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Quantum State Reduction: Generalized Bipartitions from Algebras of Observables

Kabernik, Oleg and Pollack, Jason and Singh, Ashmeet (2020) Quantum State Reduction: Generalized Bipartitions from Algebras of Observables. Physical Review A, 101 (3). Art. No. 032303. ISSN 2469-9926.

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Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity.

Item Type:Article
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URLURL TypeDescription Paper
Kabernik, Oleg0000-0001-7358-0657
Pollack, Jason0000-0003-4754-4905
Singh, Ashmeet0000-0002-4404-1416
Additional Information:© 2020 American Physical Society. Received 14 October 2019; accepted 3 February 2020; published 5 March 2020. We thank Sean Carroll, Olivia Di Matteo, Ben Michel, Eric Minton, Philippe Sabella-Garnier, Mark Van Raamsdonk, Robert Raussendorf, and Benson Way for useful discussions. We thank two anonymous reviewers for providing useful feedback on the manuscript. We would also like to acknowledge the help of an anonymous user on the Mathematics Stackexchange website with the proof of Proposition II C. J.P. would like to thank the organizers of the workshop “Gravity—New perspectives from strings and higher dimensions” held at the Centro de Ciencias de Benasque Pedro Pascual, where part of this work was completed. A.S. would like to thank the High Energy Physics group at University of British Columbia at Vancouver for hosting and supporting his visit where part of this work was carried out. O.K. was supported by the Natural Sciences and Engineering Research Council of Canada. J.P. was supported in part by the Simons Foundation and in part by the Natural Sciences and Engineering Research Council of Canada. A.S. was funded in part by the Walter Burke Institute for Theoretical Physics at Caltech, by DOE Grant No. DE-SC0011632, and by the Foundational Questions Institute.
Group:Walter Burke Institute for Theoretical Physics
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Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Simons FoundationUNSPECIFIED
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
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Issue or Number:3
Record Number:CaltechAUTHORS:20191028-151222041
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:99505
Deposited By: Joy Painter
Deposited On:28 Oct 2019 22:51
Last Modified:05 Mar 2020 19:48

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