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The paper under discussion is T. Pernambuco and L. C. Céleri, Geometry of restricted information: the case of quantum thermodynamics, arXiv:2602.06716v1 (6 Feb 2026), CC BY 4.0. Its Appendix D introduces the Bures angle $\mathcal{L}(\rho_\tau^E,\sigma_\tau)$ as a geometric tightening of a Clausius inequality, observes that it measures a distance between two sections of the associated bundle of Ref. [7], and then stops:
That is the only sentence in the paper flagged as unresolved. §2–§5 answer it. The answer is not difficult; it is the standard bundle construction, and the only thing that has to be checked is an invariance. What is worth the page is which invariance, what the resulting length actually measures, and where it breaks.
Fix the setting of the paper. $\xi=(\mathbb{R}\times \mathrm{U}(d),\pi,\mathrm{U}(d), \mathbb{R})$ is the trivial principal bundle over time; $A_t=\dot u_t u_t^\dagger$ is the connection; $E=\xi\times_{\mathrm{Ad}}\mathrm{Herm}(d)$ is the associated bundle in which $\rho_t$ is a section; $\nabla_t(\cdot)=\partial_t(\cdot)+[A_t,\cdot]$.
Let $\mathcal{D}_+\subset\mathrm{Herm}(d)$ be the full-rank density operators and $E_+=\mathbb{R}\times\mathcal{D}_+$ the corresponding sub-bundle. Let $g^B$ be the Bures (quantum Fisher) metric on $\mathcal{D}_+$ and fix any $c>0$. Then
$$ G \;=\; c^2\,\pi^*(dt\otimes dt)\;\oplus\;g^B\!\left(P_V\,\cdot\,,P_V\,\cdot\,\right) $$is a smooth Riemannian metric on the total space $E_+$, where $P_V$ is the vertical projection determined by $A$. It is invariant under the gauge action of $\mathrm{U}(d)$, hence in particular under any thermodynamic group $\mathcal{G}_T(t)\subset\mathrm{U}(d)$.
The proof is four observations, none of them new individually.
Item (4) requires the inhomogeneous rule $A\mapsto VAV^\dagger-\dot VV^\dagger$. With the opposite sign, $\nabla_t\rho$ acquires a stray $2[\dot VV^\dagger,\rho]$ and is not covariant. The cited paper does not print a transformation rule for $A_t$; this is the one its $\nabla_t$ requires.
The construction of §2 is available for any invariant fibre metric, so on its own it does not answer the "physically meaningful" half of the question. What answers that half is item (3). Define the gauge-covariant thermodynamic length of a process
$$ \mathcal{L}_{\rm cov}[\rho] \;=\; \int_0^\tau \!\! dt\; \sqrt{\,g^B_{\rho_t}\!\left(\nabla_t\rho_t,\,\nabla_t\rho_t\right)\,}. $$$\mathcal{L}_{\rm cov}$ is invariant under $\mathcal{G}_T$, and $\mathcal{L}_{\rm cov}[\rho]=0$ if and only if the section is horizontal, i.e. $\nabla_t\rho_t=0$ — which is, in the paper's own reading, parallel transport, hence a path of zero invariant heat, hence (for a closed system) zero coherent heat, hence adiabatic.
So the vertical direction of the bundle metric is not an arbitrary decoration: it is the coherent-heat direction. The metric assigns length zero to exactly the processes the paper calls adiabatic and positive length to exactly the processes in which information leaks into the degrees of freedom the agent cannot resolve. That is the physical content the open question asks for, and it comes from the connection, not from the choice of $g^B$.
Two consequences worth recording, because they make the abstract quantities concrete.
From $W_{\rm inv}=\int \mathrm{Tr}(\rho_t^E\dot H_t)\,dt$ and $\mathrm{Tr}(\Pi_{n^k_t}\dot H_t)=n_t^k\dot\epsilon^k_t$, invariant work collapses to $W_{\rm inv}=\int\sum_k p^k_t\,\dot\epsilon^k_t\,dt$: population-weighted level shift, the textbook classical expression. Hence $W_{\rm inv}\equiv 0$ whenever the spectrum is constant, however violently the eigenbasis rotates, and in that case all the power is coherent heat. MEASURED for a four-level $H_t$ with fixed spectrum and rotating eigenbasis the verifier finds $\dot W_{\rm inv}=-1.2\times10^{-11}$ against $\dot W_u=\dot Q_c=0.102973386$ (Part 5).
Eq. (2) (via $\nabla_t$) and Eq. (13) (via $\dot\rho^E$) are not obviously the same object. For a closed system they are: $\mathrm{Tr}(\rho[H_t,A_t])=\mathrm{Tr}(H_t\,\partial_t\rho^E_t)$, confirmed to $10^{-9}$ (Part 5). The internal consistency the paper claims for its heat definitions is real. Eq. (3), which is supposed to be the same quantity again, is not — see defect D3.
Theorem 1 is stated on $E_+$, over full-rank states, and the restriction is not cosmetic. There are exactly two obstructions, and the paper has already met both under other names.
In an eigenbasis of $\rho$ the Bures metric is $g^B=\tfrac12\sum_{jk}|\langle j|d\rho|k\rangle|^2/(\lambda_j+\lambda_k)$. The denominators vanish as the state loses rank, so $g^B$ is a smooth Riemannian metric on $\mathcal{D}_+$ and degenerates on every boundary stratum. $\mathcal{D}$ is a stratified space, not a manifold, and no invariant fibre metric fixes this — it is a property of the fibre.
Now take the paper's own third-law limit: $\rho_{\beta\to\infty}=\Pi_{n^0_t}/n^0_t$, of rank $n^0_t\lt d$. That state sits on the boundary. The "singular zero-temperature limit" of their §"The third law" is, in this language, exactly the singularity of the Bures metric at the rank-deficient stratum. The unattainability statement they give geometrically — that the singular region cannot be reached by finite physically allowed transport — is then the metric-space statement that the Riemannian structure does not extend there, rather than an extra postulate.
$\mathcal{G}_T(t)\simeq \mathrm{U}(n^1_t)\times\cdots\times\mathrm{U}(n^{p_t}_t)$ depends on $t$ through the degeneracies. The gauge-invariant states form the fixed-point set
$$ \mathrm{Fix}\,\mathcal{G}_T(t)\;=\;\Big\{\textstyle\bigoplus_k (p_k/n^k_t)\, \mathbb{I}_{n^k_t}\Big\}\;\cong\;\Delta^{p_t-1}, $$whose dimension is $p_t-1$, the number of distinct eigenvalues minus one. At a level crossing $p_t$ jumps and the fibre changes dimension. A family of fibres of varying dimension over $\mathbb{R}$ is not a fibre bundle. This is not a pathology invented here: it is the paper's Curie–Weiss run, where the degeneracy structure changes at $B_5=0$ and the cost $TS_\Gamma$ appears. Their remark that $TS_\Gamma$ is "an energy cost related to changing the topology of the thermodynamic group" is, precisely, the dimension jump of the reduced fibre.
Yes for the associated bundle with structure group $\mathrm{U}(d)$, restricted to full-rank states: the extension exists, is canonical given a base scale, and its vertical part measures coherent heat (§2–§3). No for the gauge-reduced family, which is not a bundle at all once degeneracies move (§4b), and no at the boundary strata, where the fibre metric itself degenerates (§4a). Both failures coincide with phenomena the paper already identifies as singular, which is the sense in which the answer is physical rather than merely available.
The paper's tightened inequality carries the term $\tfrac{8T}{\pi^2}\mathcal{L}^2(\rho^E_\tau,\sigma_\tau)$ and leaves it abstract. On the reduced space it is a one-line formula.
$\mathcal{G}_T$ acts on $(\mathcal{D}_+,g^B)$ by isometries (fidelity invariance), so by Cartan's theorem the connected components of its fixed-point set are totally geodesic submanifolds. Hence the Bures geodesic between two gauge-invariant states stays gauge-invariant, and $\mathcal{L}(\rho^E_\tau,\sigma_\tau)$ is the same number whether measured in the ambient state space or intrinsically in the reduced space. MEASURED the geodesic between two gauge-invariant states of a $(3,2,1)$-degenerate six-level system deviates from the fixed-point set by $2.5\times10^{-16}$, and its length $0.475825415$ matches the angle $0.475825408$ (Part 2).
On $\mathrm{Fix}\,\mathcal{G}_T$ the Bures metric restricts to the Fisher–Rao metric $\tfrac14\sum_k dp_k^2/p_k$ of the simplex of block weights, and the Bures angle is the Bhattacharyya angle:
$$ \mathcal{L}(\rho^E_\tau,\sigma_\tau)\;=\;\arccos\Big(\textstyle\sum_l\sqrt{p^l_\tau q^l_\tau}\Big), \qquad q^l_\tau=n^l_\tau e^{-\beta\epsilon^l_\tau}/Z_\tau . $$Both states are block-diagonal and maximally mixed within blocks, so they commute and the root fidelity is the classical Bhattacharyya coefficient. Verified to $10^{-12}$ (Part 2).
A second simplification comes free. The paper's Eq. (30) splits $S_{\mathcal{G}_T}$ into a von Neumann term, a relative entropy of coherence and a Holevo asymmetry. Those last two are one object:
$$ C_{\rm rel}[\rho]+S_\Gamma[f] \;=\; S_{\mathcal{G}_T}[\rho]-S_{\rm vN}[\rho] \;=\; S\!\left(\rho\,\|\,\rho^E\right) \;=\; \min_{\tau\in\mathrm{Fix}\,\mathcal{G}_T} S(\rho\|\tau), $$because $\log\tau$ lies in the commutant for gauge-invariant $\tau$, giving the Pythagorean identity $S(\rho\|\tau)=S(\rho\|\rho^E)+S(\rho^E\|\tau)$. So $\rho^E$ is the relative-entropy projection onto the reduced space and the coherence-plus-asymmetry cost is simply the divergence from the state to that space. Putting this into Eq. (51):
The first correction is a distance off the gauge-reduced space; the second is a distance along it, to equilibrium. Every term is now computable from the block weights and the spectrum alone.
The two corrections are measured by different projections. $\rho^E$ minimises relative entropy to $\mathrm{Fix}\,\mathcal{G}_T$, but it is not the Bures-nearest gauge-invariant state. MEASURED for a random full-rank six-level state the Bures distance to the nearest point of $\mathrm{Fix}\,\mathcal{G}_T$ is $0.502044$, attained at block weights $(0.6426,0.3149,0.0425)$, while the distance to the twirl $\rho^E$, at weights $(0.6033,0.3390,0.0577)$, is $0.503652$ (Part 2). The two terms of the boxed inequality are therefore additive as printed but do not form legs of a right triangle in a single metric. Any attempt to read them as an orthogonal decomposition is a vocabulary correspondence, not a theorem.
Found while checking the above, not sought. Each is reproduced by the verifier.
| # | Where | What is wrong | Counterexample |
|---|---|---|---|
| D1 | Eq. (8), main text | $\langle\sigma_{\rm inv}\rangle$ is identified with $S_{\mathcal{G}_T}(\rho_R)-S_{\mathcal{G}_T}(\rho_F)$. That is a cross-entropy misread: the forward average runs over the actual final distribution $p^l_\tau=\sum_k p_F(k,l)$, not over $p_R$. | $\langle\sigma_{\rm inv}\rangle=2.4887$ while the printed middle term is $-0.3636$ — negative, so as printed the chain asserts a negative quantity is $\ge 0$. |
| D2 | Eq. (49) | $\mathcal{L}=\arccos(F)$ with $F=[\mathrm{Tr}\sqrt{\sqrt{\rho_1}\rho_2\sqrt{\rho_1}}]^2$. The Deffner–Lutz Bures angle is $\arccos$ of the root of that quantity. Since $F\le\sqrt F$, the printed $\mathcal{L}$ is too large and the bound (50) becomes false. | 197 violations in 4000 random gauge-invariant pairs; 0 with $\arccos\sqrt F$. |
| D3 | Eq. (3) | With the stated convention $h_t=u_tH_tu_t^\dagger$, the printed expression $\mathrm{Tr}(\rho\,\dot u_th_tu_t^\dagger+\rho\,u_th_t\dot u_t^\dagger)$ is not the $Q_c$ it defines. Exchanging $u$ and $u^\dagger$ repairs it. | printed $=0.639547$; repaired $=0.102973$; the $Q_c$ rate from Eqs. (2)/(13) $=0.102973$. |
D1 is the consequential one. Eq. (8) is the paper's statement of the second law, and it is also the equation the Conclusions lean on ("the second law with the positivity of the relative entropy between forward and backward path measures"). The paper's own Appendix C already contains the correct statement at Eq. (44), $\langle\sigma_{\rm inv}\rangle=\Delta S_{\mathcal{G}_T}+S(\rho^E_\tau\|\sigma_\tau)$, which the verifier confirms to $10^{-12}$. So the repair is local: Eq. (8) should carry Eq. (44)'s right-hand side, or none at all. The inequality $\langle\sigma_{\rm inv}\rangle\ge0$ survives untouched.
D2 does not threaten the paper's conclusions either — Eq. (50) is a quotation from Ref. [31], where the convention is the Bures angle — but as printed §Appendix D states a false inequality and Fig. 1 and Fig. 2 plot a bound computed from some convention that the text does not pin down. Which one was plotted cannot be determined from the preprint.
The paper calls Eq. (6) a "detailed fluctuation relation", but Eq. (6) is the definition of $\sigma_{\rm inv}$ written out. The statement that carries the weight — the one that makes $e^{-\sigma_{\rm inv}}$ a Radon–Nikodym derivative, that yields Eq. (7), and that makes Crooks "a direct consequence" — is never written. It is true, and it has a two-line proof.
With $\rho_k=\Pi_{n^k_0}/n^k_0$, write $T_{lk}=\mathrm{Tr}(\Pi_{n^l_\tau}U_\tau \Pi_{n^k_0}U^\dagger_\tau)$. Then
$$ n^k_0\,p(l|k)\;=\;T_{lk}\;=\;n^l_\tau\,p(k|l) \qquad\text{(gauge-invariant microreversibility)}, $$and therefore, for any gauge-invariant $\rho_F,\rho_R$,
$$ \frac{p_F(k,l)}{p_R(l,k)}\;=\;\frac{p^k_F}{p^l_R}\cdot\frac{p(l|k)}{p(k|l)} \;=\;\frac{p^k_F}{p^l_R}\cdot\frac{n^l_\tau}{n^k_0}\;=\;e^{\,\sigma_{\rm inv}} . $$Eq. (7) follows by summing $p_R(l,k)$ over both indices, and $\langle\sigma_{\rm inv}\rangle=D_{\rm KL}\!\left(p_F\|p_R\right)\ge0$ is then immediate — which is the geometric reading the paper asserts but does not derive. MEASURED for a six-level system whose degeneracies change from $(3,2,1)$ to $(2,1,2,1)$ under a random unitary: microreversibility holds to $10^{-16}$, $\langle e^{-\sigma_{\rm inv}}\rangle=1.000000000000000$, $\sigma_{\rm inv}=\ln[p_F/p_R]$ to $4.4\times10^{-16}$, and $D_{\rm KL}=2.4887=\langle\sigma_{\rm inv}\rangle$ (Part 1).
Note where the degeneracy factors enter: $p(l|k)$ is computed from the twirled conditional state, which is what makes the ratio of conditionals equal to a ratio of degeneracies. An agent who could resolve within a degenerate eigenspace would not get this identity. The degeneracy term in Eq. (6) is thus forced by the gauge postulate rather than added to it — that much of the paper's reading is correct, and Proposition 4 is what establishes it.
§2 fails if the Bures metric is not $\mathrm{U}(d)$-invariant — one line to check. §5 fails if the Bures angle between two block-scalar states is not the Bhattacharyya angle of their block weights — one line to check. §6 fails if the verifier's transition matrix is mis-constructed; the guard is the microreversibility identity in Part 1, which would not hold for a wrong $T_{lk}$. §4 is the part with no numerical falsifier: it is a statement about dimension and about where a denominator vanishes.