Everything about Von Neumann Entropy totally explained
In
quantum statistical mechanics,
von Neumann entropy refers to the extension of classical
entropy concepts to the field of
quantum mechanics.
John von Neumann rigorously established the correct mathematical framework for quantum mechanics with his work
Mathematische Grundlagen der Quantenmechanik. He provided in this work a theory of measurement, where the usual notion of wave collapse is described as an irreversible process (the so called von Neumann or projective measurement).
Description
The
density matrix was introduced, with different motivations, by von Neumann and by
Lev Landau. The motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector. On the other hand, von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.
The density matrix formalism was developed to extend the tools of classical statistical mechanics to the quantum domain. In the classical framework we compute the partition function of the system in order to evaluate all possible thermodynamic quantities. Von Neumann introduced the density matrix in the context of states and operators in a Hilbert space. The knowledge of the statistical density matrix operator would allow us to compute all average quantities in a conceptually similar, but mathematically different way. Let us suppose we've a set of wave functions
which depend parametrically on a set of quantum numbers
. The natural variable which we've is the amplitude with which a
particular wavefunction of the basic set participates in the actual wavefunction of the system. Let us denote the square of this amplitude by
. The goal is to turn this quantity
into the classical density function in phase space. We have to verify that
goes over into the density function in the classical limit and that it has
ergodic properties. After checking that
is a constant of motion, an ergodic assumption for the probabilities
makes
a function of the energy only .
After this procedure, one finally arrives at the density matrix formalism when seeking a form where
is invariant with respect to the representation used. In the form it's written, it'll only yield the correct expectation values for quantities which are diagonal with respect to the quantum numbers
.
Expectation values of operators which are not diagonal involve the phases of the quantum amplitudes. Suppose we encode the quantum numbers
into the single index
or
. Then our wave function has the form
»
The expectation value of an operator
which isn't diagonal in these wave functions, so
»
This is a much more difficult theorem and was proved in 1973 by
Elliott H. Lieb and
Mary Beth Ruskai, using a
matrix inequality of
Elliott H. Lieb proved in
1973.
The von Neumann entropy is being extensively used in different forms (conditional entropies, relative entropies, etc.) in the framework of quantum information theory. Entanglement measures are based upon some quantity directly related to the von Neumann entropy. However, there have appeared in the literature several papers dealing with the possible inadequacy of the Shannon information measure, and consequently of the von Neumann entropy as an appropriate quantum generalization of Shannon entropy. The main argument is that in classical measurement the Shannon information measure is a natural measure of our ignorance about the properties of a system, whose existence is independent of measurement. Conversely, quantum measurement can't be claimed to reveal the properties of a system that existed before the measurement was made. This controversy has encouraged some authors to introduce the non-additivity property of Tsallis' entropy (a generalization of the standard Boltzmann-Gibbs entropy) as the main reason for recovering a true quantal information measure in the quantum context, claiming that non-local correlations ought to be described because of the particularity of Tsallis' entropy.
In 2004
A. Stotland,
A.A. Pomeransky,
E. Bachmat and
D. Cohen have introduced a new definition of entropy that reflects the inherent uncertainty of quantum mechanical states. This proper definition allows to distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures. Furthermore this proper definition satisfies the basic inequalities of information theory.
Further Information
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