Title: A Multiplicative Ergodic Theorem for Repeated Bistochastic Quantum | Interactions with Applications to Entanglement
Domain: Math

Question: How does randomness in bistochastic completely positive maps shape the long-term behavior of entanglement in quantum dynamics, and under what conditions does entanglement become asymptotically broken? In particular, how can one formulate a Multiplicative Ergodic Theorem for random bcp maps via multiplicative domains with convergence rates, explain why the presence of occasional PPT maps forces asymptotic breaking, and characterize the limiting regime by extending Kuperberg’s theorem to the random i.i.d. setting? Furthermore, how can finite-time behavior be quantified, including explicit breaking times, probabilistic bounds, and the transition from finite to asymptotic regimes?

Background: 1. Quantum channel compositions connect quantum information with dynamical systems. | 2. Classical framework: open quantum system on \mathbb{C}^d; channels as CPTP maps. | 3. Previous results: Horodecki on entanglement breaking, Christandl’s PPT² conjecture, later partial resolutions. | 4. Modern focus: | • Asymptotic behavior (entanglement breaking, convergence, stability). | • Random dynamics (ergodicity, disorder effects). | • Structural properties (multiplicative domains, algebraic/topological features). | 5. Central question: how randomness in bistochastic CP maps governs long-term entanglement preservation and breaking.

Definitions: 1. Matrix space \mathcal{M}_d: complex d\times d matrices; operations include conjugate transpose, spectrum, trace. | 2. Norms and inner product: operator norm \|\cdot\|\infty; Hilbert–Schmidt inner product \langle a,b\rangle{HS}=\mathrm{Tr}(a^*b). | 3. Quantum channels: completely positive trace-preserving (CPTP) maps. | 4. Bistochastic CP maps (bcp): CPTP maps that are also unital (\psi(I)=I). | 5. Entanglement breaking maps: channels whose output is separable for all inputs. | 6. Asymptotic entanglement breaking: convergence of channel iterates toward the EB class. | 7. Multiplicative domain: \mathcal{M}_\psi = \{a: \psi(ab)=\psi(a)\psi(b),\; \psi(ba)=\psi(b)\psi(a)\}. | 8. Measure-theoretic tools: Fell \sigma-algebra, filtrations, stopping times.

Methodology: 1. Multiplicative Ergodic Theorem (MET): ergodic theory applied to random bcp maps; decomposition into stable domain and contracting complement. | 2. Key tools: Lyapunov exponents for convergence rates; measurability of multiplicative domains; stopping time analysis of multiplicative index. | 3. Entanglement breaking: equivalence with abelian multiplicative domains; use of PPT maps to force asymptotic EB. | 4. I.i.d. case: random compositions yield constant asymptotic domains almost surely; spectral characterization of limit behavior. | 5. Finite-time analysis: eventual vs asymptotic EB; probabilistic bounds on breaking times; primitive maps ensure exponential convergence. | 6. Geometric framework: Grassmannian metrics and Fell \sigma-algebra used for measurability and structural stability proofs.

Answer: 1. Multiplicative Ergodic Theorem (MET) for random bcp maps | Random quantum channels are modeled as bistochastic completely positive (bcp) maps \Phi: \mathcal{M}d \to \mathcal{M}d, preserving both identity and trace. The multiplicative domain of a channel, | \mathcal{M}\Phi = \{a \in \mathcal{M}d : \Phi(ab) = \Phi(a)\Phi(b), \;\Phi(ba)=\Phi(b)\Phi(a), \;\forall b\}, | captures invariant algebraic substructures. The MET asserts that under ergodic driving, the algebra decomposes as | \mathcal{M}d = \mathcal{M}\infty^\Phi \oplus \mathcal{M}\infty^{\Phi,\perp}, | with \dim \mathcal{M}\infty^\Phi constant almost surely, enforced by ergodicity. On the complementary space, iterates contract exponentially: for a \in \mathcal{M}\infty^{\Phi,\perp}, | \|\Phi^{(n)}\omega(a)\|{HS} \leq C e^{n\kappa}\|a\|{HS}, \quad \kappa < 0. | Moreover, the restriction of \Phi_\omega to \mathcal{M}_\infty^\Phi defines isometric isomorphisms between successive domains, ensuring stability of the asymptotic algebra. This structural decomposition parallels Oseledets’ theorem in classical ergodic theory. | | 2. Entanglement breaking under disorder | A channel is entanglement breaking (EB) if its image lies in an abelian \text{C}^*-subalgebra. For random bcp maps, asymptotic EB is equivalent to the asymptotic multiplicative domain \mathcal{M}\infty^\Phi being abelian. This can be expressed via Hilbert–Schmidt convergence: | \lim{n \to \infty} d_{HS}(\Phi^{(n)}\omega, \mathcal{E}) = 0, \quad \text{a.s.}, | where \mathcal{E} is the EB set. Importantly, if the driving dynamics include occasionally PPT maps (completely positive under partial transpose), then ergodicity forces \mathcal{M}\infty^\Phi to be abelian almost surely. Thus, randomness ensures that even partial presence of PPT maps drives the system towards entanglement breaking. This result provides explicit conditions and links algebraic structure to physical EB behavior, while ergodic averages yield quantitative timescales for the onset of entanglement loss. | | 3. Extension of Kuperberg’s theorem to random i.i.d. setting | In the i.i.d. random case, the asymptotic multiplicative domain is non-random almost surely, collapsing to a fixed \text{C}^*-subalgebra \mathcal{A}. This algebra can be characterized spectrally: elements v \in \mathcal{A} evolve as eigenvectors, | \Phi^{(n)}_\omega(v) = \lambda^n v, \quad |\lambda|=1. | Thus, the asymptotic structure is determined by a deterministic algebra carrying phase rotations. This generalizes Kuperberg’s theorem on deterministic bcp maps to the random setting, showing that ergodic randomness preserves structural regularity in the limit. The extension provides necessary and sufficient conditions for convergence of random compositions, and gives an explicit description of their limiting algebraic behavior, clarifying how random fluctuations average out in the long run. | | 4. Finite-time entanglement breaking and probabilistic bounds | Beyond asymptotics, finite-time analysis investigates explicit stopping times when a random composition becomes EB. Define the eventual EB time | \iota(\omega) = \inf\{n : \Phi^{(n)}_\omega \in \mathcal{E}\}. | Ergodicity imposes a 0–1 law: with probability one, EB either eventually occurs or never occurs. For primitive maps (those driving all states to full rank), eventual EB coincides with asymptotic EB, and exponential convergence bounds ensure rapid approach to the maximally mixed channel. Probabilistic estimates quantify the likelihood that EB occurs by time n, yielding explicit upper and lower bounds. This links finite-time breaking to asymptotic EB and provides concrete timescales for entanglement decay in random environments. | |

Checklist: 1. Defines the framework of quantum channels, bistochastic CP maps, and multiplicative domains. | 2. Establishes the connection between ergodic theory and random channel compositions. | 3. Proves the Multiplicative Ergodic Theorem: decomposition \mathcal{M}d = \mathcal{M}\infty^\Phi \oplus \mathcal{M}_\infty^{\Phi,\perp} with exponential convergence in the complement. | 4. Demonstrates that the multiplicative index is a stopping time and proves measurability of the stopped process. | 5. Shows asymptotic entanglement breaking is equivalent to abelian structure of the asymptotic domain. | 6. Proves that presence of PPT maps implies asymptotic EB almost surely. | 7. Characterizes the i.i.d. case by showing the asymptotic domain is constant and describes it spectrally. | 8. Establishes measurability of multiplicative domain maps using Fell \sigma-algebra and Grassmannian analysis. | 9. Proves isometric properties and algebra isomorphisms of \Phi restricted to multiplicative domains. | 10. Analyzes convergence in the complementary space with explicit exponential bounds. | 11. Links primitive maps to fast convergence and asymptotic EB behavior. | 12. Distinguishes between eventual and asymptotic entanglement breaking using ergodicity and stopping times.