Abstract
The recent article ‘Entropic Updating of Probability and Density Matrices’ [] derives and demonstrates the inferential origins of both the standard and quantum relative entropies in unison. Operationally, the standard and quantum relative entropies are shown to be designed for the purpose of inferentially updating probability distributions and density matrices, respectively, when faced with incomplete information. We call the inferential updating procedure for density matrices the ‘quantum maximum entropy method’. Standard inference techniques in probability theory can be criticized for lacking concrete physical consequences in physics; but here, because we are updating quantum mechanical density matrices, the quantum maximum entropy method has direct physical and experimental consequences. The present article gives a new derivation of the Quantum Bayes Rule, and some generalizations, using the quantum maximum entropy method while discuss some of the limitations the quantum maximum entropy method puts on the measurement process in Quantum Mechanics.
Highlights
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
We provide a new derivation of the Quantum Bayes Rule (QBR), discuss the physical implications entropic methods puts on the measurement process in Quantum Mechanics (QM), and briefly discuss how the quantum maximum entropy method provides some simple generalizations of the QBR
The Lagrange multiplier technique is used in the maximum entropy method community ([23, 25, 24] 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
Summary
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 Lagrange multiplier technique is used in the maximum entropy method community ([23, 25, 24] 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 As both forms of the standard and quantum relative entropy resemble one another, they inevitably share analogous solutions and face similar limitations; because we are dealing with density matrices, these limitations have physical consequences. A special case of the PDMT insists that the detection of an observable from a pure state (the collapse) is impossible without first decohering (or partially decohering) the pure state This is a rediscovery of Luders’ notion [10] that the action of a measurement device is to project the pure state into a mixed state ρ → PiρPi, except our argument is from purely entropic and inferential arguments. We will introduce the PDMT and apply the quantum maximum entropy method to derive the aforementioned cases of interest
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