A Multiplicative Ergodic Theorem for Repeated Bistochastic Quantum Interactions with Applications to Entanglement

• URL: http://arxiv.org/abs/2502.14997v1
• Date: Thu, 20 Feb 2025 19:40:07 GMT
• Title: A Multiplicative Ergodic Theorem for Repeated Bistochastic Quantum Interactions with Applications to Entanglement
• Authors: Owen Ekblad,
• Abstract summary: We consider compositions of random bistochastic completely positive (bcp) maps driven by an ergodic dynamical system.<n>By applying our main theorem, we classify when compositions of random bcp maps areally entanglement breaking.<n>We prove a generalization of a theorem of Kuperberg for compositions of i.i.d. random bcp maps.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider compositions of random bistochastic completely positive (bcp) maps driven by an ergodic dynamical system. Our main result is the explicit description of Oseledets's Multiplicative Ergodic Theorem (MET) for such compositions in terms of the random multiplicative domains corresponding to the random bcp maps in the compositions. By applying our main theorem, we classify when compositions of random bcp maps are asymptotically entanglement breaking, and use this classification to show that occasionally PPT maps are asymptotically entanglement breaking, i.e., if compositions of random bcp maps contain PPT bcp maps with positive probability, then they are asymptotically entanglement breaking almost surely. We then strengthen our results in the case of compositions of i.i.d. random bcp maps, and, by an application of the theory of random walks on linear groups, we prove a generalization of a theorem of Kuperberg for compositions of i.i.d. random bcp maps. We conclude with a cursory discussion of how to use our methods to move beyond asymptotics and find random finite times at which compositions of random bcp maps are entanglement breaking.

Related papers

• On the Contractivity of Stochastic Interpolation Flow [1.90365714903665]
  We investigate, a recently introduced framework for high dimensional sampling which bears many similarities to diffusion modeling. We show that for a base distribution and a strongly log-concave target distribution, the flow map is Lipschitz with a sharp constant which matches that of Caffarelli's theorem for optimal transport maps. We are further able to construct Lipschitz transport maps between non-Gaussian distributions, generalizing some recent constructions in the literature on transport methods for establishing functional inequalities.
  arXiv  Detail & Related papers  (2025-04-14T19:10:22Z)
• Radiative transport in a periodic structure with band crossings [47.82887393172228]
  We derive the semi-classical model for the Schr"odinger equation in arbitrary spatial dimensions. We consider both deterministic and random scenarios. As a specific application, we deduce the effective dynamics of a wave packet in graphene with randomness.
  arXiv  Detail & Related papers  (2024-02-09T23:34:32Z)
• Integral fluctuation theorems and trace-preserving map [2.492884361833709]
  We rewrite the generating function by integrating measurements and evolution into a constructed mapping. This mapping is completely positive, and the original integral FT is determined by the trace-preserving property of these constructed maps.
  arXiv  Detail & Related papers  (2023-07-06T00:47:52Z)
• Dynamical chaos in nonlinear Schr\"odinger models with subquadratic power nonlinearity [137.6408511310322]
  We deal with a class of nonlinear Schr"odinger lattices with random potential and subquadratic power nonlinearity. We show that the spreading process is subdiffusive and has complex microscopic organization. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border.
  arXiv  Detail & Related papers  (2023-01-20T16:45:36Z)
• On the use of total state decompositions for the study of reduced dynamics [0.0]
  We show that a decomposition of any possibly correlated bipartite state allows one to fix reduced-system evolution via a finite set of time-dependent CPTP maps. In particular, we show that such a decomposition always exists, also for infinite dimensional Hilbert spaces, and that the number of resulting CPTP maps is bounded by the Schmidt rank of the initial global state. For two simple qubit models, we identify the positivity domain defined by the initial states that are mapped into proper states at any time of the evolution fixed by the CPTP semigroups.
  arXiv  Detail & Related papers  (2022-09-06T08:33:42Z)
• Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits [68.8204255655161]
  We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems can be employed to assess the simulatability.
  arXiv  Detail & Related papers  (2022-03-21T17:57:02Z)
• Mean hitting time formula for positive maps [0.0]
  We present an analogous construction for the setting of irreducible, positive, trace preserving maps. The problem at hand is motivated by questions on quantum information theory.
  arXiv  Detail & Related papers  (2022-03-21T01:25:25Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)
• Non-Markovianity criteria for mixtures of noninvertible Pauli dynamical maps [0.0]
  We analyze the connections between the non-Markovianity degree of the most general phase-damping qubit maps and their legitimate mixtures. Using the results for image non-increasing dynamical maps, we formulate the necessary and sufficient conditions for the Pauli maps to satisfy specific divisibility criteria.
  arXiv  Detail & Related papers  (2021-04-13T20:07:03Z)
• Hilbert-space geometry of random-matrix eigenstates [55.41644538483948]
  We discuss the Hilbert-space geometry of eigenstates of parameter-dependent random-matrix ensembles. Our results give the exact joint distribution function of the Fubini-Study metric and the Berry curvature. We compare our results to numerical simulations of random-matrix ensembles as well as electrons in a random magnetic field.
  arXiv  Detail & Related papers  (2020-11-06T19:00:07Z)
• On the Universality of the Double Descent Peak in Ridgeless Regression [0.0]
  We prove a non-asymptotic distribution-independent lower bound for the expected mean squared error caused by label noise in ridgeless linear regression. Our lower bound generalizes a similar known result to the overized (interpolating) regime.
  arXiv  Detail & Related papers  (2020-10-05T08:30:25Z)
• Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps [91.3755431537592]
  We show that any positive product of a qubit map with itself is decomposable. We characterize the cone of decomposable ququart Pauli diagonal maps.
  arXiv  Detail & Related papers  (2020-06-25T16:39:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.