The Shannon-McMillan theorem for ergodic quantum lattice systems

The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems

We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on Z-lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.

The Shannon-McMillan theorem and related results for ergodic quantum spin lattice systems and applications in quantum information theory

The aim of this thesis is to formulate and prove quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the Shannon-McMillan-Breiman theorem is one of the fundamental limit theorems for classical discrete dynamical systems. It can be interpreted as a special case of the individual ergodic theorem. In this work, we consider spin latti...

A Bilateral Version of the Shannon-McMillan-Breiman Theorem

We give a new version of the Shannon-McMillan-Breiman theorem in the case of a bijective action. For a finite partition α of a compact set X and a measurable action T on X, we denote by CT n,m,α(x) the element of the partition α ∨ T 1α ∨ . . . ∨ Tmα ∨ T−1α ∨ . . . ∨ T−nα which contains a point x. We prove that for μ-almost all x, lim n+m→∞ ( −1 n+m ) logμ(C n,m,α(x)) = hμ(T, α), where μ is a T ...

A Bilateral version of Shannon-Breiman-McMillan Theorem

We give a new version of the Shannon-McMillan-Breiman theorem in the case of a bijective action. For a finite partition α of a compact set X and a measurable action T on X, we denote by C n,m,α(x) the element of the partition α ∨ T 1α ∨ . . . ∨ Tα ∨ T−1α ∨ . . . ∨ Tα which contains a point x. We prove that for μ-almost all x, lim n+m→∞ (

Quantum Reverse Shannon Theorem