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Open Physics

Measurement interpretation and information measures in general probabilistic theory

Zhaoqi Wu, Chuanxi Zhu, Xiaozhi Zhang

Published Feb 21, 2013• 5 citations
Publisher

Abstract

Abstract The incompatibility of dynamics postulate (unitary evolution) and the measurement postulate (wave-packet collapse) of quantum mechanics has recently been solved by Zurek from an information transfer perspective. Luo gave his derivation by relaxing the repeatability postulate. In this paper, we reconsider Luo’s derivation in the setting of general probabilistic theory (GPT). We also introduce the concept of sub- and super-fidelity in GPT and discuss their properties.

DOI
10.2478/s11534-012-0169-x

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Abstract The incompatibility of dynamics postulate (unitary evolution) and the measurement postulate (wave-packet collapse) of quantum mechanics has recently been solved by Zurek from an information transfer perspective. Luo gave his derivation by relaxing the repeatability postulate. In this paper, we reconsider Luo’s derivation in the setting of general probabilistic theory (GPT). We also introduce the concept of sub- and super-fidelity in GPT and discuss their properties.
We investigate the main features of a measure of fidelity between states in a general family of probabilistic theories admitting classical probability theory and standard quantum theory as particular instances. We apply the aforementioned measure to investigate informationtheoretical \nfeatures of these theories related to the conservation of information during the evolution \nof closed physical systems. In particular, we derive a generalization of a fundamental result in \nquantum theory relevant for the measurement problem: Zurek’s recent extension of the no-cloning theorem.
Abstract Information transfer in generalized probabilistic theories (GPT) is an important problem. We have dealt with the problem based on repeatability postulate, which generalizes Zurek’s result to the GPT framework (Wu et al., Phys. Lett. A 379, 2694, 2015). A natural question arises: can we deduce the information transfer result under weaker assumptions? In this paper, we generalize Zurek’s result to the framework of GPT using weak repeatability postulate. We show that if distinguishable information can be transferred from a physical system to a series of apparatuses under the weak repeatability postulate in GPT, then the initial states of the physical system must be completely distinguishable. Moreover, after each step of invertible transformation, the composite states of the composite system composed of the physical systems and the apparatuses must also be completely distinguishable.
Generalized probabilistic theories (GPTs) provide a unified framework for describing probabilistic physical theories, encompassing classical and quantum theories as well as hypothetical theories beyond quantum mechanics. Since most GPTs are highly unrealistic and far removed from known physical theories, it is important to constrain them by physically reasonable principles. One of the most important such principles is consistency with thermodynamics, which has been extensively studied through toy models involving semipermeable membranes (SPMs) implementing measurements. On the other hand, information thermodynamics, which plays a central role in understanding the relationship between measurement and thermodynamics in classical and quantum theory, has remained largely undeveloped in GPTs. In this work, we construct information thermodynamics in GPTs and provide a unified framework for analyzing the relationship between measurement, feedback, information erasure, and the second law of thermodynamics. We also formulate a general framework for SPM models and analyze the thermodynamic cost of measurements implemented by SPMs. As a result, we show that no work can be extracted in contradiction with the second law as long as the measurement processes are consistent with entropy nondecrease, and derive sufficient conditions for this property for several entropy definitions proposed in GPTs. Moreover, by considering measurement processes violating these conditions, we construct explicit GPT systems realizing isothermal SPM cycles from which positive work can be extracted. These results demonstrate that violations of the second law can arise from the lack of fundamental entropy properties or discrepancies between entropy definitions, and provide a unified and model-independent foundation for understanding the relationship between thermodynamics and measurement in GPTs.
Abstract Information transfer in generalized probabilistic theories (GPT) is an important problem. We have dealt with the problem based on repeatability postulate, which generalizes Zurek’s result to the GPT framework (Wu et al., Phys. Lett. A 379 , 2694, 2015). A natural question arises: can we deduce the information transfer result under weaker assumptions? In this paper, we generalize Zurek’s result to the framework of GPT using weak repeatability postulate. We show that if distinguishable information can be transferred from a physical system to a series of apparatuses under the weak repeatability postulate in GPT, then the initial states of the physical system must be completely distinguishable. Moreover, after each step of invertible transformation, the composite states of the composite system composed of the physical systems and the apparatuses must also be completely distinguishable.
In quantum mechanics, outcomes of measurements on a state have a probabilistic interpretation while the evolution of the state is treated deterministically. Here we show that one can also treat the evolution as being probabilistic in nature and one can measure which unitary acted. In further analogy to states, one can also choose which basis of unitaries to measure. Likewise, one can give an information-theoretic interpretation to evolutions by defining the entropy of a completely positive map. This entropy gives the rate at which the informational content of the evolution can be compressed. One cannot compress this information and still have the evolution act on an unknown state, but we demonstrate a general scheme to do so probabilistically. This allows one to generalize super-dense coding to the sending of quantum information. One can also define the ``interaction-entanglement'' of a unitary, and concentrate this entanglement.