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dc.contributor.authorHarrow, Aram W.
dc.date.accessioned2023-12-18T15:23:00Z
dc.date.available2023-12-18T15:23:00Z
dc.date.issued2023-12-16
dc.identifier.urihttps://hdl.handle.net/1721.1/153193
dc.description.abstractConsider the n! different unitary matrices that permute n d-dimensional quantum systems. If $$d\ge n$$ d ≥ n then they are linearly independent. This paper discusses a sense in which they are approximately orthogonal (with respect to the Hilbert–Schmidt inner product, $$\langle A,B\rangle = \textrm{tr}A^\dag B/\textrm{tr}I$$ ⟨ A , B ⟩ = tr A † B / tr I ) if $$d\gg n^2$$ d ≫ n 2 , or, in a different sense, if $$d\gg n$$ d ≫ n . Previous work had shown pairwise approximate orthogonality of these matrices, but here we show a more collective statement, quantified in terms of the operator norm distance of the Gram matrix to the identity matrix. This simple point has several applications in quantum information and random matrix theory: (1) showing that random maximally entangled states resemble fully random states, (2) showing that Boson sampling output probabilities resemble those from Gaussian matrices, (3) improving the Eggeling–Werner scheme for multipartite data hiding, (4) proving that the product test of Harrow–Montanaro cannot be performed using LOCC without a large number of copies of the state to be tested, (5) proving that the purity of a quantum state also cannot be efficiently tested using LOCC, and (6, published separately with Brandão and Horodecki) helping prove that poly-size random quantum circuits are poly-designs.en_US
dc.publisherSpringer Netherlandsen_US
dc.relation.isversionofhttps://doi.org/10.1007/s11005-023-01744-1en_US
dc.rightsCreative Commons Attribution-Noncommercial-Share Alikeen_US
dc.rights.urihttp://creativecommons.org/licenses/by-nc-sa/4.0/en_US
dc.sourceSpringer Netherlandsen_US
dc.titleApproximate orthogonality of permutation operators, with application to quantum informationen_US
dc.typeArticleen_US
dc.identifier.citationLetters in Mathematical Physics. 2023 Dec 16;114(1):1en_US
dc.contributor.departmentMassachusetts Institute of Technology. Center for Theoretical Physics
dc.eprint.versionAuthor's final manuscripten_US
dc.type.urihttp://purl.org/eprint/type/JournalArticleen_US
eprint.statushttp://purl.org/eprint/status/PeerRevieweden_US
dc.date.updated2023-12-17T04:09:26Z
dc.language.rfc3066en
dc.rights.holderThe Author(s), under exclusive licence to Springer Nature B.V.
dspace.embargo.termsY
dspace.date.submission2023-12-17T04:09:26Z
mit.licenseOPEN_ACCESS_POLICY
mit.metadata.statusAuthority Work and Publication Information Neededen_US


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