Min entropy

The min entropy, in information theory, is the smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the most likely outcome. The various Rényi entropies are all equal for a uniform distribution, but measure the unpredictability of a nonuniform distribution in different ways. The min entropy is never greater than the ordinary or Shannon entropy (which measures the average unpredictability of the outcomes) and that in turn is never greater than the Hartley or max entropy, defined as the logarithm of the number of outcomes.

As with the classical Shannon entropy and its quantum generalization, the von Neumann entropy, one can define a conditional versions of min entropy. The conditional quantum min entropy is a one-shot, or conservative, analog of conditional quantum entropy.

To interpret a conditional information measure, suppose Alice and Bob were to share a bipartite quantum state ${\displaystyle \rho _{AB}}$. Alice has access to system A and Bob to system B. The conditional entropy measures the average uncertainty Bob has about Alice's state upon sampling from his own system. The min entropy can be interpreted as the distance of a state from a maximally entangled state.

This concept is useful in quantum cryptography, in the context of privacy amplification (See for example ).

Definition: Let ${\displaystyle \rho _{AB}}$ be a bipartite density operator on the space ${\displaystyle {\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}}$. The min-entropy of A conditioned on B is defined to be

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