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 for classical distributions 
If  is a classical finite probability distribution, its min-entropy can be defined as One way to justify the name of the quantity is to compare it with the more standard definition of entropy, which reads , and can thus be written concisely as the expectation value of  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 .

From an operational perspective, the min-entropy equals the negative logarithm of the probability of successfully guessing the outcome of a random draw from .
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.

Definition for quantum states 

A natural way to generalize "min-entropy" from classical to quantum states is to leverage the simple observation that quantum states define classical probability distributions when measured in some basis. There is however the added difficulty that a single quantum state can result in infinitely many possible probability distributions, depending on how it is measured. A natural path is then, given a quantum state , to still define  as , but this time defining  as the maximum possible probability that can be obtained measuring , maximizing over all possible projective measurements.
Using this, one gets the operational definition that the min-entropy of  equals the negative logarithm of the probability of successfully guessing the outcome of any measurement of .

Formally, this leads to the definition where we are maximizing over the set of all projective measurements ,  represent the measurement outcomes in the POVM formalism, and  is therefore the probability of observing the -th outcome when the measurement is .

A more concise method to write the double maximization is to observe that any element of any POVM is a Hermitian operator such that , and thus we can equivalently directly maximize over these to get In fact, this maximization can be performed explicitly and the maximum is obtained when  is the projection onto (any of) the largest eigenvalue(s) of . We thus get yet another expression for the min-entropy as: remembering that the operator norm of a Hermitian positive semidefinite operator equals its largest eigenvalue.

Conditional entropies 
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 

Since the conditional min-entropy is always smaller than the conditional Von Neumann entropy, it follows that

Min-entropy as overlap with the maximally entangled state 

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 .

See also
 von Neumann entropy
 Generalized relative entropy
 max-entropy

References

Quantum mechanical entropy