arXiv subjects

Lubashan Pathirana

Publications and source records attributed to Lubashan Pathirana.

8 recordsLinked to original sources

arXiv2026

Julia Sets for Sequences of Monomials

We study Julia sets arising from non-autonomous iteration of monomials on the Riemann sphere. We give a precise description of the Julia set associated to an arbitrary sequence of monomials in terms of their coefficients and degrees. Combined with the basic invariance relation for non-autonomous Julia sets, this yields a plethora of striking examples in polynomial non-autonomous dynamics: finite Julia sets of arbitrary cardinality, non-trivial Julia sets with non-empty interior, Julia sets that are perfect but not uniformly perfect, and Julia sets with empty interior but positive area. We also describe a connection between autonomous and non-autonomous Julia sets of monomial sequences through Kuratowski limits.

math.DS↗

arXiv2026

Asymptotic Replacement for Quantum Channel Products with Applications to Inhomogeneous Matrix Product States

We develop a product-level trace-Dobrushin theory for finite-dimensional quantum channel products and apply it to deterministic and stationary random inhomogeneous matrix product states in left-canonical CPTP gauge. For a product of channels, the centered trace-Dobrushin coefficient quantifies the residual dependence on the input state, and its decay is the criterion for trace-norm forgetting. In the deterministic setting, this decay is equivalent to asymptotic replacement by a moving replacement channel. For two-sided products, pullback forgetting produces a unique boundary state, which determines the canonical replacement family. For stationary random CPTP cocycles, submultiplicativity of the product coefficient yields a trace-Dobrushin Lyapunov exponent. We prove that the almost sure negativity of this exponent is equivalent to quenched trace-norm memory loss and gives exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. When the $\varrho$-mixing profile of the channel environment tends to zero, we obtain annealed super-polynomial estimates, while independence gives annealed exponential estimates. Finally, we transfer these estimates to inhomogeneous matrix product states whose auxiliary transfer maps are CPTP. These channel estimates transfer to deterministic and stationary random inhomogeneous MPS, giving infinite-volume limits of trace-closed finite-volume states, quantitative boundary stability, and correlation bounds governed by the same auxiliary product coefficients.

quant-ph↗

arXiv2026

Central Limit Theorems for Outcome Records in Disordered Quantum Trajectories

We prove annealed central limit theorems for finite pattern counts in the measurement record of discrete-time quantum trajectories generated by repeated measurements in a disordered environment. Under summable mixing assumptions on the environment and an annealed trace-norm forgetting property for the associated non-selective channel cocycle, we first establish the CLT under the annealed law determined by the dynamically stationary state. This part applies to general disordered quantum instruments and, in particular, is not restricted to the perfect-measurement regime; it complements both the corresponding law of large numbers for disordered measurement records and the homogeneous central limit theorem. We then introduce a coupling-based notion of admissibility for initial states and show that the same Gaussian limit extends to every admissible initial law, with unchanged centering and asymptotic variance. In the perfect-measurement setting, we further identify a general condition ensuring admissibility for every initial state, and hence obtain a universal annealed central limit theorem. We also provide practical sufficient criteria for this condition and verify the assumptions across a broad family of examples, including disordered walk-type models generated by finite group actions

math-ph↗

arXiv2025

Correlation Lengths for Stochastic Matrix Product States

We introduce a general model of stochastically generated matrix product states (MPS) in which the local tensors share a common distribution and form a strictly stationary sequence, without requiring spatial independence. Under natural conditions on the associated transfer operators, we prove the existence of thermodynamic limits of expectations of local observables and establish almost-sure exponential decay of two-point correlations. In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially in the distance between the two sites, with a deterministic rate. In the i.i.d. case, the exponential decay still holds with a deterministic rate, with the probability approaching one exponentially fast in the distance. For strictly stationary ensembles with decaying spatial dependence, the correlation decay quantitatively reflects the mixing profile: ($\rho$)-mixing yields polynomial bounds with high probability, while stretched-exponential (resp. exponential) decay in ($\rho$) (resp. ($\beta$)) yields stretched-exponential (resp. exponential) decay of the two-point function, again with correspondingly strong high-probability guarantees. Altogether, the framework unifies and extends recent progress on stationary ergodic and Gaussian translation-invariant ensembles, providing a transfer-operator route to typical correlation decay in random MPS.

quant-ph↗

arXiv2025

Asymptotic Behavior of Random Time-Inhomogeneous Markovian Quantum Dynamics

We study the asymptotic behavior of continuous-time, time-inhomogeneous Markovian quantum dynamics in a stationary random environment. Under mild faithfulness and eventually positivity-improving assumptions, the normalized evolution converges almost surely to a stationary family of full-rank states, and the normalized propagators converge almost surely to a rank-one family determined by these states. Beyond a disorder-dependent threshold, these convergences occur at exponential rates that may depend on the disorder; when the environment is ergodic, the rate itself is deterministic. When the dynamical propagators display vanishing maximal temporal stochastic correlation, convergence in stochastic expectations for the above limits is faster than any power of the time separation, and improves to exponential rates when the dynamical propagators display stochastically independent increments. These expectation bounds yield disorder-uniform high-probability estimates. The framework does not require complete positivity or trace preservation and encompasses random Lindbladian evolutions and collision-model dynamics.

quant-ph↗

arXiv2025

Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements

We consider quantum trajectories arising from disordered, repeated generalized measurements, which have the structure of Markov chains in random environments (MCRE) with dynamically-defined transition probabilities; we call these disordered quantum trajectories. Under the assumption that the underlying disordered open quantum dynamical system approaches a unique equilibrium in time averages, we establish a strong law of large numbers for measurement outcomes arising from disordered quantum trajectories, which follows after we establish general annealed ergodic theorems for the corresponding MCRE. The type of disorder our model allows includes the random settings where the disorder is i.i.d. or Markovian, the periodic (resp. quasiperiodic) settings where the disorder has periodic (resp. quasiperiodic) structure, and the nonrandom setting where the disorder is constant through time. In particular, our work extends the earlier noise-free results of K\"ummerer and Maassen to the present disordered framework.

math-ph↗

arXiv2024

Asymptotic Purification of Quantum Trajectories under Random Generalized Measurements

We develop a general framework to study quantum trajectories resulting from repeated random measurements subject to stationary noise, and generalize results of Kümmerer and Maassen to this setting. The resulting trajectory of quantum states is a time-inhomogeneous Markov chain in a random environment. Kümmerer and Maassen introduced the concept of dark subspaces for noise-free processes, establishing that their absence is equivalent to asymptotic purification of the system state. We clarify the notion of dark subspaces in the disordered setting by defining a measurable correspondence consisting of a collection of random subspaces satisfying a darkness condition. We further prove that asymptotic purification occurs if and only if this collection of random dark subspaces is empty. Several examples of these phenomena are provided.

quant-ph↗

arXiv2023

Law of large numbers and central limit theorem for ergodic quantum processes

A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum operation with noise. Such ergodic quantum processes generalize independent quantum processes. An ergodic theorem describing convergence to equilibrium for a general class of such processes was recently obtained by Movassagh and Schenker. Under irreducibility and mixing conditions, we obtain a central limit type theorem describing fluctuations around the ergodic limit.

math-ph↗