Coarse-grained distinguishability of field interactions

Cédric Bény

doi:10.22331/q-2018-05-24-67

Coarse-grained distinguishability of field interactions

Cédric Bény

Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
Department of Applied Mathematics, Hanyang University (ERICA), 55 Hanyangdaehak-ro, Ansan, Gyeonggi-do, 426-791, Korea

Published:	2018-05-24, volume 2, page 67
Eprint:	arXiv:1509.03249v5
Doi:	https://doi.org/10.22331/q-2018-05-24-67
Citation:	Quantum 2, 67 (2018).

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Abstract

Information-theoretical quantities such as statistical distinguishability typically result from optimisations over all conceivable observables. Physical theories, however, are not generally considered valid for all mathematically allowed measurements. For instance, quantum field theories are not meant to be correct or even consistent at arbitrarily small lengthscales. A general way of limiting such an optimisation to certain observables is to first coarse-grain the states by a quantum channel. We show how to calculate contractive quantum information metrics on coarse-grained equilibrium states of free bosonic systems (Gaussian states), in directions generated by arbitrary perturbations of the Hamiltonian. As an example, we study the Klein-Gordon field. If the phase-space resolution is coarse compared to h-bar, the various metrics become equal and the calculations simplify. In that context, we compute the scale dependence of the distinguishability of the quartic interaction.

Featured image: The basketball diagram is a tensor network computing the distinguishability of a quartic interaction to lowest order.

► BibTeX data

@article{Beny2018coarsegrained,
  doi = {10.22331/q-2018-05-24-67},
  url = {https://doi.org/10.22331/q-2018-05-24-67},
  title = {Coarse-grained distinguishability of field interactions},
  author = {B{\'{e}}ny, C{\'{e}}dric},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {2},
  pages = {67},
  month = may,
  year = {2018}
}

► References

[1] Jens O Andersen, Eric Braaten, and Michael Strickland. The massive thermal basketball diagram. 2000. 10.1103/PhysRevD.62.045004. (arXiv:hep-ph/0002048).
https://doi.org/10.1103/PhysRevD.62.045004
arXiv:hep-ph/0002048

[2] Vijay Balasubramanian, Jonathan J Heckman, and Alexander Maloney. Relative entropy and proximity of quantum field theories. Journal of High Energy Physics, 2015 (5): 1–14, 2015. 10.1007/JHEP05(2015)104. (arXiv:1410.6809).
https://doi.org/10.1007/JHEP05(2015)104
arXiv:1410.6809

[3] Leonardo Banchi, Samuel L Braunstein, and Stefano Pirandola. Quantum fidelity for arbitrary gaussian states. Physical review letters, 115 (26): 260501, 2015. 10.1103/PhysRevLett.115.260501. (arXiv:1507.01941).
https://doi.org/10.1103/PhysRevLett.115.260501
arXiv:1507.01941

[4] H Barnum and E Knill. Reversing quantum dynamics with near-optimal quantum and classical fidelity. Journal of Mathematical Physics, 43 (5): 2097–2106, 2002. 10.1063/1.1459754. (arXiv:quant-ph/0004088).
https://doi.org/10.1063/1.1459754
arXiv:quant-ph/0004088

[5] C. Bény and T. J. Osborne. Information geometric approach to the renormalisation group. Phys. Rev. A, 92: 022330, 2015a. 10.1103/PhysRevA.92.022330. (arXiv:1206.7004).
https://doi.org/10.1103/PhysRevA.92.022330
arXiv:1206.7004

[6] Cédric Bény and Tobias J Osborne. Renormalisation as an inference problem. arXiv:1310.3188, 2013.
arXiv:1310.3188

[7] Cédric Bény and Tobias J Osborne. The renormalisation group via statistical inference. New J. Phys., 17: 083005, 2015b. 10.1088/1367-2630/17/8/083005. (arXiv:1402.4949).
https://doi.org/10.1088/1367-2630/17/8/083005
arXiv:1402.4949

[8] Samuel L Braunstein and Carlton M Caves. Statistical distance and the geometry of quantum states. Physical Review Letters, 72 (22): 3439–3443, 1994. 10.1103/PhysRevLett.72.3439.
https://doi.org/10.1103/PhysRevLett.72.3439

[9] Sergey Bravyi. Lagrangian representation for fermionic linear optics. arXiv:quant-ph/0404180, 2004.
arXiv:quant-ph/0404180

[10] Shi-Jian Gu. Fidelity approach to quantum phase transitions. International Journal of Modern Physics B, 24 (23): 4371–4458, 2010. 10.1142/S0217979210056335. (arXiv:0811.3127).
https://doi.org/10.1142/S0217979210056335
arXiv:0811.3127

[11] Anna Jenčová. Geodesic distances on density matrices. Journal of Mathematical Physics, 45 (5): 1787–1794, 2004. 10.1063/1.1689000. (arXiv:math-ph/0312044).
https://doi.org/10.1063/1.1689000
arXiv:math-ph/0312044

[12] Masamichi Miyaji, Tokiro Numasawa, Noburo Shiba, Tadashi Takayanagi, and Kento Watanabe. Gravity dual of quantum information metric. 10.1103/PhysRevLett.115.261602. arXiv:1507.07555, 2015.
https://doi.org/10.1103/PhysRevLett.115.261602
arXiv:1507.07555

[13] Hui Khoon Ng and Prabha Mandayam. Simple approach to approximate quantum error correction based on the transpose channel. Physical Review A, 81 (6): 062342, 2010. 10.1103/PhysRevA.81.062342. (arXiv:0909.0931).
https://doi.org/10.1103/PhysRevA.81.062342
arXiv:0909.0931

[14] M. Ohya and D. Petz. Quantum entropy and its use. Springer Verlag, 2004.

[15] D. Petz. Monotone metrics on matrix spaces. Linear algebra and its applications, 244: 81–96, 1996. 10.1016/0024-3795(94)00211-8.
https://doi.org/10.1016/0024-3795(94)00211-8

[16] Dénes Petz. Covariance and fisher information in quantum mechanics. Journal of Physics A: Mathematical and General, 35 (4): 929, 2002. 10.1088/0305-4470/35/4/305. (arXiv:quant-ph/0106125).
https://doi.org/10.1088/0305-4470/35/4/305
arXiv:quant-ph/0106125

[17] Dénes Petz and Csaba Sudár. Geometries of quantum states. Journal of Mathematical Physics, 37 (6): 2662–2673, 1996. 10.1063/1.531535. (arXiv:quant-ph/0102132).
https://doi.org/10.1063/1.531535
arXiv:quant-ph/0102132

[18] Shinsei Ryu and Tadashi Takayanagi. Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence. Physical review letters, 96 (18): 181602, 2006. 10.1103/PhysRevLett.96.181602. (arXiv:hep-th/0603001).
https://doi.org/10.1103/PhysRevLett.96.181602
arXiv:hep-th/0603001

[19] K. G. Wilson. The renormalization group: Critical phenomena and the kondo problem. Rev. Mod. Phys., 47: 773, 1975. 10.1103/RevModPhys.47.773.
https://doi.org/10.1103/RevModPhys.47.773

[20] Paolo Zanardi, Paolo Giorda, and Marco Cozzini. Information-theoretic differential geometry of quantum phase transitions. Physical review letters, 99 (10): 100603, 2007. 10.1103/PhysRevLett.99.100603. (arXiv:quant-ph/0701061).
https://doi.org/10.1103/PhysRevLett.99.100603
arXiv:quant-ph/0701061

Cited by

[1] Cédric Bény, "Inferring relevant features: From QFT to PCA", International Journal of Quantum Information 16 08, 1840012 (2018).

[2] Vincent Lahoche, Dine Ousmane Samary, and Mohamed Tamaazousti, "Signal Detection in Nearly Continuous Spectra and ℤ2-Symmetry Breaking", Symmetry 14 3, 486 (2022).

[3] Amit Gordon, Aditya Banerjee, Maciej Koch-Janusz, and Zohar Ringel, "Relevance in the Renormalization Group and in Information Theory", Physical Review Letters 126 24, 240601 (2021).

[4] Vincent Lahoche, Mohamed Ouerfelli, Dine Ousmane Samary, and Mohamed Tamaazousti, "Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra II: Tensorial Data", Entropy 23 7, 795 (2021).

[5] Vincent Lahoche, Dine Ousmane Samary, and Mohamed Tamaazousti, "Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra I: Matricial Data", Entropy 23 9, 1132 (2021).

[6] H Erbin, V Lahoche, and D Ousmane Samary, "Non-perturbative renormalization for the neural network-QFT correspondence", Machine Learning: Science and Technology 3 1, 015027 (2022).

[7] Ludovico Lami, Siddhartha Das, and Mark M. Wilde, "Approximate reversal of quantum Gaussian dynamics", Journal of Physics A Mathematical General 51 12, 125301 (2018).

[8] John B. DeBrota, "A Quantum Information Geometric Approach to Renormalization", arXiv:1609.09440, (2016).

[9] Vincent Lahoche, Dine Ousmane Samary, and Mohamed Tamaazousti, "Signal detection in nearly continuous spectra and symmetry breaking", arXiv:2011.05447, (2020).

[10] Cédric Bény, "Quantum deconvolution", Quantum Information Processing 17 2, 26 (2018).

[11] Cédric Bény, "Learning relevant features for statistical inference", arXiv:1904.10387, (2019).

[12] Cédric Bény, "Inferring effective field observables from a discrete model", New Journal of Physics 19 1, 013013 (2017).

[13] Cédric Bény, "Inferring effective field observables from a discrete model", arXiv:1511.05090, (2015).

The above citations are from Crossref's cited-by service (last updated successfully 2023-09-28 01:42:05) and SAO/NASA ADS (last updated successfully 2023-09-28 01:42:06). The list may be incomplete as not all publishers provide suitable and complete citation data.

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