diff --git a/book/The-Incompleteness-of-Observation-FULL.md b/book/The-Incompleteness-of-Observation-FULL.md index c46f2d40a..5b94af0bd 100644 --- a/book/The-Incompleteness-of-Observation-FULL.md +++ b/book/The-Incompleteness-of-Observation-FULL.md @@ -5882,7 +5882,7 @@ $$\Phi\!\left(X^{a}Z^{b}\right) = \big[\,Ca = 0\,\big]\,\big[\,B^{T}A^{-T}b = 0\ *Proof.* The framework's $\Phi_t$ are permutation dilations with uniform ancilla, so they map computational-diagonal states to exactly diagonal states (a permutation unitary conjugates a diagonal state to a diagonal state; the partial trace preserves diagonality). If $\Phi_{t_2} = \Lambda \circ \Phi_{t_1}$ with $\Lambda$ CPTP, then $M_{kj} := \langle j|\Lambda(|k\rangle\langle k|)|j\rangle$ is stochastic and, by the diagonality of $\Phi_{t_1}(|i\rangle\langle i|)$, $T(t_2) = T(t_1)\,M$ — P-divisibility of the population process. The contrapositive gives the result. For a general CPTP family the diagonal-input reduction fails — a Hadamard pair is CP-divisible with P-indivisible populations — so the diagonal-preservation property is what licenses the step. $\square$ -CP-indivisibility is completely witnessed by ancilla-assisted distinguishability ([Main §3.4]); ordinary system-only trace-distance revival — the Breuer-Laine-Piilo *information backflow* signature — is sufficient evidence of non-Markovianity, not a necessary consequence of CP-indivisibility. The framework's content is therefore that the C1–C4/readback sector is necessarily memory-bearing — non-Markovian, with information backflow at observable timescales — while fixed-basis quantum representability itself is universal, not confined to that sector ([Main §3.4]). +For bijective (invertible) dynamical maps, CP-indivisibility is completely witnessed by ancilla-assisted distinguishability ([Main §3.2]); ordinary system-only trace-distance revival — the Breuer-Laine-Piilo *information backflow* signature — is sufficient evidence of non-Markovianity, not a necessary consequence of CP-indivisibility. The framework's content is therefore that the C1–C4/readback sector is necessarily memory-bearing — non-Markovian, with information backflow at observable timescales — while fixed-basis quantum representability itself is universal, not confined to that sector ([Main §3.4]). **Approximate unitarity.** On observable timescales $t \ll \tau_B$, the hidden-sector state is approximately frozen (conditions C2-C3). At $t=0$ the channel generator is exactly Hamiltonian, $$\frac{d\Phi_t}{dt}\bigg|_{t=0}(\rho_V) = -i[\hat{H}_{\text{eff}}, \rho_V],$$ diff --git a/book/The-Incompleteness-of-Observation-FULL.pdf b/book/The-Incompleteness-of-Observation-FULL.pdf index ae1cb7b2f..1f721bb4d 100644 Binary files a/book/The-Incompleteness-of-Observation-FULL.pdf and b/book/The-Incompleteness-of-Observation-FULL.pdf differ diff --git a/book/The-Incompleteness-of-Observation-FULL.tex b/book/The-Incompleteness-of-Observation-FULL.tex index 75f5c0a84..9c0672d16 100644 --- a/book/The-Incompleteness-of-Observation-FULL.tex +++ b/book/The-Incompleteness-of-Observation-FULL.tex @@ -26315,15 +26315,16 @@ \subsection{B.2 The Stinespring construction for emergent quantum the diagonal-preservation property is what licenses the step. \(\square\) -CP-indivisibility is completely witnessed by ancilla-assisted -distinguishability ({[}Main §3.4{]}); ordinary system-only -trace-distance revival --- the Breuer-Laine-Piilo \emph{information -backflow} signature --- is sufficient evidence of non-Markovianity, not -a necessary consequence of CP-indivisibility. The framework's content is -therefore that the C1--C4/readback sector is necessarily memory-bearing ---- non-Markovian, with information backflow at observable timescales ---- while fixed-basis quantum representability itself is universal, not -confined to that sector ({[}Main §3.4{]}). +For bijective (invertible) dynamical maps, CP-indivisibility is +completely witnessed by ancilla-assisted distinguishability ({[}Main +§3.2{]}); ordinary system-only trace-distance revival --- the +Breuer-Laine-Piilo \emph{information backflow} signature --- is +sufficient evidence of non-Markovianity, not a necessary consequence of +CP-indivisibility. The framework's content is therefore that the +C1--C4/readback sector is necessarily memory-bearing --- non-Markovian, +with information backflow at observable timescales --- while fixed-basis +quantum representability itself is universal, not confined to that +sector ({[}Main §3.4{]}). \textbf{Approximate unitarity.} On observable timescales \(t \ll \tau_B\), the hidden-sector state is approximately frozen @@ -30537,4 +30538,4 @@ \subsection{Historical and philosophical {[}Origin of the molecular clock hypothesis.{]} \end{document} -% source-sha256: 4f3591df482d5bd8978047945c2aad8ea6784018e030d4ed620c3ccf004fd21c +% source-sha256: b641c6f7fa54c0b508a17da7b0941c0ab59f70877d940a36d3d86d3b1270da59 diff --git a/book/appendix-b-derivations.md b/book/appendix-b-derivations.md index 8c7c6cd91..d963a1c46 100644 --- a/book/appendix-b-derivations.md +++ b/book/appendix-b-derivations.md @@ -93,7 +93,7 @@ $$\Phi\!\left(X^{a}Z^{b}\right) = \big[\,Ca = 0\,\big]\,\big[\,B^{T}A^{-T}b = 0\ *Proof.* The framework's $\Phi_t$ are permutation dilations with uniform ancilla, so they map computational-diagonal states to exactly diagonal states (a permutation unitary conjugates a diagonal state to a diagonal state; the partial trace preserves diagonality). If $\Phi_{t_2} = \Lambda \circ \Phi_{t_1}$ with $\Lambda$ CPTP, then $M_{kj} := \langle j|\Lambda(|k\rangle\langle k|)|j\rangle$ is stochastic and, by the diagonality of $\Phi_{t_1}(|i\rangle\langle i|)$, $T(t_2) = T(t_1)\,M$ — P-divisibility of the population process. The contrapositive gives the result. For a general CPTP family the diagonal-input reduction fails — a Hadamard pair is CP-divisible with P-indivisible populations — so the diagonal-preservation property is what licenses the step. $\square$ -CP-indivisibility is completely witnessed by ancilla-assisted distinguishability ([Main §3.4]); ordinary system-only trace-distance revival — the Breuer-Laine-Piilo *information backflow* signature — is sufficient evidence of non-Markovianity, not a necessary consequence of CP-indivisibility. The framework's content is therefore that the C1–C4/readback sector is necessarily memory-bearing — non-Markovian, with information backflow at observable timescales — while fixed-basis quantum representability itself is universal, not confined to that sector ([Main §3.4]). +For bijective (invertible) dynamical maps, CP-indivisibility is completely witnessed by ancilla-assisted distinguishability ([Main §3.2]); ordinary system-only trace-distance revival — the Breuer-Laine-Piilo *information backflow* signature — is sufficient evidence of non-Markovianity, not a necessary consequence of CP-indivisibility. The framework's content is therefore that the C1–C4/readback sector is necessarily memory-bearing — non-Markovian, with information backflow at observable timescales — while fixed-basis quantum representability itself is universal, not confined to that sector ([Main §3.4]). **Approximate unitarity.** On observable timescales $t \ll \tau_B$, the hidden-sector state is approximately frozen (conditions C2-C3). At $t=0$ the channel generator is exactly Hamiltonian, $$\frac{d\Phi_t}{dt}\bigg|_{t=0}(\rho_V) = -i[\hat{H}_{\text{eff}}, \rho_V],$$ diff --git a/papers/Main.md b/papers/Main.md index dff869152..d59e5b5f8 100644 --- a/papers/Main.md +++ b/papers/Main.md @@ -329,7 +329,7 @@ $$\Phi\!\left(X^{a}Z^{b}\right) = \big[\,Ca = 0\,\big]\,\big[\,B^{T}A^{-T}b = 0\ *Remark (scope: P-indivisibility is not quantum non-Markovianity).* For a general CPTP family the reduction fails: $\Phi_{t_1} = \mathrm{Ad}_H$ (Hadamard), $\Phi_{t_2} = \mathrm{id}$ is CP-divisible with a unitary intermediate map, while the computational-basis populations $\tfrac{1}{2}J \to I$ are P-indivisible — the intermediate map consumes coherences invisible to $T(t_1)$. Stochastic P-indivisibility must therefore not be equated with open-system non-Markovianity outside the diagonal-preserving class; the framework's channels are in that class, which is what licenses the theorem. (Certified exactly over $\mathbb{Q}(\sqrt{2})$ in `c1_cp_scope_probes.py`.) -By the Breuer-Laine-Piilo criterion [63], CP-indivisibility is witnessed by non-monotonic distinguishability, but the correct witness is the *ancilla-assisted* one [4]: monotonicity of system-plus-ancilla distinguishability characterizes CP-divisibility, while ordinary system trace distance is a strictly weaker witness and can stay monotone on a CP-indivisible family. The classical analogue is exact and is certified here: a family can be P-indivisible while total-variation distance is non-increasing for every pair of visible distributions (`translation_probes.py`, §2.3). Backflow in the system alone is therefore sufficient evidence of indivisibility, never necessary. +By the Breuer-Laine-Piilo criterion [63], CP-indivisibility is witnessed by non-monotonic distinguishability, but the correct witness is *ancilla-assisted*: the entanglement of the system with a shielded, maximally entangled ancilla decreases monotonically under CP-divisible evolution, so its increase witnesses CP-indivisibility [4], and for bijective (invertible) dynamical maps on $\mathbb{C}^d$, CP-divisibility is equivalent to monotonic non-increase of the trace distance between every pair of equiprobable states of the system and a $(d+1)$-dimensional ancilla [9]; ordinary system trace distance is a strictly weaker witness and can stay monotone on a CP-indivisible family. The classical analogue is exact and is certified here: a family can be P-indivisible while total-variation distance is non-increasing for every pair of visible distributions (`translation_probes.py`, §2.3). Backflow in the system alone is therefore sufficient evidence of indivisibility, never necessary. *Remark (three notions, kept apart).* Three distinct things travel under nearby names, and this paper uses them in fixed senses. (1) **Accessible finite-horizon memory**, equivalently information backflow in the visible law: this is the framework's own notion, what the characterization theorem is about, what $M_t > 0$ measures, and what (C1), (C3) and (C4) diagnose on a finite horizon. Where this paper writes "accessible non-Markovianity" it means exactly this and says nothing about a quantum channel. (2) **Stochastic P-indivisibility**: failure of stochastic divisibility of the visible population law, which on one fixed finite reversible representative is what the recurrence-scale theorem delivers — resting on recurrence together with a non-permutation rooted map, not on (C1), (C3) and (C4) alone. (3) **Generic quantum non-Markovianity**: an open-system notion whose content depends on which quantum divisibility or backflow criterion is adopted. No unconditional implication among the three is asserted here. The two preceding remarks give the exact conditions under which (2) constrains a quantum intermediate map, and the Hadamard family shows that outside the diagonal-preserving class (2) and (3) come apart. @@ -603,7 +603,7 @@ is **false**: the process-dilation construction of §3.4 realizes Markov laws as *Remark (which condition is primitive).* Of the three structural conditions, only (C4) is logically independent. (C1) follows from it — a readback gap requires hidden mediation, and zero-coupling realizations exhibit no gap at any order — and (C3) follows by data processing, with the bound $\log_2|\mathcal{C}_H| \geq I^*$ saturated by realizations that attain it; both are certified by exhaustive enumeration in `primitive_probes.py`. The equivalence could therefore be stated with (C4) alone. It is not, deliberately: (C1) and (C3) are independently *checkable* — Bell-class dynamics force (C1) directly, capacity arguments use (C3) directly — and a characterization that lists checkable consequences alongside its primitive is more useful than a minimal one, in the way that axiom lists for groups are conventionally redundant. The dependency is recorded at each condition's definition (§1.3).* -*Remark (two notions of memory).* Non-Markovianity of the process — condition (ii) — and P-indivisibility of the transition-matrix family are distinct, and the distinction mirrors the quantum-side gap between process-tensor non-Markovianity and CP-divisibility (§3.2). Markov implies P-divisible (chain factorization), so P-indivisibility is the strictly stronger property; the strictness is witnessed by an explicit family — $X_1$ uniform independent of $X_0$, $X_2 = X_0 \oplus X_1$ — with maximal conditional memory ($I(X_0; X_2 \mid X_1) = 1$ bit) whose matrix family factors through $\Lambda = I$ (`review4_probes.py`). The framework's accessible-timescale claim is (ii). P-indivisibility enters as the stronger property in two ways: unconditionally at the recurrence scale from a non-permutation one-step witness (§2.3), and empirically — P-indivisible dynamics are documented in open-system experiments, the imported correspondence reads Bell-class dynamics as indivisible, and (ii) itself is certified by directly observed memory effects. +*Remark (two notions of memory).* Non-Markovianity of the process — condition (ii) — and P-indivisibility of the transition-matrix family are distinct, and the distinction mirrors the quantum-side gap between process-tensor non-Markovianity and CP-divisibility (§3.2). Markov implies P-divisible (chain factorization), so P-indivisibility is the strictly stronger property; the strictness is witnessed by an explicit family — $X_1$ uniform independent of $X_0$, $X_2 = X_0 \oplus X_1$ — with maximal conditional memory ($I(X_0; X_2 \mid X_1) = 1$ bit) whose matrix family factors through $\Lambda = I$ (`review4_probes.py`; up to flipping the final bit, the process of [5], Eq. (15)). The framework's accessible-timescale claim is (ii). P-indivisibility enters as the stronger property in two ways: unconditionally at the recurrence scale from a non-permutation one-step witness (§2.3), and empirically — P-indivisible dynamics are documented in open-system experiments, the imported correspondence reads Bell-class dynamics as indivisible, and (ii) itself is certified by directly observed memory effects. *Proof.* (iii) $\implies$ (ii): the readback lemma of §2.3 — (iii) includes (C4), whose gap forces $I(X_{ 0$ at an accessible order. (ii) $\implies$ (iii): existence, per accessible horizon containing the witness order $k$, by the finite-horizon process-dilation theorem above, which reproduces the full multi-time law; the realization inherits (C4) at $k$ directly — the ledger-conditioned kernels reproduce $\mathcal{S}$'s differing past-conditionals exactly (`c4_backflow_probes.py`) — and satisfies the remaining conditions: (C1) because a Markov-breaking kernel dependence requires hidden influence at that step (with none anywhere, every visible transition is a fixed function of the visible present and the process is Markov, contradicting (ii)); (C2) trivially in the construction — the bath is static, $\tau_B = \infty$, satisfying $\tau_S \ll \tau_B$; for *physical* hidden sectors the physical-memory theorem's contrapositive (§2.3, the C2 process form) supplies the necessity direction — observed conditional memory bounds the mixing residual below — conditional on the conditional-mixing-curve hypothesis with finite $\tau_B(\varepsilon_0)$ (Remark, §3.4; ETH [17, 18] the motivation; no smallness of $\varepsilon$ at the coupling time is assumed); (C3) from the data-processing bound in per-process form ($m \geq 2^{I^*}$). Both directions are proved here in prose and neither is formalized in the kernel; since (C4) coincides with (ii) in every faithful realization (Remark, *the conditions are diagnostics, not hypotheses*), what (ii) $\implies$ (iii) adds to (ii) is the existence of a realization, and (iii) $\implies$ (ii) is the readback lemma. Finally, a unitarily evolving representation is available for $\mathcal{S}$ regardless — internally by the permutation unitary of any realization (the equivalence above), and under (T) by the imported correspondence [11, 12], which represents it by a unitarily evolving quantum system; under the time-homogeneity restriction the representation takes the fixed-$\hat{H}$ ancilla-marginal form of (i) (scope Remark) — the paper's single external import, sitting outside the equivalence. $\square$ diff --git a/papers/Main.pdf b/papers/Main.pdf index 6ce448cb7..0ffedc6cd 100644 Binary files a/papers/Main.pdf and b/papers/Main.pdf differ diff --git a/papers/Main.tex b/papers/Main.tex index da8de9b47..e10dac70c 100644 --- a/papers/Main.tex +++ b/papers/Main.tex @@ -1956,15 +1956,19 @@ \subsubsection{3.2 Independent Derivation via Stinespring By the Breuer-Laine-Piilo criterion {[}63{]}, CP-indivisibility is witnessed by non-monotonic distinguishability, but the correct witness -is the \emph{ancilla-assisted} one {[}4{]}: monotonicity of -system-plus-ancilla distinguishability characterizes CP-divisibility, -while ordinary system trace distance is a strictly weaker witness and -can stay monotone on a CP-indivisible family. The classical analogue is -exact and is certified here: a family can be P-indivisible while -total-variation distance is non-increasing for every pair of visible -distributions (\texttt{translation\_probes.py}, §2.3). Backflow in the -system alone is therefore sufficient evidence of indivisibility, never -necessary. +is \emph{ancilla-assisted}: the entanglement of the system with a +shielded, maximally entangled ancilla decreases monotonically under +CP-divisible evolution, so its increase witnesses CP-indivisibility +{[}4{]}, and for bijective (invertible) dynamical maps on +\(\mathbb{C}^d\), CP-divisibility is equivalent to monotonic +non-increase of the trace distance between every pair of equiprobable +states of the system and a \((d+1)\)-dimensional ancilla {[}9{]}; +ordinary system trace distance is a strictly weaker witness and can stay +monotone on a CP-indivisible family. The classical analogue is exact and +is certified here: a family can be P-indivisible while total-variation +distance is non-increasing for every pair of visible distributions +(\texttt{translation\_probes.py}, §2.3). Backflow in the system alone is +therefore sufficient evidence of indivisibility, never necessary. \emph{Remark (three notions, kept apart).} Three distinct things travel under nearby names, and this paper uses them in fixed senses. (1) @@ -3898,7 +3902,8 @@ \subsubsection{3.4 The Characterization Theorem: Necessity of the explicit family --- \(X_1\) uniform independent of \(X_0\), \(X_2 = X_0 \oplus X_1\) --- with maximal conditional memory (\(I(X_0; X_2 \mid X_1) = 1\) bit) whose matrix family factors through -\(\Lambda = I\) (\texttt{review4\_probes.py}). The framework's +\(\Lambda = I\) (\texttt{review4\_probes.py}; up to flipping the final +bit, the process of {[}5{]}, Eq. (15)). The framework's accessible-timescale claim is (ii). P-indivisibility enters as the stronger property in two ways: unconditionally at the recurrence scale from a non-permutation one-step witness (§2.3), and empirically --- @@ -5326,4 +5331,4 @@ \subsection{REFERENCES}\label{references}} \emph{Phys. Rev.~Lett.} \textbf{103}, 210401 (2009); arXiv:0908.0238. \end{document} -% source-sha256: 09dad693a9f5dfc44b485c08280d44e93eef8a4f8ca2b015284ac5c1293ab503 +% source-sha256: f694baeb3b9d59b2098ad87ef61172841ca9fa0bc89088150030527ef23098d3