Quenched large deviations of Birkhoff sums along random quantum measurements

We prove a quenched version of the large deviation principle for Birkhoff-like sums along a sequence of random quantum measurements driven by an ergodic process. We apply the result to the study of entropy production in the two-time measurement framework.

💡 Research Summary

The paper establishes a quenched large‑deviation principle (LDP) for Birkhoff‑type sums generated by a sequence of random quantum measurements whose statistics are driven by an ergodic dynamical system. The authors consider a finite‑dimensional quantum system (Hilbert space ℂ^d) subjected at each discrete time step to a measurement instrument chosen from a finite outcome set A. For each environment state ω in an ergodic probability space (Ω, θ, P), a family of completely positive maps ψ_{ω,a} (a∈A) is prescribed; their sum ϕ_ω = Σ_a ψ_{ω,a} is a completely positive trace‑preserving (CPTP) channel. The key structural assumptions are that each ϕ_ω is irreducible (the quantum analogue of a primitive stochastic matrix) and that a suitable power of the random composition becomes positivity‑improving with positive probability.

Given a real‑valued observable f_ω(a) attached to each outcome, the Birkhoff sum after n steps is Σ_n(ω;a_1,…,a_n)=∑{j=1}^n f{θ^{j-1}ω}(a_j). The probability of a particular outcome string is expressed via the trace of the composed instruments acting on an initial density matrix ρ. The moment generating function (MGF) of the sum is M_n^{ω,ρ}(α)=∑_{a_1,…,a_n} e^{-α Σ_n} p_n^{ω,ρ}(a_1,…,a_n) and can be rewritten as a trace of a product of deformed channels ϕ^{(α)}ω = Σ_a e^{-α f_ω(a)} ψ{ω,a}.

Three technical conditions are imposed: (A1) the minimal 1‑norm of the dual map ϕ_ω^* belongs to L¹(Ω); (A2) there exists an integer N₀ such that the composition ϕ_{θ^{N₀-1}ω}⋯ϕ_ω is positivity‑improving with positive P‑probability; (A3) the random variable F_ω = max_{a∈A}|f_ω(a)| satisfies e^{αF_ω}∈L¹(Ω) for all α∈ℝ. These ensure that Kingman’s subadditive ergodic theorem applies to the operator norms of the products of ϕ^{(α)}ω, yielding a deterministic Lyapunov exponent λ(α)=lim{n→∞} (1/n) log‖ϕ^{(α)}_{θ^{n-1}ω}⋯ϕ^{(α)}ω‖{op}, which exists for P‑almost every ω and is differentiable in α.

The main result (Theorem 2.3) states that for almost every ω and any Borel set E⊂ℝ, \