nLab
min-entropy

Contents

Contents

Idea

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)

Properties

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

References

Review:

  • Yi-Hsiu Chen, Computational Notions of Entropy: Classical, Quantum, and Applications, 2019 (pdf)

  • Robert Koenig, Renato Renner, Christian Schaffner, The operational meaning of min- and max-entropy, IEEE Trans. Inf. Th., vol. 55, no. 9 (2009) (arXiv:0807.1338)

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 May 28, 2021 at 07:38:31. See the history of this page for a list of all contributions to it.