Quantum diffusions on the algebra of all bounded operators on a hilbert space

Luigi Accardi, Wilhelm von Waldenfels

Cecchini's transition expectations and markov chains.- Central limits of squeezing operators.- On the weak coupling limit problem.- On multi-dimensional markovian cocycles.- Quantum stop times.- Quantum random walks.- Unitary dilation of a nonlinear quantum boltzmann equation.- On isometries of non associative Lp-spaces.- Convolution semigroups in quantum probability and quantum stochastic calculus.- Stochastic couplings for von neumann algebras.- The covering property in a causal logic.- Temperature states of spin-boson models.- Bernoulli fields.- The relations of the non-commutative coefficient algebra of the unitary group.- Orthogonal series and strong laws of large numbers in von neumann algebras.- Limit theorems for repeated measurements and continuous measurement processes.- Quantum diffusions on the algebra of all bounded operators on a hilbert space.- Asymptotic formula for normal operators in non-commutative L2-spaces.- Convergences in W*-algebras - Their strange behaviour and tools for their investigation.- Positive mappings on matrix algebras.- Fluctuations of the dicke maser.- Tangent bimodule and locality for dissipative operators on C*-algebras.- Noncommutative stochastic processes with independent and stationary additive increments.- The atom in the radiation field as a quantum stochastic process.

David Applebaum

Klaas Landsman

In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B ( H ) be the set of all bounded operators on H . Here a notable point is that linear operators on finite - dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B ( H ) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform , as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

R. L. Hudson

Sanjay Moudgalya, Olexei I. Motrunich

Symmetry algebras of quantum many-body systems with locality can be understood using commutant algebras, which are defined as algebras of operators that commute with a given set of local operators. In this work, we show that these symmetry algebras can be expressed as frustration-free ground states of a local superoperator, which we refer to as a “super-Hamiltonian.” We demonstrate this for conventional symmetries such as Z 2 , U ( 1 ) , and S U ( 2 ) , where the symmetry algebras map to various kinds of ferromagnetic ground states, as well as for unconventional ones that lead to weak ergodicity-breaking phenomena of Hilbert-space fragmentation (HSF) and quantum many-body scars. In addition, we show that the low-energy excitations of this super-Hamiltonian can be understood as approximate symmetries, which in turn are related to slowly relaxing hydrodynamic modes in symmetric systems. This connection is made precise by relating the super-Hamiltonian to the superoperator that governs the operator relaxation in noisy symmetric Brownian circuits and this physical interpretation also provides a novel interpretation for Mazur bounds for autocorrelation functions. We find examples of gapped (gapless) super-Hamiltonians indicating the absence (presence) of slow modes, which happens in the presence of discrete (continuous) symmetries. In the gapless cases, we recover hydrodynamic modes such as diffusion, tracer diffusion, and asymptotic scars in the presence of U ( 1 ) symmetry, HSF, and a tower of quantum scars, respectively. In all, this demonstrates the power of the commutant-algebra framework in obtaining a comprehensive understanding of exact symmetries and associated approximate symmetries and hydrodynamic modes, and their dynamical consequences in systems with locality. Published by the American Physical Society 2024

Werner Timmermann