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.
Contents
- Definitions
- Operational interpretation of smoothed min entropy
- Min entropy as uncertainty about classical information
- Min entropy as distance from maximally entangled state
- Proof of operational characterization of min entropy
- References
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
This concept is useful in quantum cryptography, in the context of privacy amplification (See for example ).
Definitions
Definition: Let
where the infimum ranges over all density operators
The smooth min entropy is defined in terms of the min entropy.
where the sup and inf range over density operators
where
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. The smoothed entropies share many interesting properties with the von Neumann entropy. For example, the smooth min entropy satisfy a data-processing inequality:
Operational interpretation of smoothed min entropy
Henceforth, we shall drop the subscript
Min-entropy as uncertainty about classical information
Suppose an agent had access to a quantum system B whose state
where
where
If the state
Min-entropy as distance from maximally entangled state
The maximally entangled state
where
where the maximum is over all CPTP operations
Proof of operational characterization of min-entropy
The proof is from a paper by König, Schaffner, Renner '08. It involves the machinery of semidefinite programs,. Suppose we are given some bipartite density operator
This can be re-written as
subject to the conditions
We notice that the infimum is taken over compact sets and hence can be replaced by a minimum. This can then be expressed succinctly as a semidefinite program. Consider the primal problem
This primal problem can also be fully specified by the matrices
We can express the dual problem as a maximization over operators
Using the Choi Jamiolkowski isomorphism, we can define the channel
where the bell state is defined over the space AA'. This means that we can express the objective function of the dual problem as
as desired.
Notice that in the event that the system A is a partly classical state as above, then the quantity that we are after reduces to
We can interpret