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Ergodicity of quantum cellular automata

S. Richter, R. F. Werner

arXiv:cond-mat/9504001v1cond-mat

TL;DR

Quantum cellular automata extend probabilistic cellular automata to quasi-local quantum observable algebras, where ergodicity means uniqueness of the invariant state. The paper develops oscillation norms and completely bounded local contraction criteria, obtaining ergodicity and exponential correlation decay while noting limitations in explicit norm characterization and spectral optimization.

  • Problem

    The paper asks for sufficient local-data criteria ensuring ergodicity and correlation decay in quantum cellular automata.

    Source: §2 · pp. 5, 7, §5.3 · p. 33

  • Method

    It introduces quantum oscillation norms, tensorable constructions, and completely bounded oscillation estimates for local transition operators.

    Source: §1 · p. 2, §4.1 · p. 20

  • Results

    A completely bounded local contraction criterion, |||P1|||cb < 1, implies a unique invariant state and exponential decay of correlations.

    Source: §5.1 · p. 29, §5.3 · p. 33

  • Takeaways & Limitations

    The criterion expresses ergodicity as sufficiently large local disorder and does not depend on the propagation maps distinguishing interacting from non-interacting QCAs.

    Source: §5.1 · p. 29

    The authors do not obtain an explicit expression for the completely bounded oscillation norm in their finite-dimensional example, and suitable oscillation norms cannot generally be optimized to the nontrivial spectral modulus.

    Source: §4.1 · p. 25, §5.1 · p. 29

Abstract

from arXiv · show

We define a class of dynamical maps on the quasi-local algebra of a quantum spin system, which are quantum analogues of probabilistic cellular automata. We develop criteria for such a system to be ergodic, i.e., to possess a unique invariant state. Intuitively, ergodicity obtains if the local transition operators exhibit sufficiently large disorder. The ergodicity criteria also imply bounds for the exponential decay of correlations in the unique invariant state. The main technical tool is a quantum version of oscillation norms, defined in the classical case as the sum over all sites of the variations of an observable with respect to local spin-flips.

1. Introduction

The paper introduces quantum cellular automata as observable-algebra analogues of probabilistic cellular automata and develops ergodicity criteria based on local transitions. Its main technical tool is a general class of quantum oscillation norms supporting ergodicity and correlation-decay results.

Motivation

  • Quantum cellular automata generalize probabilistic cellular automata from classical configurations to quantum observable algebras, with ergodicity meaning a unique invariant state.The construction is motivated by finitely correlated states and allows generally non-unitary evolution.Source: §1 · p. 2

Technical contribution

  • The paper introduces general oscillation norms on quantum lattice systems as its main technical contribution.These norms extend classical tools used to prove probabilistic cellular-automaton ergodicity criteria.Source: §1 · p. 2
  • Stabilized oscillation-norm bounds are necessary because transition-operator bounds can increase when the system is enlarged, even classically with more than two states per site.The paper addresses this through completely bounded oscillation norms.Source: §1 · p. 3

Results

  • The paper develops local-to-infinite-system estimates, derives ergodicity criteria, relates them to classical PCAs, and obtains exponential correlation decay.The organization explicitly links local contraction estimates to uniqueness of the invariant state and decay of correlations.Source: §1 · p. 3

2. Definition of Quantum Cellular Automata

The paper defines QCAs on quasi-local observable algebras by combining local completely positive transitions with disjoint propagation ranges. This construction ensures well-defined global dynamics and frames ergodicity and correlation decay as problems controlled by local data.

Definition of Quantum Cellular Automata

  • Quantum generalization requires commuting transition-operator ranges because products of positive noncommuting elements need not remain positive.The paper enforces this condition by refining cells into subcells with separate tensor factors.Source: §2 · pp. 5–6
  • A QCA consists of a countable lattice, subcell algebras, injective propagation maps, and a completely positive unit-preserving local operator.Global transition operators are then assembled from these local ingredients.Source: §2 · p. 6
  • Disjoint propagated subcell ranges make the global transition operator completely positive, norm-continuous, and extendable to the quasi-local algebra.This yields a well-defined operator on the infinite system.Source: §2 · p. 6

Ergodicity

  • A QCA is ergodic when it has one invariant state, and the paper seeks criteria for this property and for convergence toward that state using local data.It also studies decay of correlations in the invariant state.Source: §2 · p. 7

Motivation

  • Direct tensor-product estimates accumulate errors with the number of sites, so infinite-lattice analysis requires a norm whose contraction is controlled by the worst local factor.This motivates the quantum oscillation norms developed later.Source: §2 · p. 7

3. Oscillation norms on C*-algebras

The paper generalizes classical oscillation norms to noncommutative C*-algebras using operations that annihilate constants and measure local variation. It compares constructions, establishes their basic norm properties, and identifies tensor-compatible examples and limitations.

3.1. Definition and basic properties

  • Quantum oscillation norms are built from operations that annihilate the identity and quantify an observable’s response to local transformations.The construction extends the classical interpretation based on local spin flips.Source: §3.1 · p. 8
  • Several local constructions are considered, including normalized-trace oscillations, sup-oscillation norms, state-space metrics, and sums over elementary oscillations.The paper adopts the elementary-oscillation approach because it matches the QCA subcell structure.Source: §3.1 · pp. 8–9
  • An oscillation seminorm vanishes only on scalar multiples of the identity under the stated conditions, thereby defining a norm on the quotient by constants.The paper also discusses reference states and equivalent formulations.Source: §3.1 · pp. 9, 11
  • For finite collections of oscillation operators, the resulting norm constructions are equivalent, while translation-based examples show that the required global bound can fail.The failure occurs when no finite constant controls distance from scalars by the oscillation norm.Source: §3.1 · p. 10
  • Group-conjugation oscillations satisfy Matsui’s condition; for irreducible representations, the reference state is the normalized trace, including the Pauli-matrix example.The paper records this as a concrete tensor-compatible construction.Source: §3.1 · p. 11

5 Lemma. Afin/

The paper proves that contraction in an oscillation norm yields a unique invariant state and convergence in the C*-norm, then constructs tensorable norms for quasi-local systems. Tensorability transfers local estimates to infinite tensor products without deterioration.

5 Lemma. Afin/

  • Oscillation-norm contraction with factor ǫ < 1 implies a unique invariant state and convergence of iterates toward it in the C*-norm.The result applies to norm-contractive, unit-preserving operators and extends from a dense finite domain.Source: §5 · pp. 12–13
  • The proof considers iterates An = P n(A), chooses scalars λn, for example λn = η(An), and extends the resulting functional continuously to ρ with ∥ρ∥ = 1 and ρ(1I) = 1I, hence a state.Source: §5 · p. 13

3.2. Tensorable oscillation norms

  • The infinite-system norm is assembled by taking the union of local oscillation operators across sites, with local observables dense in the resulting oscillation space.The construction uses reference-state projections to approximate quasilocal observables.Source: §3.2 · pp. 14, 16, 17
  • Tensorable oscillation norms remain compatible with adjoining matrix algebras, making them suitable for completely bounded estimates.In the commutative case every oscillation norm is tensorable, while the noncommutative case requires additional conditions.Source: §3.2 · pp. 14, 15, 18
  • The construction reduces infinite-system tensorability questions to one-site algebras and supports the product estimates needed for QCA ergodicity analysis.The paper identifies Matsui’s condition as a sufficient criterion for tensorability.Source: §3.2 · pp. 17, 19

4. Contractivity of tensor product operators

The paper shows that completely bounded oscillation norms make tensor-product contractivity dimension-independent, while several alternative norms fail to support ergodicity estimates for interacting infinite systems.

4.1. Oscillation norm estimate

  • The stabilized norm is necessary because ordinary oscillation norms can increase after coupling a system to an external algebra, even when the original map contracts.The completely bounded formulation tests stability under system enlargement.Source: §4.1 · p. 20
  • |||P|||cb ≤ ǫ whenever every completely positive, unital factor Px satisfies |||Px|||cb ≤ ǫ, so tensor products inherit the worst single-site contraction rate.This is the central tensor-product estimate underlying the non-interacting ergodicity criterion.Source: §4.1 · p. 21
  • For non-interacting systems, |||Px|||cb ≤ ǫ < 1 for every site suffices to establish ergodicity.The criterion combines the tensor-product estimate with the paper’s general contractivity-to-ergodicity result.Source: §4.1 · p. 21
  • Small completely bounded oscillation norm represents large disorder that nearly erases prior-state memory, and sufficiently small mixtures with a constant map remain ergodic.For sufficiently small λ, (1 − λ)Pω + λP has a unique invariant state close to, but not equal to, ω.Source: §4.1 · p. 22
  • In classical systems, stabilization may coincide with the ordinary norm for restricted cases, but can be strictly larger when sites have more than two spin values.For abelian algebras with a two-dimensional target or a single oscillation generator, |||P|||cb = |||P|||; beyond such cases, strict inequality can occur.Source: §4.1 · pp. 23–24

4.2. Estimates in other norms

  • Alternative quotient, spectral, and sup-oscillation norms either require known invariant states, grow with region size, fail for propagation maps, or lack convergence in the quasi-local algebra.A two-cell example has one-site rate 1/2 but three invariant states globally, while sup-oscillation Cauchy sequences need not converge.Source: §4.2 · pp. 26–28

5. Applications to Cellular Automata

The applications establish an ergodicity criterion for QCAs, recover classical PCAs within the framework, and derive exponential correlation decay for the unique invariant state.

5.1. Ergodicity

  • If |||P1|||cb < 1, then the QCA is ergodic and has a unique P-invariant state ρ.The total transition operator satisfies |||P|||cb = |||P1|||cb.Source: §5.1 · p. 28
  • The criterion expresses sufficiently large disorder and is independent of the propagation maps distinguishing interacting from non-interacting QCAs.It holds when P1 is sufficiently close to a map A ↦ ω(A)1I.Source: §5.1 · p. 29

5.2. The classical case

  • The classical PCA transition operator can be recovered completely from the QCA construction through the multiplication map and its factorization.The commutative one-site algebras make the quantum label formal in this embedding.Source: §5.2 · pp. 30–31
  • For finite-dimensional abelian one-site algebras, the nth-order diagonal M has completely bounded oscillation norm at most 1, with sharp constant 1.Sharpness follows because MJ = idA and J preserves the oscillation norm.Source: §5.2 · pp. 31–32
  • The construction suggests replacing the multiplication map by a completely positive unital map, yielding a QCA generalization without subcell decompositions but without commuting ranges.Such systems would no longer update cells independently.Source: §5.2 · p. 32

5.3. Decay of correlations in the invariant state

  • The correlation distance is the last time step during which two regions remain independent, as determined by iterated propagation of their supports.This geometric quantity enters the correlation estimate, unlike the ergodicity criterion itself.Source: §5.3 · p. 33
  • Under the same ergodicity criterion, correlations decay exponentially with a rate determined by the correlation distance and |||P1|||cb.For translational propagation, the rate also depends on the separation direction.Source: §5.3 · pp. 33–34

Appendix: Complete boundedness

The appendix explains why complete positivity and complete boundedness are required for transition operators on composite quantum systems. It defines stabilized positivity and norm control and gives their main structural properties.

Appendix: Complete boundedness

  • Ordinary positivity and boundedness can fail after tensoring a map with an identity, so composite quantum systems require stabilized notions.Transposition on Mn is positive and contractive, but idMn ⊗ P can have norm at least n and fail to preserve positivity.Source: Appendix: Complete boundedness · p. 34
  • A map is completely positive when idMn ⊗ P is positive for every n, and completely bounded when these amplified norms remain uniformly bounded.The completely bounded norm is the supremum of the amplified operator norms.Source: Appendix: Complete boundedness · p. 34
  • When ||P||cb ≤ 1, P is a complete contraction, and finite-dimensional target algebras permit checking complete positivity or boundedness at a finite matrix size.The required size is the largest matrix dimension appearing as a direct summand.Source: Appendix: Complete boundedness · pp. 34–35
  • Completely positive maps are completely bounded, and unital completely positive maps satisfy ||P(1I)|| = ||P|| = ||P||cb.They admit a Stinespring representation P(A) = V*π(A)V.Source: Appendix: Complete boundedness · p. 35
  • Complete contractions admit dilation-style representations, but unlike completely positive maps their corresponding decompositions generally lack uniqueness.They can also be realized as off-diagonal corners of completely positive maps.Source: Appendix: Complete boundedness · p. 35
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