Sufficiency in quantum statistical inference: a survey with examples. (English) Zbl 1108.46045
Summary: This paper attempts to give an overview about sufficiency in the setting of quantum statistics. The basic concepts are treated in parallel to the measure theoretic case. It turns out that several classical examples and results have a noncommutative analogue. Some of the results are presented without proof (but with exact references) and the presentation is intended to be self-contained. The main examples discussed in the paper are related to the Weyl algebra and to the exponential family of states. The characterization of sufficiency in terms of quantum Fisher information is a new result.
MSC:
| 46L53 | Noncommutative probability and statistics |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
| 62B05 | Sufficient statistics and fields |
Keywords:
quantum statistics; coarse-graining; factorization theorem; exponential family; perturbation of states; sufficient subalgebra; quantum Fisher informationReferences:
| [1] | DOI: 10.1016/0022-1236(82)90022-2 · Zbl 0483.46043 · doi:10.1016/0022-1236(82)90022-2 |
| [2] | Barndorff-Nielsen O., Wiley Series in Probability and Mathematical Statistics, in: Information and Exponential Families in Statistical Theory (1978) · Zbl 0387.62011 |
| [3] | DOI: 10.1111/1467-9868.00415 · Zbl 1059.62128 · doi:10.1111/1467-9868.00415 |
| [4] | Blank J., Hilbert Space Operators in Quantum Physics (1994) · Zbl 0873.46038 |
| [5] | Bratteli O., Texts and Monographs in Physics, in: Operator Algebras and Quantum Statistical Mechanics. 1. C*- and W*-Algebras, Symmetry Groups, Decomposition of States (1987) |
| [6] | DOI: 10.1007/s00220-005-1510-7 · Zbl 1156.81325 · doi:10.1007/s00220-005-1510-7 |
| [7] | Nielsen M. A., Quantum Computation and Quantum Information (2000) · Zbl 1049.81015 |
| [8] | Nielsen M. A., Quantum Inf. Comput. 6 pp 507– |
| [9] | Ohya M., Quantum Entropy and Its Use (2004) |
| [10] | DOI: 10.1007/BF01212345 · Zbl 0597.46067 · doi:10.1007/BF01212345 |
| [11] | Petz D., Rep. Math. Phys. 21 pp 57– |
| [12] | Petz D., Quart. J. Math. 39 pp 907– |
| [13] | Petz D., An Invitation to the Algebra of the Canonical Commutation Relation (1990) · Zbl 0704.46045 |
| [14] | DOI: 10.1063/1.530611 · Zbl 0804.53101 · doi:10.1063/1.530611 |
| [15] | DOI: 10.1006/jfan.1994.1024 · Zbl 0808.46096 · doi:10.1006/jfan.1994.1024 |
| [16] | DOI: 10.1088/0305-4470/35/4/305 · Zbl 1168.81317 · doi:10.1088/0305-4470/35/4/305 |
| [17] | DOI: 10.1142/S0129055X03001576 · Zbl 1134.82303 · doi:10.1142/S0129055X03001576 |
| [18] | DOI: 10.1063/1.1497701 · Zbl 1060.82006 · doi:10.1063/1.1497701 |
| [19] | Strătilă Ş., Modular Theory of Operator Algebras (1981) |
| [20] | DOI: 10.1515/9783110850826 · Zbl 0594.62017 · doi:10.1515/9783110850826 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
