efficient compiler for the witness reflection is claimed. | -| Ambient-sphere and projective phase metrics are distinct conventions | Supporting geometric clarification | `ambient_phase_metric`, `projective_phase_metric` | `test_phase_metric_conventions_and_support_rank` | Hopf-QBP uses the ambient-sphere convention. | +| Raw-coordinate accuracy does not imply normalized-frame accuracy without metric conditioning | Main-text interpretation with Appendix-B proof and exact executable support | `swap_reflection`, `STATISTICAL_ACCURACY.md` | `test_coordinate_frame_separation_with_hopf_split` | Real-chart existence result; no efficient compiler for the witness reflection is claimed. | +| The ambient-sphere phase metric is the adopted convention; the projective block is an unused comparison | Main-text convention with Appendix-B derivation and executable support | `ambient_phase_metric`, `projective_phase_metric` | `test_phase_metric_conventions_and_support_rank` | The projective quotient remains low-profile and is not used by the theorem. | | The magnitude output objects obey the stated sufficient norm hierarchy | Supporting analytic result | norm-bound helpers, `STATISTICAL_ACCURACY.md` | `test_magnitude_norm_hierarchy` | Upper bounds for the displayed rescaling decoder, not converses. | | The direct-angle assigned CNOT ledger matches the manuscript's compiler model | **Paper-level resource claim** | `conventions.py`, `native_schedule.py`, `qbp_resource_ledger.py` | `test_resource_ledger.py` | Finite assigned counts, not global optimality or routing. | -| The same forward and frame objects admit one `O(N)` multiplexed realization | **Appendix-B ideal-model compiler result** | `optimized_compiler.py`, `qbp_optimized_resource_ledger.py` | `test_optimized_compiler.py` | One reusable clean flag; elementary angles generally recombine Hopf coordinates. | -| A reflection sum can be estimated by coefficient-one-norm term sampling | Supporting algebraic extension | `supporting_analysis.py` | `test_supporting_analysis.py` | Portable upper bound, not an optimal Hamiltonian strategy. | +| One exact multiplexed realization preserves the estimator and the `O(N)` matched forward/frame scaling | **Appendix-B exact compiler result with detailed executable support** | `optimized_compiler.py`, `qbp_optimized_resource_ledger.py` | `test_optimized_compiler.py` | One reusable clean flag; elementary angles generally recombine Hopf coordinates. | +| A reflection sum admits unbiased coefficient-one-norm term sampling with a `Lambda**2` sufficient-shot factor | Main-text extension with executable support | `supporting_analysis.py`, `OBSERVABLES_AND_READOUT.md` | `test_supporting_analysis.py` | Portable upper bound, not an optimal Hamiltonian strategy. | +| Independent symmetric readout errors transform records by attenuation and bin mixing | Supporting analytic readout model | `supporting_analysis.py` | `test_supporting_analysis.py` | Readout-only; no coherent gate-noise or mitigation claim. | + +## Scope relative to the first paper + +The first Hopf paper supplie²È="27hase. | +| The global complex magnitude circuit decodes all magnitude derivatives | Paper-level exact complete-distribution check | `circuits.py`, `decoders.py`, `reference.py` | `test_complex_magnitude.py` | General implementation uses separated phase/frame blocks. | +| The direct complex phase circuit decodes all leaf-phase derivatives | Paper-level exact signed one-hot check | `circuits.py`, `decoders.py`, `reference.py` | `test_complex_phase.py` | Expectation objectives are invariant under a uniform leaf-phase shift. | +| A checkpoint circuit returns every derivative at a selected depth | Paper-level exact depth-block check | `circuits.py`, `decoders.py`, `reference.py` | `test_checkpoints.py` | Each selected depth uses its own circuit stream. | +| Integrated complex checkpoint substitution is valid on the active interface | Paper-level projected-matrix and decoded-mean checks | reference.py`, `circuits.py`, `conventions.py` | `test_operator_contracts.py`, `test_four_qubit_example.py` | It need not preserve the full unitary or complete distribution. | +| Singular magnitude and zero-amplitude phase coordinates are handled without division | Paper-level exact singular checks | reference.py`, `circuits.py`, `decoders.py` | `test_singular_cases.py`, `test_complex_phase.py` | The derivative and ordinary record vanish when the differential vanishes. | +| Every single-depth global, checkpoint, and direct-phase record has norm 2 | Paper-level algebraic record checks | `decoders.py`, `reference.py` | `test_decoders.py`, `test_checkpoints.py`, `test_complex_phase.py` | Finite premise of the concentration argument. | +| Global coordinatewise execution scaling is O((1 + log(n/delta))/epsilon^2) | Paper-level analytic deduction | Record definitions and norm tests | Norm and decoder tests above | Absolute raw-coordinate error. | +| The concatenated magnitude record has norm 2*sqrt(n) and fixed l_2 accuracy costs O(n/epsilon_2^2) up to confidence | Appendix-B analytic result with executable support | `supporting_analysis.py`, `SHATISTICAL_ACCURACY.md` | `test_supporting_analysis.py` | Raw magnitude vector; the complex phase stream is separate. | +| Relative/directional control and magnitude-block natural-gradient conditioning follow from the l_2 bound and metric factors | Appendix-B conditional result | `supporting_analysis.py`, `STATISTICAL_ACCURACY.md` | `test_supporting_analysis.py` | Requires a nonzero-gradient margin; damping is an optimizer choice. | +| Raw-coordinate accuracy does not imply normalized-frame accuracy without metric conditioning | Main-text interpretation with Appendix-B proof and exact executable support | `swap_reflection`, `STATISTICAL_ACCURACY.md` | `test_coordinate_frame_separation_with_hopf_split` | Real-chart existence result; no efficient compiler for the witness reflection is claimed. | +| The ambient-sphere phase metric is the adopted convention; the projective block is an unused comparison | Main-text convention with Appendix-B derivation and executable support | `ambient_phase_metric`, `projective_phase_metric` | `test_phase_metric_conventions_and_support_rank` | The projective quotient remains low-profile and is not used by the theorem. | +| The magnitude output objects obey the stated sufficient norm hierarchy | Supporting analytic result | norm-bound helpers, `STATISTICAL_ACCURACY.md` | `test_magnitude_norm_hierarchy` | Upper bounds for the displayed rescaling decoder, not converses. | +| The direct-angle assigned CNOT ledger matches the manuscript's compiler model | Paper-level resource claim | `conventions.py`, `native_schedule.py`, `qbp_resource_ledger.py` | `test_resource_ledger.py` | Finite assigned counts, not global optimality or routing. | +| One exact multiplexed realization preserves the estimator and the O(N) matched forward/frame scaling | Appendix-B exact compiler result with detailed executable support | `optimized_compiler.py`, `qbp_optimized_resource_ledger.py` | `test_optimized_compiler.py` | One reusable clean flag; elementary angles generally recombine Hopf coordinates. | +| A reflection sum admits unbiased coefficient-one-norm term sampling with a Lambda**2 sufficient-shot factor | Main-text extension with executable support | `supporting_analysis.py`, `OBSERVABLES_AND_READOUT.md` | `test_supporting_analysis.py` | Portable upper bound, not an optimal Hamiltonian strategy. | | Independent symmetric readout errors transform records by attenuation and bin mixing | Supporting analytic readout model | `supporting_analysis.py` | `test_supporting_analysis.py` | Readout-only; no coherent gate-noise or mitigation claim. | ## Scope relative to the first paper @@ -66,7 +86,7 @@ The present project supplies the computationally addressed differential frame, one magnitude record shared across coordinates and depths, Walsh decoding, the direct complete phase record, complete-gradient concentration, checkpoint suffixes and active-interface contracts, and the supporting compiler, -statistical, observable, and readout analyses listed above. +statistical, observable, readout, and method-positioning analyses listed above. ## Claim 1: inherited conventions and forward states @@ -74,8 +94,7 @@ For `N = 2**n`: - the real magnitude block has length `N - 1`; - the complex phase block has length `N`; -- the combined complex order is - `(theta_1, ..., theta_{N-1}, theta_N, ..., theta_{2N-1})`; +- the combined complex order is `(theta_1, ..., theta_{N-1}, theta_N, ..., theta_{2N-1})`; - native `HopfReal` and `HopfComplex` schedules reproduce the recursive state. Evidence: `native_schedule.py`, `reference.py`, `test_native_schedule.py`, and @@ -209,13 +228,13 @@ but its scale changes: ```math q_j=\partial_{\theta_j}E_O, \qquad -c_j=\frac{q_j}{\sqrt{g_{j,j}}}, +\chi_j=\frac{q_j}{\sqrt{g_{j,j}}}, \qquad \nu_j=\frac{q_j}{g_{j,j}}. ``` -The exact swap-reflection example shows that `q_k < epsilon` can coexist with -`c_k = 2` when `g[k,k]` is small. Therefore raw-coordinate accuracy does not +The exact swap-reflection example shows that `|partial_{theta_k} E| < epsilon` can coexist with +`chi_k = 2` when `g[k,k]` is small. Therefore raw-coordinate accuracy does not imply normalized-frame accuracy without metric conditioning. The complete raw magnitude record has norm `2*sqrt(n)`, whereas the complete @@ -229,7 +248,7 @@ its boundaries. For leaf probabilities `p`, Hopf-QBP follows the ambient-sphere phase block ```math -G_{\mathrm{ph}}^{\mathrm{sphere}}=\mathrm{diag}(p). +G_{\mathrm{ph}}^{\mathrm{sph}}=\mathrm{diag}(p). ``` The optional projective convention is @@ -260,13 +279,13 @@ model and does not establish routed or noisy-device performance. For ```math -H=\sum_\alpha c_\alpha O_\alpha, +H=\sum_\alpha a_\alpha O_\alpha, \qquad -\Lambda=\sum_\alpha|c_\alpha|, +\Lambda=\sum_\alpha|a_\alpha|, ``` -sampling term `alpha` with probability `|c_alpha|/Lambda` and scaling its QBP -record by `Lambda*sign(c_alpha)` is unbiased. Record norms gain a factor +sampling term `alpha` with probability `|a_alpha|/Lambda` and scaling its QBP +record by `Lambda*sign(a_alpha)` is unbiased. Record norms gain a factor `Lambda`, and sufficient shot counts gain `Lambda**2`. Under independent symmetric readout flips, global parities are attenuated by diff --git a/docs/OBSERVABLES_AND_READOUT.md b/docs/OBSERVABLES_AND_READOUT.md index e2c5a7b..ae547f7 100644 --- a/docs/OBSERVABLES_AND_READOUT.md +++ b/docs/OBSERVABLES_AND_READOUT.md @@ -1,102 +1,96 @@ -# Observable extensions and analytic readout sensitivity +# Reflection-sum objectives and analytic readout sensitivity -The validated core contract uses one known Hermitian unitary `O`, exact -phase-calibrated controlled access to `O`, and exact-logical circuit execution. -This page gives two portable extensions that do not alter that contract: -reflection-sum term sampling and an analytic independent-readout-error model. -It is not a hardware benchmark. +The paper's validated core uses one known Hermitian-unitary objective, exact +phase-calibrated controlled access, and exact-logical circuit execution. This +page expands the main-text reflection-sum formula and records an independent +symmetric-readout model. It is not a hardware benchmark. ## 1. Core controlled-reflection contract The objective is ```math -E_O(\theta) +E_O(\boldsymbol\theta) = -\langle\psi(\theta)|O|\psi(\theta)\rangle, -``` - -with - -```math +\langle\psi(\boldsymbol\theta)|O|\psi(\boldsymbol\theta)\rangle, +\qquad O^\dagger=O, \qquad O^2=I. ``` The controlled branch must have a known relative phase. If the implemented -branch is `exp(i gamma) O`, the measured interference quadrature is rotated by +branch is `exp(i gamma) O`, the measured interference components are rotated by `gamma`. A known phase can be compensated; an unknown phase invalidates the -direct decoder. +fixed decoder. A classical description of `O`, or uncontrolled access to it, +does not itself supply this coherent interface. -The repository does not claim to compile an arbitrary nonunitary observable, -block encoding, or application-specific binary test into this interface. +## 2. Reflection-sum objectives -## 2. Reflection-sum Hamiltonians - -Suppose +Let ```math H = -\sum_{\alpha=1}^{L}c_\alpha O_\alpha, +\sum_{\alpha=1}^{L}a_\alpha O_\alpha, \qquad O_\alpha^\dagger=O_\alpha, \qquad O_\alpha^2=I, +\qquad +\Lambda=\sum_\alpha|a_\alpha|>0. ``` -with real coefficients. Define - -```math -\Lambda -= -\sum_{\alpha=1}^{L}|c_\alpha|. -``` - -For `Lambda > 0`, sample term `alpha` with +Suppose every term has the calibrated controlled access above. Sample term +`alpha` with ```math p_\alpha = -\frac{|c_\alpha|}{\Lambda}. +\frac{|a_\alpha|}{\Lambda}. ``` -If `Z^(alpha)` is any unbiased Hopf-QBP record for `O_alpha`, output +If `Z^(alpha)` is any unbiased Hopf-QBP record for `O_alpha`, return ```math \widetilde Z = -\Lambda\,\mathrm{sgn}(c_\alpha)Z^{(\alpha)}. +\Lambda\,\mathrm{sgn}(a_\alpha)Z^{(\alpha)}. ``` Then ```math +\begin{aligned} \mathbb E[\widetilde Z] -= -\sum_\alpha c_\alpha\mathbb E[Z^{(\alpha)}] +&= +\sum_\alpha +\frac{|a_\alpha|}{\Lambda} +\Lambda\,\mathrm{sgn}(a_\alpha) +\mathbb E[Z^{(\alpha)}]\\ +&= +\sum_\alpha a_\alpha\nabla E_{O_\alpha} = \nabla\langle H\rangle. +\end{aligned} ``` -A norm-`2` depth or phase record becomes a norm-at-most-`2 Lambda` sampled -record. The sufficient execution count therefore gains a factor -`Lambda**2`. For the complete concatenated magnitude record, the norm bound is -`2 Lambda sqrt(n)`. +A base norm bound `B` becomes `Lambda*B`, so the corresponding sufficient +execution count gains a factor `Lambda**2`. For a norm-`2` depth or phase +record the sampled norm is at most `2*Lambda`; for the concatenated magnitude +record it is at most `2*Lambda*sqrt(n)`. -The expected controlled-operation cost of one term-sampled record is +The expected controlled-term charge of one sampled record is ```math \sum_\alpha p_\alpha C(\mathrm{ctrl}(O_\alpha)). ``` -A matched scalar comparator can use the same term sampling and controlled-term -cost. This is a portable upper bound, not a claim that independent term -sampling is optimal for every Hamiltonian. Commuting-group measurements, -classical shadows, coefficient-aware shot allocation, or application-specific -block encodings may provide better scalar and gradient interfaces and must be -compared under their own access assumptions. +A matched scalar comparator uses the same term distribution and expected +controlled-term charge. This is a portable upper bound, not a claim that term +sampling is optimal for every Hamiltonian. Commuting groups, shadow methods, +coefficient-aware allocation, or application-specific block encodings require +their own access and normalization models. ## 3. Global parity under independent readout flips @@ -107,8 +101,8 @@ For internal node `j`, the global magnitude sign is ``` Let the interference-ancilla bit flip independently with probability `p_c` and -system bit `k` flip with probability `p_k`. Conditional on the ideal outcome, -the observed sign is attenuated by +system bit `k` with probability `p_k`. Conditional on the ideal outcome, the +mean sign is attenuated by ```math \kappa_j @@ -117,10 +111,8 @@ the observed sign is attenuated by \prod_{k:\lambda(j)_k=1}(1-2p_k). ``` -Only marker-supported system bits enter this product. The record is not -necessarily a parity of the entire measured string. - -For uniform error probability `p`, with node `j = 2**d + r`, +Only marker-supported system bits enter this product. For a uniform error rate +`p` and node `j=2**d+r`, ```math \kappa_j @@ -130,26 +122,20 @@ For uniform error probability `p`, with node `j = 2**d + r`, (1-2p)^{2+\mathrm{wt}(r)}. ``` -The largest sign-parity weight at depth `d` is therefore `d+2`, including the +Thus the largest sign-parity weight at depth `d` is `d+2`, including the interference ancilla. ## 4. Checkpoint and direct-phase records -The checkpoint sign is - -```math -(-1)^{b_c+b_t}. -``` - -Independent ancilla and target readout errors attenuate it by +The checkpoint sign is `(-1)^(b_c+b_t)`. Independent branch and target errors +attenuate it by ```math (1-2p_c)(1-2p_t). ``` -Errors in the measured prefix do not add to this sign parity; instead they mix -the one-hot address bins through the classical independent-bit-flip channel. -For bit error probabilities `p_1, ..., p_d`, the bin channel is +Prefix errors do not add to the sign parity; they mix the one-hot address bins +through ```math T @@ -161,52 +147,43 @@ p_k & 1-p_k \end{pmatrix}. ``` -The direct complex phase record behaves similarly: +For the direct complex phase record, the branch error attenuates the sign by +`1-2p_c`, while system `Z`-readout errors mix leaf bins through the analogous +independent-bit-flip channel. -- the ancilla error attenuates the sign by `1-2p_c`; -- system `Z`-readout errors mix the leaf bins through the corresponding - independent-bit-flip channel. +If calibrated transfer factors are nonsingular, one may invert or regularize +these classical channels. Such correction amplifies variance and is not part of +the exact-logical theorem. -If the error rates are calibrated and the transfer factors are nonsingular, -one may invert or regularize these classical channels. Such correction -amplifies variance and is not included in the exact-logical theorem. +## 5. Boundary of this analysis -## 5. What this analysis does and does not establish +The formulas above establish unbiased reflection-sum sampling and exact mean +transformations under independent symmetric readout flips. They do not model: -It establishes the exact mean transformation under independent symmetric -readout flips. It does not model: - -- coherent gate errors; +- coherent state-preparation, frame, or controlled-reflection errors; - two-qubit depolarizing noise; - correlated readout; -- device connectivity and SWAP insertion; -- controlled-observable synthesis noise; -- error mitigation; or +- device routing and SWAP insertion; +- approximate synthesis; +- mitigation; or - optimizer behavior. -A meaningful comparison of global, checkpoint, separate-tangent, and -parameter-shift methods under those effects must fix a device topology, -transpiler, observable compiler, noise channel, mitigation method, parameter -ensemble, shot allocation, and optimizer. Those choices define a separate -hardware study rather than a validation requirement for the present -exact-logical interface. +A hardware comparison must fix the topology, transpiler, controlled-objective +compiler, noise channel, mitigation method, parameter ensemble, shot +allocation, and requested output norm. ## 6. Executable support -The reflection-sum and readout transfer formulas are implemented in: +The formulas are implemented and tested in: ```text qbp_validation/supporting_analysis.py qbp_validation/tests/test_supporting_analysis.py ``` -The tests verify: - -- unbiased one-norm term sampling; -- the `Lambda` record-norm factor; -- marker-supported global attenuation; -- checkpoint and direct-phase sign attenuation; and -- stochasticity of the independent bin-mixing channel. +The tests verify coefficient-one-norm probabilities, unbiased scaled records, +the `Lambda` norm factor, marker-supported parity attenuation, checkpoint and +phase attenuation, and stochasticity of the bin-mixing channel. Run: diff --git a/docs/STATISTICAL_ACCURACY.md b/docs/STATISTICAL_ACCURACY.md index 2d5396a..099f68d 100644 --- a/docs/STATISTICAL_ACCURACY.md +++ b/docs/STATISTICAL_ACCURACY.md @@ -1,11 +1,11 @@ # Statistical accuracy, output geometry, and conditioning The manuscript's primary finite-shot target is simultaneous absolute accuracy -of the **raw Hopf-coordinate gradient**. Appendix B also gives the complete -magnitude-vector `l_2` bound, conditional relative and directional guarantees, -and magnitude-block natural-gradient conditioning. This page makes the output -objects, metric conventions, separation examples, and executable test map -explicit. +of the **raw Hopf-coordinate gradient**. Its main text distinguishes coordinate +and geometric outputs, adopts the ambient-sphere phase convention, and gives the +reflection-sum extension; Appendix B supplies the complete-vector, conditional +directional, separation, conditioning, and exact-recompilation derivations. +This page retains the full output hierarchy and executable test map. ## 1. Raw coordinatewise target @@ -120,8 +120,8 @@ so `-v_hat` is a descent direction. The normalized-direction error obeys \frac{2\xi}{\mathcal G}. ``` -Choosing `xi = rho*||v||_2` gives relative `l_2` error at most `rho`, direction -error at most `2*rho`, and the sufficient magnitude-stream count +Choosing `xi = rho*||v||_2` gives relative `l_2` error at most `rho`, direction error +at most `2*rho`, and the sufficient magnitude-stream count ```math S_{\mathrm{rel}} @@ -138,8 +138,8 @@ hold at a stationary point. Shared readout removes the coordinate-count penalty at fixed absolute accuracy; it does not remove signal-to-noise conditioning when the complete gradient norm -is small. If `||v||_2` is exponentially small, the displayed relative-error -count is correspondingly large. +is small. If `||v||_2` is exponentially small, the displayed relative-error count is +correspondingly large. ## 4. Raw coordinate accuracy does not imply frame accuracy @@ -151,60 +151,52 @@ For an active magnitude coordinate `k`, \sqrt{g_{k,k}}\,|e_k\rangle. ``` -Define +Following the paper, define the rescaled outputs ```math -q_k +\chi_k = -\partial_{\theta_k}E_O, -\qquad -c_k -= -\frac{q_k}{\sqrt{g_{k,k}}}, +\frac{\partial_{\theta_k}E_O}{\sqrt{g_{k,k}}}, \qquad \nu_k = -\frac{q_k}{g_{k,k}}. +\frac{\partial_{\theta_k}E_O}{g_{k,k}}. ``` -These are respectively the raw coordinate derivative, normalized-frame -coefficient, and inverse-metric coordinate. - -Choose `0 < g[k,k] < (epsilon/2)**2` and define the Householder reflection +These are the normalized-frame and inverse-metric coordinates, respectively. +For an active real coordinate, choose +`0 < g^R[k,k] < (epsilon/2)**2` and define ```math O_k = I- -\bigl(|\psi\rangle-|e_k\rangle\bigr) -\bigl(\langle\psi|-\langle e_k|\bigr). +\bigl(|\psi^R\rangle-|e_k^R\rangle\bigr) +\bigl(\langle\psi^R|-\langle e_k^R|\bigr). ``` -Because `|psi>` and `|e_k>` are orthonormal, +Orthonormality gives ```math O_k^\dagger=O_k, \qquad O_k^2=I, \qquad -O_k|\psi\rangle=|e_k\rangle. +O_k|\psi^R\rangle=|e_k^R\rangle, ``` -Therefore +and therefore ```math -q_k +\partial_{\theta_j}E_{O_k} = -2\sqrt{g_{k,k}} -< -\varepsilon, +2\sqrt{g^R_{k,k}}\,\delta_{jk}, \qquad -c_k=2, +\chi_k=2. ``` -and all other normalized-frame coefficients vanish. The zero estimate is thus -`epsilon`-accurate in raw coordinate `l_infinity` error while its -normalized-frame error is `2`. +The complete zero estimate is thus `epsilon`-accurate in raw coordinate +`l_infinity` error while its normalized-frame error is exactly `2`. This is an exact separation of output tasks, not a defect in the raw-coordinate theorem. Converting raw-coordinate guarantees into normalized-frame or natural @@ -291,11 +283,11 @@ p_\ell=|x_\ell|^2, \sum_\ell p_\ell=1. ``` -The complex Hopf chart follows the ambient round-sphere convention of the first +The complex Hopf chart follows the ambient round-sphere convention used in the paper. Its phase block is ```math -G_{\mathrm{ph}}^{\mathrm{sphere}} +G_{\mathrm{ph}}^{\mathrm{sph}} = \mathrm{diag}(p). ``` @@ -319,9 +311,8 @@ G_{\mathrm{ph}}^{\mathrm{FS}}\mathbf 1=0. ``` If `s` leaves have positive probability, the projective block has rank `s-1`; -zero-probability leaves add further null coordinate directions. This projective -quotient is a valid alternative geometry, but it is not the metric convention -used by Hopf-QBP. +zero-probability leaves add further null coordinate directions. This projective quotient is included only to distinguish the alternative +geometry; it is not used by Hopf-QBP. ## 8. Uniform-phase objective invariance