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 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.
Definitions
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. 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 from the min-entropy when it is obvious from the context on what state it is evaluated.
Min-entropy as uncertainty about classical information
Suppose an agent had access to a quantum system whose state depends on some classical variable. Furthermore, suppose that each of its elements is distributed according to some distribution. This can be described by the following state over the system. where form an orthonormal basis. We would like to know what the agent can learn about the classical variable. Let be the probability that the agent guesses when using an optimal measurement strategy where is the POVM that maximizes this expression. It can be shown that this optimum can be expressed in terms of the min-entropy as If the state is a product state i.e. for some density operators and, then there is no correlation between the systems and. In this case, it turns out that
The maximally entangled state on a bipartite system is defined as where and form an orthonormal basis for the spaces and respectively. For a bipartite quantum state, we define the maximum overlap with the maximally entangled state as where the maximum is over all CPTP operations and is the dimension of subsystem. This is a measure of how correlated the state is. It can be shown that. If the information contained in is classical, this reduces to the expression above for the guessing probability.
Proof of operational characterization of min-entropy
The proof is from a paper by König, Schaffner, Renner in 2008. It involves the machinery of semidefinite programs. Suppose we are given some bipartite density operator. From the definition of the min-entropy, we have 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 where is the adjoint of the partial trace over. The action of on operators on can be written as We can express the dual problem as a maximization over operators on the space as Using the Choi–Jamiołkowski isomorphism, we can define the channel such that where the bell state is defined over the space. This means that we can express the objective function of the dual problem as as desired. Notice that in the event that the system is a partly classical state as above, then the quantity that we are after reduces to We can interpret as a guessing strategy and this then reduces to the interpretation given above where an adversary wants to find the string given access to quantum information via system.
