Distance between quantum states, statistical inference and the projection postulate (original) (raw)
On Quantum Statistical Inference
Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, theoretical developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates a number of new concepts for an extension of classical statistical inference to the quantum context.
On Quantum Statistical Inference, II
Eprint Arxiv Quant Ph 0307191, 2003
Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, theoretical developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates a number of new concepts for an extension of classical statistical inference to the quantum context. (An earlier version of the paper containing material on further topics is available as quant-ph/0307189).
On Quantum Statistical Inference Ole E
2003
Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, theoretical developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates a number of new concepts for an extension of classical statistical inference to the quantum context. (An earlier version of the paper containing material on further topics is available as
2020
In this paper, inspired by the "Minimum Description Length Principle" in classical Statistics, we introduce a new method for predicting the outcomes of a quantum measurement and for estimating the state of a quantum system with minimum quantum complexity, while, at the same time, avoiding overfitting.
Principles of Quantum Inference
Annals of Physics, 1991
A new approach to quantum state determination is developed using data in the form of observed eigenvectors. An exceedingly natural inversion of such data results when the quantum probability rule is recognised as a conditional. The reversal of this conditional via Bayesian methods results in an inferred probability density over states which readily reduces to a density matrix estimator. The inclusion of concepts drawn from communication theory then defines an Optimal State Determination Problem which is explored on Hilbert spaces of arbitrary finite dimensionality.
Geometry of Quantum Statistical Inference
Physical Review Letters, 1996
An efficient geometric formulation of the problem of parameter estimation is developed, based on Hilbert space geometry. This theory, which allows for a transparent transition between classical and quantum statistical inference, is then applied to the analysis of exponential families of distributions (of relevance to statistical mechanics) and quantum mechanical evolutions. The extension to quantum theory is achieved by the introduction of a complex structure on the given real Hilbert space. We find a set of higher order corrections to the parameter estimation variance lower bound, which are potentially important in quantum mechanics, where these corrections appear as modifications to Heisenberg uncertainty relations for the determination of the parameter. [S0031-9007(96)01153-2]
Underlining some limitations of the statistical formalism in quantum mechanics
2011
In a paper of us, it is showed that Density Matrices do not provide a complete description of ensembles of states in quantum mechanics, since they lack measurable information concerning the preparation of the ensembles. Bodor and Diósi have later posted a comment on that article, which agrees on some points of it but disagrees on some others. This reply is intended to clarify the discussion. ¶ In Ref. [1], the Variance has been computed by taking the quantum mechanical prediction on the single-particles measurements and by then applying Classical Statistics.
Some trends and problems in quantum probability
0.) From the experimental point of view probability enters quantum theory just like classical statistical physics, i.e. as an expected relative frequency. However it is well known that the statistical formalism of quantum theory is quite different from the usual Kolmogorovian one involving, for example, complex numbers, amplitudes, Hilbert spaces .... The quantum statistical formalism has been described, developed~ applied, generalized with the contributions of many authors; however its theoretical status remained, until recently, quite obscure, as shown by the widely contrasting statements that one can find in the vast literature concerning the following questions. Question I.) Is it possible to justify the choice of the classical or the quantum statistical formalism, for the description of a given set of statistical data, on rigorous mathematical criteria rather than on empirical ones? In particular, is the quantum statistical formalism in some sense necessary, or (as some authors...
On asymptotic quantum statistical inference
We study asymptotically optimal statistical inference concerning the unknown state of N identical quantum systems, using two complementary approaches: a "poor man's approach" based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam's theory of Local Asymptotic Normality. * URL: www.math.leidenuniv.nl/∼gill.