Quantum decimation in Hilbert space: Coarse graining without structure

Singh, Ashmeet; Carroll, Sean M.

doi:10.1103/PhysRevA.97.032111

Title: Quantum decimation in Hilbert space: Coarse graining without structure

Journal Article · Wed Mar 14 00:00:00 EDT 2018 · Physical Review A

DOI:https://doi.org/10.1103/PhysRevA.97.032111· OSTI ID:1597497

Singh, Ashmeet ^[1]; Carroll, Sean M. ^[1]

California Inst. of Technology (CalTech), Pasadena, CA (United States)

We introduce a technique to coarse grain quantum states in a finite-dimensional Hilbert space. Our method is distinguished from other approaches by not relying on structures such as a preferred factorization of Hilbert space or a preferred set of operators (local or otherwise) in an associated algebra. Rather, we use the data corresponding to a given set of states, either specified independently or constructed from a single state evolving in time. Our method is based on principle component analysis (PCA), and the resulting coarse-grained quantum states live in a lower-dimensional Hilbert space whose basis is defined using the underlying (isometric embedding) transformation of the set of fine-grained states we wish to coarse grain. Physically, the transformation can be interpreted to be an “entanglement coarse-graining” scheme that retains most of the global, useful entanglement structure of each state, while needing fewer degrees of freedom for its reconstruction. This scheme could be useful for efficiently describing collections of states whose number is much smaller than the dimension of Hilbert space, or a single state evolving over time.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Cite

Export

Save

Research Organization:: California Institute of Technology (CalTech), Pasadena, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), High Energy Physics (HEP)

Grant/Contract Number:: SC0011632

OSTI ID:: 1597497

Alternate ID(s):: OSTI ID: 1426014

Journal Information:: Physical Review A, Vol. 97, Issue 3; ISSN 2469-9926

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 3 works

Citation information provided by
Web of Science

References (42)

Quantum Coarse-Graining: An Information-theoretic Approach to thermodynamics Faist, Philippe ETH Zurich https://doi.org/10.3929/ethz-a-010695790	text	January 2016
Angular Quantization and the Density Matrix Renormalization Group Gaite, J. Modern Physics Letters A, Vol. 16, Issue 17 https://doi.org/10.1142/S0217732301004364	journal	June 2001
One-Shot Lossy Quantum Data Compression Datta, Nilanjana; Renes, Joseph M.; Renner, Renato IEEE Transactions on Information Theory, Vol. 59, Issue 12 https://doi.org/10.1109/TIT.2013.2283723	journal	December 2013
Efficient compression of quantum information Plesch, Martin; Bužek, Vladimír Physical Review A, Vol. 81, Issue 3 https://doi.org/10.1103/PhysRevA.81.032317	journal	March 2010
Glimmers of a Pre-geometric Perspective Piazza, Federico Foundations of Physics, Vol. 40, Issue 3 https://doi.org/10.1007/s10701-009-9387-5	journal	December 2009
Static Phenomena Near Critical Points: Theory and Experiment Kadanoff, Leo P.; GÖTze, Wolfgang; Hamblen, David Reviews of Modern Physics, Vol. 39, Issue 2 https://doi.org/10.1103/RevModPhys.39.395	journal	April 1967
Concepts of coarse graining in quantum mechanics Busch, Paul; Quadt, Ralf International Journal of Theoretical Physics, Vol. 32, Issue 12 https://doi.org/10.1007/BF00672998	journal	December 1993
Extensions of Lipschitz mappings into a Hilbert space Johnson, William B.; Lindenstrauss, Joram Conference on Modern Analysis and Probability https://doi.org/10.1090/conm/026/737400	book	January 1984
Class of Quantum Many-Body States That Can Be Efficiently Simulated Vidal, G. Physical Review Letters, Vol. 101, Issue 11 https://doi.org/10.1103/PhysRevLett.101.110501	journal	September 2008
Coarse-grained quantum state estimation for noisy measurements Teo, Yong Siah; Řeháček, Jaroslav; Hradil, Zdeněk Physical Review A, Vol. 88, Issue 2 https://doi.org/10.1103/PhysRevA.88.022111	journal	August 2013
The renormalization group: Critical phenomena and the Kondo problem Wilson, Kenneth G. Reviews of Modern Physics, Vol. 47, Issue 4 https://doi.org/10.1103/RevModPhys.47.773	journal	October 1975
Teaching the renormalization group Maris, Humphrey J.; Kadanoff, Leo P. American Journal of Physics, Vol. 46, Issue 6 https://doi.org/10.1119/1.11224	journal	June 1978
The density-matrix renormalization group Schollwöck, U. Reviews of Modern Physics, Vol. 77, Issue 1 https://doi.org/10.1103/RevModPhys.77.259	journal	April 2005
Quantum decimation for spin-(1/2) chains in a magnetic field Xi-Yao, Chen; Tuthill, G. F. Physical Review B, Vol. 32, Issue 11 https://doi.org/10.1103/PhysRevB.32.7280	journal	December 1985
Density-matrix algorithms for quantum renormalization groups White, Steven R. Physical Review B, Vol. 48, Issue 14 https://doi.org/10.1103/PhysRevB.48.10345	journal	October 1993
Quantum coding Schumacher, Benjamin Physical Review A, Vol. 51, Issue 4 https://doi.org/10.1103/PhysRevA.51.2738	journal	April 1995
Entanglement Renormalization Vidal, G. Physical Review Letters, Vol. 99, Issue 22 https://doi.org/10.1103/PhysRevLett.99.220405	journal	November 2007
Consciousness as a state of matter Tegmark, Max Chaos, Solitons & Fractals, Vol. 76 https://doi.org/10.1016/j.chaos.2015.03.014	journal	July 2015
Decimation approach in quantum systems Castellani, C.; Di Castro, C.; Ranninger, J. Nuclear Physics B, Vol. 200, Issue 1 https://doi.org/10.1016/0550-3213(82)90057-8	journal	January 1982
Energetics of Domain Walls in the 2D $t - J$ Model White, Steven R.; Scalapino, D. J. Physical Review Letters, Vol. 81, Issue 15 https://doi.org/10.1103/PhysRevLett.81.3227	journal	October 1998
The renormalization group in the theory of critical behavior Fisher, Michael E. Reviews of Modern Physics, Vol. 46, Issue 4 https://doi.org/10.1103/RevModPhys.46.597	journal	October 1974
Scaling and Renormalization in Statistical Physics Cardy, John Cambridge University Press https://doi.org/10.1017/CBO9781316036440	book	January 2015
Quantum cluster decimation Matsubara, T.; Totsuji, C.; Thompson, C. J. Journal of Physics A: Mathematical and General, Vol. 24, Issue 19 https://doi.org/10.1088/0305-4470/24/19/023	journal	October 1991
Coarse graining and the quantum—Classical connection Quadt, Ralf; Busch, Paul Open Systems & Information Dynamics, Vol. 2, Issue 2 https://doi.org/10.1007/BF02228961	journal	June 1994
Density matrix formulation for quantum renormalization groups White, Steven R. Physical Review Letters, Vol. 69, Issue 19 https://doi.org/10.1103/PhysRevLett.69.2863	journal	November 1992
Coarse graining the phase space of $N$ qubits Di Matteo, Olivia; Sánchez-Soto, Luis L.; Leuchs, Gerd Physical Review A, Vol. 95, Issue 2 https://doi.org/10.1103/PhysRevA.95.022340	journal	February 2017
Renormalization group theory: Its basis and formulation in statistical physics Fisher, Michael E. Reviews of Modern Physics, Vol. 70, Issue 2 https://doi.org/10.1103/RevModPhys.70.653	journal	April 1998
An elementary proof of a theorem of Johnson and Lindenstrauss Dasgupta, Sanjoy; Gupta, Anupam Random Structures and Algorithms, Vol. 22, Issue 1 https://doi.org/10.1002/rsa.10073	journal	November 2002
Erratum: The renormalization group in the theory of critical behavior Fisher, Michael E. Reviews of Modern Physics, Vol. 47, Issue 2 https://doi.org/10.1103/RevModPhys.47.543	journal	April 1975
Quantum Data Compression of a Qubit Ensemble Rozema, Lee A.; Mahler, Dylan H.; Hayat, Alex Physical Review Letters, Vol. 113, Issue 16 https://doi.org/10.1103/PhysRevLett.113.160504	journal	October 2014
Density Matrix Renormalization Group Study of the Striped Phase in the 2D $t - J$ Model White, Steven R.; Scalapino, D. J. Physical Review Letters, Vol. 80, Issue 6 https://doi.org/10.1103/PhysRevLett.80.1272	journal	February 1998
Space from Hilbert space: Recovering geometry from bulk entanglement Cao, ChunJun; Carroll, Sean M.; Michalakis, Spyridon Physical Review D, Vol. 95, Issue 2 https://doi.org/10.1103/PhysRevD.95.024031	journal	January 2017
Efficient compression of quantum information Plesch, Martin; Buzek, Vladimir arXiv https://doi.org/10.48550/arxiv.0907.1764	text	January 2009
Coarse-grained quantum state estimation for noisy measurements Teo, Yong Siah; Rehacek, Jaroslav; Hradil, Zdenek arXiv https://doi.org/10.48550/arxiv.1304.1866	text	January 2013
Consciousness as a State of Matter Tegmark, Max arXiv https://doi.org/10.48550/arxiv.1401.1219	text	January 2014
Quantum Data Compression of a Qubit Ensemble Rozema, Lee A.; Mahler, Dylan H.; Hayat, Alex arXiv https://doi.org/10.48550/arxiv.1410.3941	text	January 2014
Space from Hilbert Space: Recovering Geometry from Bulk Entanglement Cao, ChunJun; Carroll, Sean M.; Michalakis, Spyridon arXiv https://doi.org/10.48550/arxiv.1606.08444	text	January 2016
Coarse graining the phase space of $N$ qubits Di Matteo, Olivia; Sanchez-Soto, Luis L.; Leuchs, Gerd arXiv https://doi.org/10.48550/arxiv.1701.08630	text	January 2017
Angular quantization and the density matrix renormalization group Gaite, Jose arXiv https://doi.org/10.48550/arxiv.cond-mat/0106049	text	January 2001
The density-matrix renormalization group Schollwoeck, Ulrich arXiv https://doi.org/10.48550/arxiv.cond-mat/0409292	text	January 2004
Entanglement renormalization Vidal, Guifre arXiv https://doi.org/10.48550/arxiv.cond-mat/0512165	text	January 2005
Energetics of Domain Walls in the 2D t-J model White, Steven R.; Scalapino, D. J. arXiv https://doi.org/10.48550/arxiv.cond-mat/9801274	text	January 1998

Cited By (2)

Quantum coarse graining, symmetries, and reducibility of dynamics Kabernik, Oleg Physical Review A, Vol. 97, Issue 5 https://doi.org/10.1103/physreva.97.052130	journal	May 2018
Quantum Coarse-Graining, Symmetries and Reducibility of Dynamics Kabernik, Oleg arXiv https://doi.org/10.48550/arxiv.1801.09770	text	January 2018

Similar Records

Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space

Journal Article · Tue Apr 03 00:00:00 EDT 2018 · Physical Review. D. · OSTI ID:1597497

Cao, ChunJun; Carroll, Sean M.

Space from Hilbert space: Recovering geometry from bulk entanglement

Journal Article · Fri Jan 27 00:00:00 EST 2017 · Physical Review D · OSTI ID:1597497

Cao, ChunJun; Carroll, Sean M.; Michalakis, Spyridon

Quantum mereology: Factorizing Hilbert space into subsystems with quasiclassical dynamics

Journal Article · Tue Feb 16 00:00:00 EST 2021 · Physical Review A · OSTI ID:1597497

Carroll, Sean M.; Singh, Ashmeet

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS

Title: Quantum decimation in Hilbert space: Coarse graining without structure

Citation Formats

References (42)

Cited By (2)

Similar Records

Related Subjects