In the case of the von Neumann entropy, this minimum determines also the quantum discord. This quantity is a measure of the average mixedness of A after such measurement, and its minimum over all local measurements is shown to be the associated entanglement of formation between A and a purifying third system C. We analyze, for a general concave entropic form, the associated conditional entropy of a quantum system A + B, obtained as a result of a local measurement on one of the systems (B). ДЄlthough the specific-conditional entropy,, can be either less or greater than, can never exceed. Where is the mutual information between and. The latter can take negative values, unlike its classical counterpart.Д«ayes' rule does not hold for conditional quantum entropy, since. In quantum information theory, the conditional entropy is generalized to the conditional quantum entropy.

If Y is conditional independent of Z given X we have: Generalization to quantum theory Subtracting the two equations implies Bayes' rule. Bayes' ruleД«ayes' rule for conditional entropy states It has a similar form to Chain rule (probability) in probability theory, except that addition instead of multiplication is used. In general, a chain rule for multiple random variables holds: The chain rule follows from the above definition of conditional entropy: This quantity is exactly, which gives the chain rule of conditional entropy: Once is known, we only need bits to describe the state of the whole system. Now if we first learn the value of, we have gained bits of information. Chain ruleДЄssume that the combined system determined by two random variables X and Y has joint entropy, that is, we need bits of information to describe its exact state. Conversely, if and only if and are independent random variables. If and only if the value of is completely determined by the value of. Note: It is understood that the expressions 0 log 0 and 0 log ( c/0) for fixed c>0 should be treated as being equal to zero. Given discrete random variables with Image and with Image, the conditional entropy of given is defined as: (Intuitively, the following can be thought as the weighted sum of for each possible value of, using as the weights) If is the entropy of the variable conditioned on the variable taking a certain value, then is the result of averaging over all possible values that may take. The entropy of conditioned on is written as. Here, information is measured in shannons, nats, or hartleys. In information theory, the conditional entropy (or equivocation) quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known.

The violet is the mutual information I(X Y). The circle on the right (blue and violet) is H(Y), with the blue being H(Y |X). The circle on the left (red and violet) is the individual entropy H(X), with the red being the conditional entropy H(X |Y). The area contained by both circles is the joint entropy H(X,Y). Venn diagram for various information measures associated with correlated variables X and Y.

Conditional entropy