Trace-Dobrushin Lyapunov Exponent

• Trace-Dobrushin Lyapunov exponent is a product-level invariant that measures the exponential decay rate of the residual trace-norm memory in random quantum channel products.
• It quantifies the exact contraction on the traceless subspace, ensuring that channel products forget input state differences and converge to unique replacement channels.
• The exponent plays a crucial role in determining boundary stability and spatial clustering in both deterministic and random inhomogeneous matrix product states.

Searching arXiv for the cited paper and closely related context papers. The trace-Dobrushin Lyapunov exponent is a product-level asymptotic invariant for stationary random cocycles of finite-dimensional quantum channels, introduced in the study of trace-norm forgetting and asymptotic replacement for products of completely positive trace-preserving (CPTP) maps. It is defined from the decay of the centered trace-Dobrushin coefficient of channel products, where the latter measures the exact trace-norm contraction on the self-adjoint trace-zero subspace. In the framework developed in "Asymptotic Replacement for Quantum Channel Products with Applications to Inhomogeneous Matrix Product States" (Pathirana, 30 Apr 2026), the exponent governs quenched memory loss, forward and pullback convergence to a unique dynamically stationary random replacement channel, and quantitative consequences for deterministic and stationary random inhomogeneous matrix product states (MPS).

1. Centered trace-Dobrushin framework

The exponent is built on the centered trace-Dobrushin coefficient of a positive trace-preserving map $\Phi : M_d \to M_d$, where $M_d$ denotes the space of $d \times d$ complex matrices equipped with the trace norm $\|X\|_1 := \operatorname{Tr}|X|$, and

$H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$

For such a map,

$$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$

which is the induced $1 \to 1$ operator norm of $\Phi$ restricted to $(H,\|\cdot\|_1)$ (Pathirana, 30 Apr 2026).

Equivalent variational characterizations identify $D_c(\Phi)$ as

$M_d$0

with $M_d$1. Moreover, $M_d$2 if and only if $M_d$3 is a replacement channel of the form $M_d$4 for some state $M_d$5. Thus $M_d$6 is the trace-norm diameter of the output state space $M_d$7.

For a sequence of channels $M_d$8, the chronological product is defined for $M_d$9 by

$d \times d$0

Then $d \times d$1 is the residual trace-norm memory after the block, since

$d \times d$2

This centered viewpoint is operational: trace preservation leaves the hyperplane $d \times d$3 invariant, and all state differences $d \times d$4 lie in that hyperplane. The coefficient therefore quantifies residual dependence on the input state rather than mere contractivity of the channel on the full operator space.

A central structural fact is submultiplicativity:

$d \times d$5

for positive trace-preserving maps $d \times d$6 and $d \times d$7. Consequently, for $d \times d$8,

$d \times d$9

which yields logarithmic subadditivity along products. This product-level property is the basis for the Lyapunov exponent.

2. Definition and ergodic formulation

The trace-Dobrushin Lyapunov exponent is defined for a stationary random CPTP cocycle over an invertible measure-preserving system $\|X\|_1 := \operatorname{Tr}|X|$0, with a measurable assignment $\|X\|_1 := \operatorname{Tr}|X|$1 of CPTP maps on $\|X\|_1 := \operatorname{Tr}|X|$2. For $\|X\|_1 := \operatorname{Tr}|X|$3,

$\|X\|_1 := \operatorname{Tr}|X|$4

Writing

$\|X\|_1 := \operatorname{Tr}|X|$5

submultiplicativity gives

$\|X\|_1 := \operatorname{Tr}|X|$6

hence

$\|X\|_1 := \operatorname{Tr}|X|$7

The trace-Dobrushin Lyapunov exponent is then

$\|X\|_1 := \operatorname{Tr}|X|$8

whenever the limit exists (Pathirana, 30 Apr 2026).

By Kingman’s subadditive ergodic theorem, the limit exists almost surely and defines a $\|X\|_1 := \operatorname{Tr}|X|$9-invariant random variable $H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$0. If $H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$1 is ergodic, then $H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$2 is almost surely constant and satisfies

$H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$3

The same asymptotic quantity arises from pullback products over the past interval $H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$4,

$H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$5

with

$H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$6

so the forward and pullback formulations share the same almost-sure Lyapunov exponent.

Conceptually, $H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$7 records the asymptotic exponential decay rate of the exact residual memory coefficient of random channel products. Because it is defined from $H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$8 rather than from a coarser one-step sufficient criterion, it is an intrinsic product-level quantity. This suggests a noncommutative analogue of the role played by classical Dobrushin contraction rates for random Markov products, but the framework is formulated directly in trace norm on the traceless subspace.

3. Negativity, forgetting, and asymptotic replacement

The decisive threshold is the almost sure negativity of $H := \{X \in M_d^{sa} : \operatorname{Tr}X = 0\}.$9. In the stationary random setting, the following are equivalent: $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$0 for $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$1-almost every $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$2; $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$3 almost surely, where $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$4; and $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$5 almost surely. These equivalences identify negative trace-Dobrushin Lyapunov exponent with quenched forward forgetting and quenched pullback forgetting (Pathirana, 30 Apr 2026).

This equivalence is the random counterpart of a deterministic replacement theorem. For fixed $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$6, decay $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$7 is equivalent to strong asymptotic replacement from $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$8: there exist states $$D_c(\Phi) := \sup\left\{\frac{\|\Phi(X)\|_1}{\|X\|_1} : X \in H,\ X \neq 0\right\},$$9 such that the replacement channels

$1 \to 1$0

satisfy

$1 \to 1$1

Equivalently, for every $1 \to 1$2,

$1 \to 1$3

The basic quantitative comparison is

$1 \to 1$4

In the two-sided deterministic setting, pullback forgetting produces a unique boundary state sequence $1 \to 1$5 with $1 \to 1$6 such that, for each fixed $1 \to 1$7,

$1 \to 1$8

Moreover,

$1 \to 1$9

for every $\Phi$0. In the random stationary setting, negative $\Phi$1 yields the corresponding random object: a unique dynamically stationary random state $\Phi$2 satisfying

$\Phi$3

The associated random replacement channel is

$\Phi$4

At the state level, the bounds become

$\Phi$5

while at the operator level,

$\Phi$6

Accordingly, the exponent does not merely detect convergence of a preferred state; it governs convergence of the entire channel product to a rank-one replacement map.

4. Quenched and annealed rates

When $\Phi$7 almost surely, the product coefficients decay exponentially along almost every realization. More precisely, there exist a measurable $\Phi$8-invariant random variable $\Phi$9, a full-measure $(H,\|\cdot\|_1)$0-invariant set $(H,\|\cdot\|_1)$1, and almost surely finite measurable random variables $(H,\|\cdot\|_1)$2 and $(H,\|\cdot\|_1)$3 such that for all $(H,\|\cdot\|_1)$4 and all $(H,\|\cdot\|_1)$5,

$(H,\|\cdot\|_1)$6

and

$(H,\|\cdot\|_1)$7

The corresponding state-level bounds hold with factor $(H,\|\cdot\|_1)$8 instead of $(H,\|\cdot\|_1)$9 (Pathirana, 30 Apr 2026).

These are quenched statements: they hold almost surely for each environment. Annealed estimates require additional assumptions on the channel environment. Defining

$D_c(\Phi)$0

the following bounds hold. If the environment is $D_c(\Phi)$1-mixing in the maximal correlation sense, with $D_c(\Phi)$2, then for every $D_c(\Phi)$3 there exists $D_c(\Phi)$4 such that

$D_c(\Phi)$5

If the one-step channels are jointly independent, then there exist $D_c(\Phi)$6 and $D_c(\Phi)$7 such that

$D_c(\Phi)$8

The deterministic theory contains analogous quantitative clocks. If $D_c(\Phi)$9 with $M_d$00, then

$M_d$01

If $M_d$02 counts indices $M_d$03 with $M_d$04, then

$M_d$05

A further good-block criterion states that if, for some $M_d$06, $M_d$07, and $M_d$08, every window $M_d$09 contains a subblock $M_d$10 with $M_d$11, then

$M_d$12

yielding exponential bounds at rate $M_d$13. The significance of this statement is explicit in the source: multi-step contraction may be present even when every one-step coefficient equals $M_d$14.

5. Criteria, comparisons, and examples

Several sufficient criteria certify $M_d$15 without requiring strict positivity at every step. In the ergodic case, if there exists $M_d$16 with

$M_d$17

then $M_d$18. If, with positive probability, the block $M_d$19 is strictly positive, then $M_d$20 as well. A Doeblin-type block minorization also suffices: if for some $M_d$21 there exist $M_d$22 and $M_d$23 such that

$M_d$24

and $M_d$25 in the extended sense, then $M_d$26; it is enough that $M_d$27. Likewise, a Markov-Dobrushin block lower bound with

$M_d$28

and $M_d$29 implies negativity (Pathirana, 30 Apr 2026).

The framework is explicitly related to classical Dobrushin coefficients. For a classical Markov chain with transition matrix $M_d$30, the classical coefficient is half the total-variation diameter of the image of the probability simplex, equivalently the contraction factor on the centered $M_d$31 subspace. The quantum coefficient $M_d$32 is the exact noncommutative analogue:

$M_d$33

and equals the sharp one-step trace-distance contraction coefficient $M_d$34. The paper contrasts this exact product-level quantity with one-step sufficient criteria such as Markov-Dobrushin and quantum Doeblin minorization; for a CPTP map $M_d$35,

$M_d$36

and the bound can be strict.

Three examples clarify what the exponent does and does not encode. In alternating dephasing, $M_d$37 but $M_d$38, so forgetting can be a genuinely multi-step effect. For the qubit amplitude-damping channel $M_d$39 with $M_d$40,

$M_d$41

and the powers $M_d$42 converge to a pure replacement channel, giving

$M_d$43

This shows replacement mixing without eventual strict positivity. Conversely, there exists a deterministic channel with unique fixed point $M_d$44 but $M_d$45 for all $M_d$46, so uniqueness of a stationary state does not imply $M_d$47.

For unital CPTP maps, Appendix B provides a Hilbert-Schmidt sufficient criterion. If $M_d$48 is the largest singular value on the complex trace-zero Hilbert-Schmidt subspace, then

$M_d$49

hence

$M_d$50

If $M_d$51 (or the corresponding block condition holds), then $M_d$52 and the convergence is to the completely depolarizing replacement channel

$M_d$53

6. Role in inhomogeneous matrix product states

The trace-Dobrushin Lyapunov exponent enters the theory of inhomogeneous MPS through the left-canonical CPTP gauge. In the Schrödinger picture, placing MPS tensors in left-canonical form means

$M_d$54

so the auxiliary transfer map

$M_d$55

is CPTP. This places the auxiliary dynamics exactly in the centered trace-Dobrushin framework, with product coefficients controlling boundary stability, thermodynamic limits, and correlations (Pathirana, 30 Apr 2026).

In the deterministic setting, let

$M_d$56

If for every fixed $M_d$57, $M_d$58 as $M_d$59, then there exists a unique right-boundary sequence $M_d$60 with

$M_d$61

and a unique infinite-volume state $M_d$62 on the quasi-local algebra such that for local $M_d$63,

$M_d$64

If $M_d$65 is the trace-closed finite-volume MPS vector and $M_d$66 the normalized finite-volume state whenever $M_d$67, then for fixed local $M_d$68,

$M_d$69

Quantitatively, for all large $M_d$70,

$M_d$71

where $M_d$72.

The same coefficient governs spatial clustering. If $M_d$73 and $M_d$74 with a gap $M_d$75, then

$M_d$76

Thus the auxiliary product coefficient across the gap is the direct control parameter for correlation decay.

In the stationary random setting, the right-tail MPS product becomes a pullback product,

$M_d$77

If $M_d$78 almost surely, then there is a unique stationary random boundary state $M_d$79 with

$M_d$80

and a random infinite-volume state

$M_d$81

for local $M_d$82. For fixed local $M_d$83,

$M_d$84

for all large $M_d$85, with $M_d$86 and $M_d$87 finite almost surely. Correlations also cluster exponentially:

$M_d$88

where $M_d$89. Annealed high-probability and expectation bounds follow from the channel-product estimates: under $M_d$90-mixing, super-polynomial; under independence, exponential in the gap.

A plausible implication is that the trace-Dobrushin Lyapunov exponent functions as the asymptotic boundary-stability rate for random left-canonical MPS with CPTP auxiliary transfer maps. In the source formulation, the same product coefficient that measures trace-norm forgetting for the auxiliary channels also governs infinite-volume limits and spatial clustering.

7. Scope, assumptions, and interpretation

The theory is formulated in finite dimension throughout, for CPTP maps in the Schrödinger picture. Measurability of $M_d$91 is assumed, and for Kingman’s theorem no integrability beyond $M_d$92 is needed. Ergodicity of $M_d$93 is used only to make $M_d$94 almost surely constant and for certain positive-probability criteria; nonergodic variants are handled by two-sided saturation arguments. Mixing assumptions such as $M_d$95-mixing or independence are required only for annealed bounds, not for quenched convergence (Pathirana, 30 Apr 2026).

For MPS applications, the left-canonical CPTP gauge is the structural condition ensuring that the auxiliary transfer maps lie in the trace-preserving centered framework. No primitivity or strict positivity is assumed. Eventual strict positivity is sufficient for $M_d$96, but the amplitude-damping example shows that it is not necessary.

Within this framework, the trace-Dobrushin Lyapunov exponent has a precise interpretation. It is not merely a spectral quantity attached to a single channel, nor simply a certificate of existence of a stationary state. Rather, it is the almost-sure asymptotic exponential rate of the exact residual input-state dependence of random channel products. Negative exponent means that the state-space diameter of the evolved image collapses exponentially, and this collapse is equivalent to convergence of the full product to a unique dynamically stationary random replacement channel. In that sense, the exponent is the random product-level invariant that links centered trace-norm contraction, asymptotic replacement, and boundary formation in inhomogeneous MPS.