From time-reversal symmetry to quantum Bayes' rules
arXiv:2212.08088 · doi:10.1103/PRXQuantum.4.020334
Abstract
Bayes' rule 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.
arXiv commentSome adjustments and organizational changes, typos fixed, new tables added; 24 pages total
References in corpus (53)
- Quantum data processing and error correction
- Notes on Some Entanglement Properties of Quantum Field Theory
- From Classical to Quantum Shannon Theory
- Quantum coherence, time-translation symmetry and thermodynamics
- Continuous quantum measurement of a double dot
- Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
- Reversing quantum dynamics with near-optimal quantum and classical fidelity
- Quantum conditional mutual information and approximate Markov chains
- Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories
- Selective quantum evolution of a qubit state due to continuous measurement
- Monotonicity of quantum relative entropy revisited
- Quantum Bayes rule
- Holographic Pseudo Entropy
- Quantum Operation Time Reversal
- Entanglement Wedge Reconstruction via Universal Recovery Channels
- Recoverability in quantum information theory
- Is a time symmetric interpretation of quantum theory possible without retrocausality?
- Pseudo Entropy in dS/CFT and Time-like Entanglement Entropy
- Universal recovery maps and approximate sufficiency of quantum relative entropy
- Multiple-time states and multiple-time measurements in quantum mechanics
- On a Time Symmetric Formulation of Quantum Mechanics
- Quantum Dynamics as an analog of Conditional Probability
- A simple approach to approximate quantum error correction based on the transpose channel
- Multivariate Trace Inequalities
- Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map
- Pseudo Entropy in Free Quantum Field Theories
- Aspects of Pseudo Entropy in Field Theories
- Superdensity Operators for Spacetime Quantum Mechanics
- Fluctuation Theorems for a Quantum Channel
- Can a quantum state over time resemble a quantum state at a single time?
- Entanglement Wedge Reconstruction using the Petz Map
- Picturing classical and quantum Bayesian inference
- Fluctuation theorems from Bayesian retrodiction
- Holographic spacetime, black holes and quantum error correcting codes: A review
- How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem
- Quantum operations with indefinite time direction
- The arrow of time in operational formulations of quantum theory
- From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality
- Axioms for retrodiction: achieving time-reversal symmetry with a prior
- Preservation of a quantum Renyi relative entropy implies existence of a recovery map
- Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability
- Jordan products of quantum channels and their compatibility
- Symmetries of quantum evolutions
- A non-commutative Bayes' theorem
- Real-space RG, error correction and Petz map
- Universal Optimal Quantum Correlator
- State retrieval beyond Bayes' retrodiction
- Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
- Quantum Causal Influence
- Conditional expectation and Bayes' rule for quantum random variables and positive operator valued measures
- The information loss of a stochastic map
- Fluctuation Theorems for multitime processes
- Bayesian inversion and the Tomita-Takesaki modular group
Cited by in corpus (29)
- 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
- Non-commutative disintegrations: existence and uniqueness in finite dimensions
- Role of Dilations in Reversing Physical Processes: Tabletop Reversibility and Generalized Thermal Operations
- Path Integrals from Spacetime Quantum Actions
- 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
- Bipartite quantum states admitting a causal explanation
- Stochastic Schrödinger Equations for Quantum Reverse Diffusion
- A Retrodictive Approach to Quantum State Smoothing
- Virtual phase-covariant quantum broadcasting for qubits