nLab min-entropy




A notion of entropy.

In the context of probability theory, the min-entropy of a discrete probability distribution is the negative logarithm of the probability of the most likely outcome. It is never greater than the ordinary entropy of that distribution.

For a quantum system with density matrix ρ\rho, the min-entropy is minus the logarithm of the maximum of its operator spectrum, hence (since density matrices are self-adjoint operators) of its operator norm (which in the case of a finite-dimensional space of states this is the maximum eigenvalue):

S min(ρ)ln(max(Spectrum(ρ)))=ln(A). S_{min}(\rho) \;\coloneqq\; - ln \Big( max \big( Spectrum(\rho) \big) \Big) \;=\; - ln \big( \left\Vert A \right\Vert \big) \,.

(e.g. Chen 19, Def. 5.2.2)


Relation to Reny entropy

Min-entropy is the limit of Rényi entropy at order α\alpha as α\alpha \to \infty.

order0\phantom{\to} 01\to 12\phantom{\to}2\to \infty
Rényi entropyHartley entropy\geqShannon entropy\geqcollision entropy\geqmin-entropy



See also:

Generalization to condition entropies:

  • Fabian Furrer, Johan Åberg, Renato Renner, Min- and Max-Entropy in Infinite Dimensions, Communications in Mathematical Physics volume 306, pages 165–186 (2011) (doi:10.1007/s00220-011-1282-1)

In the context of holographic entanglement entropy:

Last revised on March 27, 2023 at 12:45:53. See the history of this page for a list of all contributions to it.