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Zeno Dynamics of von Neumann Algebras

Andreas U. Schmidt

Published Aug 17, 2002• 7 citations
OpenAlex

Abstract

ABSTRACT. The dynamical quantum Zeno effect is studied in the context of von Neumann algebras. We identify a localized subalgebra on which the Zeno dynamics acts by automorphisms. The Zeno dynamics coincides with the mod-ular dynamics of that subalgebra, if an additional assumption is satisfied. This relates the modular operator of that subalgebra to the modular operator of the original algebra by a variant of the Kato–Lie–Trotter product formula. 1.

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Zeno dynamics of von Neumann algebras

Andreas U Schmidt

The dynamical quantum Zeno effect is studied in the context of von Neumann algebras. We identify a localized subalgebra on which the Zeno dynamics acts by automorphisms. The Zeno dynamics coincides with the modular dynamics of that subalgebra, if an additional assumption is satisfied. This relates the modular operator of that subalgebra to the modular operator of the original algebra by a variant of the Kato–Lie–Trotter product formula.

arXiv:math-ph/0203008•Journal of Physics A Mathematical and General•Aug 28, 2002•10 citations

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If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finite-rank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily bounded from below) generator as a generalized mean value Hamiltonian.

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ABSTRACT. The dynamical quantum Zeno effect is studied in the context of von Neumann algebras. We identify a localized subalgebra on which the Zeno dynamics acts by automorphisms. The Zeno dynamics coincides with the mod-ular dynamics of that subalgebra, if an additional assumption is satisfied. This relates the modular operator of that subalgebra to the modular operator of the original algebra by a variant of the Kato–Lie–Trotter product formula. 1.
The problem of characterizing GKLS-generators and CP-maps with an invariant von Neumann algebra [Formula: see text] appeared in different guises in the literature. We prove two unifying results, which hold even for weakly closed *-algebras: first, we show how to construct a normal form for [Formula: see text]-invariant GKLS-generators, if a normal form for [Formula: see text]-invariant CP-maps is known — rendering the two problems essentially equivalent. Second, we provide a normal form for [Formula: see text]-invariant CP-maps if [Formula: see text] is atomic (which includes the finite-dimensional case). As an application we reproduce several results from the literature as direct consequences of our characterizations and thereby point out connections between different fields.
We study the quantum Zeno effect in quantum statistical mechanics within the operator algebraic framework. We formulate a condition for the appearance of the effect in W *-dynamical systems, in terms of the short-time behaviour of the dynamics. Examples of quantum spin systems show that this condition can be effectively applied to quantum statistical mechanical models. Furthermore, we derive an explicit form of the Zeno generator, and use it to construct Gibbs equilibrium states for the Zeno dynamics. As a concrete example, we consider the X – Y model, for which we show that a frequent measurement at a microscopic level, e.g. a single lattice site, can produce a macroscopic effect in changing the global equilibrium.
We analyze the entropy behaviour of a discrete quantum channel defined by the composition of a unitary automorphism and a projective conditional expectation onto a finite-dimensional subalgebra. Let [Formula: see text], where [Formula: see text] is unitary and [Formula: see text] is the conditional expectation associated with an orthogonal projector decomposition of the identity. We note that the von Neumann entropy is monotone nondecreasing under repeated application of [Formula: see text]. Equality is characterized explicitly in terms of the invariant algebra of the channel. We further identify the fixed-point space as the intersection of the commutant of [Formula: see text] with the block-diagonal von Neumann algebra defined by the coarse-graining projectors. Finally, we describe the asymptotic structure under repeated application of the channel via spectral properties of completely positive trace-preserving maps. The results demonstrate that entropy growth in this setting arises solely from algebraic noncommutativity between unitary action and conditional expectation, without invoking thermodynamic limits, statistical ensembles, or environmental assumptions.
If frequent measurements ascertain whether a quantum system is still in a given subspace, it remains in that subspace and a quantum Zeno effect takes place. The limiting time evolution within the projected subspace is called quantum Zeno dynamics. This phenomenon is related to the limit of a product formula obtained by intertwining the time evolution group with an orthogonal projection. By introducing a novel product formula, we will give a characterization of the quantum Zeno effect for finite-rank projections in terms of a spectral decay property of the Hamiltonian in the range of the projections. Moreover, we will also characterize its limiting quantum Zeno dynamics and exhibit its (not necessarily bounded from below) generator as a generalized mean value Hamiltonian.
An important object appearing in the framework of the Tomita–Takesaki theory is an invariant cone under the modular automorphism group of von Neumann algebras. As a result of the connection between von Neumann algebras and quantum field theory, von Neumann algebras have become increasingly important for (higher) category theory and topology. We show explicitly how an example of a class of cones discovered by Connes–Araki–Haagerup (CAH), invariant under the modular automorphism group, are related to Wishart laws and information geometry. Given its relation to 2D quantum field theory this highlights new relations between (quantum) information geometry and quantum geometry. Additionally, our results have applications to qubits. The cones and the Wishart laws defined on them carry interesting monoidal structures, revealing the possibility of using a more abstract formalism of monoidal categories. Such a language is interesting as it opens a deeper possibility for generalizations of results.