Musings on SVD and pseudo entanglement entropies

Creators: Caputa, Paweł^{1, 2}; Purkayastha, Souradeep¹; Saha, Abhigyan¹; Sułkowski, Piotr^{1, 3}

1. University of Warsaw
2. Kyoto University
3. California Institute of Technology

Abstract

Pseudo-entropy and SVD entropy are generalizations of the entanglement entropy that involve post-selection. In this work we analyze their properties as measures on the spaces of quantum states and argue that their excess provides useful characterization of a difference between two (i.e. pre-selected and post-selected) states, which shares certain features and in certain cases can be identified as a metric. In particular, when applied to link complement states that are associated to topological links via Chern-Simons theory, these generalized entropies and their excess provide a novel quantification of a difference between corresponding links. We discuss the dependence of such entropy measures on the level of Chern-Simons theory and determine their asymptotic values for certain link states. We find that imaginary part of the pseudo-entropy is sensitive to, and can diagnose chirality of knots. We also consider properties of entropy measures for simpler quantum mechanical systems, such as generalized SU(2) and SU(1,1) coherent states, and tripartite GHZ and W states.

Copyright and License

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Acknowledgement

We would like to thank Onkar Parrikar and Tadashi Takayanagi for discussions and comments, Siddharth Dwivedi and Jackson Fliss for discussions and sharing Mathematica codes, and Anuvab Kayaal for the knot/link graphics. We especially thank Pichai Ramadevi for several helpful discussions and comments on the draft. SP thanks the Galileo Galilei Institute of Theoretical Physics and the École de Physique des Houches for hospitality, and the INFN Italy and the Université Grenoble-Alpes for partial support during the completion of this work. This work is supported by the OPUS grant no. 2022/47/B/ST2/03313 “Quantum geometry and BPS states” and NCN Sonata Bis 9 grant no. 2019/34/E/ST2/00123 funded by the National Science Centre, Poland, and NAWA “Polish Returns 2019” grant no. PPN/PPO/2019/1/00010/U/0001. This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). The work of PS is also supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632.

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