Hamiltonian Extraction from Antisymmetric GENERIC Flow

The antisymmetric sector of the GENERIC decomposition must be unitary — a consequence of the categorical axioms on reversible, entropy-conserving morphisms. This page explains how that unitary sector is explicitly mapped to an effective Hamiltonian \(H_\text{eff}(\theta)\) such that the reversible density-matrix dynamics take the von Neumann form

\[\dot{\rho}_\text{rev} = -i[H_\text{eff}(\theta), \rho(\theta)].\]

Gibbs-Locked Frames

The central object of the Hamiltonian-emergence paper is a Gibbs-locked frame \(K_0 = \beta H\). A Gibbs-locked frame is a fixed point of the linearised reversible dynamics: because \([K_0, H] = [H, H] = 0\) by construction, the commutator term \(i[\delta K, K_0]\) in the linearised GENERIC equation generates only phase rotation in the eigenbasis of \(H\), without displacing \(K_0\) itself.

The background Gibbs state is

\[\rho_0 = \frac{e^{-\beta H}}{\operatorname{tr}(e^{-\beta H})},\]

and the Bohr gaps are the eigenvalue differences of \(H\):

\[\Delta\epsilon_{ij} = \epsilon_i - \epsilon_j, \qquad H \,|\!\epsilon_i\rangle = \epsilon_i\,|\!\epsilon_i\rangle.\]

Linearised dynamics and the uniform decay rate \(\mu_0\)

Near a Gibbs-locked background the linearised GENERIC equation decouples element-by-element in the eigenbasis of \(H\):

\[\frac{d}{dt}(\delta\rho)_{ij} = \bigl(i\,\beta\,\Delta\epsilon_{ij} - \mu_0\bigr)\,(\delta\rho)_{ij},\]

with analytical solution

\[(\delta\rho)_{ij}(t) = (\delta\rho)_{ij}(0)\, \exp\!\bigl[(i\,\beta\,\Delta\epsilon_{ij} - \mu_0)\,t\bigr].\]

Here \(\mu_0 \geq 0\) is the uniform decay rate that controls how quickly off-diagonal coherences decay. It can be inferred from a trajectory via qig.gibbs_lock.infer_mu0(), which strips the oscillatory phase factor and fits the mean off-diagonal magnitude to \(e^{-\mu_0 t}\).

Loewner-kernel pushforward

In modular-generator coordinates, perturbations are described by \(\delta K\). The corresponding first-order change in the density matrix is given by the Loewner (Kubo-Mori) divided-difference map:

\[(\delta\rho)_{ij} = c(\lambda_i, \lambda_j)\,(\delta K)_{ij}, \qquad c(\lambda_i, \lambda_j) = \frac{\lambda_i - \lambda_j}{\log\lambda_i - \log\lambda_j},\]

where \(\lambda_i\) are the eigenvalues of \(\rho_0\) and the off-diagonal formula is extended by L’Hôpital’s rule to \(c(\lambda, \lambda) = \lambda\) on the diagonal.

Iso-marginal perturbations are those for which the first-order change in every subsystem marginal is zero:

\[\operatorname{tr}_{\neq k}\!\bigl(J_{\rho_0}(\delta K)\bigr) = 0 \quad \forall\, k.\]

The GibbsLockedFrame class exposes both the kernel and the iso-marginal test:

from qig import GibbsLockedFrame

frame = GibbsLockedFrame(H, beta=2.0, dims=[3, 3])

# Loewner kernel at rho_0
C, vals, vecs = frame.loewner_kernel()

# Map a modular perturbation to density-matrix coordinates
delta_rho = frame.loewner_map(delta_K)

# Check iso-marginal condition
print(frame.is_iso_marginal(delta_K))   # True / False

Mathematical Background

GENERIC Decomposition

The Jacobian of the constrained flow \(\dot{\theta} = F(\theta)\) splits as

\[M(\theta) = S(\theta) + A(\theta),\]

where \(S = \tfrac{1}{2}(M + M^T)\) is the symmetric (irreversible, dissipative) part and \(A = \tfrac{1}{2}(M - M^T)\) is the antisymmetric (reversible, entropy-conserving) part.

The antisymmetric part generates circulation on the constraint manifold and, in finite-dimensional examples, must correspond to a Hamiltonian flow.

Lie-Algebraic Structure

Let \(\{F_a\}\) be a Hermitian operator basis that is Lie-closed:

\[[F_a, F_b] = 2i \sum_c f_{abc} F_c,\]

where \(f_{abc}\) are the real structure constants of the corresponding Lie algebra (e.g. \(\mathfrak{su}(d)\) for qutrits, \(\mathfrak{su}(4)\) for qubit pairs).

An antisymmetric flow \(A\) acting on natural parameters \(\theta\) induces a density-matrix vector field. For this to equal the commutator flow of some Hamiltonian \(H_\text{eff} = \sum_c \eta_c F_c\), the coefficients \(\eta_c\) must satisfy the linear system

\[(A \theta)_r = \sum_{b,c} f_{rbc}\, \theta_b\, \eta_c(\theta),\]

which can be written in matrix form as

\[F\,\eta(\theta) = A\,\theta,\]

where \(F_{rc} = \sum_b f_{rbc}\, \theta_b\) is the adjoint-action matrix built from the structure constants and the current parameter point.

Note

BCH approximation — known limitation. The formula above is derived under the assumption that the Kubo-Mori derivative \(\partial\rho/\partial\theta_a\) reduces analytically to the commutator \([F_a, \rho]\) via Baker-Campbell-Hausdorff identities (the strong BCH identity, see below). This identity does not hold in general; the full Kubo-Mori kernel introduces corrections of \(O(\|\theta\|^2)\). Consequently, the vector \(A\theta\) has a component outside the column space of \(F\) of magnitude \(\sim 0.1\,\|\theta\|^2\), so the system \(F\eta = A\theta\) is inconsistent: no choice of \(\eta\) achieves a residual smaller than this BCH floor. For \(\|\theta\|\sim 0.05\) the achievable residual is \(\sim 10^{-3}\), not \(10^{-6}\). The matrix \(F\) is also typically rank-deficient (rank 12 of 15 for the qubit-pair system), reflecting the null directions of the adjoint action at finite \(\theta\).

Extraction Algorithm

Given a point \(\theta^*\) on the constraint manifold:

  1. Compute the Jacobian \(M(\theta^*)\) and extract \(A = \tfrac{1}{2}(M - M^T)\).

  2. Solve the linear system \(F\,\eta = A\,\theta\) for the coefficient vector \(\eta \in \mathbb{R}^{\dim\mathfrak{g}}\), where \(F_{rc} = \sum_b f_{rbc}\,\theta_b\). Because \(F\) is rank-deficient and the system is inconsistent (BCH residual \(\sim 0.1\|\theta\|^2\)), the solver uses Tikhonov regularisation.

  3. Build the effective Hamiltonian:

    \[H_\text{eff}(\theta) = \sum_c \eta_c(\theta)\, F_c.\]

  4. Verify that \(H_\text{eff}\) is Hermitian and traceless (as it should be for an element of \(\mathfrak{su}(d^2)\)).

Implementation

The extraction is available in qig.generic_decomposition:

from qig.generic_decomposition import (
    effective_hamiltonian_coefficients,
    effective_hamiltonian_operator,
    verify_antisymmetric_flow_equals_commutator,
)
from qig.structure_constants import structure_constants
from qig.core import generic_decomposition

# At a parameter point theta_star:
M = exp_fam.jacobian(theta_star)
S, A = generic_decomposition(M)

# Extract structure constants for the chosen basis
f_abc = structure_constants(operators)           # shape (n_ops, n_ops, n_ops)

# Solve F @ eta = A @ theta for eta  (F[r,c] = sum_b f[r,b,c] * theta[b])
eta = effective_hamiltonian_coefficients(A, theta_star, f_abc)

# Build H_eff = sum_c eta_c F_c
H_eff = effective_hamiltonian_operator(eta, operators)

print(f"H_eff Hermitian: {np.allclose(H_eff, H_eff.conj().T)}")
print(f"H_eff traceless: {abs(np.trace(H_eff)) < 1e-10}")

A complete worked example is in examples/effective_hamiltonian_derivation.ipynb and the qutrit-pair case is demonstrated in examples/origin_two_qutrit_worked_example.ipynb.

Validation and Limitations

Structural properties that hold exactly:

  • \(H_\text{eff}\) is Hermitian (verified to machine precision).

  • \(H_\text{eff}\) is traceless (by construction from \(\mathfrak{su}\) basis).

  • The extraction formula \(F\,\eta \approx A\,\theta\) is \(O(\|\theta\|^2)\) accurate; the residual floor is \(\sim 0.1\,\|\theta\|^2\) (e.g. \(\sim 10^{-3}\) for \(\|\theta\| \sim 0.05\)), not \(10^{-6}\). This arises because \(A\,\theta\) lies partly outside \(\operatorname{col}(F)\); the exact magnitude of the BCH correction is verified in tests/test_generic_hamiltonian.py::test_extraction_consistency_multiple_points.

A subtlety about the Kubo-Mori kernel

A natural (but incorrect) expectation is that the induced density-matrix flow should satisfy the strong BCH identity

\[\sum_a \eta_a \frac{\partial\rho}{\partial\theta_a} \stackrel{?}{=} -i[H_\text{eff}, \rho].\]

Numerical investigation shows this is false — with \(\sim 14\times\) relative error for both numerical and analytical (spectral) Duhamel methods. The left-hand side includes the Kubo-Mori kernel \(K_\rho = f(\operatorname{ad}_H)\) where \(f(z) = (e^z - 1)/z\), while the right-hand side is a pure commutator. The two differ by a factor of \(\sim 7\)–\(8\times\) due to the operator-ordered integral structure of \(\partial\rho/\partial\theta_a\).

The correct statement is weaker: the antisymmetric parameter-space flow corresponds to a Hamiltonian in the sense that \(H_\text{eff}\) generates the correct symplectic structure on the manifold, but the naive identification of Kubo-Mori derivatives with commutator brackets fails because the Kubo-Mori inner product and the standard Hilbert-Schmidt inner product differ.

This is a valuable theoretical clarification, not a limitation of the implementation. Tests in tests/test_generic_hamiltonian.py guard against reintroduction of the over-strong assumption.

See Also

References

  • Paper: The Inaccessible Game — sections on GENERIC decomposition, categorical forcing of unitarity, and Hamiltonian reconstruction from \(A\) and \(f_{abc}\).

  • Paper: Gibbs-Lock and the Emergence of Hamiltonian Structure in the Inaccessible Game — Gibbs-locked frames, Bohr gaps, iso-marginal perturbations, and the uniform decay rate \(\mu_0\).

  • CIP-0009: Explicit Hamiltonian Extraction from Antisymmetric GENERIC Flow.

  • CIP-000C: Gibbs-Lock API and \(\mu_0\) Inference.

  • CIP-000D: Hamiltonian Paper End-to-End Companion notebook.

  • Notebook: examples/effective_hamiltonian_derivation.ipynb — symbolic derivation and numerical validation.