# nLab min-entropy

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# 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 $\rho$, 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):

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

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

## Properties

### Relation to Reny entropy

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

order$\phantom{\to} 0$$\to 1$$\phantom{\to}2$$\to \infty$
Rényi entropyHartley entropy$\geq$Shannon entropy$\geq$collision entropy$\geq$min-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)