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 with nonzero probability. As with the classical Shannon entropy and its quantum generalization, the von Neumann entropy, one can define a conditional version 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 . Alice has access to system and Bob to system . 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 be a bipartite density operator on the space . The min-entropy of conditioned on is defined to be where the infimum ranges over all density operators on the space . The measure is the maximum relative entropy defined as The smooth min-entropy is defined in terms of the min-entropy. where the sup and inf range over density operators which are -close to . This measure of -close is defined in terms of the purified distance where is the fidelity measure. These quantities can be seen as generalizations of the von Neumann entropy. Indeed, the von Neumann entropy can be expressed as This is called the fully quantum asymptotic equipartition theorem.