Metric on the space of quantum states from relative entropy. Tomographic reconstruction

Abstract

In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2, N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states. By using the tomographic picture of quantum mechanics we have obtained the Fisher- Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.

31 pages. No figures. Abstract and Introduction rewritten. Minor corrections. References added

References in corpus (9)

• Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics
• Geometry of quantum systems: density states and entanglement
• Multilevel superconducting circuits as two-qubit systems: Operations, state preparation, and entropic inequalities
• Star products, duality and double Lie algebras
• Orbits of quantum states and geometry of Bloch vectors for $N$-level systems
• Duality symmetry for star products
• Inverse spin-s portrait and representation of qudit states by single probability vectors
• On the $\star$-product quantization and the Duflo map in three dimensions
• A pedagogical presentation of a $C^{\star }$-algebraic approach to quantum tomography

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• Fisher Metric, Geometric Entanglement and Spin Networks
• A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the $q$-$z$ Family
• God plays coins or superposition principle for classical probabilities in quantum suprematism representation of qubit states
• From the Jordan product to Riemannian geometries on classical and quantum states
• Correlations in a system of classical--like coins simulating spin-1/2 states in the probability representation of quantum mechanics
• Probability representation of quantum states as a renaissance of hidden variables -- God plays coins
• Quantum suprematism picture of Malevich's squares triada for spin states and the parametric oscillator evolution in the probability representation of quantum mechanics
• Geometric approach to quantum statistical inference
• Lie groupoids in information geometry
• Geometry of the Pauli maps and Pauli channels
• Quantum States, Groups and Monotone Metric Tensors
• Tomographic Reconstruction of Quantum Metrics
• Nonlinear Dynamics from Linear Quantum Evolutions
• Parametric models and information geometry on W*-algebras
• Generalised Potential Functions in Differential Geometry and Information Geometry
• Spin kinetic equations in the probability representation of quantum mechanics
• Group actions and Monotone Quantum Metric Tensors
• Dynamics and Geometry of Entanglement in Many-Body Quantum Systems
• Probability representation of photon states and tomography
• Geometry from divergence functions and complex structures