nLab min-entropy

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 entropy		Hartley entropy	$\geq$	Shannon entropy	$\geq$	collision entropy	$\geq$	min-entropy

Related entries

• max-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, doi:10.1109/TIT.2009.2025545]

See also:

• Wikipedia, Min-entropy

• Scholarpedia, Quantum entropies – Min entropy

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:

• Ning Bao, Geoffrey Penington, Jonathan Sorce, Aron C. Wall, Section 2.3 of: Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT, JHEP 2019:69 (arXiv:1812.01171)