Introduction to Quantum-State Estimation

Contents\nPreface\nAcknowledgments\nAcronyms\nSymbols and Notations\n1. Preliminaries of Quantum-State Estimation\n	1.1 Overview of quantum-state estimation\n	1.2 Basic aspects of estimation theory\n		1.2.1 Cost functions\n			1.2.1.1 Cost for Bayesian-mean estimation\n			1.2.1.2 Cost for maximum-likelihood (ML) estimation\n		1.2.2 Superoperators\n			1.2.2.1 Formalism\n			1.2.2.2 Dual operators and quantum-state estimation\n			1.2.2.3 Symmetric informationally complete measurements\n		1.2.3 A simple ML estimator for minimally complete measurements\n		1.2.4 Linear inversion\n			1.2.4.1 Minimally complete measurements\n			1.2.4.2 Arbitrary measurements — with emphasis on overcomplete ones\n		1.2.5 Geometry of the quantum state space\n	1.3 Uncertainties and information\n		1.3.1 Cramer–Rao limit for unbiased quantum-state estimation\n		1.3.2 Asymptotic efficiency\n	Further reading\n2. Informationally Complete Estimation\n	2.1 ML scheme — perfect detections\n		2.1.1 A brief recap on basic concepts\n		2.1.2 The extremal equations\n		2.1.3 A simple example with the von Neumann measurement — first digression\n		2.1.4 Solution to the extremal equations\n			2.1.4.1 Operator calculus\n			2.1.4.2 Complex vector calculus — second digression\n			2.1.4.3 Operator calculus revisited — a steepest-ascent algorithm\n	2.2 Generalized ML scheme — imperfect detections\n		2.2.1 Imperfect measurements\n		2.2.2 ML estimation of the unknown number of copies\n		2.2.3 The extremal probabilities\n		2.2.4 Statistical consistency of the generalized ML estimator\n		2.2.5 A steepest-ascent algorithm\n	2.3 Another way of accounting for imperfections\n	2.4 Bayesian-mean estimation — some brief statements\n	Further reading\n3. Informationally Incomplete Estimation\n	3.1 Likelihood and entropy — perfect detections\n		3.1.1 Informationally incomplete measurements\n		3.1.2 Properties of ML convex sets\n		3.1.3 Measurement-subspace decomposition\n		3.1.4 Maximum-likelihood–maximum-entropy (MLME) method\n			3.1.4.1 The von Neumann entropy\n			3.1.4.2 The maximum-entropy estimator\n			3.1.4.3 The extremal equations\n			3.1.4.4 A steepest-ascent algorithm\n			3.1.4.5 Simple closed-form expressions\n	3.2 Likelihood and entropy — imperfect detections\n		3.2.1 The extremal probabilities\n		3.2.2 Multiple ML convex sets\n		3.2.3 A steepest-ascent algorithm\n	3.3 An easy trick for constrained function optimization\n	3.4 Some remarks\n	Further reading\n4. Practical Aspects of State Estimation\n	4.1 Accuracy of unbiased state estimation\n	4.2 Experimental schemes\n		4.2.1 Discrete-variable measurements — photon polarization\n			4.2.1.1 Six-outcome measurement\n			4.2.1.2 Trinemeasurement\n			4.2.1.3 Tetrahedron measurement\n		4.2.2 Discrete-variable measurements — photon counting and its difficulties\n			4.2.2.1 Photon-number statistics\n			4.2.2.2 Characteristic functions\n			4.2.2.3 The physics of photodetection\n			4.2.2.4 Indirect photon counting\n		4.2.3 Continuous-variable measurements\n			4.2.3.1 Displaced Fock-state measurement\n			4.2.3.2 Quadrature-eigenstate measurement\n	Further reading\n5. Quasi-Probability Distributions\n	5.1 Distribution functions\n		5.1.1 Wigner function\n			5.1.1.1 Definitions and properties\n			5.1.1.2 Quadrature eigenstates\n			5.1.1.3 Coherent states\n			5.1.1.4 Odd and even coherent states\n			5.1.1.5 Fock states\n		5.1.2 Glauber-Sudarshan P function\n			5.1.2.1 Definitions and properties\n			5.1.2.2 Coherent states\n			5.1.2.3 Odd and even coherent states\n			5.1.2.4 Fock states\n		5.1.3 Husimi Q function\n	5.2 Connections among quasi-probability distribution functions\n		5.2.1 Formalism\n		5.2.2 Applications\n	5.3 An equivalence theorem and generalized quasi-probability distributions\n	Further reading\nHints to All Problems\nSample Solutions to All Problems\nAppendix: Squeezed Coherent States\nIndex