From time-reversal symmetry to quantum Bayes' rules

Abstract

Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes' rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes' rules appearing in the literature are all instances of our definition. Moreover, our Bayes' rule is based upon a simple relationship between the notions of state over time and a time-reversal symmetry map, both of which are introduced here.

Some adjustments and organizational changes, typos fixed, new tables added; 24 pages total

References in corpus (12)

• Quantum coherence, time-translation symmetry and thermodynamics
• Quantum Operation Time Reversal
• Pseudo Entropy in dS/CFT and Time-like Entanglement Entropy
• Quantum Dynamics as an analog of Conditional Probability
• Pseudo Entropy in Free Quantum Field Theories
• Aspects of Pseudo Entropy in Field Theories
• Axioms for retrodiction: achieving time-reversal symmetry with a prior
• Universal Optimal Quantum Correlator
• State retrieval beyond Bayes' retrodiction
• Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
• Fluctuation Theorems for multitime processes
• Bayesian inversion and the Tomita-Takesaki modular group

Cited by in corpus (28)

• Time-like Entanglement Entropy in AdS/BCFT
• SVD Entanglement Entropy
• Virtual quantum broadcasting
• Inferring the arrow of time in quantum spatiotemporal correlations
• Causal classification of spatiotemporal quantum correlations
• Uniqueness of quantum state over time function
• Quantum space-time marginal problem: global causal structure from local causal information
• Experimental demonstration of input-output indefiniteness in a single quantum device
• Unification of spatiotemporal quantum formalisms: mapping between process and pseudo-density matrices via multiple-time states
• Quantum Causal Inference with Extremely Light Touch
• Certifying Quantum Temporal Correlation via Randomized Measurements: Theory and Experiment
• Bayesian inversion and the Tomita-Takesaki modular group
• Operational meanings of a generalized conditional expectation in quantum metrology
• Quantum Bayes' rule and Petz transpose map from the minimum change principle
• Path Integrals from Spacetime Quantum Actions
• Role of Dilations in Reversing Physical Processes: Tabletop Reversibility and Generalized Thermal Operations
• Quantum dynamics as a pseudo-density matrix
• Thermalization in open many-body systems and KMS detailed balance
• Multipartite quantum states over time from two fundamental assumptions
• Quantum Onsager relations
• Experimental Virtual Quantum Broadcasting
• Measuring temporal entropies in experiments
• No Practical Quantum Broadcasting: Even Virtually
• Proper and improper mixed states serve as different prior beliefs for quantum state retrodiction
• A Retrodictive Approach to Quantum State Smoothing
• Bipartite quantum states admitting a causal explanation
• Stochastic Schrödinger Equations for Quantum Reverse Diffusion
• Virtual phase-covariant quantum broadcasting for qubits