Contents
What are the properties of the von Neumann entropy?
Some properties of the von Neumann entropy: S(ρ) is zero if and only if ρ represents a pure state. S(ρ) is maximal and equal to ln N for a maximally mixed state, N being the dimension of the Hilbert space. S(ρ) is invariant under changes in the basis of ρ, that is, S(ρ) = S(UρU†), with U a unitary transformation.
Which is the right hand inequality of entropy?
This right hand inequality is known as subadditivity. The two inequalities together are sometimes known as the triangle inequality. They were proved in 1970 by Huzihiro Araki and Elliott H. Lieb.
Why is the vanishing entropy of a pure state important?
In other words, it codifies the degree of mixing of the state describing a given finite system. Measurement decoheres a quantum system into something noninterfering and ostensibly classical; so, e.g., the vanishing entropy of a pure state
How is entropy cancelled by an equal amount?
The left-hand inequality can be roughly interpreted as saying that entropy can only be cancelled by an equal amount of entropy. If system A and system B have different amounts of entropy, the smaller can only partially cancel the greater, and some entropy must be left over.
How did John von Neumann contribute to quantum mechanics?
John von Neumann established a rigorous mathematical framework for quantum mechanics in his 1932 work Mathematical Foundations of Quantum Mechanics. In it, he provided a theory of measurement, where the usual notion of wave-function collapse is described as an irreversible process (the so-called von Neumann or projective measurement).
When is the entropy of a composite system maximized?
Likewise, the right-hand inequality can be interpreted as saying that the entropy of a composite system is maximized when its components are uncorrelated, in which case the total entropy is just a sum of the sub-entropies. This may be more intuitive in the phase space formulation]