跡不等式與量子熵相關問題之探討

Abstract

量子理論是當今最重要的領域之一。其中，von Neumann熵為特定量子系統的不確定性提供了一種度量，其被定義為 S(rho) = -tr(f(rho))，rho為量子系統上的密度矩陣，f(x)=x log x。人們發現熵具有數個性質，例如凹凸性和某種單調性等。我們將證明相關的跡不等式後，在一般情況下找出某些也具有這些性質的函數 f: R to R 。

我們將介紹運算子單調性以及運算子的(聯合)凹凸性，同時介紹子系統（它們是ast子代數）上的條件期望值與張量積，以及它們之間的關係。然後，我們將探討具有更多變量的運算子的跡相關函數的單調性和凹凸性。我們將定義矩陣的L_p範數，並用刻畫數個矩陣範數相關函數的性質。最後，我們將證明熵的次可加性和強次可加性，推廣到非交換代數，並以此證明其上的廣義楊氏不等式。

此篇探討主要參考Eric Carlen的論文，並以鄭皓中教授提出的相關猜想的討論作為結尾。

Quantum theory has been one of the most important fields these days. Most of all, the von Neumann entropy gives a measurement to the uncertainty of a specific quantum system, which is defined as S(rho) = -tr(f(rho)) where f(x) = x log x for density matrices ``rho'' on the quantum system. It has been discovered that the entropy has several propositions, such as concavity a specific kind of monotonicity, etc. We state and proof several related trace inequalities and find out which functions f: R to R have the former properties in general.

We introduce the operator monotonocity and operator (joint) concavity for several functions; also, we introduce the conditional expectation and tensor products on subsystems, which are ast-subalgebras, and the relation between them. After that, we discover the monotonicity and concavity for trace associated functions on operators with more variables. Moreover, we deifine the L_p norm for matrices and characterize several functions related with the matrix norm. Last, we give proof to the subadditivity and strong subadditivity for the entropy, generalize to non-commutative algebras, and prove the generalized Young's Inequality on them.

We mainly follows the essay of Eric Carlen, and ended with the discussion on a related conjecture which has been announced by Professor Hau-Chung Cheng.