Shadow tomography on general measurement frames

Abstract

We provide a new perspective on shadow tomography by demonstrating its deep connections with the general theory of measurement frames. By showing that the formalism of measurement frames offers a natural framework for shadow tomography -- in which ``classical shadows'' correspond to unbiased estimators derived from a suitable dual frame associated with the given measurement -- we highlight the intrinsic connection between standard state tomography and shadow tomography. Such perspective allows us to examine the interplay between measurements, reconstructed observables, and the estimators used to process measurement outcomes, while paving the way to assess the influence of the input state and the dimension of the underlying space on estimation errors. Our approach generalizes the method described in [H.-Y. Huang {\it et al.}, Nat. Phys. 16, 1050 (2020)], whose results are recovered in the special case of covariant measurement frames. As an application, we demonstrate that a sought-after target of shadow tomography can be achieved for the entire class of tight rank-1 measurement frames -- namely, that it is possible to accurately estimate a finite set of generic rank-1 bounded observables while avoiding the growth of the number of the required samples with the state dimension.

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References in corpus (33)

• On the Measurement of Qubits
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Unknown Quantum States and Operations, a Bayesian View
• On mutually unbiased bases
• The randomized measurement toolbox
• Tight informationally complete quantum measurements
• The SIC Question: History and State of Play
• Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
• Learning Quantum Systems
• Theory of quantum system certification: a tutorial
• Informationally complete measurements and groups representation
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Learning to Measure: Adaptive Informationally Complete Generalized Measurements for Quantum Algorithms
• Quantum Semiparametric Estimation
• Experimental single-setting quantum state tomography
• Quantum State Tomography with Joint SIC POMs and Product SIC POMs
• Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
• Quantum state estimation with nuisance parameters
• Quantum state estimation with informationally overcomplete measurements
• Potential and limitations of quantum extreme learning machines
• The Weingarten Calculus
• Optimising shadow tomography with generalised measurements
• Measuring Arbitrary Physical Properties in Analog Quantum Simulation
• Optimal data processing for quantum measurements
• A Bayesian analysis of classical shadows
• Optimal quantum tomography
• Dynamical learning of a photonics quantum-state engineering process
• Enhanced detection techniques of Orbital Angular Momentum states in the classical and quantum regimes
• Quantum verification and estimation with few copies
• Regression of high dimensional angular momentum states of light
• Mutually unbiased frames
• Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements

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• Enhanced observable estimation through classical optimization of informationally over-complete measurement data -- beyond classical shadows
• Optimising quantum tomography via shadow inversion
• Quantum Extreme Learning of molecular potential energy surfaces and force fields
• Quantum reservoir computing for photonic entanglement witnessing
• Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate
• Completeness Stability of Quantum Measurements
• Low variance estimations of many observables with tensor networks and informationally-complete measurements
• Chasing shadows with Gottesman-Kitaev-Preskill codes
• Classification of quantum states of light using random measurements through a multimode fiber
• Classical Shadows with Improved Median-of-Means Estimation
• Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography
• Improving shadow estimation with locally-optimal dual frames
• Reducing Complexity of Shadow Process Tomography with Generalized Measurements