Entropic Updating of Probabilities and Density Matrices

Abstract

Abstract:We find that the standard relative entropy and the Umegaki entropy are designed for the purpose of inferentially updating probabilities and density matrices, respectively. From the same set of inferentially guided design criteria, both of the previously stated entropies are derived in parallel. This formulates a quantum maximum entropy method for the purpose of inferring density matrices in the absence of complete information.

The quantum Bayes rule and generalizations from the quantum maximum entropy method

“…The recent article 'Entropic Updating of Probability and Density Matrices' [1] derives and demonstrates the inferential origins of both the standard and quantum relative entropies in unison. The derivations of the standard and quantum relative entropies in [1] were not rudimentary; rather, a set of inferentially guided design criteria were proposed to design a function capable of accurately updating probability distributions when faced with incomplete information. The solution has the functional form of the standard relative entropy and thus the standard relative entropy is the functional designed for the purpose of probability updating.…”

“…The solution has the functional form of the standard relative entropy and thus the standard relative entropy is the functional designed for the purpose of probability updating. Similar (design) derivations exist [2][3][4][5][6][7][8], but the number of required design criteria was reduced in [1]. What is particularly pleasant in [1] is the equal implementation of the same design criteria to design a functional capable of updating density matrices.…”

“…Similar (design) derivations exist [2][3][4][5][6][7][8], but the number of required design criteria was reduced in [1]. What is particularly pleasant in [1] is the equal implementation of the same design criteria to design a functional capable of updating density matrices. This parallel derivation shows the quantum relative entropy is designed to update density matrices-formulating an inferentially oriented quantum maximum entropy method.…”

“…This parallel derivation shows the quantum relative entropy is designed to update density matrices-formulating an inferentially oriented quantum maximum entropy method. Not only was the quantum relative entropy found in [1], but we also learned how to use it. This discussion saturates the previous article.…”

“…Our result further perpetuates the interpretation that entropy may be used to inferentially collapse the wavefunction using projectors [19,20], and our result is generalized to the QBR using POVM's (and a Quantum Jeffrey's Rule (QJR)). Rather than appealing to group theoretic arguments [19,20], our derivations are seemingly simpler as they require solving the Lagrange multiplier problem following [1] (the derivation parallels [21,22] but using density matrices) and naturally avoids the infinite entropy problem that that appears in the 'strong Lüders rule' derivation in [19]. The Lagrange multiplier technique is used in the maximum entropy method community ( [23][24][25] and the works and conferences that have followed) for updating probability distributions, so we refer to the method of inference capable of update density matrices as the quantum maximum entropy method.…”