Introduction: What is the Free Energy Principle?

• The Free Energy Principle (FEP) suggests that systems try to minimize prediction error or “surprise”.
• This principle is based on Bayesian inference – updating beliefs based on new information.

Quantum Extension of the FEP

• This paper extends the FEP to generic quantum systems.
• Quantum systems are described without a fixed spacetime background, viewing them as active observers.
• They update their “beliefs” through measurement, much like we adjust our expectations in everyday life.

Key Concepts and Definitions

• Variational Free Energy (VFE): An upper bound on “surprise”, measuring how unexpected a state is compared to a model.
• Surprise (Surprisal): A measure of how unexpected an event is; lower surprise means better prediction.
• Bayesian Prediction Error: The difference between what is expected and what is observed.
• Quantum Reference Frames (QRFs): Systems that provide a measurement basis – similar to choosing a coordinate system for understanding observations.
• Unitarity: A fundamental principle of quantum mechanics ensuring that the total probability (and thus information) is conserved over time.
• Markov Blankets (MB): Conceptual boundaries that separate a system’s internal states from its environment, much like a protective shell.

Reformulating the FEP in Quantum Terms

• The paper redefines the FEP using quantum information theory concepts.
• It removes the need for a fixed spacetime backdrop and objective randomness; uncertainty arises from the process of quantum measurement.
• Interactions between systems are viewed as exchanges of information through a “holographic screen”—a conceptual boundary where information is encoded.
• Agents (or observers) are defined by their ability to break symmetry at this boundary using QRFs, effectively “choosing” how to measure the world.

Measurement, Memory, and System Identification

• Systems record observations in a step-by-step process, similar to following a detailed recipe.
• Coarse-graining: Simplifying complex data to capture the essential behavior over time, much like summarizing detailed notes.
• Measurements are made using operators that allow the system to compare its predictions with observed outcomes.
• This iterative process helps minimize prediction errors over repeated cycles.

Noncommutativity and Context-Switching

• In quantum mechanics, some measurements do not commute, meaning the order in which they are performed matters.
• This leads to context-switching, where changing the measurement basis can temporarily increase prediction error.
• Aligning reference frames (QRFs) between interacting systems minimizes this error—similar to synchronizing watches for coordinated action.

Asymptotic Behavior: Entanglement and Unitarity

• As prediction errors are minimized over time, the observer’s internal model becomes closely aligned with the actual system state.
• This alignment results in entanglement, where systems share information completely.
• Thus, in the long run, the FEP becomes equivalent to the Principle of Unitarity – a core rule of quantum mechanics that ensures information conservation.

Implications for Biological Cognition

• The framework suggests that biological systems may use quantum coherence as a resource for efficient information processing.
• Living organisms might balance classical communication (clear, distinct signals) with quantum entanglement (deep, shared information).
• This balance could help explain efficient cellular processes and even aspects of consciousness.

Discussion and Predictions

• The paper unifies quantum mechanics and the FEP, indicating that all quantum systems naturally perform active inference.
• It predicts that even at the cellular or organismal level, quantum effects could play a significant role in cognition.
• Future experimental work is needed to test these predictions in biological and artificial systems.

Conclusion

• The FEP for quantum systems shows that minimizing prediction errors leads to entanglement and aligns with the principle of unitarity.
• This work provides a foundation for viewing all physical systems as active agents that continuously update their internal models.
• It bridges ideas from quantum physics, information theory, and biology to explain how systems maintain their identity.

引言：自由能原理是什么？

• 自由能原理（FEP）认为系统会尽量最小化预测误差或“惊讶度”。
• 这一原理基于贝叶斯推理，即根据新信息不断更新“信念”。

自由能原理在量子领域的扩展

• 本文将自由能原理扩展到通用的量子系统中。
• 在这种描述中，量子系统不依赖固定的时空背景，而是被视为主动的观察者。
• 它们通过测量不断更新自己的“信念”，类似于我们在日常生活中调整预期。

关键概念和定义

• 变分自由能（VFE）： 对“惊讶度”的上界，用来衡量状态与模型之间的匹配程度。
• 惊讶度： 衡量事件发生的意外程度；惊讶度越低，表示预测越准确。
• 贝叶斯预测误差： 期望结果与实际观测之间的差距，即预测误差。
• 量子参考系（QRFs）： 为测量提供基准的系统，类似于在空间中选择坐标系，但用于量子状态。
• 幺正性： 量子力学中的基本原则，确保总概率（信息）守恒；在此保证信息不会丢失。
• 马尔可夫毛毯（MB）： 将系统内部状态与外部环境分离的“保护屏障”。

用量子语言重新表述自由能原理

• 论文利用量子信息理论的概念重新构建了自由能原理。
• 它不再依赖固定的时空背景和固有随机性，而是不确定性源于量子测量中的信息交换。
• 系统间的相互作用被视为通过“全息屏”（一种概念边界）进行信息传递。
• “代理”或观察者通过使用量子参考系打破边界对称性，从而决定如何进行测量。

测量、记忆与系统识别

• 系统像烹饪一样，按照详细步骤记录观测结果（记忆）。
• 粗粒化： 简化复杂信息以捕捉长时间内的本质行为，就像总结长篇笔记中的关键点。
• 通过测量算子比较预测与实际结果，逐步减少预测误差。
• 这一反复过程帮助系统不断改进其内部模型。

非对易性与情境切换

• 在量子力学中，有些测量不满足对易性，即测量顺序会影响结果。
• 这种现象会导致情境切换，如果观察者的参考系不匹配，就会增加预测误差。
• 论文指出，通过对齐这些参考系，可以最小化误差，就像同步时钟以确保顺畅沟通一样。

渐近行为：纠缠与幺正性

• 当预测误差趋于零时，观察者的内部模型与实际系统状态趋于一致。
• 这种一致性会导致系统间的纠缠，即信息的完全共享。
• 因此，从长远来看，自由能原理与幺正性原理是等价的。

对生物认知的启示

• 该框架暗示生物系统可能利用量子相干性来实现高效的信息处理。
• 生物体可能在经典通信（清晰、独立的信号）与量子纠缠（深度共享信息）之间取得平衡。
• 这种平衡可能解释细胞过程的高效性，甚至与意识的某些方面有关。

讨论与预测

• 论文统一了量子力学与自由能原理，表明所有量子系统都自然地进行主动推理。
• 它预测，即使在细胞或有机体层面，量子效应也可能在认知中发挥重要作用。
• 未来需要通过实验在生物和人工系统中验证这些预测。

结论

• 对于量子系统，自由能原理表明最小化预测误差会导致纠缠，并与幺正性保持一致。
• 这项工作为将所有物理系统视为通过主动推理维持自身身份的“代理”提供了理论基础。
• 它连接了量子物理、信息理论与生物学的思想，帮助解释系统如何维持其独特性。