First exit time: A theory of stopping times in quantum processes

Luigi Accardi, Wilhelm von Waldenfels

A note on Meyer's note.- Stochastic integration.- Quantum stochastic parallel transport on non-commutative vector bundles.- Input and output channels in quantum systems and quantum stochastic differential equations.- Some noncommutative Radon-Nikodym Theorems for von Neumann algebras.- Multidimensional quantum diffusions.- An application of de finetti's theorem.- The quantum Langevin equation from the independent-oscillator model.- Quantum poisson processes: Physical motivations and applications.- A noncommutative generalization of conditionally positive definite functions.- Contraction semigroups in L2 over a von neumann algebra.- Survey on a theory of non-commutative stationary markov processes.- Dynamical entropy for quantum systems.- An integral kernel approach to noise.- A note on shifts and cocycles.- Local measures in fock space stochastic calculus and a generalized ito-tanaka formula.- Representation of a class of quantum martingales II.- Conditional expectation in quantum probability.- Mutual quadratic variation and ito's table in quantum stochastic calculus.- On mixing properties of automorphisms of von neumann algebras related to measure space transformations.- First exit time: A theory of stopping times in quantum processes.- A central limit theorem on the free lie group.- Entropy, observability and the generalised second law of thermodynamics.- Remarks on asymptotic properties of groups of the bogoliubov transformations of car c*-algebras.- Linear and non-linear stochastic processes.- Detailed balance and critical slowing down.- Quantum martingales and stochastic integrals.

Lawrence S. Schulman

R. L. Hudson

We review Fock space based quantum probability and in particular, the theory of stop times based on it. In the Fock space , a stop time may be defined as a positive self-adjoint operator whose spectral resolution is adapted to the natural filtration based on the splittings Then if K is one of the basic quantum martingales (creation, preservation or annihilation) the corresponding process beginning anew at time T can be defined unambiguosly using functional calculus for the self-adjoint operator T with an operator-valued integrand since the integrand commutes with the integrator. An isometric operator , the forward shift through T, is defined by a similar spectral integral of sure forward shifts and conjugates with . An optional stopping theorem is formulated and proved using similar operator valued spectral integrals.

Juan Gonzalo Muga, C. Richard Leavens

Rong-Tao Qiu, Wu-Sheng Dai, Mi Xie

Angela Sara Cacciapuoti, Jessica Illiano, Michele Viscardi +1 more

Multipartite entanglement distribution is a key functionality of the Quantum Internet. However, quantum entanglement is very fragile, easily degraded by decoherence, which strictly constraints the time horizon within the distribution has to be completed. This, coupled with the quantum noise irremediably impinging on the channels utilized for entanglement distribution, may imply the need to attempt the distribution process multiple times before the targeted network nodes successfully share the desired entangled state. And there is no guarantee that this is accomplished within the time horizon dictated by the coherence times. As a consequence, in noisy scenarios requiring multiple distribution attempts, it may be convenient to stop the distribution process early. In this paper, we take steps in the direction of knowing when to stop the entanglement distribution by developing a theoretical framework, able to capture the quantum noise effects. Specifically, we first prove that the entanglement distribution process can be modeled as a Markov decision process. Then, we prove that the optimal decision policy exhibits attractive features, which we exploit to reduce the computational complexity. The developed framework provides quantum network designers with flexible tools to optimally engineer the design parameters of the entanglement distribution process.