17 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 \(X\).
17.1 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 \(\sqrt{\sigma }\cdot 1\cdot \sqrt{\sigma }=\sigma \).
17.2 Petz’s equality theorem
The Petz map already recovers \(\sigma \) unconditionally (Theorem 17.4); 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.
17.2.1 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 17.6 to \((V,X_{\mathrm{bd}})\).
17.2.2 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 \(\operatorname{tr}(\rho (\log \rho -\log \sigma ))=S(\rho \| \sigma )\).
17.2.3 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 17.8; 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 17.10, 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 17.10 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 17.11) gives the resolvent intertwining. (The general finite square index follows by an equivFin reindexing.)
Under the hypotheses of Theorem 17.12, 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 17.12 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 17.13 with \(g=\cos (t\log \cdot )\) and \(g=\sin (t\log \cdot )\); the scalar saturation input is Theorem 17.9. 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 17.14, 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 17.16 (entropy equality \(\Rightarrow \) modular \(it\)-intertwining) with Theorem 17.17 (intertwining \(\Rightarrow \) recovery). This is the full-generality form of Petz (2003, Thm 2). Together with Theorem 17.5 it closes the equivalence: for faithful states, saturation of the data-processing inequality holds if and only if the Petz map recovers the input.
17.3 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 17.6) 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 §17.2 had to dispense with, replacing the \(Y^{-1}\) bridge by the cancelling decomposition of Lemma 17.10.
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 17.6 the operator \(B-A\ge 0\), where \(B=W^{\dagger }(-\log )(\Delta )W\) and \(A=(-\log )(W^{\dagger }\Delta W)\). Using Lemma 17.8 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 17.19 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 17.15).
17.4 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 17.21) 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 17.24) 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 17.27) and a projection partition (Definition 17.28), 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 17.30) 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 17.31), 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 17.33) turns this into the claimed bound.
17.4.1 Finite-dimensional vanishing of the CNT rate (issue #26)
The one-sided bound above (Theorem 17.34) is completed in two complementary directions. First, in finite dimension the CNT/ALF dynamical entropy of any dynamics vanishes identically: the correlation matrices are Gram matrices factoring through the fixed \(d^{2}\)-dimensional matrix space \(\operatorname {Matrix}_d(\mathbb {C})\), so their rank — and hence their entropy — is bounded independently of the resolution \(n\), squeezing the rate to \(0\). This is the well-known degeneracy of quantum dynamical entropy in finite dimension (Alicki–Fannes, Quantum Dynamical Systems, OUP 2001, where the AF entropy vanishes precisely because \(\operatorname {rank}(\rho [X^{(n)}])\le (\dim \mathcal H)^{2}\); Neshveyev–Størmer, Dynamical Entropy in Operator Algebras, Springer 2006, that the CNT entropy of a finite-dimensional algebra is \(0\)). We disclose it honestly: the system-level equalities are degenerate \(0 = 0\) collapses, and the substantive content is the per-resolution rank/entropy bound below — and, on the abelian corner, the collapse \(S(\mathrm{corrMatrix}\, n) = H(\text{join}_n)\) already recorded in Theorem 17.30.
For a dynamics \(\Phi \), a state \(\rho \) on \(\mathbb {C}^{d}\) and an operational partition \(X\), the depth-\(n\) Gram vectors are the Hilbert–Schmidt vectors \(w_f = (\mathrm{refine}\, \Phi \, X\, n\, f)\cdot \sqrt{\rho }\), reshaped into the matrix \(\mathrm{gramVec}\, \Phi \, \rho \, X\, n\) with rows indexed by \(\operatorname {Fin}d\times \operatorname {Fin}d\) and columns by the classical index set \((\operatorname {Fin}k)^{\operatorname {Fin}n}\); here \(\sqrt{\rho } = \texttt{CFC.sqrt}\, \rho \) is the continuous-functional-calculus positive-semidefinite square root.
The CNT correlation matrix is the Gram matrix of its Hilbert–Schmidt vectors:
Entrywise this is the trace identity \(\operatorname{tr}\bigl((A\sqrt\rho )^{\dagger }(B\sqrt\rho )\bigr) = \operatorname{tr}(\rho \, A^{\dagger }B)\), valid for the Hermitian square root \(\sqrt\rho \) (\(\sqrt\rho ^{\dagger } = \sqrt\rho \), \(\sqrt\rho \, \sqrt\rho = \rho \)) by trace cyclicity, applied with \(A = \mathrm{refine}(g)\), \(B = \mathrm{refine}(f)\): the left side is the \((g,f)\) Gram entry \(\langle w_g, w_f\rangle \), the right side is the \((g,f)\) correlation entry.
The correlation density matrix of an \(n\)-fold refinement has rank at most \(d^{2}\), uniformly in \(n\):
A Gram matrix \(V^{\dagger }V\) has the same rank as \(V\) (\(\texttt{rank\_ conjTranspose\_ mul\_ self}\)), and by Theorem 17.36 the correlation matrix is such a Gram matrix with \(V = \mathrm{gramVec}\) having only \(\# (\operatorname {Fin}d\times \operatorname {Fin}d) = d^{2}\) rows; a matrix rank is at most its number of rows.
At every resolution \(n\), the von Neumann entropy of the correlation matrix is bounded by \(\log (d^{2})\), a constant independent of \(n\):
For every operational partition \(X\), the CNT partition entropy is \(0\): \(h_\Phi (\rho ,X) = \inf _{n\ge 1}S(\mathrm{corrMatrix}\, n)/n = 0\).
The rate is nonnegative, and by the uniform bound (Theorem 17.38) each averaged term satisfies \(0\le S(\mathrm{corrMatrix}\, n)/n\le \log (d^{2})/n\); since \(\log (d^{2})/n\to 0\), the infimum is \(0\).
For every operational partition, the entropy rate converges to the partition entropy, \(S(\mathrm{corrMatrix}\, n)/n\to h_\Phi (\rho ,X)\); the \(\inf \)-definition captures a true limit here because \(0\le S(\mathrm{corrMatrix}\, n)/n\le \log (d^{2})/n\to 0\).
Squeeze between \(0\) and \(\log (d^{2})/n\to 0\); the limit is \(0\), which Theorem 17.39 identifies with \(h_\Phi (\rho ,X)\).
The full CNT/ALF dynamical entropy of any finite-dimensional dynamics vanishes, \(h_\Phi (\rho ) = 0\).
Every term of the defining supremum over operational partitions is \(0\) (Theorem 17.39); the trivial one-cell partition \(x = 1\) witnesses nonemptiness at \(k = 1\), so the supremum is \(0\). This is the Alicki–Fannes / Neshveyev–Størmer finite-dimensional degeneracy.
17.4.2 The abelian corner as a Fekete limit and the KS equality (issue #26)
On the abelian corner the per-resolution collapse \(S(\mathrm{corrMatrix}\, n) = H(\text{join}_n)\) (Theorem 17.30) endows the CNT sequence with structure the general construction lacks: it inherits the classical Fekete subadditivity, so its \(\inf \)-rate is a genuine limit; and the abelian supremum upgrades to a literal equality with the Kolmogorov–Sinai entropy, promoting the one-sided Theorem 17.34 to a full identity. Both sides are again \(0\) — a permutation of a finite set has zero KS entropy — so the system-level equalities are degenerate; we state them as such, with the non-vacuous content residing in the per-resolution collapse.
On the diagonal corner the correlation-entropy sequence \(n\mapsto S\bigl(\mathrm{corrMatrix}\, (\mathrm{adPerm}\, \sigma )\, \rho _\mu \, (\mathrm{projPartition}\, c)\, n\bigr)\) is subadditive, \(S(m+n)\le S(m) + S(n)\).
By the per-resolution diagonal collapse (Theorem 17.30) the sequence coincides term by term with the classical iterated-join entropy sequence \(\texttt{ksEntropySeq}\) of the system \((\operatorname {Fin}d,\mu ,\sigma )\), whose subadditivity is the classical Fekete inequality (Theorem 10.10). In the non-commutative case this subadditivity fails — see §17.4.3.
On the diagonal corner the averaged entropies converge to the per-partition CNT rate,
The a priori infimum is thus a true limit on the abelian corner.
The full abelian-corner CNT dynamical entropy is the supremum of the partition rate over all operational partitions whose operators are diagonal (soft / POVM partitions of the diagonal subalgebra), a wider family than the sharp projection partitions of \(\mathrm{cntDynamicalEntropyAbelian}\) (Definition 17.32).
In finite dimension the full diagonal supremum vanishes, \(h^{\mathrm{ab,full}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) = 0\), as does the sharp projection supremum \(h^{\mathrm{ab}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) = 0\) .
Each diagonal (resp. projection) partition already has entropy rate \(0\) (Theorem 17.39); the diagonal projection partition of the constant cell map \(\operatorname {Fin}d\to \operatorname {Fin}1\) witnesses nonemptiness, so each supremum is \(0\).
The sharp abelian-corner entropy is dominated by the full diagonal version, \(h^{\mathrm{ab}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) \le h^{\mathrm{ab,full}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu )\), since every projection partition \(\mathrm{projPartition}\, c\) is diagonal.
Each term of the projection supremum is a term of the wider diagonal supremum (\(\mathrm{projPartition}\, c\) has diagonal operators, \(\texttt{Matrix.isDiag\_ diagonal}\)), so the smaller sup is \(\le \) the larger.
For a \(\sigma \)-invariant state with every mass positive, the full diagonal CNT dynamical entropy equals the classical Kolmogorov–Sinai entropy of the permutation system, \(h^{\mathrm{ab,full}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) = h_\mu (\sigma )\). Both sides are \(0\).
Under the same positivity hypothesis, the classical Kolmogorov–Sinai entropy of \(\sigma \) equals the full CNT dynamical entropy — the supremum over all operational partitions, not merely the diagonal ones:
This upgrades the one-sided bound Theorem 17.34 to an equality. Both sides are \(0\): the KS entropy of a finite-set permutation, and (by Theorem 17.41) the finite-dimensional CNT dynamical entropy — so the equality, while honest, is degenerate.
17.4.3 Failure of subadditivity: an explicit counterexample (issue #26)
The Fekete subadditivity of §17.4.2 is a special feature of the abelian corner, not a general property of the CNT construction. For a genuinely non-commutative operational partition the sequence \(n\mapsto S(\rho [X^{(n)}]) = S(\mathrm{corrMatrix}\, \Phi \, \rho \, X\, n)\) need not be subadditive — which is exactly why the CNT/ALF entropy of a partition is defined as an infimum rate rather than a Fekete limit of a subadditive sequence (Alicki–Fannes use a \(\limsup \) for the same reason). We exhibit an explicit witness in the simplest possible dynamical setting: the identity dynamics \(\Phi = \mathrm{id}\) on \(\operatorname {Matrix}_2(\mathbb {C})\), the invariant pure state \(\rho = |0\rangle \langle 0| = \operatorname {diag}(1,0)\), and the two-element operational partition
For the pure state \(\rho \), the length-\(1\) correlation matrix is the classical pure state \(\operatorname {diag}(1,0)\), so \(S(\rho [X^{(1)}]) = 0\).
Since \(\Phi = \mathrm{id}\), the depth-\(1\) refinement along \(f\) is just \(x_{f(0)}\); the \(\rho \)-weighted pairing \(\operatorname{tr}(\rho \, x_g^{\dagger }x_f)\) reads off column \(0\) of the pure state, giving \(\operatorname {diag}(1,0)\) on the classical index. A pure classical distribution has Shannon entropy \(0\).
For the same pure state, the length-\(2\) correlation matrix is not idempotent (its diagonal \(w_{00}\)-entry violates \(M^{2} = M\): the computation yields \(3/8\ne 1/2\)), hence \(S(\rho [X^{(2)}]) {\gt} 0\).
The depth-\(2\) refinement along \(f\) is \(x_{f(0)}x_{f(1)}\); computing the \((w_{00},w_{00})\) entry of \(M^{2}\) against \(M\) gives \(3/8\) versus \(1/2\), so \(M^{2}\ne M\). Strict positivity of the entropy for a non-idempotent state (Theorem 16.12) then yields \(S(\rho [X^{(2)}]) {\gt} 0\).
The CNT/ALF entropy sequence \(n\mapsto S(\rho [X^{(n)}])\) of the partition above, under the identity dynamics and the invariant pure state, is not subadditive: it violates \(u_2\le u_1 + u_1\), since
This is a feature, not a defect: it is precisely why the CNT rate is an infimum (an \(\texttt{sInf}\)), not a Fekete limit of a subadditive sequence.
17.5 Genuinely non-commutative sealed dynamics (issue #59)
The finite-dimensional CNT rate vanishes identically (Theorem 17.41), so the issue’s literal tier-1 goal — a system-level strictly positive quantum dynamical entropy in finite dimension — is provably impossible. What survives, and is genuinely non-commutative, is a pair of honest certificates: a per-resolution correlation-entropy dichotomy that strictly separates the non-commuting operational partition from every abelian one, and a relative-entropy seal whose strict data-processing drop forecloses any Stinespring recovery. (Connes–Narnhofer –Thirring, Comm. Math. Phys. 112 (1987) 691–719; Alicki–Fannes, Quantum Dynamical Systems, OUP (2001); Neshveyev–Størmer, Dynamical Entropy in Operator Algebras, Springer (2006); Petz 1986/2003; Wilde, Quantum Information Theory, on recoverability.)
17.5.1 Shared spectral lemmas
If a density matrix is a projection (\(\rho ^2 = \rho \)) then \(S(\rho ) = 0\): transporting to the eigenbasis forces every eigenvalue into \(\{ 0,1\} \), on which \(\operatorname {negMulLog}\) vanishes. This is the converse of \(\operatorname {vonNeumannEntropy\_ pos\_ of\_ sq\_ ne}\).
\(D(\rho \, \| \, I/d) = \log d - S(\rho )\); the cross term collapses through the doubly-stochastic row sum to \(-\log d\cdot \operatorname {Tr}\rho = -\log d\). Unitary invariance of the von Neumann entropy () is proved alongside, via \(\operatorname {charpoly\_ mul\_ comm}\).
17.5.2 The dephasing recovery seal
The dephasing (pinching) channel \(\Delta \) on a qubit deletes off-diagonal coherences. Fed a coherent state it produces a strict drop of the Umegaki relative entropy against a faithful diagonal fixed point \(\sigma = \operatorname {diagState} s = \operatorname {diag}((1+s)/2, (1-s)/2)\) (\(0 {\lt} s {\lt} 1\)); by Petz’s theorem no faithful-ancilla Stinespring section can undo it.
The maximally coherent pure state \(|{+}\rangle \langle {+}|\) dephases to \(I/2\), a strict relative-entropy drop of \(S(I/2) - S(|{+}\rangle \langle {+}|) = \log 2 {\gt} 0\) against \(\operatorname {diagState} s\); consequently no faithful-ancilla Stinespring section recovers it.
The faithful one-parameter family \(\rho _r = \tfrac 12\! \left(\begin{smallmatrix} 1 & r \\ r & 1 \end{smallmatrix}\right)\) (\(0 {\lt} r {\lt} 1\), positive definite) dephases to \(I/2\) with a strict drop \(\log 2 - h_2\! \big((1+r)/2\big) {\gt} 0\), where \(h_2\) is Mathlib’s binary entropy (whose strict maximum at \(1/2\) supplies the strict inequality); again no faithful-ancilla Stinespring section recovers it.
QA note — the reference state must be off the dephasing image. Taking \(\sigma := I/2\) is degenerate: then \(\Delta \rho = I/2 = \Delta \sigma \), so the two Stinespring-section hypotheses \(R(\Delta \rho ) = \rho \) and \(R(\Delta \sigma ) = \sigma \) share an identical left-hand side and force \(\rho = \sigma \) set-theoretically — False with no entropy input, the no-recovery content vacuous. The honest packaging therefore uses a faithful reference whose dephasing image differs from \(\Delta \rho \); \(\operatorname {diagState} s\) does this while remaining a faithful dephasing fixed point. The \(\sigma = I/2\) strict-DPI drops are still recorded honestly as standalone lemmas (\(\operatorname {relEntropy\_ drop\_ plusState\_ mm}\), \(\operatorname {relEntropy\_ drop\_ rhoR\_ mm}\)). The Petz-map corollary (that the explicit Petz recovery map fails on \(\rho _r\)) would need a general-\(\operatorname {KrausChannel}\) data-processing inequality — absent from the repo, only the faithful-ancilla Stinespring family is available — so it is intentionally omitted.
17.5.3 The per-resolution non-commutativity certificate
On the same witness as Section 17.4.3 — identity dynamics \(\operatorname {cexEndo}\) on \(\operatorname {Matrix}_2(\mathbb {C})\) and invariant pure state \(|0\rangle \langle 0|\) — the correlation entropy sees non-commutativity at fixed resolution, even though the system-level rate vanishes.
For every abelian (diagonal) operational partition the correlation density matrix is a rank-one idempotent at every resolution \(n\) — a Gram collapse onto the column-\(0\) diagonal entry — so its von Neumann entropy is \(0\) (); while for the non-commuting partition \(\operatorname {cexPartition}\) the length-\(2\) correlation matrix is not idempotent, so its entropy is strictly positive. This is issue #59’s “entropy strictly above every abelian restriction”, stated honestly at the per-resolution level: the system-level CNT/ALF rate is the disclosed \(0 = 0\) of Theorem 17.41, so the non-commutativity lives in the per-resolution correlation matrices.
17.5.4 The canonical-MASA incompatibility certificate
By Skolem–Noether every unital \(*\)-endomorphism of \(\operatorname {Matrix}_d\) is inner, hence conjugation by a unitary, and always preserves some maximal abelian subalgebra (MASA) — the eigenbasis of that unitary. So no obstruction can be about a single map; the certificate is about the pair (dynamics, seal). The dynamics \(\operatorname {qDynamics} = \operatorname {Ad}(U)\) is built from a Pythagorean \((3,4,5)\) rotation of the eigenbasis composed with the phase \(\operatorname {diag}(1,i)\), so every entry lies in \(\mathbb {Q}(i)\) and each claim reduces to \(\operatorname {norm\_ num}\).
The seal’s diagonal MASA is not dynamics-invariant — \(U\cdot \operatorname {diag}(1,0)\cdot U^\dagger \) has a nonzero \((1,0)\) entry () — and the dynamics’ MASA (the eigenbasis, \(=\) commutant of the eigenprojection \(\operatorname {eigProj}\)) is not seal-invariant: the dephasing map moves \(\operatorname {eigProj}\) out of that MASA, \(E(\operatorname {eigProj})\) failing to commute with \(\operatorname {eigProj}\) ().
A documented landmine and a disclosed frontier. The natural Hadamard/rotation choices fail to witness this: the circular (mutually unbiased) basis is simultaneously dephasing-invariant and swap-invariant, hence a common MASA. The Pythagorean tilt is chosen precisely to dodge that degeneracy (\(|V_{00}|^2 = 16/25\ne 1/2\)). The genuinely strong statement — that there is no common invariant MASA over all unitary conjugates of the diagonal — is true for this \(U\) but its formalization over the full unitary group is deferred (it is large); what is formalized is the concrete pairwise failure on the two canonical candidate MASAs, the honest operational core of the obstruction.
17.6 The finite reservoir: cap, generic rate engine, and Pauli saturation (issue #69)
The finite-dimensional vanishing \(h_\Phi (\rho ) = 0\) (Theorem 17.41) is driven by a single uniform cap on the correlation entropy — the correlation matrix lives on a \(d^{2}\)-dimensional Gram factorization, so its von Neumann entropy never exceeds \(\log (d^{2})\) regardless of the resolution \(n\). This section records three complementary refinements. First, the cap read in the \(2\log d\) reservoir form, together with the boundedness of the correlation-entropy sequence it produces. Second, the generic pigeonhole engine behind the collapse, isolated as a domain-neutral lemma: any nonnegative sequence under a fixed ceiling has vanishing per-step rate. Third, and the substantive point of issue #69, the cap is a genuine saturation rather than an identically-zero degeneracy: at \(d = 2\) the Pauli operational partition fills the reservoir completely at a single step, so the per-step rate dies only because the finite reservoir is uniform in \(n\), not because individual steps carry no information. Weyl/Pauli-operator saturation of the CNT reservoir is folklore in this framework (Connes–Narnhofer–Thirring, Comm. Math. Phys. 112 (1987) 691–719; Ohya–Petz, Quantum Entropy and Its Use, Springer).
17.6.1 The reservoir cap and the boundedness of the entropy sequence
For every unital \(*\)-endomorphism \(\Phi \), state \(\rho \), operational partition \(X\), and resolution \(n\), the correlation entropy obeys the uniform-in-\(n\) reservoir ceiling
The reservoir is \(\log (d^{2}) = 2\log d\), not the naive single-copy ceiling \(\log d\): the correlation-matrix construction lives on a \(d^{2}\)-dimensional Gram factorization, so the honest cap is \(\log (d^{2})\).
Restate the uniform bound \(S \le \log (d^{2})\) (Theorem 17.38) through \(\log (d^{2}) = 2\log d\) (\(\texttt{Real.log\_ pow}\), valid also at \(d = 0\) since \(\log 0 = 0\)).
For every partition the sequence \(n\mapsto S\bigl(\mathrm{corrMatrix}\, \Phi \, \rho \, X\, n\bigr)\) is bounded above by the fixed reservoir \(\log (d^{2})\), uniformly in the step count \(n\). Monotonicity in \(n\) is deliberately not claimed — it is unproven in general and forced flat in the saturated regime; only the uniform reservoir cap is asserted.
The constant \(\log (d^{2})\) is an upper bound of the range by the per-resolution cap (Theorem 17.38).
17.6.2 The generic cumulative-bounded rate engine
Let \(a:\mathbb {N}\to \mathbb {R}\) be nonnegative and uniformly bounded above by a fixed constant \(C\) (a reservoir). Then the per-step rate vanishes,
Squeeze between \(0 \le a(n)/n\) and \(a(n)/n \le C/n \to 0\).
This tiny lemma is the pigeonhole face of the whole finiteness doctrine, kept domain-neutral (namespace \(\texttt{ErgodicTheory}\), not \(\texttt{CNT}\)) so classical instantiations can reuse it. Instantiated at \(a(n) = S(\mathrm{corrMatrix}\, \Phi \, \rho \, X\, n)\) and \(C = \log (d^{2})\) it re-derives the finite-dimensional CNT collapse: the correlation rate is squeezed to \(0\), which is exactly \(h_\Phi (\rho ,X) = 0\) (Theorem 17.39) and hence \(h_\Phi (\rho ) = 0\) (Theorem 17.41). The reservoir cap (Theorem 17.58) supplies the hypothesis; the engine supplies the squeeze.
17.6.3 Pauli saturation at \(d = 2\): the cap is attained
The Pauli operational partition of \(\operatorname {Matrix}_2(\mathbb {C})\) is the family of four operators \((\tfrac 12 P_i)_{i{\lt}4}\), where \(P_0 = 1\), \(P_1 = X\), \(P_2 = Y\), \(P_3 = Z\) are the Pauli matrices. Each \(P_i\) is Hermitian and unitary, so \(\sum _i (\tfrac 12 P_i)^{\dagger }(\tfrac 12 P_i) = \sum _i \tfrac 14 P_i^{2} = 4\cdot \tfrac 14\cdot 1 = 1\), an operational partition of unity of size \(4 = 2^{2}\).
For every unital \(*\)-endomorphism \(\Phi \) of \(\operatorname {Matrix}_2(\mathbb {C})\), the depth-\(1\) correlation matrix of the Pauli partition in the maximally mixed state \(I/2\) is itself the maximally mixed state on the \(4\)-element classical index set \((\operatorname {Fin}4)^{\operatorname {Fin}1}\):
The depth-\(1\) refinement along a word \(f\) is just the selected operator \((\tfrac 12 P_{f_0})\). Pauli trace-orthogonality \(\operatorname{tr}(P_i^{\dagger }P_j) = 2\, \delta _{ij}\), scaled by the state \(I/2\) and the \((\tfrac 12)^{2}\) normalisation, makes the correlation entry \(\operatorname{tr}\bigl((I/2)\, (\tfrac 12 P_g)^{\dagger }(\tfrac 12 P_f)\bigr)\) equal \(\tfrac 14\, \delta _{gf}\); reading this off on the classical index gives the diagonal \((\tfrac 14)\cdot 1\) density matrix.
For every unital \(*\)-endomorphism \(\Phi \) of \(\operatorname {Matrix}_2(\mathbb {C})\), the Pauli partition in the maximally mixed state attains the uniform cap at a single step:
By Theorem 17.62 the correlation matrix is maximally mixed on \(4\) points, whose von Neumann entropy is \(\log 4 = \log (2^{2})\).
Saturation, not identically zero. Theorem 17.63 realises issue #69’s distinction: the mechanism behind \(h_\Phi (\rho ) = 0\) is a finite reservoir \(\log (d^{2})\) that single steps can fill completely, not an absence of information at each step. The per-step rate still dies (Theorem 17.60) purely because the cap is uniform in \(n\). The witness also exhibits, honestly, that the naive single-copy ceiling \(\log d\) is false for the correlation-matrix construction: a genuinely \(d^{2}\)-dimensional density matrix occurs, so \(\log (d^{2})\) — and not \(\log d\) — is the correct reservoir.
17.7 The growing-finite quantum world (issue #70)
The finite-dimensional CNT rate vanishes identically at a fixed algebra (Theorem 17.41), so any strictly positive entropy production must come from a system whose algebra grows. This section documents the growing-finite qubit tower \(\operatorname {Matrix}_2 \hookrightarrow \operatorname {Matrix}_4 \hookrightarrow \operatorname {Matrix}_8 \hookrightarrow \cdots \): at step \(n\) the carrier is the \(n\)-fold qubit block of dimension \(2^{n}\), the distinguished state is the \(n\)-fold product \(\rho ^{\otimes n}\) of a fixed single-qubit state \(\rho \), and each step enlarges the capacity by adjoining one fresh qubit. The tower carries a spatial (per-step) entropy rate equal to the single-qubit entropy \(S(\rho )\), a Kronecker-lifted dephasing seal that survives uniformly in the block, and a base-factor non-commutativity certificate; these three faces are then bundled into a single witnessed object. The underlying additivity of the von Neumann entropy under tensor products, \(S(\rho ^{\otimes n}) = n\, S(\rho )\), is textbook (Nielsen–Chuang, Quantum Computation and Quantum Information, §11.3; Ohya–Petz, Quantum Entropy and Its Use), as is the data-processing / Petz-recovery seal (Petz 1986, 2003; Ohya–Petz) and the Connes–Narnhofer– Thirring non-commutativity against a canonical MASA (Connes–Narnhofer–Thirring, Comm. Math. Phys. 112 (1987) 691–719). The new content is the growing carrier, the capacity-enlargement embedding, the marginal-consistency of the finite blocks, the packaged rate, and the bundling.
17.7.1 The growing carrier, the product state, and the linear entropy law
The length-\(n\) qubit block index \(\mathrm{Qbits}\, n\) is defined by recursion, \(\mathrm{Qbits}\, 0 = \operatorname {Fin}1\) and \(\mathrm{Qbits}\, (n+1) = \operatorname {Fin}2\times \mathrm{Qbits}\, n\); it has cardinality \(2^{n}\), so the block algebra is \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\).
For a single-qubit state \(\rho \), the product state \(\rho ^{\otimes n}\) on the length-\(n\) block is defined by \(\rho ^{\otimes 0} = I/1\) (the unique state on \(\operatorname {Fin}1\), of zero entropy) and \(\rho ^{\otimes (n+1)} = \rho \otimes \rho ^{\otimes n}\).
The block entropy of the tower at level \(n\) is the von Neumann entropy of the product state, \(\mathrm{blockEntropy}\, \rho \, n = S(\rho ^{\otimes n})\).
The block entropy grows exactly linearly in the level:
Induct on \(n\): the von Neumann entropy is additive under the Kronecker product, \(S(\rho \otimes \rho ^{\otimes n}) = S(\rho ) + S(\rho ^{\otimes n})\), and the base block on \(\operatorname {Fin}1\) has zero entropy. Summing the single-qubit contribution \(n\) times gives \(n\cdot S(\rho )\).
17.7.2 Capacity enlargement and marginal consistency
The capacity-enlargement step is the embedding \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\hookrightarrow \operatorname {Matrix}_{2^{n+1}}(\mathbb {C})\) adjoining one fresh qubit on the left, \(A\mapsto 1\otimes A\). It is unital, multiplicative, and \(*\)-preserving — a unital \(*\)-homomorphism realising the tower inclusion — and it is injective, since \(1\otimes A\) determines \(A\) through the entry identity \((1\otimes A)_{(i,a),(i,b)} = A_{a,b}\).
The finite blocks form an honest consistent family of marginals: tracing out the fresh qubit of the length-\((n+1)\) block recovers the length-\(n\) block, \(\operatorname {Tr}_A(\rho ^{\otimes (n+1)}) = \rho ^{\otimes n}\) (Lean: \(\texttt{rhoPow\_ partialTraceLeft}\)), because \(\rho \) has unit trace. So \(\rho ^{\otimes \bullet }\) is a marginal-consistent product-state family along the capacity-enlargement embedding.
17.7.3 The spatial entropy rate and its positivity
At the maximally mixed single-qubit state \(I/2\) the linear law reads \(\mathrm{blockEntropy}\, (I/2)\, n = n\cdot \log 2\), recovering issue #70’s \(n\cdot \log 2\) claim as an equality — the maximal case, where \(S(\rho ) = \log 2\) is largest; for a general \(\rho \) the honest law is \(n\cdot S(\rho )\le n\cdot \log 2\).
Specialise Theorem 17.67 at \(\rho = I/2\), where \(S(I/2) = \log 2\).
The per-step entropy rate converges to the single-qubit entropy,
as an eventually-constant sequence (equal to \(S(\rho )\) for every \(n\ge 1\)).
By the linear law (Theorem 17.67), for \(n\ge 1\) the ratio is \(n\, S(\rho )/n = S(\rho )\); a constant sequence converges to its value.
For the concrete faithful family \(\rho _r = \tfrac 12\! \left(\begin{smallmatrix} 1 & r \\ r & 1 \end{smallmatrix}\right)\) (\(0 {\lt} r {\lt} 1\)) every nonempty block carries strictly positive entropy, \(\mathrm{blockEntropy}\, \rho _r\, n {\gt} 0\) for \(n\ge 1\); its single-qubit entropy is the binary entropy \(h_2\! \big((1+r)/2\big) {\gt} 0\), so the spatial rate is a positive constant — the tower is genuinely alive.
By the linear law the block entropy is \(n\cdot S(\rho _r)\); the single-qubit entropy equals the binary entropy \(h_2\! \big((1+r)/2\big)\), strictly positive for \(0 {\lt} r {\lt} 1\), so each nonempty block entropy is positive.
17.7.4 The Kronecker-lifted dephasing seal, uniform in the block
The single-qubit dephasing (pinching) seal of §17.5 (Theorem 17.55) is lifted from \(\operatorname {Matrix}_2(\mathbb {C})\) to \(\operatorname {Matrix}_2(\mathbb {C})\otimes \operatorname {Matrix}_{\mathrm{blk}}(\mathbb {C})\) for an arbitrary block algebra: dephase the first qubit, leave the block untouched. The strict relative-entropy drop, hence the no-recovery obstruction, survives the tensoring uniformly in the block — in particular at every level \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\) of the tower.
For a block index \(\mathrm{blk}\), the partial-dephasing channel \(\Delta \otimes \mathrm{id}_{\mathrm{blk}}\) on \(\operatorname {Matrix}_2(\mathbb {C})\otimes \operatorname {Matrix}_{\mathrm{blk}}(\mathbb {C})\) dephases the first qubit (killing its coherences) and leaves the block untouched; its Kraus operators are \(K_i\otimes 1\) for the qubit-dephasing Kraus operators \(K_i\) (the diagonal projections). It factorises on product states, \((\Delta \otimes \mathrm{id})(x\otimes \beta ) = (\Delta x)\otimes \beta \).
For a faithful block state \(\beta \) and the reference \(\sigma = \operatorname {diagState} s\) (\(0 {\lt} s {\lt} 1\)), partial dephasing of \(\rho _r\otimes \beta \) against \(\operatorname {diagState} s\otimes \beta \) strictly lowers the Umegaki relative entropy:
The block \(\beta \) is a common faithful ancilla, so the relative entropy is unchanged by it (ancilla invariance); the channel factorises on product states, reducing the claim to the qubit-level strict drop \(\log 2 - h_2\! \big((1+r)/2\big) {\gt} 0\) (the strict maximum of the binary entropy at \(1/2\)), which carries through. The reference pair has distinct dephasing images (\((I/2)\otimes \beta \ne \operatorname {diagState} s\otimes \beta \)), so the pair is non-degenerate.
For an arbitrary block \(\mathrm{blk}\) (in particular every tower level \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\)) and a faithful block state \(\beta \), no faithful-ancilla Stinespring channel on the enlarged system \(\operatorname {Matrix}_2(\mathbb {C})\otimes \operatorname {Matrix}_{\mathrm{blk}}(\mathbb {C})\) inverts the partial dephasing on \(\rho _r\otimes \beta \) and \(\operatorname {diagState} s\otimes \beta \) simultaneously.
The strict relative-entropy drop (Theorem 17.73) is uniform in the block; a Stinespring section simultaneously recovering both states would saturate the data-processing inequality, contradicting the strict drop (Petz’s equality theorem). “Uniform in \(n\)” means the block \(\mathrm{blk}\) is arbitrary, and the recovery is quantified over all faithful-ancilla Stinespring dilations on the enlarged system.
17.7.5 The bundled growing quantum world
A growing quantum world bundles, for one concrete faithful local state \(\rho _r\) and one diagonal reference \(\operatorname {diagState} s\), the three faces of the growing-finite tower:
aliveness — the single-qubit entropy \(S(\rho _r) {\gt} 0\) is positive and the per-step spatial rate \(\mathrm{blockEntropy}\, \rho _r\, n / n\) converges to it (Theorem 17.70);
the per-stage dephasing seal at the world’s own block states — at every level \(n\), no faithful-ancilla Stinespring recovery inverts the partial dephasing on both \(\rho _r\otimes \rho _r^{\otimes n}\) and \(\operatorname {diagState} s\otimes \rho _r^{\otimes n}\), the block being the world’s own length-\(n\) block \(\rho _r^{\otimes n}\) (Theorem 17.74);
a base-factor non-commutativity certificate — the single-qubit dynamics does not preserve the canonical MASA and the seal’s eigenprojection is off-diagonal (Theorem 17.57).
A concrete alive-and-sealed growing quantum world exists. Take the balanced faithful local state \(\rho _{1/2}\) and the balanced diagonal reference \(\operatorname {diagState}(1/2)\): aliveness holds because \(h_2(3/4) {\gt} 0\), the per-stage seal is Theorem 17.74 instantiated at the world’s own block state \(\rho _{1/2}^{\otimes n}\) (positive definite), and the base-factor certificate is Theorem 17.57.
Assemble the three faces at \(r = s = 1/2\): positivity of the single-qubit entropy from \(h_2(3/4) {\gt} 0\); the per-stage seal from Theorem 17.74 with the block taken to be the world’s own \(\rho _{1/2}^{\otimes n}\) (positive definite as a Kronecker product of positive-definite factors); and the base-factor non-commutativity certificate directly.
Growing-finite, not the completed chain (issue #71). This is deliberately the growing-finite tower — one fresh qubit adjoined per step — and not the completed thermodynamic-limit spin chain \(\bigotimes _{\mathbb {Z}}\operatorname {Matrix}_2(\mathbb {C})\); that idealisation is the subject of issue #71 and is not formalised here.
Spatial versus temporal rate — no tension with Theorem 17.41. The rate \(S(\rho )\) above is a spatial (per-step capacity-growth) rate: the algebra grows by one fresh qubit per step. It does not contradict the temporal fixed-dimension vanishing \(h_\Phi (\rho ) = 0\) (Theorem 17.41), which concerns a fixed algebra under iteration of a fixed dynamics. Fixed algebra with trivial dynamics gives zero temporal rate; growing algebra gives positive spatial rate — two different rates of two different systems.
The base-factor certificate is transported, not re-proved per level. The non-commutativity certificate (Theorem 17.57) certifies the base qubit factor. Via the capacity-enlargement embedding \(\mathrm{shiftAdjoinQubit}\) (\(A\mapsto 1\otimes A\), Definition 17.68) it embeds into every level of the tower, but it is not a re-proved per-level MASA statement — it is the base-factor certificate transported by the tower embedding, the honest reading of the bundled world’s third face.
17.8 The quantum Bernoulli shift and the finite modular clock (issue #71)
The growing-finite tower of §17.7 enlarges the algebra one qubit per step; here the same fixed hierarchy of finite qubit blocks \(\operatorname {Matrix}_2(\mathbb {C})\hookrightarrow \operatorname {Matrix}_4(\mathbb {C})\hookrightarrow \operatorname {Matrix}_8(\mathbb {C})\hookrightarrow \cdots \) is read as the carrier of the one-sided quantum Bernoulli shift — the shift endomorphism of the qubit chain — and its temporal entropy rate is extracted. The two structural maps on the levels \(\mathrm{Qbits}\, n\) (of cardinality \(2^{n}\), local algebra \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\)) are the inclusion \(\iota _n:A_n\hookrightarrow A_{n+1}\), \(x\mapsto x\otimes 1\), adjoining a fresh site at the far end, and the capacity-enlargement shift \(\mathrm{shiftAdjoinQubit}:A_n\hookrightarrow A_{n+1}\), \(A\mapsto 1\otimes A\) (Definition 17.68), adjoining one fresh qubit on the left. These commute, so the finite levels form a genuine directed system carrying the shift; the tracial (maximally mixed) state is compatible with both maps and shift-invariant; the site-window filtration along the shift has temporal rate \(\log 2\); the dephasing seal survives at every level; and the finite modular clock \(\sigma _t(a)=\rho ^{it}\, a\, \rho ^{-it}\) exhibits an intrinsic-clock dichotomy between the tracial and non-tracial states. All of this is assembled into one witnessed object. The shift entropy of the qubit chain equals the entropy density \(\log 2\) (Connes–Narnhofer–Thirring, Dynamical entropy of \(C^{*}\) algebras and von Neumann algebras, Comm. Math. Phys. 112 (1987) 691–719); the quasi-local (inclusion \(+\) shift) framework of the chain is standard (Bratteli–Robinson, Operator Algebras and Quantum Statistical Mechanics II, §5.3, Prop. 5.3.7; Ohya–Petz, Quantum Entropy and Its Use, §1.3), the hyperfinite II\(_1\) and the Powers-type type-III product-state factors going back to Powers (R. T. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. 86 (1967)). The new content is the packaging on the finite growing tower: the commuting inclusion/shift, the shift-invariant tracial hierarchy, the site-window rate, the per-level seal, the finite modular clock with its dichotomy, and the bundled witness.
This is the directed system of finite levels with compatible tracial marginals and the shift endomorphism, not the completed quasi-local \(C^{*}\)-algebra \(\bigotimes _{\mathbb {Z}}\operatorname {Matrix}_2(\mathbb {C})\) or its GNS representation. Mathlib has no noncommutative inductive limit (Ring.DirectLimit is CommRing-only, and the matrix algebras here are noncommutative), so the completed chain is deliberately not formed; the directed local system is all that the entropy-rate and modular statements below require.
17.8.1 The directed local system: commuting inclusion and shift
The inclusion \(A_n\hookrightarrow A_{n+1}\) appends a fresh qubit at the far end, \(x\mapsto x\otimes 1\), reindexed along \(\mathrm{Equiv.prodComm}\) because \(\mathrm{Qbits}\, (n+1)\) places the fresh factor on the left. It is unital, multiplicative and \(*\)-preserving — a unital \(*\)-homomorphism realising the tower inclusion.
The far-end inclusion is injective: \(x\otimes 1\) (reindexed) determines \(x\) through the entry identity \((\mathrm{appendQubit}\, x)_{(0,a),(0,b)} = x_{a,b}\).
Read off the \((0,a),(0,b)\) entry of \(x\otimes 1\): the identity factor contributes \(1_{0,0}=1\), leaving \(x_{a,b}\). Two inclusions with equal images therefore have equal entries.
The capacity-enlargement shift \(\mathrm{shiftAdjoinQubit}\) (\(A\mapsto 1\otimes A\)) and the far-end inclusion \(\mathrm{appendQubit}\) (\(x\mapsto x\otimes 1\)) commute:
Hence the family of finite levels is a genuine directed system carrying the shift as an endomorphism.
Both sides realise the same entrywise triple Kronecker product \(1\otimes M\otimes 1\): the shift amplifies on the left and the inclusion on the right, and these two amplifications act on disjoint index factors, so their composites agree slot by slot.
17.8.2 The shift-invariant product-state hierarchy
The maximally mixed state is closed under the Kronecker product, \(\mathrm{mm}\otimes \mathrm{mm}=\mathrm{mm}\): concretely \((a^{-1}\cdot 1)\otimes (b^{-1}\cdot 1)=(a\, b)^{-1}\cdot 1\) with \(a,b\) the factor cardinalities.
Expand both maximally mixed factors as normalized identities; the Kronecker product of identities is the identity, and the scalar prefactors multiply to \((a\, b)^{-1}\), which is exactly the normalizing constant of the maximally mixed state on the product carrier.
The \(n\)-fold product of the maximally mixed single-qubit state is the maximally mixed state on the whole length-\(n\) block, \(\mathrm{rhoPow}\, \mathrm{mm}\, n=\mathrm{mm}\): the tracial state is a fixed point of the tower construction.
Writing \(\tau _m(y)=\operatorname{tr}\bigl((\mathrm{rhoPow}\, \rho \, m)\cdot y\bigr)\), pairing the enlarged state against an observable \(\mathrm{shiftAdjoinQubit}\, x=1\otimes x\) that acts trivially on the fresh qubit returns the previous-block pairing:
The family \((\tau _n)\) thus fits together into one shift-invariant state of the one-sided qubit chain.
The product state factors as \(\rho \otimes (\mathrm{rhoPow}\, \rho \, n)\) and the observable as \(1\otimes x\); the Kronecker product multiplies factorwise and the trace factorises, so the fresh qubit contributes \(\operatorname{tr}(\rho \cdot 1)=\operatorname{tr}\rho =1\), leaving \(\operatorname{tr}((\mathrm{rhoPow}\, \rho \, n)\cdot x)\).
The far-end site adjoined by \(\mathrm{appendQubit}\) is a fresh, maximally mixed degree of freedom, so pairing the \((n+1)\)-level maximally mixed state against \(\mathrm{appendQubit}\, x\) reproduces the \(n\)-level pairing against \(x\):
This is the compatible-marginals condition for the tracial state along the inclusion tower.
The trace of \(x\otimes 1\) is \(2\, \operatorname{tr}x\) (the fresh far-end factor contributes \(\operatorname{tr}1=2\)), and the maximally mixed normalizations differ by exactly the factor \(2 = 2^{n+1}/2^{n}\); the two cancel, leaving the level-\(n\) tracial pairing.
The \(k\)-fold shift \(A_n\hookrightarrow A_{n+k}\) iterates the capacity-enlargement embedding \(\mathrm{shiftAdjoinQubit}\) (\(A\mapsto 1\otimes A\)) \(k\) times, by the recursion \(\mathrm{shiftIter}\, 0=\mathrm{id}\) and \(\mathrm{shiftIter}\, (k+1)=\mathrm{shiftAdjoinQubit}\circ \mathrm{shiftIter}\, k\). It is a unital \(*\)-homomorphism and is injective.
The fixed state \(\mathrm{rhoPow}\, \rho \), read through \(k\) shift-iterates of a level-\(n\) observable \(x\), returns the level-\(n\) marginal:
This is the shift-invariance of the whole hierarchy — the temporal stationarity of the quantum Bernoulli shift (“the past slides out”): a level-\(n\) observable, however far the shift carries its window, is still evaluated by the level-\(n\) state.
Induct on \(k\): each shift step peels off one application of Theorem 17.83, lowering the level by one at unchanged value, and the inductive hypothesis carries the level-\(n+k\) pairing down to the level-\(n\) marginal.
17.8.3 The temporal site-window rate
The window entropy \(\mathrm{windowEntropy}\, \rho \, k\) is the von Neumann entropy of the length-\(k\) window of the chain, generated by the first \(k\) sites. Numerically it equals the block entropy of §17.7 (\(\mathrm{windowEntropy}\, \rho \, k=\mathrm{blockEntropy}\, \rho \, k\)); the temporal content is the reading — the window of the shift filtration on the fixed, shift-invariant state hierarchy (Theorem 17.86) — not the value.
At the tracial (maximally mixed) state the length-\(k\) window carries entropy \(\mathrm{windowEntropy}\, \mathrm{mm}\, k = k\cdot \log 2\).
The per-window rate \(\mathrm{windowEntropy}\, \mathrm{mm}\, k / k\) is the constant \(\log 2\) for every \(k\ge 1\), hence converges to \(\log 2\):
This is the site-window entropy rate of the quantum Bernoulli shift at the tracial state.
By Theorem 17.88 the numerator is \(k\cdot \log 2\), so for \(k\ge 1\) the ratio is the constant \(\log 2\); a constant sequence converges to its value.
\(\mathrm{windowEntropy}\) is numerically the block entropy of issue #70 (Definition 17.87); the temporal content is the fixed state hierarchy, its shift-invariance (Theorem 17.86), and the shift-window filtration — not the numerical value. Correspondingly the rate of Theorem 17.89 is the site-window filtration rate, a bespoke CNT-style definition, and not the full CNT dynamical entropy (a supremum over all finite operational partitions, Definition 17.26).
17.8.4 The per-level dephasing seal at the tracial blocks
For the faithful local state \(\rho _r\) and diagonal reference \(\operatorname {diagState} s\) (\(0{\lt}r,s{\lt}1\)), the predicate \(\mathrm{ChainSealed}\) asserts that at every level \(n\) the partial-dephasing channel on \(\operatorname {Matrix}_2(\mathbb {C})\otimes \operatorname {Matrix}_{2^{n}}(\mathbb {C})\) admits no faithful-ancilla Stinespring recovery that simultaneously inverts it on the coherent state \(\rho _r\otimes \mathrm{mm}_n\) and on the reference \(\operatorname {diagState} s\otimes \mathrm{mm}_n\), where the block is the tracial (maximally mixed) block state \(\mathrm{mm}_n\). This is the chain-level, per-stage (channel-level) seal: single-step per stage, not a flow seal, with the recovery quantified over all faithful-ancilla Stinespring dilations.
For all faithful parameters \(0{\lt}r,s{\lt}1\), the chain-seal predicate holds: \(\mathrm{ChainSealed}\, r\, s\). Dephasing site \(0\) of the chain at the tracial block admits no faithful-ancilla Stinespring recovery, at every level \(n\).
Instantiate the uniform-in-the-block seal (Theorem 17.74) at the tracial (maximally mixed) block state \(\mathrm{mm}_n\) on \(\mathrm{Qbits}\, n\), which is positive definite; the strict relative-entropy drop of the partial dephasing then obstructs any simultaneous Stinespring recovery at that level.
17.8.5 The finite modular clock
For a faithful (positive-definite) density matrix \(\rho \), the modular automorphism \(\sigma _t(a)=\rho ^{it}\, a\, \rho ^{-it}\) is built from the unitary power \(\rho ^{it}\) of the Tomita–Takesaki / Petz-equality infrastructure. It is the finite-dimensional shadow of the type-III modular flow of the qubit chain.
The modular flow is a one-parameter group, \(\sigma _s\circ \sigma _t=\sigma _{s+t}\):
Together with \(\sigma _0=\mathrm{id}\), multiplicativity \(\sigma _t(a\, b)=\sigma _t(a)\, \sigma _t(b)\) and the adjoint law \(\sigma _t(a^{\dagger })=\sigma _t(a)^{\dagger }\), this makes \((\sigma _t)_t\) a one-parameter \(*\)-automorphism group.
Expand \(\sigma _s(\sigma _t(a))=\rho ^{is}\rho ^{it}\, a\, \rho ^{-it}\rho ^{-is}\); the unitary powers add, \(\rho ^{is}\rho ^{it}=\rho ^{i(s+t)}\) and \(\rho ^{-it}\rho ^{-is}=\rho ^{-i(s+t)}\), giving \(\sigma _{s+t}(a)\).
For faithful \(\rho \), with the Bratteli–Robinson II §5.3 convention \(\sigma _{-i}(y)=\rho \, y\, \rho ^{-1}\),
Substitute \(\sigma _{-i}(y)=\rho \, y\, \rho ^{-1}\) and use trace cyclicity together with \(\rho ^{-1}\rho =1\): the left side is \(\operatorname{tr}(\rho \, x\, \rho \, y\, \rho ^{-1})\), and cycling \(\rho ^{-1}\) to the front cancels one \(\rho \), leaving \(\operatorname{tr}(\rho \, y\, x)\).
The identity of Theorem 17.95 is nothing more than trace cyclicity together with \(\rho \, \rho ^{-1}=1\), so it holds for every invertible \(\rho \); it is the boundary form of the KMS condition, not the substance of modular theory. Not delivered here: the Tomita–Takesaki uniqueness of the KMS one-parameter group, and the genuine type-III modular theory of the completed \(C^{*}\)-chain (GNS is Mathlib-absent). The modular-theoretic content of this section lives in the group / tower-compatibility laws and in the intrinsic-clock dichotomy below.
The intrinsic clock is consistent along the chain: on the embedded block \(\mathrm{shiftAdjoinQubit}\, a=1\otimes a\), the modular flow of the \((n+1)\)-fold product state acts by the modular flow of the \(n\)-fold state on \(a\),
because \(\rho ^{it}\) acts trivially on the fresh qubit.
The unitary power of a Kronecker product factorises, \((\rho \otimes \rho ^{\otimes n})^{it} =\rho ^{it}\otimes (\rho ^{\otimes n})^{it}\); conjugating \(1\otimes a\) by it, the first-factor \(\rho ^{it}\) cancels against \(\rho ^{-it}\) around the identity block, leaving \(1\otimes \bigl((\rho ^{\otimes n})^{it}\, a\, (\rho ^{\otimes n})^{-it}\bigr)\).
The maximally mixed (tracial) state has trivial modular flow, \(\sigma _t=\mathrm{id}\) for all \(t\):
In particular this holds at every level \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\) of the chain — the type II\(_1\) shadow.
The maximally mixed state is a scalar multiple of the identity, \(\rho =c\cdot 1\), so its unitary power \(\rho ^{it}\) is again a scalar (a constant diagonal), hence central; it commutes past \(a\), and \(\rho ^{it}\rho ^{-it}=1\) leaves \(a\).
A faithful non-tracial (Powers-type) product state \(\operatorname {diagState} s\) (\(0{\lt}s{\lt}1\), \(\rho =\operatorname {diag}((1+s)/2,(1-s)/2)\)) has a nontrivial modular flow: the modular group is not the trivial action. Concretely, at \(t_0=\pi /\log \bigl((1+s)/(1-s)\bigr)\) the flow sends the off-diagonal unit \(E_{01}\) to \(-E_{01}\).
On the off-diagonal unit \(E_{01}\) the flow multiplies by the phase \(\exp \! \bigl(it\, \log ((1+s)/(1-s))\bigr)\); at \(t_0=\pi /\log ((1+s)/(1-s))\) this phase is \(e^{i\pi }=-1\ne 1\), so \(\sigma _{t_0}(E_{01})=-E_{01}\ne E_{01}\). This is the finite shadow of the type-III character of the Powers factors; nontriviality is transported up the tower by combining it with Theorem 17.97.
17.8.6 The bundled quantum Bernoulli shift
A quantum Bernoulli shift bundles the fixed hierarchy of finite marginals of the one-sided qubit chain into one certificate:
the directed system — the inclusion \(\mathrm{appendQubit}\) and the shift \(\mathrm{shiftAdjoinQubit}\) commute (Theorem 17.80);
the inclusion- and shift-compatibility of the tracial state (Theorems 17.84 and 17.83);
the temporal entropy rate \(\log 2\) along the shift-window filtration (Theorem 17.89);
the per-stage dephasing seal at the tracial blocks, for some faithful parameters \(r,s\) (Definition 17.91, Theorem 17.92);
the trivial tracial modular clock at every level (Theorem 17.98).
The maps are definitionally the fixed \(\mathrm{appendQubit}\) / \(\mathrm{shiftAdjoinQubit}\), so only their laws are recorded.
A concrete quantum Bernoulli shift exists. The witness assembles the already-proved theorems: the inclusion/shift commutation, the tracial inclusion- and shift-compatibility, the temporal rate \(\log 2\), the per-stage seal at the balanced parameters \(r=s=1/2\), and the trivial tracial clock.