Quantum Dynamics as an analog of Conditional Probability

Abstract

Quantum theory can be regarded as a non-commutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the well-known isomorphism between completely positive maps and bipartite density operators is derived, which makes this connection much more explicit. The new isomorphism is given an operational interpretation in terms of statistical correlations between ensemble preparation procedures and outcomes of measurements. Finally, the isomorphism is applied to elucidate the connection between no-cloning/no-broadcasting theorems and the monogamy of entanglement, and a simplified proof of the no-broadcasting theorem is obtained as a byproduct.

16 pages, 3 figures. v2 Presentation greatly improved, references updated and typos fixed

References in corpus (18)

• Distributed Entanglement
• General Monogamy Inequality for Bipartite Qubit Entanglement
• Noncommuting mixed states cannot be broadcast
• Monogamy of entanglement and other correlations
• Negative entropy and information in quantum mechanics
• Programmable quantum gate arrays
• Is Entanglement Monogamous?
• Quantum Information Theory of Entanglement and Measurement
• Entangled webs: Tight bound for symmetric sharing of entanglement
• Quantum conditional operator and a criterion for separability
• Translation of Lueders' "Uber die Zustandsanderung durch den Messprozess"
• Imprinting a complete information about a quantum channel on its output state
• On Quantum Operations as Quantum States
• Superbroadcasting of mixed states
• Entanglement sharing among qudits
• Channel kets, entangled states, and the location of quantum information
• Superbroadcasting of continuous variables mixed states
• Superbroadcasting of conjugate quantum variables

Cited by in corpus (62)

• Einstein, incompleteness, and the epistemic view of quantum states
• Quantum-Bayesian Coherence: The No-Nonsense Version
• Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
• Quantum causal modelling
• Quantum common causes and quantum causal models
• Quantum Graphical Models and Belief Propagation
• Distinct Quantum States Can Be Compatible with a Single State of Reality
• Inferring causal structure: a quantum advantage
• Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction
• Quantum Cloning Machines and the Applications
• On modifications of Reichenbach's principle of common cause in light of Bell's theorem
• Cyclic Quantum Causal Models
• Quartic quantum theory: an extension of the standard quantum mechanics
• Can a quantum state over time resemble a quantum state at a single time?
• Updating the Born rule
• From time-reversal symmetry to quantum Bayes' rules
• Picturing classical and quantum Bayesian inference
• Quantum-coherent mixtures of causal relations
• Negative Conditional Entropy of Post-Selected States
• Quantum States as Ordinary Information
• Quantum speedup in the identification of cause-effect relations
• Why initial system-environment correlations do not imply the failure of complete positivity: a causal perspective
• New Slant on the EPR-Bell Experiment
• Axioms for retrodiction: achieving time-reversal symmetry with a prior
• Von Neumann Was Not a Quantum Bayesian
• Quantum probabilities of composite events in quantum measurements with multimode states
• Quantum instruments as a foundation for both states and observables
• Quantum-Bayesian Coherence
• A quantum causal discovery algorithm
• On quantum states over time
• Quantum teleportation with relativistic communication from first principles
• A non-commutative Bayes' theorem
• State retrieval beyond Bayes' retrodiction
• Observational entropy, coarse quantum states, and Petz recovery: information-theoretic properties and bounds
• Information-theoretic limitations on approximate quantum cloning and broadcasting
• Action Duality: A Constructive Principle for Quantum Foundations
• A general framework for cyclic and fine-tuned causal models and their compatibility with space-time
• Inferring the arrow of time in quantum spatiotemporal correlations
• Quantum Causal Models
• Quantum inferring acausal structures and the Monty Hall problem
• Embedding cyclic information-theoretic structures in acyclic spacetimes: no-go results for indefinite causality
• Impossibility of superluminal signalling in Minkowski space-time does not rule out causal loops
• Quantum-correlation breaking channels, quantum conditional probability and Perron-Frobenius theory
• Determining complementary properties using weak-measurement: uncertainty, predictability, and disturbance
• Fluctuation Theorems for multitime processes
• Quantum Bayes' rule and Petz transpose map from the minimum change principle
• Inverses, disintegrations, and Bayesian inversion in quantum Markov categories
• Non-commutative disintegrations: existence and uniqueness in finite dimensions
• A Bayesian Analogue of Gleason's Theorem
• Quantum to classical transitions in causal relations
• Quantum observation scheme universally identifying causalities from correlations
• Quantum dynamics as a pseudo-density matrix
• On state vs. channel quantum extension problems: exact results for UxUxU symmetry
• The Operational Choi-Jamiolkowski Isomorphism
• Conditional Distributions for Quantum Systems
• Quantum Bayesianism: A Study
• Time Symmetry in Operational Theories
• Reality, Causality, and Quantum Theory
• Measuring temporal entropies in experiments
• Quantum conditional probabilities
• Revealing quantum operator scrambling via measuring Holevo information on digital quantum simulators
• Causal Emergence in Quantum Mechanics