矩阵算子 A 与矩阵算子 B 的 相对熵

In the context of matrix operators (linear operators on finite-dimensional inner product spaces), the concept of relative entropy typically arises in quantum information theory and statistical mechanics , where matrices represent density operators (positive semidefinite, trace 1).

If AA and BB are positive definite matrices (or operators) acting on a Hilbert space, the relative entropy of AA with respect to BB is defined as:

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Key properties:

1. Domain:

   • and are positive semidefinite (often with for finiteness).

   • If is a density operator (trace 1), then , with equality iff .

2. Relation to quantum relative entropy :

   This is the Umegaki relative entropy, fundamental in quantum information (e.g., quantum Stein's lemma, data processing inequality).

3. When A, B are classical :

   If and are diagonal in the same basis with entries and ​, then

   

   which is the classical Kullback--Leibler divergence.

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Example:

Let

, .

Then

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Generalizations:

• For positive semidefinite with , the relative entropy is .

• In von Neumann algebras, there's an extension to general states.

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If your operators and are not necessarily density matrices (trace ≠ 1 or not positive), the "relative entropy" may refer to other divergences (e.g., Bregman divergence for matrix functions), but the standard term in matrix analysis / quantum information is the quantum relative entropy defined above.

More to see its differentiability , joint convexity, or an application (e.g., in quantum hypothesis testing)