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Which is an extension of the von Neumann entropy?
Mathematically, is a positive-semidefinite Hermitian matrix with unit trace. Given the density matrix ρ, von Neumann defined the entropy as which is a proper extension of the Gibbs entropy (up to a factor kB) and the Shannon entropy to the quantum case.
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.
How is the entropy in one spin cancelled?
The entropy in one spin can be “cancelled” by being correlated with the entropy of the other. The left-hand inequality can be roughly interpreted as saying that entropy can only be cancelled by an equal amount of entropy.
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
Who was the first person to invent entropy?
Surprisingly, von Neumann entropy was introduced by von Neumann, (1932), almost 20 years before Shannon entropy was in (Shannon 1948). Several entropy measures are discussed in this article: von Neumann entropy, Rényi entropy, Tsallis entropy, Min entropy, Max entropy, and Unified entropy .
Which is the entropy of a density function?
One of the most studied and frequently used entropy functions is the von Neumann entropy, which is defined as follows: For a density operator ho\\in\\mathcal {D} (\\mathcal {H}) the von Neumann entropy is defined as follows S (ho)=-\\mathrm {Tr} (ho \\logho)\\ . This entropy is a quantum generalization of the classical Shannon entropy.
Continuity: The entropy S (ho) is continuous with respect to the trace norm. That is, if two density operators are close with respect to the trace norm, then so are their entropies. Monotonicity: Given a family of entropies \\ {S_\\alpha (ho)\\}_\\alpha, where \\alpha is some index, the entropies are monotonic in \\alpha\\in\\mathbb {R}.