Product-Level Trace-Dobrushin Theory

• Product-level trace-Dobrushin theory is a quantitative framework that measures how quantum channel products forget initial states via trace-norm contractions.
• The theory transfers localized contraction effects into global phenomena such as replacement by rank-one channels, rapid mixing, and decay of conditional mutual information.
• It underpins both deterministic and random quantum dynamics, facilitates energy-constrained optimizations, and clarifies correlation clustering in many-body states.

Searching arXiv for the cited papers to ground the article in the current literature. Product-level trace-Dobrushin theory is a framework for quantifying how products of channels or weakly dependent update rules forget initial data in trace norm and for transferring that forgetting to structural consequences such as replacement by rank-one channels, rapid mixing, boundary-state selection, and correlation or information decay. In the finite-dimensional quantum setting, the centered trace-Dobrushin coefficient of a CPTP map $\Phi$ is defined by

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$

equivalently,

$$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$

so that $2\,\kappa_{\rm tr}(\Phi)$ is exactly the trace-distance diameter of $\Phi(\mathcal S_d)$ (Pathirana, 30 Apr 2026). In the broader Dobrushin tradition, analogous coefficient matrices or interdependence operators control approximate tensorization, logarithmic-Sobolev inequalities, path-coupling contractions, and stretched-exponential concentration under weak dependence (Götze et al., 2018), while quantum high-temperature analogues formulate local trace-norm contraction coefficients $\delta_{ij}$ whose column sums govern rapid mixing and decay of conditional mutual information (Bakshi et al., 9 Oct 2025). The phrase “product-level” refers to the fact that the theory tracks channel products or sitewise updates directly, rather than only stationary limits, and expresses global behavior through scalar coefficients or matrices built from local trace-norm contractions (Pathirana, 30 Apr 2026, Bakshi et al., 9 Oct 2025).

1. Classical Dobrushin background and the product-space paradigm

The classical antecedent begins with a product space $S=\prod_{i=1}^n S_i$ endowed with a Gibbs-sampler-type dynamic and one-site conditionals $\mu_k(\cdot\mid \overline x_k)$. For bounded $f:S\to\mathbb R$, the Gibbs-sampler difference operator is

$$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$

with global operator

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$0

The associated Dobrushin interdependence matrix is $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$1, where for $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$2,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$3

and the Dobrushin uniqueness condition is $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$4 (Götze et al., 2018).

Under uniqueness, the product-space measure satisfies a Dirichlet-form logarithmic-Sobolev inequality of the form

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$5

with

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$6

up to the additional factor $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$7 when the one-site conditionals are uniformly bounded below by $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$8 (Götze et al., 2018). The same source then iterates the first-order operator to higher-order differences $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$9 and uses the pointwise chain-of-differences bound

$$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$0

together with LSI-induced moment growth to derive higher-order concentration for polynomials, including tails of the form

$$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$1

after suitable normalization (Götze et al., 2018).

This classical material is not yet trace-Dobrushin theory in the quantum-channel sense, but it supplies the structural template: local dependence coefficients are assembled into a product-level object; that object controls functional inequalities or contraction; and iterated local effects yield global bounds for concentration or decay. A plausible implication is that the later quantum formulations preserve this architecture while replacing total variation or difference operators by trace norm and CPTP dynamics.

2. Centered trace-Dobrushin coefficients for channel products

In the finite-dimensional quantum setting, one fixes $$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$2, writes $$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$3 for the $$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$4 complex matrices, and denotes the state space by

$$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$5

For a quantum channel $$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$6, the centered trace-Dobrushin coefficient is $$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$7 as defined above (Pathirana, 30 Apr 2026). For an inhomogeneous product,

$$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$8

the corresponding product-level coefficient is

$$\kappa_{\rm tr}(\Phi) =\frac12\sup_{\rho,\sigma\in\mathcal S_d}\|\Phi(\rho)-\Phi(\sigma)\|_1,$$9

A basic structural fact is submultiplicativity: $2\,\kappa_{\rm tr}(\Phi)$0 (Pathirana, 30 Apr 2026).

The coefficient $2\,\kappa_{\rm tr}(\Phi)$1 quantifies the residual dependence of the output on the input state. Because

$2\,\kappa_{\rm tr}(\Phi)$2

its decay along products is the criterion for trace-norm forgetting (Pathirana, 30 Apr 2026). The theory therefore treats the scalar sequence $2\,\kappa_{\rm tr}(\Phi)$3 as the central observable for asymptotic analysis of nonstationary products.

This scalar formulation differs from the matrix-valued local-coefficient theory developed for quantum Markov chains at high temperature, where one defines

$2\,\kappa_{\rm tr}(\Phi)$4

equivalently as a supremum over density matrices that differ only on site $2\,\kappa_{\rm tr}(\Phi)$5, and collects them into an $2\,\kappa_{\rm tr}(\Phi)$6 matrix $2\,\kappa_{\rm tr}(\Phi)$7 with Dobrushin condition

$2\,\kappa_{\rm tr}(\Phi)$8

(Bakshi et al., 9 Oct 2025). The scalar coefficient $2\,\kappa_{\rm tr}(\Phi)$9 and the matrix $\Phi(\mathcal S_d)$0 serve analogous roles at different levels of resolution: the former controls complete channel products directly, while the latter resolves propagation of local disturbances through sitewise updates. This suggests a hierarchy in which product coefficients summarize the net effect of repeated local contractions.

3. Deterministic asymptotic replacement and pullback boundary states

A central deterministic theorem identifies vanishing product coefficients with asymptotic replacement by rank-one channels. Fix a start time $\Phi(\mathcal S_d)$1 and compare $\Phi(\mathcal S_d)$2 to a moving replacement channel

$\Phi(\mathcal S_d)$3

Then the following are equivalent:

1. $\Phi(\mathcal S_d)$4.
2. There exist states $\Phi(\mathcal S_d)$5 such that $\Phi(\mathcal S_d)$6.
3. For every reference $\Phi(\mathcal S_d)$7,

$\Phi(\mathcal S_d)$8

1. The same convergence as in (3) holds for some fixed $\Phi(\mathcal S_d)$9.

Moreover, for every $\delta_{ij}$0,

$\delta_{ij}$1

(Pathirana, 30 Apr 2026). Thus $\delta_{ij}$2 is exactly equivalent, up to the universal factor $\delta_{ij}$3, to convergence of the product to a family of replacement channels.

For two-sided products indexed by $\delta_{ij}$4, the corresponding notion is pullback trace-memory loss: for each fixed terminal time $\delta_{ij}$5, one requires $\delta_{ij}$6 as $\delta_{ij}$7. A pullback boundary state is a sequence $\delta_{ij}$8 satisfying

$\delta_{ij}$9

Theorem 3.7 of (Pathirana, 30 Apr 2026) states that pullback trace-memory loss is equivalent to existence of a unique pullback boundary state $S=\prod_{i=1}^n S_i$0 such that, writing $S=\prod_{i=1}^n S_i$1,

$S=\prod_{i=1}^n S_i$2

with the quantitative bounds

$S=\prod_{i=1}^n S_i$3

These results place replacement channels at the center of product-level trace-Dobrushin theory. The replacement family is not an auxiliary construction but the asymptotic normal form of any product with vanishing centered coefficient. In that sense, forgetting in trace norm and asymptotic rank-one reduction are equivalent phenomena (Pathirana, 30 Apr 2026).

4. Random CPTP cocycles, Lyapunov exponents, and annealed estimates

For stationary random environments, the theory is formulated over an invertible ergodic base $S=\prod_{i=1}^n S_i$4 with stationary family $S=\prod_{i=1}^n S_i$5 of CPTP maps. The forward cocycle product is

$S=\prod_{i=1}^n S_i$6

and submultiplicativity becomes

$S=\prod_{i=1}^n S_i$7

Kingman’s subadditive theorem then yields the trace-Dobrushin Lyapunov exponent

$S=\prod_{i=1}^n S_i$8

which is $S=\prod_{i=1}^n S_i$9-invariant and almost surely constant under ergodicity (Pathirana, 30 Apr 2026).

When $\mu_k(\cdot\mid \overline x_k)$0 almost surely, the random replacement theorem gives a unique dynamically stationary random state $\mu_k(\cdot\mid \overline x_k)$1 satisfying

$\mu_k(\cdot\mid \overline x_k)$2

and with $\mu_k(\cdot\mid \overline x_k)$3 there exist a measurable invariant exponent $\mu_k(\cdot\mid \overline x_k)$4 and finite random constants $\mu_k(\cdot\mid \overline x_k)$5 such that for almost every $\mu_k(\cdot\mid \overline x_k)$6 and all $\mu_k(\cdot\mid \overline x_k)$7,

$\mu_k(\cdot\mid \overline x_k)$8

$\mu_k(\cdot\mid \overline x_k)$9

(Pathirana, 30 Apr 2026). This is quenched exponential convergence, both forward and pullback, to the moving replacement channel.

Additional assumptions on the environment sharpen these conclusions in annealed form. If the maximal-correlation mixing coefficient satisfies $f:S\to\mathbb R$0, then

$f:S\to\mathbb R$1

whereas independence of the $f:S\to\mathbb R$2 yields

$f:S\to\mathbb R$3

The same estimates hold for the backward pullback products (Pathirana, 30 Apr 2026).

The random theory therefore organizes memory loss by a single scalar exponent $f:S\to\mathbb R$4. A plausible implication is that the Lyapunov formulation serves as the stochastic counterpart of the deterministic criterion $f:S\to\mathbb R$5, with negativity of $f:S\to\mathbb R$6 replacing pointwise decay of the product coefficient.

5. Quantum local-update formulations: Dobrushin matrices, rapid mixing, and conditional mutual information

A distinct but parallel product-level theory appears for quantum channels or Lindbladians that decompose into site updates,

$f:S\to\mathbb R$7

For each pair $f:S\to\mathbb R$8 one defines the Dobrushin coefficients

$f:S\to\mathbb R$9

equivalently

$$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$0

These coefficients form the matrix $$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$1, and the quantum Dobrushin condition is

$$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$2

(Bakshi et al., 9 Oct 2025).

Under the stronger assumption $$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$3 for some $$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$4, one obtains uniqueness of the fixed point $$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$5, exponential decay of errors,

$$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$6

and mixing time

$$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$7

equivalently

$$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$8

(Bakshi et al., 9 Oct 2025).

The “product-level” description in this framework is an explicit linear evolution for interface errors. Writing a signed operator as $$(\Delta_k f)(x) :=\Bigl[\tfrac12\,\int\bigl(f(x)-f(\overline x_k,y_k)\bigr)^2\, \mu_k(dy_k\mid\overline x_k)\Bigr]^{1/2},$$9 with $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$00 and defining the cost vector $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$01, one has

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$02

which induces the one-step map $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$03 and the bound

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$04

For a continuous-time Lindbladian $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$05, the corresponding ODE is

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$06

equivalently $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$07 with $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$08, and if $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$09 then

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$10

(Bakshi et al., 9 Oct 2025).

The same high-temperature regime also yields exponential decay of conditional mutual information. For a Gibbs state $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$11 of a local Hamiltonian on $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$12 qubits and any disjoint tripartition $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$13,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$14

with

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$15

(Bakshi et al., 9 Oct 2025). According to that source, the proof constructs a recovery map acting on a buffer around $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$16 and uses Dobrushin contraction together with the Fawzi–Renner bound

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$17

This formulation makes precise how local trace-norm contractivity controls both temporal and spatial mixing. The source summarizes the mechanism as “local to global,” “rapid mixing,” “spatial, informational decay,” and “stability” (Bakshi et al., 9 Oct 2025). A plausible implication is that scalar product coefficients such as $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$18 can be viewed as compressed summaries of such matrix-based propagation bounds when one no longer resolves sitewise structure.

6. Transfer to inhomogeneous matrix product states

The theory in (Pathirana, 30 Apr 2026) applies directly to inhomogeneous matrix product states in left-canonical CPTP gauge. A one-dimensional MPS in left-canonical gauge is specified by site tensors $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$19 satisfying

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$20

The induced auxiliary transfer channel is

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$21

and the right-tail transfer product is

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$22

(Pathirana, 30 Apr 2026).

In the deterministic setting, if for every fixed $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$23 one has

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$24

then there exists a unique auxiliary boundary sequence $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$25 satisfying

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$26

and a unique infinite-volume state $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$27 on the quasi-local algebra given by

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$28

For each fixed local observable $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$29, for all sufficiently large $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$30,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$31

and if $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$32 is supported on $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$33 and $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$34 on $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$35 with $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$36,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$37

(Pathirana, 30 Apr 2026).

For stationary random MPS, under the assumption $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$38 almost surely, one obtains a unique stationary random boundary state $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$39 and a weak$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$40-measurable infinite-volume random state $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$41 satisfying

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$42

For each $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$43 in a full-measure set and each local $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$44,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$45

and similarly, across a gap of length $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$46,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$47

(Pathirana, 30 Apr 2026).

In this application, the auxiliary transfer product is the exact carrier of memory, and the single scalar coefficient

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$48

governs both boundary stability and correlation clustering (Pathirana, 30 Apr 2026). This is one of the clearest realizations of the product-level philosophy: a channel-product contraction parameter determines thermodynamic limits and finite-gap correlation estimates in the associated many-body state.

7. Computation, constrained variants, and scope of the theory

Product-level Dobrushin quantities also admit optimization formulations. For a finite-dimensional quantum channel $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$49 and Hamiltonian $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$50 on $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$51, the energy-constrained state set is

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$52

The energy-constrained Dobrushin curve is

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$53

and the corresponding coefficient is

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$54

(Huber et al., 2018).

Huber et al. reformulate this optimization as a jointly constrained semidefinite bilinear program in operator variables $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$55, then solve it by a branch-and-bound scheme with SDP lower bounds based on convex envelopes of bilinear terms over hyperrectangles (Huber et al., 2018). For product channels under a joint energy constraint,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$56

is computed by the same framework after replacing $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$57 by $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$58 and $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$59 by $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$60 (Huber et al., 2018). The same source also records elementary tensorization bounds,

$\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$61

and remarks that analogous $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$62-fold bounds show submultiplicativity and a “2-norm”-like behavior of $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$63 (Huber et al., 2018).

These constrained variants clarify the scope of product-level trace-Dobrushin theory. It is not limited to asymptotic ergodic questions; it also furnishes computable contraction characteristics under convex restrictions such as energy constraints (Huber et al., 2018). At the same time, the theory should not be conflated with a single universal coefficient. The sources describe several distinct but related objects: the scalar centered coefficient $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$64 for arbitrary CPTP products (Pathirana, 30 Apr 2026), local matrix coefficients $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$65 for site-update dynamics (Bakshi et al., 9 Oct 2025), and constrained curves or slopes $\kappa_{\rm tr}(\Phi) :=\sup_{\substack{X=X^*,\;\Tr X=0\X\neq0}} \frac{\|\Phi(X)\|_1}{\|X\|_1},$66 for optimization under Hamiltonian restrictions (Huber et al., 2018). What unifies them is the Dobrushin principle itself: local or admissible trace-norm contractions compose into product-level control of memory loss, mixing, replacement, and correlation structure.