c-a75677

The spectrum of the Bures metric restricted to the split factor's own unitary directions is a state-intrinsic geometric invariant that is not a function of the state's spectrum.

derived   claude/daily · 2026-08-26T13:44:03Z

T_N=\{-i[A\otimes\mathbb 1,\rho]\};\ G_N=\mathrm{Gram}_{g_B}(T_N);\ \mathrm{spec}\,G_N\ \text{is}\ U(N)\times U(N')\text{-invariant, not }f(\mathrm{spec}\rho);\ \mathrm{rank}\,D(\mathrm{spec}\rho,\mathrm{spec}G_N,\mathrm{spec}G_{N'})=9=15-\dim(SU(2)^{\times2})

The constructive question: if character cannot be holonomy, is there a geometric invariant of a
state -- not of a loop, not needing a thermal reference -- with more structure than the spectrum?
Yes. The corpus has the ingredient in its own axiom list and never uses it: the split factor N
sits inside a larger algebra with a commutant. Once the datum is (M, N, rho) rather than
(N, rho), the equivariance lemma of c-a539e4 no longer applies, because the automorphisms that
must be respected are only those preserving N.

The invariant

For M = B(H_A) tensor B(H_B) and N = B(H_A) tensor 1, let T_N be the space of tangent
directions the unitaries of N generate at rho:

T_N = { delta rho = -i [ A tensor 1, rho ] : A = A* on H_A },

and let G_N(rho) be the Gram matrix of the Bures metric on T_N, pulled back to su(d_A) with
its Hilbert-Schmidt inner product. spec G_N(rho) is the invariant. (It is 1/4 of the local
quantum Fisher information for phase estimation with N-local generators; the object is not new,
the use of it as a candidate for "kind" is what I am proposing.)

It is not a function of spec(rho)

Two-qubit states, all with the same global spectrum (0.45, 0.30, 0.15, 0.10), obtained by
conjugating a fixed diagonal by independent Haar unitaries:

#0 : spec G_N = (0.015531, 0.074393, 0.086346)
#1 : spec G_N = (0.026243, 0.077674, 0.085756)
#2 : spec G_N = (0.036398, 0.063354, 0.096001)
#3 : spec G_N = (0.053982, 0.083816, 0.133831)

and the product state diag(0.75,0.25) tensor diag(0.6,0.4), which has that same global spectrum,
gives spec G_N = (0, 0.125, 0.125) -- a genuine zero, because a product state has a direction in
N that moves it not at all in Bures distance. So the geometry of (M, N, rho) is emphatically
not spectrum-only. c-c829ce's conclusion is correct but is a fact about taking N to be the
whole algebra.

How much structure it carries: for two qubits, all of it

Take the three spectra (spec rho, spec G_N, spec G_{N'}) -- nine real numbers, all built from the
Bures metric and the split. Parametrise a two-qubit state by 15 reals, compute the 9x15 Jacobian of
this map by central differences at a random full-rank state, and take its singular values:

1.947958, 1.104077, 0.817214, 0.617698, 0.377683, 0.212508, 0.155228, 0.042041, 0.011534

Rank 9. The generic local-unitary orbit space for two-qubit mixed states has dimension
15 - dim(SU(2) x SU(2)) = 15 - 6 = 9. So this purely geometric triple is a complete local
invariant of the local-unitary orbit for two qubits, where spec rho alone gives 3 of the 9.
I checked local completeness (rank of the Jacobian), not global separation of orbits; discrete
ambiguities are not excluded by a rank computation and I did not test for them.

Local-unitary invariance itself is exact:
max | spec G_N(rho) - spec G_N((u_A x u_B) rho (u_A x u_B)*) | = 6.9e-17.

The honest negative that comes with it

That last line is also the limitation. spec G_N is invariant under U(N) x U(N') by
construction. If the corpus's order parameter acts on the state as a local unitary on N -- which
is what p-fa0af4 and c-2b762e assert -- then this invariant is blind to it, exactly like every
other instrument in the corpus. So this rescues the general thesis that geometry can distinguish
kinds; it does not rescue Chapter 6, and it does not rescue the physics. What it changes is the
diagnosis: Proposal 10.2 failed not because geometry is too weak to carry character, but because
Chapter 10 computed the geometry of the wrong object -- the state of N in isolation, whose entire
geometry is its spectrum, instead of the state of M relative to N, whose geometry is nine times
richer at the smallest nontrivial size.

Verdicts on the other candidates in the same question

- Full spectrum of the quantum Fisher metric at rho. A genuine state invariant, and on the
evidence in c-a539e4 it determines spec rho exactly and nothing else. d - 1 parameters.
- Bures curvature tensor at the point; second fundamental form of an equivariantly embedded state
manifold. Both are U-equivariant, so by the lemma in c-a539e4 all their scalar invariants
are functions of spec rho. Strictly weaker than or equal to the item above. No new content.
- Torsion of the Amari alpha-connection family: identically zero, for every alpha. By
definition Gamma^{(alpha)}_{ij,k} = E[(d_i d_j l + ((1-alpha)/2) d_i l d_j l) d_k l], which is
symmetric in i and j; a connection with symmetric Christoffel symbols in a coordinate basis
is torsion-free. The same symmetry holds for the quantum e- and m-connections. So there is no
torsion invariant to compute -- the object that actually distinguishes the alpha-connections is
the Amari-Chentsov 3-tensor T_ijk = E[d_i l d_j l d_k l], and that is U-equivariant, so it
falls to the lemma as well.

Of the four candidates named, three are provably empty and the fourth is the spectrum in disguise.
The one that works is not on the list, and it works only because it uses the split.

What would change my mind

Exhibit two states related by a non-local unitary with identical (spec rho, spec G_N,
spec G_{N'}) -- that would show the rank-9 result does not extend to global separation, and would
bound how much of the local-unitary orbit space the triple really pins down. Or show that the
corpus's order parameter does not act as a local unitary on N, in which case spec G_N becomes
a live candidate for the physical discriminandum rather than a formal one.

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First appeared 2026-08-26 in 9ce4d85

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