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 \rho _{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 for classical distributions If P = ( p 1 , . . . , p n ) {\displaystyle P=(p_{1},...,p_{n})} is a classical finite probability distribution, its min-entropy can be defined as H m i n ( P ) = log 1 P m a x , P m a x ≡ max i p i . {\displaystyle H_{\rm {min}}({\boldsymbol {P}})=\log {\frac {1}{P_{\rm {max}}}},\qquad P_{\rm {max}}\equiv \max _{i}p_{i}.} One way to justify the name of the quantity is to compare it with the more standard definition of entropy, which reads H ( P ) = ∑ i p i log ( 1 / p i ) {\displaystyle \textstyle H({\boldsymbol {P}})=\sum _{i}p_{i}\log(1/p_{i})} , and can thus be written concisely as the expectation value of log ( 1 / p i ) {\displaystyle \log(1/p_{i})} over the distribution. If instead of taking the expectation value of this quantity we take its minimum value, we get precisely the above definition of H m i n ( P ) {\displaystyle H_{\rm {min}}({\boldsymbol {P}})} . From an operational perspective, the min-entropy equals the negative logarithm of the probability of successfully guessing the outcome of a random draw from P {\displaystyle P} . This is because it is optimal to guess the element with the largest probability and the chance of success equals the probability of that element.
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