Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9911067 · Journal of Physics A Mathematical and General

Quantum theory of incompatible observations

Z. Hradil, J Summhammer

Published Oct 11, 2000• 15 citations
arXivHTMLPDF

Abstract

The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.

DOI
10.1088/0305-4470/33/42/309

Related papers

Quantum theory of incompatible observations

Z. Hradil

Quantum theory allows the sequential measurement of incompatible observables in the course of repeated measurements. To get information about the observed system, all observations must be synthesized. This is the main idea of quantum tomographical methods. As shown in this paper, the maximum likelihood principle provides the best measure for relating the experimental data with predictions of quantum theory. Synthesis of incompatible observations appears to be a new quantum measurement described by a positive operator-valued measure. Besides this, the procedure finds the optimal state of the system, which best fits such a measurement.

Optics and Spectroscopy•Sep 1, 2001•1 citation

Quantum measurement and estimation

Z. Hradil, Jaroslav Řeháček

Information about quantum systems is inferred from a sequence of measurements made on it. The maximum likelihood (ML) estimation gives an arguably natural optimum approach in quantum theory. The feasibility is shown on valuable examples including quantum phase or entangled spin states.

Optical Fiber Communication Conference•Jun 13, 2001•0 citations

Quantum State Reconstruction From Incomplete Data

Vladimír Bužek, G. Drobný, R. Derka +2 more

Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.

arXiv:quant-ph/9805020•May 7, 1998•22 citations

Quantum inference of states and processes

Miroslav Ježek, Jaromı́r Fiurášek, Z. Hradil

The maximum-likelihood principle unifies inference of quantum states and processes from experimental noisy data. Particularly, a generic quantum process may be estimated simultaneously with unknown quantum probe states, provided that measurements on probe and transformed probe states are available. Drawbacks of various approximate treatments are also considered.

arXiv:quant-ph/0210146•Physical Review A•Jul 9, 2003•175 citations

Quantum-State Reconstruction by Maximizing Likelihood and Entropy

Yong Siah Teo, Huangjun Zhu, Berthold‐Georg Englert +2 more

Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.

arXiv:1102.2662•Physical Review Letters•Jul 8, 2011•107 citations

Maximum-likelihood estimation of quantum measurement

Jaromı́r Fiurášek

Maximum-likelihood estimation is applied to the determination of an unknown quantum measurement. The calibrated measuring apparatus carries out measurements on many different quantum states and the positive operator-valued measure governing the measurement statistics is then inferred from the collected data via the maximum-likelihood principle. In contrast to the procedures based on linear inversion, our approach always provides a physically sensible result. We illustrate the method on the case of the Stern-Gerlach apparatus.

arXiv:quant-ph/0101027•Physical Review A•Jul 17, 2001•175 citations
The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.
Quantum theory allows the sequential measurement of incompatible observables in the course of repeated measurements. To get information about the observed system, all observations must be synthesized. This is the main idea of quantum tomographical methods. As shown in this paper, the maximum likelihood principle provides the best measure for relating the experimental data with predictions of quantum theory. Synthesis of incompatible observations appears to be a new quantum measurement described by a positive operator-valued measure. Besides this, the procedure finds the optimal state of the system, which best fits such a measurement.
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.
Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.
Maximum-likelihood estimation is applied to the determination of an unknown quantum measurement. The calibrated measuring apparatus carries out measurements on many different quantum states and the positive operator-valued measure governing the measurement statistics is then inferred from the collected data via the maximum-likelihood principle. In contrast to the procedures based on linear inversion, our approach always provides a physically sensible result. We illustrate the method on the case of the Stern-Gerlach apparatus.