[Paper Review] A Nonconventional Ergodic Theorem for quantum "diagonal measures"

Francesco Fidaleo|arXiv (Cornell University)|Mar 23, 2012

Advanced Operator Algebra Research16 references3 citations

TL;DR

This paper establishes a nonconventional ergodic theorem for quantum dynamical systems by extending Furstenberg's classical result to non-invariant, non-normal states—specifically, quantum 'diagonal measures' arising from product states or convex combinations in the GNS representation. The key result is the strong operator topology convergence of Cesàro means of the form $\frac{1}{N}\sum_{n=0}^{N-1}U_{\omega}^{nk_1}XU_{\omega}^{n(k_2-k_1)}$, enabling the computation of three-point correlation limits in ergodic and compact quantum systems under conditions on the state's support and commutant structure.

ABSTRACT

We extend the Nonconventional Ergodic Theorem for generic measures by Furstenberg, to several situations of interest arising from quantum dynamical systems. We deal with the diagonal state canonically associated to the product state (i.e. quantum "diagonal measures"), or to convex combinations of diagonal measures for non ergodic cases. For the sake of completeness, we treat also the Nonconventional Ergodic Theorem for compact dynamical systems, that is when the unitary generating the dynamics in the GNS representation is almost periodic. The Nonconventional Ergodic Theorem allows in a natural way to determine the limit of the three-point correlations, naturally relevant for the knowledge of the ergodic properties of a dynamical system.

Motivation & Objective

• To generalize Furstenberg's nonconventional ergodic theorem to quantum dynamical systems with non-invariant, non-normal states.
• To analyze the convergence of Cesàro means involving unitary operators in the GNS representation for quantum diagonal measures.
• To determine the limit of three-point correlations in quantum systems under ergodic and non-ergodic conditions.
• To establish conditions under which the limit of the Cesàro means exists in the strong operator topology.
• To treat compact dynamical systems where the unitary operator is almost periodic, ensuring convergence via spectral decomposition.

Proposed method

• Uses the Gelfand–Naimark–Segal (GNS) construction to represent the $C^*$-dynamical system in a Hilbert space with a cyclic vector $\Omega$.
• Defines quantum 'diagonal measures' as vector states on $\pi_\omega(\mathfrak{A})'' \otimes_{\text{max}} \pi_\omega(\mathfrak{A})'$, corresponding to product states in the noncommutative setting.
• Analyzes the action of the automorphism $\gamma = \text{Ad}_{U_\omega^{k_1}} \otimes \text{Ad}_{U_\omega^{k_2}}$ on the algebra $\mathfrak{M}$.
• Applies the von Neumann ergodic theorem to spectral projections when $U_\omega$ is almost periodic (compact systems).
• Imposes conditions such as $s(\omega) \in Z(\mathfrak{A}^{**})$ and $\pi_\omega(\mathfrak{A})' \cap \{U_\omega^{k_2-k_1}\}' \subset Z(\pi_\omega(\mathfrak{A})'')$ to ensure convergence in non-ergodic cases.
• Uses reduction theory for separable $C^*$-systems to decompose states with central support into ergodic components.

Experimental results

Research questions

• RQ1 Under what conditions does the Cesàro mean $\frac{1}{N}\sum_{n=0}^{N-1}U_{\omega}^{nk_1}XU_{\omega}^{n(k_2-k_1)}$ converge in the strong operator topology for non-invariant, non-normal states in quantum systems?
• RQ2 Can the limit of three-point correlations $\frac{1}{N}\sum_{n=0}^{N-1}\omega(A_0\alpha^{nk_1}(A_1)\alpha^{nk_2}(A_2))$ be characterized in quantum dynamical systems with non-invariant states?
• RQ3 Is the condition $\pi_\omega(\mathfrak{A})' \cap \{U_\omega^{k_2-k_1}\}' \subset Z(\pi_\omega(\mathfrak{A})'')$ necessary for convergence in non-ergodic cases?
• RQ4 How does the convergence of Cesàro means behave in compact quantum dynamical systems where $U_\omega$ is almost periodic?
• RQ5 Can the nonconventional ergodic theorem be extended to cases where the commutant $\pi_\omega(\mathfrak{A})' \cap \{U_\omega\}'$ is non-Abelian but not contained in the center?

Key findings

• The Cesàro means $\frac{1}{N}\sum_{n=0}^{N-1}U_{\omega}^{nk_1}XU_{\omega}^{n(k_2-k_1)}$ converge in the strong operator topology when $X$ belongs to the norm closure of $\pi_\omega(\mathfrak{A})''$ and $\pi_\omega(\mathfrak{A})'$, under the condition $s(\omega) \in Z(\mathfrak{A}^{**})$ and $\pi_\omega(\mathfrak{A})' \cap \{U_\omega^{k_2-k_1}\}' \subset Z(\pi_\omega(\mathfrak{A})'')$.
• For compact $C^*$-dynamical systems, the limit is given by $S(A) = \sum_{\{v,w \mid v^{k_1}w^{k_2-k_1}=1\}} E^{U_\omega}_v A E^{U_\omega}_w$, which converges strongly and satisfies $\|S(A)\| \leq \|A\|$.
• The three-point correlation limit is $\left\langle S(\pi(A_1))\pi(A_2)\Omega, \pi(A_0^*)\Omega \right\rangle$, explicitly computable via spectral projections in the compact case.
• The condition $s(\omega) \in Z(\mathfrak{A}^{**})$ is necessary; a counterexample shows that dropping it leads to non-convergent three-point correlations.
• The condition $\pi_\omega(\mathfrak{A})' \cap \{U_\omega^{k_2-k_1}\}' \subset Z(\pi_\omega(\mathfrak{A})'')$ is crucial for non-ergodic cases, and its failure may prevent convergence.
• The result holds for separable $C^*$-dynamical systems in non-ergodic cases, relying on reduction theory for states with central support in the bidual.