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