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arXiv:math-ph/0203008 · Journal of Physics A Mathematical and General

Zeno dynamics of von Neumann algebras

Andreas U Schmidt

Published Aug 28, 2002• 10 citations
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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 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.

DOI
10.1088/0305-4470/35/36/309

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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.
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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.
A bstract The local two-dimensional Poincaré algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincaré algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincaré algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. Interacting quantum systems in the thermodynamic limit and large N theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.