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. 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 ρ A B {displaystyle ho _{AB}} . Alice has access to system A {displaystyle A} and Bob to system B {displaystyle 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 ρ A B {displaystyle ho _{AB}} be a bipartite density operator on the space H A ⊗ H B {displaystyle {mathcal {H}}_{A}otimes {mathcal {H}}_{B}} . The min-entropy of A {displaystyle A} conditioned on B {displaystyle B} is defined to be where the infimum ranges over all density operators σ B {displaystyle sigma _{B}} on the space H B {displaystyle {mathcal {H}}_{B}} . The measure D max {displaystyle D_{max }} is the maximum relative entropy defined as