On the categorical foundations of quantum information theory: Categories and the Cramer-Rao inequality
arXiv:2309.10428 · doi:10.1142/S0217732323500852
Abstract
An extension of Cencov's categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theory in a natural way, while also allowing to formalise the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer-Rao inequality in this setting.
arXiv comment20 pages, comments are welcome!
References in corpus (30)
- Quantum Fisher information matrix and multiparameter estimation
- Quantum Mechanics as Quantum Measure Theory
- Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces
- A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
- Covariance and Fisher information in quantum mechanics
- Wigner-Yanase information on quantum state space:the geometric approach
- Uncertainty Principle and Quantum Fisher Information - II
- Quantum state estimation with nuisance parameters
- Transition Probability (Fidelity) and Its Relatives
- Classification and characterization of quantum parametric models in quantum estimation theory
- A gentle introduction to Schwinger's formulation of quantum mechanics: The groupoid picture
- Schwinger's Picture of Quantum Mechanics I: Groupoids
- Schwinger's Picture of Quantum Mechanics III: The statistical interpretation
- Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability
- Schwinger's Picture of Quantum Mechanics II: Algebras and Observables
- Dynamical aspects in the Quantizer-Dequantizer formalism
- A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the - Family
- From the Jordan product to Riemannian geometries on classical and quantum states
- Schwinger's picture of Quantum Mechanics
- Flat connections and Wigner-Yanase-Dyson metrics
- Schwinger's Picture of Quantum Mechanics IV: Composition and independence
- Lie groupoids in information geometry
- Quantum States, Groups and Monotone Metric Tensors
- Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras
- Feynman's Propagator in Schwinger's picture of Quantum Mechanics
- A quantum route to the classical Lagrangian formalism
- Information geometry on groupoids: the case of singular metrics
- Lifting statistical structures
- Groupoid and algebra of the infinite quantum spin chain
- Group actions and Monotone Quantum Metric Tensors