14 Petz recovery and quantum dynamical entropy
This chapter documents a self-contained finite-dimensional quantum-information layer (issues #22–#28) that is built on the same matrix / continuous-functional-calculus infrastructure as the multiplicative ergodic theorem, but is logically independent of it. The objects are density matrices \(\rho ,\sigma \) on \(\mathbb {C}^d\) and completely positive trace-preserving maps (quantum channels) between matrix algebras. The central quantity is the Umegaki relative entropy
(Lean: relEntropy); its defining feature is the data-processing inequality \(S(\Lambda \rho \| \Lambda \sigma )\le S(\rho \| \sigma )\) for every channel \(\Lambda \), a consequence of Lieb’s joint-convexity theorem. Two threads are developed here. The first is the Petz recovery theorem and both directions of Petz’s equality theorem: a channel saturates the data-processing inequality on a pair of faithful states if and only if the input is exactly reconstructed by the Petz transpose recovery map. The general saturation \(\Rightarrow \) recovery direction (Petz, Monotonicity of quantum relative entropy revisited, Rev. Math. Phys. 15 (2003), Thm 2; via the operator-Jensen / \(-\log \) Loewner route of Carlen–Vershynina, Recovery map stability for the data processing inequality, 2020) is proved with no injectivity hypothesis on the channel, so it covers information-losing channels such as the completely depolarising channel. The second thread is the Connes–Narnhofer–Thirring quantum dynamical entropy (Connes–Narnhofer–Thirring, Dynamical entropy of \(C^{*}\) algebras and von Neumann algebras, Comm. Math. Phys. 112 (1987) 691–719, in the operational-partition formulation of Alicki–Fannes 1994), whose abelian corner collapses onto the classical Kolmogorov–Sinai entropy of the underlying measure-preserving system, tying the quantum layer back to the ergodic theory of the core. Every node below is formalized sorry-free and audited to depend on exactly \(\{ \texttt{propext},\ \texttt{Classical.choice},\ \texttt{Quot.sound}\} \). Throughout, \(X^{\dagger }\) denotes the conjugate transpose (Hilbert–Schmidt adjoint) of \(X\).
14.1 Kraus channels and the Petz recovery map
A Kraus channel on \(\operatorname {Matrix}_n(\mathbb {C})\) is a finite family of Kraus operators \(K : \iota \to \operatorname {Matrix}_n(\mathbb {C})\) satisfying the completeness relation \(\sum _i K_i^{\dagger }K_i = 1\). Its Schrödinger action is \(\Lambda (X)=\sum _i K_i\, X\, K_i^{\dagger }\) (Lean: toMat, restricted to states as toDM), and its Heisenberg (Hilbert–Schmidt) adjoint is \(\Lambda ^{\dagger }(X)=\sum _i K_i^{\dagger }X\, K_i\) (Lean: adj).
A Kraus channel is trace preserving, \(\operatorname{tr}(\Lambda X)=\operatorname{tr}X\) for every \(X\); consequently the adjoint is unital, \(\Lambda ^{\dagger }(1)=1\) (Lean: adj_unital).
By trace cyclicity \(\operatorname{tr}(K_i X K_i^{\dagger })=\operatorname{tr}(K_i^{\dagger }K_i\, X)\); summing over \(i\) and applying \(\sum _i K_i^{\dagger }K_i=1\) collapses the sum to \(\operatorname{tr}X\). Unitality of \(\Lambda ^{\dagger }\) is the same completeness relation read at \(X=1\).
For a state \(\sigma \) and a channel \(\Lambda \), the Petz (transpose) recovery map is
realised through the continuous functional calculus (CFC.conjSqrt for the \(\sqrt{\cdot }\)-conjugations and the ring inverse for \((\Lambda \sigma )^{-1/2}\)).
If the channel output \(\Lambda \sigma \) is positive definite, then the Petz map recovers \(\sigma \) from \(\Lambda \sigma \):
Feeding \(X=\Lambda \sigma \) into \(P_{\sigma ,\Lambda }\), the inner conjugation \((\Lambda \sigma )^{-1/2}(\Lambda \sigma )(\Lambda \sigma )^{-1/2}=1\) collapses to the identity (this is where positive definiteness of \(\Lambda \sigma \) is used); unitality \(\Lambda ^{\dagger }(1)=1\) then leaves \(\sqrt{\sigma }\cdot 1\cdot \sqrt{\sigma }=\sigma \).
14.2 Petz’s equality theorem
The Petz map already recovers \(\sigma \) unconditionally (Theorem 14.4); the content of the equality theorem is that it recovers the other state \(\rho \) exactly when the channel loses no relative entropy between \(\rho \) and \(\sigma \). One direction is elementary and rests only on monotonicity (the data-processing inequality); the other is the analytic heart of the chapter.
Let \(\Lambda \) and \(R\) be maps on states, each monotone under relative entropy (each satisfies the data-processing inequality against positive-definite second arguments). If \(R\) is a recovery section for the pair \(\rho ,\sigma \), i.e. \(R(\Lambda \rho )=\rho \) and \(R(\Lambda \sigma )=\sigma \) (with \(\sigma ,\Lambda \sigma \) positive definite), then the data-processing inequality is saturated:
Monotonicity of \(\Lambda \) gives \(S(\Lambda \rho \| \Lambda \sigma )\le S(\rho \| \sigma )\). For the reverse inequality, apply monotonicity of \(R\) to the pair \(\Lambda \rho ,\Lambda \sigma \) and rewrite through the section identities \(R(\Lambda \rho )=\rho \), \(R(\Lambda \sigma )=\sigma \); this yields \(S(\rho \| \sigma )\le S(\Lambda \rho \| \Lambda \sigma )\). Antisymmetry gives equality.
14.2.1 The Choi \(-\log \) Loewner inequality
The engine of the converse is operator convexity of \(-\log \), packaged first as a Loewner inequality for a rectangular isometry and then specialised to the Kraus column.
Let \(W:\mathbb {C}^q\to \mathbb {C}^p\) be an isometry (\(W^{\dagger }W=1\)) and \(X\) self-adjoint with spectrum in \((0,\infty )\). Then
Extend the orthonormal columns of \(W\) to a unitary \(U\); conjugating \(X\) by \(U\) and reindexing, \(W^{\dagger }XW\) is the upper-left corner of \(A=U^{\dagger }XU\). Operator convexity of \(-\log \) gives the corner inequality \((-\log )(\text{corner of }A)\le \text{corner of }(-\log )(A)\) (sum_corner_loewner applied to operatorConvexOn_neg_log), which is exactly the claim after identifying both corners.
For Kraus operators \(K\) with \(\sum _i K_i^{\dagger }K_i=1\) and \(X\) self-adjoint with spectrum in \((0,\infty )\),
Stack the Kraus operators into the column isometry \(V\) and let \(X_{\mathrm{bd}}\) be the block-diagonal amplification of \(X\). Then \(V^{\dagger }X_{\mathrm{bd}}V=\sum _i K_i^{\dagger }XK_i\) and, since \(-\log \) commutes with block diagonals, \(V^{\dagger }(-\log )(X_{\mathrm{bd}})V=\sum _i K_i^{\dagger }(-\log )(X)K_i\). Apply Theorem 14.6 to \((V,X_{\mathrm{bd}})\).
14.2.2 The modular realisation of relative entropy
For faithful states \(\rho ,\sigma \), with (vectorised) relative modular operator \(\Delta =\sigma \otimes (\rho ^{-1})^{\top }\) and cyclic vector \(\xi =\operatorname {vec}(\rho ^{1/2})\),
Compute \((-\log )(\sigma \otimes (\rho ^{-1})^{\top })=-(\log \sigma )\otimes 1+1\otimes (\log \rho )^{\top }\) (via \(\log \) of a Kronecker product, \(\log \rho ^{-1}=-\log \rho \), and the transpose law). Apply this to \(\operatorname {vec}(\rho ^{1/2})\) through the vec/Kronecker rule \((A\otimes B^{\top })\operatorname {vec}X=\operatorname {vec}(AXB)\), read off via the Hilbert–Schmidt inner product \(\langle \operatorname {vec}X,\operatorname {vec}Y\rangle =\operatorname{tr}(X^{\dagger }Y)\), and use \(\rho ^{1/2}\rho ^{1/2}=\rho \) with trace cyclicity to obtain \(\operatorname{tr}(\rho (\log \rho -\log \sigma ))=S(\rho \| \sigma )\).
14.2.3 The general channel: contraction rigidity (issue #28, no injectivity)
For a general Kraus channel the vectorised Petz map \(W=\texttt{petzWChanVec}\) is only a contraction (\(W^{\dagger }W\le 1\)), and the output modular operator \(\Delta _{\mathrm{out}}=(\Lambda \sigma )\otimes ((\Lambda \rho )^{-1})^{\top }\) is a genuinely separate operator: the whole-space Loewner bound \(W^{\dagger }(-\log \Delta )W\succeq -\log \Delta _{\mathrm{out}}\) fails for a contraction. Two adaptations carry the argument through. The \(-\log \) saturation is supplied as a scalar equality of quadratic forms, and the compression \(Y=W^{\dagger }\Delta W\) is never inverted, which is precisely what removes the injectivity hypothesis.
For a Kraus channel \(\Lambda \) with all four states \(\rho ,\sigma ,\Lambda \rho ,\Lambda \sigma \) faithful, if data processing is saturated (\(S(\rho \| \sigma )=S(\Lambda \rho \| \Lambda \sigma )\)), then the two \(-\log \) modular quadratic forms agree at the output cyclic vector \(\xi =\operatorname {vec}((\Lambda \rho )^{1/2})\):
with \(W=\texttt{petzWChanVec}\), \(\Delta =\sigma \otimes (\rho ^{-1})^{\top }\), \(\Delta _{\mathrm{out}}=(\Lambda \sigma )\otimes ((\Lambda \rho )^{-1})^{\top }\).
Since \(W\xi =\operatorname {vec}(\rho ^{1/2})\) (cyclicity of the channel contraction), the left form is \(S(\rho \| \sigma )\) and the right form is \(S(\Lambda \rho \| \Lambda \sigma )\) by Lemma 14.8; the entropy equality is precisely their coincidence.
For a contraction \(W\) with defect \((1-W^{\dagger }W)\xi =0\), positive-definite shifts \(X,\mathrm{Out}\), writing \(\eta =\mathrm{Out}^{-1}\xi \), \(b=X^{-1}(W\xi )-W\eta \), \(Y=W^{\dagger }XW\),
A pure matrix identity: the isometric proof’s \(Y^{-1}\) bridge is replaced by \(\mathrm{Out}^{-1}Y\mathrm{Out}^{-1}\), which cancels between the two summands, so the compression \(Y\) is never inverted. Expanding \(\langle b,Xb\rangle \) produces \(W^{\dagger }X^{-1}W\), two cross terms \(W^{\dagger }W\mathrm{Out}^{-1}\) and \(\mathrm{Out}^{-1}(W^{\dagger }W)\), and \(\mathrm{Out}^{-1}Y\mathrm{Out}^{-1}\); the contraction defect collapses each cross term (each contains \(W^{\dagger }W\xi =\xi \)) to \(\mathrm{Out}^{-1}\), and the second summand contributes \(\mathrm{Out}^{-1}-\mathrm{Out}^{-1}Y\mathrm{Out}^{-1}\). Adding and simplifying leaves exactly \(W^{\dagger }X^{-1}W-\mathrm{Out}^{-1}\).
In the setting of Lemma 14.10, with the compression bound \(W^{\dagger }XW\le \mathrm{Out}\), if the total resolvent gap \(\operatorname {Re}\langle \xi ,(W^{\dagger }X^{-1}W-\mathrm{Out}^{-1})\xi \rangle \) vanishes, then \(X^{-1}(W\xi )=W(\mathrm{Out}^{-1}\xi )\).
Both summands of Lemma 14.10 are nonnegative (\(X\succ 0\) and \(\mathrm{Out}-Y\succeq 0\)), and their real parts sum to the vanishing gap; hence each is zero. In particular \(\operatorname {Re}\langle b,Xb\rangle =0\), and positive definiteness of \(X\) forces \(b=0\), i.e. \(X^{-1}(W\xi )=W(\mathrm{Out}^{-1}\xi )\).
Let \(W\) be a contraction (\(W^{\dagger }W\le 1\)), \(\Delta ,\Delta _{\mathrm{out}}\) positive definite with compression bound \(W^{\dagger }\Delta W\le \Delta _{\mathrm{out}}\), cyclic-norm condition \(\left\lVert W\xi \right\rVert =\left\lVert \xi \right\rVert \), and the scalar \(-\log \) saturation \(\operatorname {Re}\langle \xi ,W^{\dagger }(-\log \Delta )W\xi \rangle =\operatorname {Re}\langle \xi ,(-\log \Delta _{\mathrm{out}})\xi \rangle \). Then for every \(t{\gt}0\),
Represent \(-\log \) by the integral \(\int _0^\infty \bigl((1+t)^{-1}-(\cdot +t)^{-1}\bigr)\, dt\) and let \(F(t)\) be the real resolvent-gap quadratic form at \(\xi \). By the shifted compression bound \(W^{\dagger }(\Delta +t)W\le \Delta _{\mathrm{out}}+t\) and the injectivity-free decomposition, \(F(t)\ge 0\) pointwise; integrating recovers the scalar \(-\log \) gap, which is \(0\) by hypothesis. A nonnegative continuous integrand with zero integral vanishes on \((0,\infty )\), so each \(F(t)=0\); per-\(t\) saturation (Lemma 14.11) gives the resolvent intertwining. (The general finite square index follows by an equivFin reindexing.)
Under the hypotheses of Theorem 14.12, for every continuous \(g\),
On the finite union of the two spectra, \(g\) is a real-coefficient combination of resolvents \(x\mapsto (x+t)^{-1}\) (finite-spectrum resolvent readoff). Both \(g(\Delta _{\mathrm{out}})\) and \(g(\Delta )\) become the corresponding operator-resolvent combinations, and the per-resolvent intertwining of Theorem 14.12 propagates linearly.
Under entropy saturation, the channel contraction intertwines the modular unitary power on the output cyclic vector, \(W\bigl(\Delta _{\mathrm{out}}^{it}\, \xi \bigr)=\Delta ^{it}\, (W\xi )\), with \(W\xi =\operatorname {vec}(\rho ^{1/2})\).
Apply Lemma 14.13 with \(g=\cos (t\log \cdot )\) and \(g=\sin (t\log \cdot )\); the scalar saturation input is Theorem 14.9. The complex power \(\Delta ^{it}\) is the functional-calculus combination \(\cos (t\log \cdot )+i\sin (t\log \cdot )\), so the two intertwinings combine to the unitary-power intertwining.
The channel adjoint intertwines the modular \(it\)-flows if, for all \(t\),
For any Kraus channel with all four states faithful, saturation \(S(\rho \| \sigma )=S(\Lambda \rho \| \Lambda \sigma )\) implies IntertwinesIt.
From Theorem 14.14, reading off the vec-action \(\Delta ^{it}\operatorname {vec}X=\operatorname {vec}(P^{it}XR^{-it})\) turns the vectorised statement into \(\Lambda ^{\dagger }\)-form directly (the vectorised Petz map already contains \(\Lambda ^{\dagger }\), so no amplification is needed). Cancelling the \((\Lambda \rho )^{\pm 1/2}\) twist on the input column and the \(\rho ^{1/2}\) factor, then taking adjoints, yields \(\Lambda ^{\dagger }((\Lambda \rho )^{it}(\Lambda \sigma )^{-it})=\rho ^{it}\sigma ^{-it}\).
If IntertwinesIt holds (all four states faithful), then the Petz map recovers the input: \(P_{\sigma ,\Lambda }(\Lambda \rho )=\rho \).
Analytic continuation of the \(it\)-intertwining to the value \(t=-i/2\) (a Kadison-type argument on the modular flow) turns the cocycle identity into \(\sqrt{\sigma }\, \Lambda ^{\dagger }((\Lambda \sigma )^{-1/2}(\Lambda \rho )(\Lambda \sigma )^{-1/2})\sqrt{\sigma }=\rho \), which is exactly \(P_{\sigma ,\Lambda }(\Lambda \rho )=\rho \).
Let \(\Lambda \) be any Kraus channel and \(\rho ,\sigma \) states with all four of \(\rho ,\sigma ,\Lambda \rho ,\Lambda \sigma \) positive definite. If data processing is saturated,
then the Petz recovery map reconstructs the input state,
No injectivity of the channel (or of the vectorised Petz map) is assumed; the result holds for information-losing channels such as the completely depolarising channel.
Compose Theorem 14.16 (entropy equality \(\Rightarrow \) modular \(it\)-intertwining) with Theorem 14.17 (intertwining \(\Rightarrow \) recovery). This is the full-generality form of Petz (2003, Thm 2). Together with Theorem 14.5 it closes the equivalence: for faithful states, saturation of the data-processing inequality holds if and only if the Petz map recovers the input.
14.3 The modular-cocycle intertwining and the injectivity-free route
The general contraction argument of the previous section is a deformation of a cleaner isometric prototype, which is worth recording on its own because it is the analytic heart of the whole equality theorem. For the partial-trace channel \(\Lambda =\operatorname {Tr}_B\) on a bipartite system with faithful dilated states \(\omega ,\tau \) (and faithful marginals \(\operatorname {Tr}_B\omega ,\operatorname {Tr}_B\tau \)), the vectorised Petz map \(W=\texttt{petzWvec}\) is a genuine isometry. There the whole-space \(-\log \) Loewner bound (Theorem 14.6) is available, so the saturation input can be taken as a full operator gap rather than a scalar one, and the compression may be inverted freely — this is exactly the injectivity that the general route of §14.2 had to dispense with, replacing the \(Y^{-1}\) bridge by the cancelling decomposition of Lemma 14.10.
If the partial-trace relative entropy is preserved, \(S(\operatorname {Tr}_B\omega \| \operatorname {Tr}_B\tau )=S(\omega \| \tau )\), then the rectangular \(-\log \) operator-Jensen gap for the Petz isometry \(W\) annihilates the output cyclic vector \(\xi =\operatorname {vec}((\operatorname {Tr}_B\omega )^{1/2})\):
By Theorem 14.6 the operator \(B-A\ge 0\), where \(B=W^{\dagger }(-\log )(\Delta )W\) and \(A=(-\log )(W^{\dagger }\Delta W)\). Using Lemma 14.8 on both sides (the compression \(W^{\dagger }\Delta W\) is exactly the output modular operator, and \(W\xi =\operatorname {vec}(\omega ^{1/2})\)), the two quadratic forms at \(\xi \) equal \(S(\operatorname {Tr}_B\omega \| \operatorname {Tr}_B\tau )\) and \(S(\omega \| \tau )\); the entropy equality makes \(\operatorname {Re}\langle \xi ,(B-A)\xi \rangle =0\). For a positive semidefinite matrix a vanishing real expectation forces \((B-A)\xi =0\).
Under the same entropy-preservation hypothesis, the amplified output modular \(it\)-cocycle equals the input one, for all \(t\in \mathbb {R}\):
The gap of Theorem 14.19 is upgraded, first to an intertwining of every resolvent \((\Delta +t)^{-1}\) (the isometric rigidity tail), then — via a finite-spectrum resolvent readoff — to the intertwining of every continuous function of \(\Delta \) on \(\xi \). Feeding the pair \(\cos (t\log \cdot ),\sin (t\log \cdot )\) assembles the unitary power \(\Delta ^{it}\), giving \(W(\Delta _A^{it}\xi )=\Delta ^{it}(W\xi )\). Reading off the vec-action \(\Delta ^{it}\operatorname {vec}X=\operatorname {vec}(P^{it}XR^{-it})\), cancelling the \(\omega _A^{\pm 1/2}\) twist on the input column and the \(\omega ^{1/2}\) factor, and taking adjoints yields the stated cocycle identity — an instance of the general IntertwinesIt property (Definition 14.15).
14.4 The Connes–Narnhofer–Thirring dynamical entropy and its abelian corner
This section documents the finite-dimensional quantum dynamical entropy of Connes–Narnhofer–Thirring, in the operational-partition formulation of Alicki–Fannes (see also Ohya–Petz, Quantum Entropy and Its Use). The dynamics is a unital \(*\)-endomorphism \(\Phi \) of the matrix algebra \(\operatorname {Matrix}_d(\mathbb {C})\), the observable is a finite operational partition of unity, and the entropy is read off from a family of correlation density matrices. The headline is that on the abelian corner — diagonal dynamics on a diagonal state — the whole construction collapses onto the classical Kolmogorov–Sinai entropy of the underlying measure-preserving system (Lean: ErgodicTheory.Entropy.ksEntropy), tying the quantum layer back to the ergodic theory of the MET core.
A finite quantum dynamics is a map \(\Phi :\operatorname {Matrix}_d(\mathbb {C})\to \operatorname {Matrix}_d(\mathbb {C})\) that is additive, multiplicative, unital (\(\Phi (1)=1\)) and \(*\)-preserving (\(\Phi (x^{\dagger })=\Phi (x)^{\dagger }\)). Additivity is carried as part of the datum: it is automatic for a \(*\)-homomorphism of matrix algebras, and it is exactly what lets \(\Phi \) commute with the finite sums appearing in the telescoping identity below.
An operational partition of unity of size \(k\) is a family \((x_i)_{i{\lt}k}\) of operators in \(\operatorname {Matrix}_d(\mathbb {C})\) satisfying the partition-of-unity relation \(\sum _i x_i^{\dagger }x_i=1\). This is the noncommutative analogue of a measurable partition of the state space.
Given a dynamics \(\Phi \) and an operational partition \(X=(x_i)\), the time-ordered refinement of depth \(n\) along a word \(f\in (\operatorname {Fin}k)^{\operatorname {Fin}n}\) is the operator
defined by the telescoping recursion \(\mathrm{refine}(n+1,f)=x_{f_0}\cdot \Phi \! \bigl(\mathrm{refine}(n,\mathrm{tail}\, f)\bigr)\) with \(\mathrm{refine}(0,\cdot )=1\). It records the observable measured along the first \(n\) steps of the orbit under \(\Phi \).
The refinement of an operational partition of unity is again an operational partition of unity: for every \(n\),
Induct on \(n\). Splitting the word as \(f=(i,g)\) with \(i=f_0\), the summand factors as \(\Phi (\mathrm{refine}(n,g))^{\dagger }\, (x_i^{\dagger }x_i)\, \Phi (\mathrm{refine}(n,g))\); summing over the leading letter \(i\) and applying \(\sum _i x_i^{\dagger }x_i=1\) removes it. What remains is \(\sum _g\Phi \! \bigl(\mathrm{refine}(n,g)^{\dagger }\mathrm{refine}(n,g)\bigr)\); pulling \(\Phi \) out of the finite sum (its additivity, Definition 14.21) and applying the inductive hypothesis leaves \(\Phi (1)=1\).
For a dynamics \(\Phi \), a state \(\rho \) (a density matrix on \(\mathbb {C}^d\)) and an operational partition \(X\), the depth-\(n\) correlation density matrix \(\mathrm{corrMatrix}\, \Phi \, \rho \, X\, n\) is the matrix on the classical index set \((\operatorname {Fin}k)^{\operatorname {Fin}n}\) with entries
It is a genuine density matrix: its trace is \(1\) by the telescoping identity (Lemma 14.24) together with \(\operatorname{tr}\rho =1\), and it is positive semidefinite because the quadratic form \(x^{\dagger }Mx\) equals \(\operatorname{tr}(T\rho T^{\dagger })\ge 0\) for \(T=\sum _f x_f\, \mathrm{refine}(n,f)\).
The entropy of a partition is the infimum von Neumann entropy rate
and the CNT dynamical entropy of \(\Phi \) in the state \(\rho \) is the supremum over all finite operational partitions, \(h_\Phi (\rho )=\sup _{k,X}h_\Phi (\rho ,X)\) (valued in \(\overline{\mathbb {R}}\)). The \(\inf _n\) form is the honest analogue of the subadditive limit \(\lim _n S(\cdot )/n\) and sidesteps an operator-Fekete argument.
For a probability vector \(\mu :\operatorname {Fin}d\to \mathbb {R}_{\ge 0}\) (\(\sum _i\mu _i=1\)), the associated diagonal state is \(\rho _\mu =\operatorname {diag}\mu \), a density matrix on \(\mathbb {C}^d\).
For a cell map \(c:\operatorname {Fin}d\to \operatorname {Fin}k\), the diagonal projection partition is \(\{ \operatorname {diag}\mathbf1_{c^{-1}(i)}\} _{i{\lt}k}\); it is an operational partition of unity encoding the measurable partition \(c^{-1}(\cdot )\) of \(\operatorname {Fin}d\).
For a permutation \(\sigma \in \mathfrak {S}_d\), the permutation dynamics \(\mathrm{adPerm}\, \sigma \) is conjugation \(x\mapsto P_\sigma \, x\, P_\sigma ^{\dagger }\) by the permutation matrix, a unital \(*\)-endomorphism; on diagonal matrices it acts by \((\mathrm{adPerm}\, \sigma )(\operatorname {diag}v)=\operatorname {diag}(v\circ \sigma )\). Paired with a \(\sigma \)-invariant diagonal state \(\rho _\mu \) (\(\mu \circ \sigma =\mu \), Definition 14.27) and a projection partition (Definition 14.28), this is the abelian corner: it mirrors the classical system \((\operatorname {Fin}d,\mu ,\sigma )\) with the measurable partition \(c^{-1}(\cdot )\).
On the abelian corner the correlation matrix is diagonal, carrying the masses of the classical \(n\)-fold join on its diagonal; hence its von Neumann entropy equals the classical iterated-join Shannon entropy of the system \((\operatorname {Fin}d,\mu ,\sigma )\):
Because \(\mathrm{adPerm}\, \sigma \) preserves diagonal matrices, the refinement of a projection partition is again diagonal, and the depth-\(n\) refinement product along a word \(f\) is exactly the indicator of the classical join cell \(\bigcap _{l{\lt}n}\sigma ^{-l}c^{-1}(f_l)\). Feeding these into the correlation entries and using that \(\rho _\mu \) is diagonal, the off-diagonal entries vanish (two distinct words disagree in some slot, forcing an orthogonal pair of indicators) and the \((f,f)\) entry is the mass \(\mu (\bigcap _{l{\lt}n}\sigma ^{-l}c^{-1}(f_l))\) of the join cell. The correlation matrix is therefore \(\operatorname {diag}\) of the join distribution, so by \(S(\operatorname {diag}p)=-\sum _j p_j\log p_j\) its von Neumann entropy is the Shannon entropy of the join — the classical join entropy at resolution \(n\).
Non-vacuity. The per-resolution collapse is not the trivial \(0=0\): for the two-level uniform state \(\mu \equiv \tfrac 12\) on \(\operatorname {Fin}2\), the identity dynamics, and the identity cell map, the resolution-\(1\) correlation matrix has von Neumann entropy \(\log 2{\gt}0\). (The library records this as an executable positivity certificate, so the equality genuinely relates a positive quantum entropy to a positive classical join entropy.)
For each projection partition \(\mathrm{projPartition}\, c\), the CNT partition entropy in the abelian corner equals the classical Kolmogorov–Sinai partition entropy of the corresponding measurable partition \(c^{-1}(\cdot )\):
Both sides are the subadditive limit of the same sequence divided by \(n\): the quantum partition rate is \(\inf _n S(\mathrm{corrMatrix}\, n)/n\) and the classical partition entropy is the corresponding limit of \(H(\bigvee _{l{\lt}n}\sigma ^{-l}c^{-1})/n\). The per-resolution collapse (Theorem 14.30) identifies the two defining sequences term by term, so the limits agree.
The abelian-corner CNT dynamical entropy of \(\mathrm{adPerm}\, \sigma \) in the state \(\rho _\mu \) is the supremum of the partition rate over all projection operational partitions, \(h^{\mathrm{ab}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) =\sup _{k,c}h_{\mathrm{adPerm}\, \sigma }(\rho _\mu ,\mathrm{projPartition}\, c)\).
Suppose every state carries positive mass (\(\mu _i{\gt}0\) for all \(i\)). Then the abelian-corner CNT dynamical entropy equals the classical Kolmogorov–Sinai entropy of the permutation system:
Two inequalities. For \(\le \), each projection partition’s rate equals a classical partition entropy (Theorem 14.31), which is dominated by the classical KS entropy (a supremum over all measurable partitions); take the supremum. For \(\ge \), positivity of every \(\mu _i\) forces the cells of any measurable partition \(P\) to be genuinely disjoint (not merely a.e.), so \(P\) is realized by an honest cell map \(c\) with \(c^{-1}(\cdot )=P\); its projection partition then has the same rate as \(P\), and this rate is one of the terms of the abelian supremum. Antisymmetry gives the equality.
Under the same positivity hypothesis, the full CNT dynamical entropy of \(\mathrm{adPerm}\, \sigma \) in the state \(\rho _\mu \) — the supremum over all operational partitions — dominates the classical Kolmogorov–Sinai entropy:
The abelian dynamical entropy is a supremum over the sub-family of projection partitions, hence is \(\le \) the full CNT dynamical entropy taken over all operational partitions. Rewriting the left endpoint by the corner equality (Theorem 14.33) turns this into the claimed bound.