Uncertainty relations revisited

• URL: http://arxiv.org/abs/2310.05039v3
• Date: Mon, 4 Dec 2023 03:56:13 GMT
• Title: Uncertainty relations revisited
• Authors: Berthold-Georg Englert
• Abstract summary: We present a unified approach for deriving all standard uncertainty relations. We try to answer why the use of variances for quantifying uncertainty is so widespread. It is common to regard the states that saturate the Robertson inequality as "minimum uncertainty states"
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Introductory courses on quantum mechanics usually include lectures on uncertainty relations, typically the inequality derived by Robertson and, perhaps, other statements. For the benefit of the lecturers, we present a unified approach -- well suited for undergraduate teaching -- for deriving all standard uncertainty relations: those for products of variances by Kennard, Robertson, and Schr\"odinger, as well as those for sums of variances by Maccone and Pati. We also give a brief review of the early history of this topic and try to answer why the use of variances for quantifying uncertainty is so widespread, while alternatives are available that can be more natural and more fitting. It is common to regard the states that saturate the Robertson inequality as "minimum uncertainty states" although they do not minimize the variance of one observable, given the variance of another, incompatible observable. The states that achieve this objective are different and can be found systematically.

Related papers

• Uncertainty relations based on state-dependent norm of commutator [0.0]
  We introduce two uncertainty relations based on the state-dependent norm of commutators, utilizing generalizations of the B"ottcher-Wenzel inequality. The first relation is mathematically proven, while the second, tighter relation is strongly supported by numerical evidence.
  arXiv  Detail & Related papers  (2024-06-18T05:16:45Z)
• Testing trajectory-based determinism via time probability distributions [44.99833362998488]
  Bohmian mechanics (BM) has inherited more predictive power than quantum mechanics (QM) We introduce a prescription for constructing a flight-time probability distribution within generic trajectory-equipped theories. We derive probability distributions that are unreachable by QM.
  arXiv  Detail & Related papers  (2024-04-15T11:36:38Z)
• It's an Alignment, Not a Trade-off: Revisiting Bias and Variance in Deep Models [51.66015254740692]
  We show that for an ensemble of deep learning based classification models, bias and variance are emphaligned at a sample level. We study this phenomenon from two theoretical perspectives: calibration and neural collapse.
  arXiv  Detail & Related papers  (2023-10-13T17:06:34Z)
• Parameterized Multi-observable Sum Uncertainty Relations [9.571723611319348]
  We study uncertainty relations based on variance for arbitrary finite $N$ quantum observables. The lower bounds of our uncertainty inequalities are non-zero unless the measured state is a common eigenvector of all the observables.
  arXiv  Detail & Related papers  (2022-11-07T04:36:07Z)
• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources [51.243733928201024]
  We show that the proof of the generalised quantum Stein's lemma is not correct due to a gap in the argument leading to Lemma III.9. This puts into question a number of established results in the literature, in particular the reversibility of quantum entanglement.
  arXiv  Detail & Related papers  (2022-05-05T17:46:05Z)
• Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices [0.0]
  We present several inequalities related to the Robertson-Schr"odinger uncertainty relation. By considering a concave roof of the bound, we obtain an improvement of the Robertson-Schr"odinger uncertainty relation. We present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states.
  arXiv  Detail & Related papers  (2021-09-14T18:00:07Z)
• Einstein-Podolsky-Rosen uncertainty limits for bipartite multimode states [0.0]
  Correlations of two-party $(N, textvs,1)$-mode states are examined by using the variances of a pair of suitable EPR-like observables. The analysis of the minimal properly normalized sums of these variances yields necessary conditions of separability and EPR unsteerability.
  arXiv  Detail & Related papers  (2021-07-02T13:11:00Z)
• R\'enyi divergence inequalities via interpolation, with applications to generalised entropic uncertainty relations [91.3755431537592]
  We investigate quantum R'enyi entropic quantities, specifically those derived from'sandwiched' divergence. We present R'enyi mutual information decomposition rules, a new approach to the R'enyi conditional entropy tripartite chain rules and a more general bipartite comparison.
  arXiv  Detail & Related papers  (2021-06-19T04:06:23Z)
• Fundamental Limits and Tradeoffs in Invariant Representation Learning [99.2368462915979]
  Many machine learning applications involve learning representations that achieve two competing goals. Minimax game-theoretic formulation represents a fundamental tradeoff between accuracy and invariance. We provide an information-theoretic analysis of this general and important problem under both classification and regression settings.
  arXiv  Detail & Related papers  (2020-12-19T15:24:04Z)
• A Weaker Faithfulness Assumption based on Triple Interactions [89.59955143854556]
  We propose a weaker assumption that we call $2$-adjacency faithfulness. We propose a sound orientation rule for causal discovery that applies under weaker assumptions.
  arXiv  Detail & Related papers  (2020-10-27T13:04:08Z)
• Experimental Test of Tight State-Independent Preparation Uncertainty Relations for Qubits [1.5749416770494706]
  We present a neutron optical test of the tight state preparation uncertainty relations for non-independent Pauli spin states with mixed spin states. The final results, obtained in a polarimetric experiment, reproduce the theoretical predictions evidently for arbitrary initial states variable degree polarization.
  arXiv  Detail & Related papers  (2020-02-25T08:23:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.