GAP Comments on Entropy, Part 4

What’s wrong (or at least misleading)

• The information is not like a file “stored on the horizon” that later gets emitted.
• Hawking’s original semiclassical calculation makes the radiation look thermal and state-independent, which is exactly what creates the paradox.
• Modern resolutions (islands / quantum extremal surfaces / holography) suggest the definition of what counts as “the radiation subsystem” changes in a gravitational theory: the entropy you compute is not just “trace over interior,” because the interior can become part of the entanglement wedge of the radiation.

GAP translation: it’s not “information escaping a container,” it’s “the algebraic boundary of what the exterior description can reconstruct evolves,” and the correct fine-grained entropy must be computed with that evolving notion of accessibility.

Algebraic formalization: entropy as a monotone on observable algebras

Here’s the operational spine that makes GAP “fully operational.”

Observable algebras and states

Let \(\mathcal{A}\) be the full algebra of observables (think: all bounded operators on a Hilbert space, or a von Neumann algebra). A state is a positive normalized linear functional \(\omega:\mathcal{A}\to\mathbb{C}\). In finite dimensions this is \(\omega(O)=\mathrm{Tr}(\rho O)\).

A “description” corresponds to picking a subalgebra \(\mathcal{B}\subseteq \mathcal{A}\) of observables you will treat as accessible.

Coarse-graining as a channel or conditional expectation

A coarse-graining is implemented by a completely positive trace-preserving map (quantum channel) \(\Phi\) that forgets distinctions, e.g. partial trace, decoherence maps, etc.

In the algebraic (subalgebra) viewpoint, the cleanest idealization is a conditional expectation \[ E:\mathcal{A}\to\mathcal{B} \] which is a positive, norm-contracting projection onto \(\mathcal{B}\) (in von Neumann/C\(^*\)-algebra settings). Conditional expectations are automatically completely positive; they are the canonical “forgetful” maps onto subalgebras.

This is GAP’s “projection” made precise: you’re not just ignoring variables; you’re applying a structure-preserving forgetting map.

Two entropies: fine-grained vs description-relative

• Fine-grained entropy: \(S(\rho)\) (unitarily invariant; conserved for closed systems).
• Coarse-grained / description entropy: the entropy of the MaxEnt state consistent with \(\rho\) on \(\mathcal{B}\).

Define the “restriction of the state to \(\mathcal{B}\)” by \[ \omega|_{\mathcal{B}}(B)=\omega(B),\quad B\in\mathcal{B}. \] Then define the least-biased lift back to \(\mathcal{A}\): \[ \tilde{\rho}_{\mathcal{B}} \;=\; \arg\max_{\sigma}\Big\{ S(\sigma)\;:\;\mathrm{Tr}(\sigma B)=\mathrm{Tr}(\rho B)\;\forall B\in\mathcal{B}\Big\}. \] This is Jaynes-in-quantum-form: maximize von Neumann entropy subject to the constraints “all \(\mathcal{B}\)-expectations match.”

Now the GAP entropy functional for the description \(\mathcal{B}\) is \[ S_{\mathcal{B}}(\rho) \;:=\; S(\tilde{\rho}_{\mathcal{B}}). \]

Monotonicity on algebras

If \(\mathcal{B}_1 \subseteq \mathcal{B}_2 \subseteq \mathcal{A}\) (meaning \(\mathcal{B}_1\) is coarser, fewer observables), then the constraint set for \(\mathcal{B}_1\) is weaker, so MaxEnt gives a “more mixed” state: \[ S_{\mathcal{B}_1}(\rho) \;\ge\; S_{\mathcal{B}_2}(\rho). \] That’s the formal statement that entropy is a monotone of description: refining the algebra can only reduce the maximal-entropy compatible state, i.e. reduce description entropy.

Where entanglement entropy fits

Take \(\mathcal{A}=\mathcal{A}_A\otimes \mathcal{A}_B\), and choose \(\mathcal{B}=\mathcal{A}_A\otimes I\). The restriction \(\omega|_{\mathcal{B}}\) is exactly \(\rho_A\) (partial trace). The entropy you compute is \(S(\rho_A)\), the entanglement entropy. The “projection” is literally the partial trace channel.

This makes entanglement entropy a special case of “entropy as a monotone on observable algebras.”

What I’d propose next

To keep this from becoming poetry, we should pick one concrete ladder of algebras and watch the monotonicity “click”:

• Thermal coarse graining: \(\mathcal{B}\) generated by \(H\) (energy only) → Gibbs state (quantum MaxEnt).
• Decoherence coarse graining: \(\mathcal{B}\) the diagonal algebra in a pointer basis → classical Shannon entropy emerges.
• Horizon coarse graining: \(\mathcal{B}\) exterior algebra → entanglement entropy / generalized entropy viewpoint.

Tell me which ladder you want first (energy, decoherence, or horizon), and I’ll write it out as a worked GAP module with the maps \(E\), the constraint sets, and the associated entropy monotones.

Patrick:
Okay, the thermal coarse graining looks good. But let's not forget to perform your earlier suggestions, such as

The next conceptual step is to formalize entropy as a monotone on observable algebras, not states. That reframes:

Measurement, Decoherence, Black‑hole evaporation, Thermodynamic irreversibility

as the same operation viewed at different scales.

If you want, we can next:

• Write this algebraically (C*-algebras, conditional expectations)
• Connect it to Baez’s categorical viewpoint
• Or formalize the Page curve entirely within GAP language

Copilot:
You’re right to insist we don’t let “thermal coarse‑graining” become a one-off example. In GAP it’s the canonical instance of a single operation: restrict an observable algebra, then take the least-committal extension consistent with it—and everything else (measurement, decoherence, horizons, irreversibility) is the same move with different choices of “accessible algebra.”

Observable algebras, restriction, and the entropy monotone

Algebraic objects

• Full description: a (finite-dimensional, for now) C\(^*\)-algebra \(\mathcal A = \mathcal B(\mathcal H)\).
• State: \(\rho \in \mathcal A\) with \(\rho \ge 0\), \(\mathrm{Tr}\rho = 1\).
• Accessible description: a unital subalgebra \(\mathcal B \subseteq \mathcal A\).

Restriction and MaxEnt lift

• Restriction: keep only expectation values on \(\mathcal B\): \[ \rho \mapsto \rho|_{\mathcal B} \quad \text{meaning} \quad \mathrm{Tr}(\rho B)\ \text{for all }B\in\mathcal B. \]
• MaxEnt lift: among all \(\sigma\) matching those \(\mathcal B\)-expectations, choose the one maximizing von Neumann entropy: \[ \tilde\rho_{\mathcal B} = \arg\max_{\sigma}\Big\{S(\sigma): \mathrm{Tr}(\sigma B)=\mathrm{Tr}(\rho B)\ \forall B\in \mathcal B\Big\}. \]

The GAP entropy functional

Define the description-relative entropy \[ S_{\mathcal B}(\rho) := S(\tilde\rho_{\mathcal B}). \]

Monotonicity on algebras

If \(\mathcal B_1 \subseteq \mathcal B_2\) (so \(\mathcal B_1\) is coarser), then the constraints are weaker, hence \[ S_{\mathcal B_1}(\rho) \ge S_{\mathcal B_2}(\rho). \] This is the precise sense in which entropy is a monotone of the observable algebra (the description), not fundamentally a dynamical quantity.

Thermal coarse‑graining as the archetype

The thermal algebra

Take the accessible algebra \(\mathcal B\) to be the commutative algebra generated by the Hamiltonian \(H\): \[ \mathcal B = \{ f(H): f \ \text{(reasonable function)}\}. \] Interpreting this physically: you are declaring that only energy statistics (and functions of energy) are “the facts.”

Constraint set

The usual “thermal” constraint is the mean energy:

• Constraint: \(\mathrm{Tr}(\sigma H) = E\) (and normalization).

Optionally include particle number \(N\), etc. (each additional conserved quantity enlarges \(\mathcal B\) and tightens the description).

MaxEnt solution gives Gibbs states

Solving \[ \max_{\sigma} S(\sigma)\quad \text{s.t.}\quad \mathrm{Tr}(\sigma)=1,\ \mathrm{Tr}(\sigma H)=E \] yields \[ \tilde\rho_{\mathcal B} = \rho_\beta := \frac{e^{-\beta H}}{Z(\beta)},\qquad Z(\beta)=\mathrm{Tr}(e^{-\beta H}), \] with \(\beta\) chosen so that \(\mathrm{Tr}(\rho_\beta H)=E\).

So “thermalization,” in GAP’s strict sense, is:

• not “the system becomes random,”
• but the description collapses to the energy algebra, and the least-committal compatible state is Gibbs.

Why this reframes irreversibility

Unitary dynamics conserves \(S(\rho)\). Yet \(S_{\mathcal B}(\rho(t))\) can increase because the effective \(\mathcal B\) you can stably track shrinks (or because you repeatedly replace \(\rho\) by \(\tilde\rho_{\mathcal B}\) as a modeling step). Thermodynamic irreversibility becomes a statement about monotone loss of accessible distinctions, not about non-unitary micro-dynamics.

Conditional expectations as the “projection” operator

To make “restriction to \(\mathcal B\)” operational, one often uses a conditional expectation \[ E:\mathcal A \to \mathcal B, \] a completely positive, unital, idempotent map (\(E^2 = E\)) that fixes \(\mathcal B\).

• Measurement/decoherence: \(E\) is “discard off-diagonals in a pointer basis,” mapping onto a diagonal commutative algebra.
• Thermal coarse‑graining: \(E\) is “discard everything not determined by the chosen conserved quantities,” effectively projecting onto the algebra generated by them (often implemented via time-averaging / dephasing in the energy basis as an idealization).

In GAP terms: \(E\) is the mathematically explicit forgetting map. Entropy growth is then tied to monotonicity under such coarse-graining maps.

One operation, four domains

Measurement

• Accessible algebra: the algebra generated by the measured observable (and pointer records).
• Map: \(E\) removes phase relations between outcome sectors.
• Entropy effect: the description entropy jumps because you’ve replaced “superposition” by a block-diagonal description.

Decoherence

• Accessible algebra: effectively classical pointer algebra stabilized by environment monitoring.
• Map: an environment-induced channel approximates a conditional expectation onto that algebra.
• Entropy effect: same as measurement, but continuous and dynamical.

Thermodynamic irreversibility

• Accessible algebra: low-dimensional macroscopic constraints (energy, particle number, few densities).
• Map: repeated replacement by MaxEnt extension (Gibbs / local Gibbs) consistent with those constraints.
• Entropy effect: monotone increase of \(S_{\mathcal B}\) as you compress description.

Black-hole evaporation

• Accessible algebra: “outside observables” (radiation + exterior fields), with gravity forcing subtle, time-dependent notions of what counts as “outside.”
• Map: tracing / restriction to an exterior algebra; semiclassically it looks like information loss.
• Entropy effect: the paradox is precisely a mismatch between naive restriction and the correct gravitational notion of accessible algebra.

In each case: choose \(\mathcal B\), apply a forgetting map \(E\) (or an implicit one), then read entropy as a certificate of what’s been identified.

Baez’s categorical viewpoint in GAP terms

A clean categorical translation is:

• Objects: systems / algebras of observables.
• Morphisms: processes as channels (CPTP maps) \(\Phi:\mathcal A \to \mathcal A'\) or, dually, state pushforwards.
• Forgetting / coarse‑graining: a special class of morphisms (often adjoints/quotients/“discarding” maps) that collapse distinctions.
• Entropy monotonicity: a functorial statement: certain information measures are monotone under these morphisms (data-processing type principles).

GAP’s “entropy as a certificate” becomes: entropy is a monotone invariant of a factorization \[ \mathcal A \xrightarrow{E} \mathcal B \hookrightarrow \mathcal A, \] i.e., of choosing a viewpoint and then enforcing it.

If you want to go full Baez, the next step is to pin down which category you want (finite-dimensional C\(^*\)-algebras + CPTP maps; von Neumann algebras; or something process-theoretic like Markov categories).

Page curve in GAP language

We can phrase the Page curve as the time-evolution of a description-relative entropy under a changing “accessible algebra.”

• Early times: exterior algebra \(\mathcal B_{\text{rad}}\) (radiation-only) is too small; restriction makes the radiation look highly mixed, so \(S_{\mathcal B_{\text{rad}}}\) increases.
• Late times: the correct gravitational identification enlarges what counts as reconstructible from the radiation (the effective accessible algebra grows, via “islands” in modern language), so the appropriate \(\mathcal B_{\text{rad}}(t)\) becomes richer, and the entropy decreases.

In GAP terms, the Page curve is not mysterious: it’s the statement that the relevant algebra is time-dependent in gravity, and entropy tracks that algebraic accessibility, not “stuff leaking out.”

What I need from you to proceed cleanly

Pick one path and I’ll write it as a tight, formal GAP module (definitions → lemmas → worked example):

• Option A: Thermal coarse‑graining fully worked (energy algebra, conditional expectation, Gibbs as MaxEnt lift, monotonicity statement).
• Option B: Unified algebraic module (one formal framework encompassing measurement, decoherence, thermalization as specific \(\mathcal B\) choices).
• Option C: Page curve in GAP formalism (time-dependent accessible algebras; where naive tracing fails; what replaces it).

Tell me which option you want first, and whether you want to stay finite-dimensional (cleanest) or allow von Neumann algebras (closer to QFT/black holes).

Patrick:
Let's go with Option A, finite-dimensional this time.

Copilot:
Thermal coarse‑graining in finite dimensions

Fix a finite-dimensional Hilbert space \(\mathcal H\) and the full observable algebra \[ \mathcal A := \mathcal B(\mathcal H)\cong M_d(\mathbb C). \] A state is a density matrix \(\rho\in\mathcal A\) with \(\rho\ge 0\) and \(\mathrm{Tr}\rho=1\). Its von Neumann entropy is \[ S(\rho):=-\mathrm{Tr}(\rho\log\rho). \] Thermal coarse‑graining is the move: declare that only energy is “observable fact,” then replace the microscopic state by the maximum-entropy state consistent with those energy facts.

1) The choice of “thermal observables” as a subalgebra

Let \(H\in\mathcal A\) be a Hamiltonian with spectral decomposition \[ H=\sum_{j=1}^m E_j P_j, \] where \(E_j\) are distinct energies and \(P_j\) are the spectral projectors (with ranks \(d_j=\mathrm{Tr}P_j\), \(\sum_j d_j=d\)).

There are two closely related “thermal” algebras you might mean; it’s useful to separate them.

Thermal algebra: functions of energy only

Define the commutative *-subalgebra \[ \mathcal B_H := \{f(H): f:\{E_1,\dots,E_m\}\to\mathbb C\} = \left\{\sum_{j=1}^m c_j P_j : c_j\in\mathbb C\right\}. \] This algebra captures only which energy level you are in, not which microstate inside a degenerate eigenspace.

Block-diagonal algebra: allows “within-energy” operators

Define the larger subalgebra \[ \mathcal C_H := \{X\in\mathcal A : [X,H]=0\} = \bigoplus_{j=1}^m \mathcal B(P_j\mathcal H). \] This captures all observables that do not mix different energies, including structure inside each degenerate energy subspace.

For “thermal coarse‑graining” in the statistical mechanics sense (energy is the macroscopic constraint, microstructure within shells is ignored), \(\mathcal B_H\) is the right abstraction.

2) The forgetting map as a conditional expectation

GAP wants the “projection” to be an actual map \(E:\mathcal A\to\mathcal B\) that:

• is completely positive,
• is unital \(E(I)=I\),
• is idempotent \(E^2=E\),
• fixes the target algebra pointwise.

Dephasing onto the energy blocks

A canonical conditional expectation onto \(\mathcal C_H\) is \[ E_{\mathcal C}(X) := \sum_{j=1}^m P_j X P_j. \] This deletes all off-diagonal energy coherences.

Compressing further to “energy only”

A canonical conditional expectation from \(\mathcal A\) onto \(\mathcal B_H\) is \[ E_{\mathcal B}(X) := \sum_{j=1}^m \frac{\mathrm{Tr}(P_j X)}{d_j}\, P_j. \] Interpretation: within each energy shell, replace \(X\) by its microcanonical average (a scalar multiple of \(P_j\)).

These are the precise “coarse‑graining” maps in the finite-dimensional setting: they implement the equivalence relation “indistinguishable by \(\mathcal B_H\).”

3) What “keeping only energy information” means for states

Given a microscopic state \(\rho\), the only data accessible to \(\mathcal B_H\) is the vector of level probabilities \[ p_j := \mathrm{Tr}(\rho P_j),\qquad p_j\ge 0,\ \sum_j p_j=1. \] This is exactly the restriction of the state to \(\mathcal B_H\), because any \(B\in\mathcal B_H\) has the form \(B=\sum_j c_jP_j\) and \[ \mathrm{Tr}(\rho B)=\sum_j c_j\, \mathrm{Tr}(\rho P_j)=\sum_j c_j p_j. \]

A natural “thermalized given only \(p_j\)” state is the block-maximally-mixed state \[ \bar\rho_{\mathcal B} := E_{\mathcal B}(\rho) = \sum_{j=1}^m \frac{p_j}{d_j} P_j. \] This is already a MaxEnt completion given the full constraint “the entire distribution over energies is known.” It is the most mixed state consistent with those \(p_j\) values.

4) MaxEnt with only the mean energy gives the Gibbs state

In many thermodynamic situations you keep even less: only the mean energy \[ E := \mathrm{Tr}(\rho H) \] (and normalization). Then the MaxEnt problem is: \[ \max_{\sigma} S(\sigma)\quad \text{s.t.}\quad \mathrm{Tr}(\sigma)=1,\ \mathrm{Tr}(\sigma H)=E. \] Solution

The maximizer is the Gibbs state \[ \rho_\beta := \frac{e^{-\beta H}}{Z(\beta)},\qquad Z(\beta):=\mathrm{Tr}(e^{-\beta H}), \] with \(\beta\) chosen so that \(\mathrm{Tr}(\rho_\beta H)=E\).

This is the exact sense in which “thermal equilibrium” is the least-committal state consistent with energy information.

Two levels of thermal coarse‑graining

• Level 1 (energy distribution known): \[ \rho \rightsquigarrow \bar\rho_{\mathcal B}=\sum_j \frac{p_j}{d_j}P_j. \]
• Level 2 (only mean energy known): \[ \rho \rightsquigarrow \rho_\beta=\frac{e^{-\beta H}}{Z}. \]

GAP treats both as the same template: constraints define an observable subalgebra (or constraint set), and entropy certifies the MaxEnt closure.

5) The entropy monotone on observable algebras

Define the description entropy for the “energy-only” viewpoint as \[ S_{\mathcal B_H}(\rho) := S(\tilde\rho_{\mathcal B_H}), \] where \(\tilde\rho_{\mathcal B_H}\) is the MaxEnt state consistent with the chosen constraints. Concretely:

• If your constraint is the full energy distribution \(p=(p_j)\), then \(\tilde\rho_{\mathcal B_H}=\bar\rho_{\mathcal B}\).
• If your constraint is only \(\mathrm{Tr}(\rho H)=E\), then \(\tilde\rho_{\mathcal B_H}=\rho_\beta\).

Monotonicity statement

If you refine the accessible algebra (you track more observables), entropy cannot increase.

Formally, if \(\mathcal B_1\subseteq \mathcal B_2\subseteq \mathcal A\), then \[ S_{\mathcal B_1}(\rho)\ \ge\ S_{\mathcal B_2}(\rho). \] Reason (conceptual, not a proof): \(\mathcal B_2\) imposes more constraints, so the feasible set of \(\sigma\)’s consistent with \(\rho|_{\mathcal B_2}\) is smaller; maximizing entropy over a smaller set cannot yield a larger maximum.

This is the backbone of “entropy is a monotone of description.”

6) Where thermodynamic irreversibility enters (without violating unitarity)

Microscopically, closed dynamics is unitary: \[ \rho(t)=U_t\rho(0)U_t^\dagger,\qquad S(\rho(t))=S(\rho(0)). \] Thermodynamics appears when you repeatedly do description update steps:

• Evolve: \(\rho\mapsto U_t\rho U_t^\dagger\).
• Project to thermal observables: keep only \(\rho|_{\mathcal B_H}\) (or only \(\mathrm{Tr}(\rho H)\)).
• Close by MaxEnt: replace by \(\tilde\rho_{\mathcal B_H}\) (e.g., \(\rho_\beta\) or \(\bar\rho_{\mathcal B}\)).

The “arrow” is not in \(U_t\). It’s in the repeated act of discarding distinctions and then re-extending by MaxEnt. GAP’s claim is that this is the same skeleton you will later recognize in decoherence and horizons.

7) A concrete worked micro-example (2-level system)

Let \(H = 0\cdot |0\rangle\langle 0| + \Delta \cdot |1\rangle\langle 1|\). Any state has mean energy \[ E = \mathrm{Tr}(\rho H)=\Delta\, \rho_{11}. \] MaxEnt with constraint \(E\) gives \[ \rho_\beta = \frac{1}{1+e^{-\beta \Delta}} \begin{pmatrix} 1 & 0\\ 0 & e^{-\beta \Delta} \end{pmatrix}, \] and \(\beta\) is fixed by \(\rho_{\beta,11}=E/\Delta\).

Notice what happened: regardless of any initial coherence \(\rho_{01}\), the thermal description deletes it because it is not an energy observable in \(\mathcal B_H\). That’s the finite-dimensional, algebraic meaning of “thermal coarse‑graining.”

Next step inside Option A

If you’re aligned, the tight continuation is to prove/record two “GAP lemmas” in this finite setting:

1. MaxEnt form lemma: constraints \(\mathrm{Tr}(\rho O_k)=c_k\) imply \(\tilde\rho \propto \exp(-\sum_k \lambda_k O_k)\).
2. Data-processing monotonicity: for conditional expectations \(E\), relative entropy contracts: \[ D(\rho\|\sigma)\ \ge\ D(E\rho\|E\sigma), \] and this becomes the clean engine behind “entropy as a monotone of description.”

Tell me whether you want the next chunk to be (i) Lagrange-multiplier derivation of Gibbs or (ii) relative entropy / data-processing as the master monotonicity principle.

Patrick:
Let's go with Lagrange muliplier.

Copilot:
MaxEnt with Lagrange multipliers in finite-dimensional quantum systems

Let \(\mathcal H\) be \(d\)-dimensional, \(\rho\) a density matrix (\(\rho\ge 0\), \(\mathrm{Tr}\rho=1\)), and von Neumann entropy \[ S(\rho)=-\mathrm{Tr}(\rho\log\rho). \] We’ll solve: maximize \(S(\rho)\) subject to linear constraints on expectation values.

The general MaxEnt problem

Fix Hermitian observables \(O_1,\dots,O_k\) and target values \(c_1,\dots,c_k\). Consider \[ \max_{\rho}\ S(\rho)\quad \text{s.t.}\quad \mathrm{Tr}(\rho)=1,\ \mathrm{Tr}(\rho O_i)=c_i\ (i=1,\dots,k). \] Lagrangian

Introduce multipliers \(\alpha\in\mathbb R\) (normalization) and \(\lambda_1,\dots,\lambda_k\in\mathbb R\) (constraints). Define \[ \mathcal L(\rho,\alpha,\lambda) = -\mathrm{Tr}(\rho\log\rho) -\alpha\big(\mathrm{Tr}(\rho)-1\big) -\sum_{i=1}^k \lambda_i\big(\mathrm{Tr}(\rho O_i)-c_i\big). \]

The key variational derivative

Take a small Hermitian perturbation \(\delta\rho\) with \(\mathrm{Tr}(\delta\rho)=0\) (we’ll enforce normalization via \(\alpha\) anyway). Use the standard functional variation \[ \delta\,\mathrm{Tr}(\rho\log\rho)=\mathrm{Tr}\big(\delta\rho(\log\rho+I)\big), \] so \[ \delta S(\rho) = -\mathrm{Tr}\big(\delta\rho(\log\rho+I)\big). \] Also, \[ \delta\,\mathrm{Tr}(\rho O_i)=\mathrm{Tr}(\delta\rho\, O_i),\qquad \delta\,\mathrm{Tr}(\rho)=\mathrm{Tr}(\delta\rho). \] Thus \[ \delta\mathcal L = -\mathrm{Tr}\big(\delta\rho(\log\rho+I)\big) -\alpha\,\mathrm{Tr}(\delta\rho) -\sum_{i=1}^k \lambda_i\,\mathrm{Tr}(\delta\rho\,O_i). \] Factor out \(\delta\rho\): \[ \delta\mathcal L = -\mathrm{Tr}\left(\delta\rho\left(\log\rho+I+\alpha I+\sum_{i=1}^k \lambda_i O_i\right)\right). \]

For a stationary point, \(\delta\mathcal L=0\) for all Hermitian \(\delta\rho\), hence the operator in parentheses must vanish: \[ \log\rho + I + \alpha I + \sum_{i=1}^k \lambda_i O_i = 0. \]

Rearrange: \[ \log\rho = -(\alpha+1)I - \sum_{i=1}^k \lambda_i O_i. \]

Exponentiate: \[ \rho = \exp\!\left(-(\alpha+1)I - \sum_{i=1}^k \lambda_i O_i\right) = \frac{1}{Z}\exp\!\left(-\sum_{i=1}^k \lambda_i O_i\right), \] where the partition function \(Z\) enforces normalization: \[ Z := \mathrm{Tr}\left(\exp\!\left(-\sum_{i=1}^k \lambda_i O_i\right)\right). \]

What you just proved

The MaxEnt state consistent with linear constraints is always of generalized Gibbs form: \[ \rho_{\lambda} = \frac{e^{-\sum_i \lambda_i O_i}}{\mathrm{Tr}(e^{-\sum_i \lambda_i O_i})}. \] This is the finite-dimensional algebraic heart of “thermalization by MaxEnt.”

Specialization: thermal Gibbs state from mean energy

Take one observable \(O_1=H\) and one constraint \(\mathrm{Tr}(\rho H)=E\). Then \[ \rho_\beta = \frac{e^{-\beta H}}{Z(\beta)},\qquad Z(\beta)=\mathrm{Tr}(e^{-\beta H}), \] with \(\beta\) chosen so that \(\mathrm{Tr}(\rho_\beta H)=E\).

How \(\beta\) is fixed

Differentiate \(\log Z(\beta)\): \[ \frac{d}{d\beta} Z(\beta) = \mathrm{Tr}(-H e^{-\beta H}), \] so \[ \frac{d}{d\beta}\log Z(\beta) = \frac{1}{Z(\beta)}\mathrm{Tr}(-H e^{-\beta H}) = -\mathrm{Tr}(\rho_\beta H) = -\langle H\rangle_{\rho_\beta}. \] Thus the map \(\beta \mapsto \langle H\rangle_{\rho_\beta}\) is obtained from \(Z\) and you solve \(\langle H\rangle_{\rho_\beta}=E\) for \(\beta\).

Why this solution is the maximizer, not just a stationary point

Two finite-dimensional facts complete the story:

A strictly concave function has at most one maximizer on a convex domain. So the generalized Gibbs form you derived is not merely “a candidate”: it is the unique maximum entropy state (when the constraints are feasible and the maximizer is full-rank).

GAP translation: what this buys you

In GAP terms, the Lagrange-multiplier derivation formalizes:

• Projection: you only keep the constraints \(\mathrm{Tr}(\rho O_i)\) (equivalently, you restrict to the subalgebra generated by the \(O_i\) in the commutative case, or to their “relevant algebra” more generally).
• Closure: you select the unique least-committal completion, which must be exponential-family / Gibbs.

So “thermal coarse‑graining” is not a physical scrambling claim; it’s the inferential closure step that converts a restricted algebra of facts into a canonical state.

Next step

To connect this directly to “entropy as a monotone on observable algebras,” the clean next move is:

• Show nesting of constraints \(\Rightarrow\) nesting of feasible sets \(\Rightarrow\) monotonicity of the MaxEnt value, i.e. if you keep energy and particle number you get lower \(S_{\mathcal B}\) than if you keep energy alone.

If you tell me what you want as the next “thermal module” constraint set—just \(H\), or \(H\) plus \(N\), or local energy densities—I’ll write the monotonicity statement in the same formal style.

Patrick: Let's go with H + N, and then call it a day, and I'll start setting this chat up in HTML.

Copilot:
MaxEnt with constraints \(H + N\) in finite dimensions

Let \(\mathcal H\) be finite-dimensional and \(\mathcal A=\mathcal B(\mathcal H)\). Fix Hermitian operators:

• Hamiltonian: \(H\)
• Number operator: \(N\) (particle number, excitation number, etc.)

Lagrange-multiplier solution: the grand canonical state

Form the Lagrangian with multipliers \(\alpha,\beta,\gamma\): \[ \mathcal L(\rho,\alpha,\beta,\gamma) = -\mathrm{Tr}(\rho\log\rho) -\alpha(\mathrm{Tr}\rho-1) -\beta(\mathrm{Tr}(\rho H)-E) -\gamma(\mathrm{Tr}(\rho N)-\bar N). \]

Stationarity with respect to \(\rho\) gives \[ \log\rho = -(\alpha+1)I - \beta H - \gamma N, \] so \[ \rho_{\beta,\gamma} = \frac{e^{-\beta H-\gamma N}}{Z(\beta,\gamma)}, \qquad Z(\beta,\gamma)=\mathrm{Tr}\!\left(e^{-\beta H-\gamma N}\right). \]

It’s conventional to write \(\gamma=-\beta\mu\), yielding the familiar grand canonical form \[ \rho_{\beta,\mu} = \frac{e^{-\beta(H-\mu N)}}{\Xi(\beta,\mu)}, \qquad \Xi(\beta,\mu)=\mathrm{Tr}\!\left(e^{-\beta(H-\mu N)}\right). \]

• \(\beta\): fixed implicitly by \(\mathrm{Tr}(\rho_{\beta,\mu}H)=E\)
• \(\mu\): fixed implicitly by \(\mathrm{Tr}(\rho_{\beta,\mu}N)=\bar N\)

No assumption that \([H,N]=0\) is needed to write this state; the exponential is well-defined regardless.

Partition function derivatives that pin down the constraints

Define \(\Xi(\beta,\mu)=\mathrm{Tr}(e^{-\beta(H-\mu N)})\). Then:

• Mean of \(H-\mu N\): \[ \frac{\partial}{\partial \beta}\log \Xi(\beta,\mu) = -\mathrm{Tr}\big(\rho_{\beta,\mu}(H-\mu N)\big). \]
• Mean of \(N\): \[ \frac{\partial}{\partial \mu}\log \Xi(\beta,\mu) = \beta\,\mathrm{Tr}\big(\rho_{\beta,\mu}N\big) = \beta\,\bar N. \]

Given \((E,\bar N)\) in the feasible range, these relations implicitly determine \((\beta,\mu)\).

GAP monotonicity: adding \(N\) refines the description, so entropy drops

Let:

• Energy-only constraints: \(\mathcal C_H := \{\rho:\mathrm{Tr}(\rho)=1,\ \mathrm{Tr}(\rho H)=E\}\)
• Energy+number constraints: \(\mathcal C_{H,N} := \{\rho:\mathrm{Tr}(\rho)=1,\ \mathrm{Tr}(\rho H)=E,\ \mathrm{Tr}(\rho N)=\bar N\}\)

Then \(\mathcal C_{H,N}\subseteq \mathcal C_H\) (strictly, unless the \(N\)-constraint is redundant). Maximizing a concave functional over a smaller feasible set cannot increase the maximum, hence \[ \max_{\rho\in \mathcal C_{H,N}} S(\rho)\ \le\ \max_{\rho\in \mathcal C_H} S(\rho). \] This is the clean “entropy as a monotone on descriptions” statement in this thermal setting:

• Keeping only \(H\) corresponds to a coarser observable algebra (fewer facts).
• Keeping \(H\) and \(N\) corresponds to a refined algebra (more facts).
• The MaxEnt “certificate entropy” can only go down when you refine what you insist on matching.

Observable-algebra viewpoint for \(H+N\)

If \([H,N]=0\), then \(H\) and \(N\) are simultaneously block-diagonalizable, and the “thermodynamic facts” algebra is the commutative algebra \[ \mathcal B_{H,N} = \{ f(H,N)\}, \] i.e. generated by their joint spectral projectors. Refining from \(\mathcal B_H\) to \(\mathcal B_{H,N}\) is literally a refinement of the partition of microstates into equivalence classes—so the monotonicity above becomes a statement about smaller equivalence classes.

If \([H,N]\neq 0\), the constraint set is still perfectly valid (MaxEnt still returns \(\rho_{\beta,\mu}\)), but the “facts” are no longer a purely classical joint spectrum. GAP-wise, it’s the first place you can feel the noncommutative nature of “keeping constraints” rather than “keeping a classical macrostate.”

Minimal takeaway for your HTML write-up

• Thermal coarse-graining = choose constraints (observable facts) + MaxEnt closure.
• With constraints \((H,N)\), MaxEnt yields \(\rho_{\beta,\mu}\propto e^{-\beta(H-\mu N)}\).
• Adding constraints refines the description, and the MaxEnt entropy is monotone decreasing under that refinement.

If you paste your HTML draft and tell me your preferred notation conventions (e.g., \(Z\) vs \(\Xi\), \(\bar N\) vs \(N_0\)), I’ll help you make it clean and internally consistent.

Patrick:
Follow-up:
You said: Quantum information is the set of distinctions preserved under unitary evolution of a quantum state.

That still seems vague. So vague, that one might be tempted to reason (falsely) that

The unitary evolution of a quantum state is that kind of operation on a Hilbert space that preserves quantum information.

Of course this is circular reasoning, but why specifically is it wrong as a definition?

Copilot:
You’re right to press on this, because the phrase “quantum information is what unitary evolution preserves” sounds dangerously close to a tautology. The reason it isn’t a definition—and why it would be wrong to treat it as one—comes down to what is defined first, and what is derived.

1️⃣ Why the circular definition is tempting

The tempting (but wrong) chain goes like this:

• Quantum information is whatever is preserved by unitary evolution.
• Unitary evolution is whatever preserves quantum information.

That collapses into a vacuous statement: “unitary evolution preserves what it preserves.” Nothing has been specified.

The problem is not just logical circularity—it’s that unitarity is not defined in informational terms at all.

2️⃣ What unitarity is actually defined by

Unitary operators are defined before any talk of information.

A linear operator \(U\) on a Hilbert space is unitary if and only if: \[ U^\dagger U = UU^\dagger = I. \]

Equivalently—and this is crucial—unitaries are exactly the maps that:

• Preserve the inner product: \[ \langle U\psi, U\phi\rangle = \langle \psi, \phi\rangle \]
• Preserve transition probabilities: \[ |\langle \psi | \phi \rangle|^2 \]
• Preserve the spectrum of density operators: \[ \rho \mapsto U\rho U^\dagger \]

None of these statements mention “information.” They are geometric and algebraic facts about Hilbert space.

So unitarity is a kinematic symmetry, not an informational principle.

3️⃣ What “quantum information” refers to (non-circularly)

Once unitarity is fixed, then we can ask what structures it preserves.

A precise, non-circular characterization is:

Quantum information consists of all distinctions encoded in a quantum state that are invariant under unitary equivalence.

• The spectrum of \(\rho\) (eigenvalues)
• Relative phases between components
• Entanglement structure
• Correlations across subsystems
• Noncommuting expectation-value relations

These are structural features of the state, not labels or bits.

Unitary evolution preserves them because it is a change of basis in Hilbert space, not because it was defined to preserve “information.”

4️⃣ Why “information” is derivative, not primitive

Here’s the key asymmetry that breaks the circularity:

• Unitarity is defined geometrically (inner products, reversibility).
• Information is defined relationally (what distinctions survive allowed transformations).

This mirrors classical mechanics:

• Canonical transformations are defined by symplectic geometry.
• “Classical information” is what survives those transformations.

In both cases, information is a derived invariant, not a defining axiom.

5️⃣ Why the circular definition fails physically

If you tried to define unitarity as “whatever preserves quantum information,” you would immediately run into ambiguity:

• Completely positive trace-preserving maps preserve some information.
• Decoherence preserves classical information.
• Measurements preserve outcome statistics.
• Partial traces preserve reduced-state information.

So which of these would count as “unitary” under that definition?

Without an independent definition of unitarity, the phrase “preserves information” has no content.

6️⃣ GAP’s sharper formulation

In GAP language, the clean statement is:

Quantum information is the fine-grained distinguishability structure of a state prior to projection onto a restricted observable algebra.

• Unitary evolution preserves that structure because it preserves the full observable algebra.
• Entropy increases only when you restrict the algebra (coarse-graining, tracing out, measurement).

So information loss is never dynamical—it is descriptive.

7️⃣ The one-sentence resolution

The circular definition fails because unitarity is defined independently of information, while information is defined as the set of distinctions invariant under that independently defined structure.

Unitarity comes first. Information is what survives it.

That asymmetry is exactly what makes the concept non-vacuous—and exactly why your instinct to challenge the phrasing was right.

Afterword: