The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems
نویسندگان
چکیده
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.
منابع مشابه
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...
متن کاملIndividual ergodic theorem for intuitionistic fuzzy observables using intuitionistic fuzzy state
The classical ergodic theory hasbeen built on σ-algebras. Later the Individual ergodictheorem was studied on more general structures like MV-algebrasand quantum structures. The aim of this paper is to formulate theIndividual ergodic theorem for intuitionistic fuzzy observablesusing m-almost everywhere convergence, where m...
متن کاملAn Ergodic Theorem for the Quantum Relative Entropy
We prove the ergodic version of the quantum Stein’s lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equip...
متن کاملPointwise Theorems for Amenable Groups
In this paper we describe proofs of the pointwise ergodic theorem and Shannon-McMillan-Breiman theorem for discrete amenable groups, along Følner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups.
متن کاملThe dimension of ergodic random sequences
Let μ be a computable ergodic shift-invariant measure over {0, 1} . Providing a constructive proof of Shannon-McMillan-Breiman theorem, V’yugin proved that if x ∈ {0, 1} is Martin-Löf random w.r.t. μ then the strong effective dimension Dim(x) of x equals the entropy of μ. Whether its effective dimension dim(x) also equals the entropy was left as an problem question. In this paper we settle this...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره math.DS/0207121 شماره
صفحات -
تاریخ انتشار 2002