Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-01T03:36:05.051623Z
Paper Citation Record · LEDGER
As of 4 August 2026, this Paper Citation Record lists 100 of 112 outbound references and 0 inbound Pith citation observations for arXiv:2607.23137.
A citation records a reference. It does not transfer a finding from one paper to another.
Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-01T03:36:05.051623Z
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-04T06:34:03.388597+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links
A source-named dated measurement, never combined with another source.
Source: cited_works
100 of 112 outbound references displayed
External citation measurements
No source-named external measurement is stored.
Observation f7846035-677c-467e-b86c-d0fc4f125aed · outbound
Reasoning about Continuous-Variable Quantum Systems 2006.Infinite dimensional analysis: a hitchhiker’s guide
Reference 1
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Observation 16d31ce0-4f06-4cf1-bc22-3bff87058d2b · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 2
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Observation 224c443e-1492-4f90-9d3e-493f6f19eed1 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 3
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Observation 20502da6-9a19-4fd8-acf4-8640c17d50a8 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 4
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Observation d2a07dc3-018d-4077-9571-c4c4f16cbbcc · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 5
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Observation 4251d8ea-c26d-473b-ad65-353e9e9ec481 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 6
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Observation 6e89f817-57f5-49fb-857d-d1b72089f340 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 7
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Observation c1b63db1-b61f-4d3e-a0a4-d7207766e4f9 · outbound
Reasoning about Continuous-Variable Quantum Systems 2020.Boundary value problems, Weyl functions, and differential operators
Reference 8
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Observation 12e15ef3-17ea-41eb-9396-33704fcd1e2f · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 9
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No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.
Observation f2fbaa25-9b0c-4ab1-b2f0-7c36b0fad3e8 · outbound
Reasoning about Continuous-Variable Quantum Systems Eli Bourassa, Rafael N
Reference 10
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Observation ea25792a-0ae1-4c3a-860f-409b336e04f1 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 11
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Observation a21808f1-5b80-46cb-ae05-f49e07ab68a6 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 12
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Observation a1446ae3-f976-4392-9976-3250b9180a04 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 13
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Observation 18901baa-48ae-4a99-b8b7-224dcc819777 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 14
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Observation ebf241ec-53ba-49a6-88a3-2892f351dd1b · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 15
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Observation c5558333-a7c1-4633-bf42-459cb83cda7d · outbound
Reasoning about Continuous-Variable Quantum Systems 1998.Multivalued linear operators
Reference 16
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Observation 8a79af37-9742-40d7-88c6-27102eaac2a6 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 17
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Observation 1c28fcf8-617f-4546-bb9f-d74aacdcaf30 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 18
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Observation 732bd928-5726-4dd4-9b3f-bd3f64381f56 · outbound
Reasoning about Continuous-Variable Quantum Systems Diestel and J
Reference 19
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Observation 1b84bffb-03ef-4c61-8d87-3158826b7c29 · outbound
Reasoning about Continuous-Variable Quantum Systems Dunford and J.T
Reference 20
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Observation 343c0698-eec5-4359-aed3-d13fd3309b8c · outbound
Reasoning about Continuous-Variable Quantum Systems End-to-End Formalization of Quantum Error Correction
Reference 21
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Observation 8135d9ab-edcd-467c-8aca-0da58e18b644 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 22
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Observation b9ae6162-7e4c-4e6b-84d9-82a169ecfc0b · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 23
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Observation 5aaedffb-e594-4091-a3c0-b683859c758a · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 24
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Observation 43e0bed2-6254-4b77-bbbb-572e48f63c74 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 25
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Observation 99e9d9da-571c-49c8-9a4d-d53cfd4f9890 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 26
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Observation 520573f8-4a8d-4257-bb11-af5831406d0c · outbound
Reasoning about Continuous-Variable Quantum Systems 2001.Statistical structure of quantum theory
Reference 27
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Observation 8e26ff52-f3d4-413e-a23b-e4978a8f0229 · outbound
Reasoning about Continuous-Variable Quantum Systems 2011.Probabilistic and statistical aspects of quantum theory
Reference 28
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Observation 32d7704a-fcc0-47ad-8389-a0d1e928b024 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 29
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Observation a4680456-8c9f-468a-b886-9ace9a7d0f3a · outbound
Reasoning about Continuous-Variable Quantum Systems 2016.Analysis in Banach spaces
Reference 30
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Observation b3e21ff5-401e-4e79-b1c7-410754ed15b1 · outbound
Reasoning about Continuous-Variable Quantum Systems 1997.Fundamentals of the Theory of Operator Algebras
Reference 31
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Observation 79a93f56-62f6-4945-adbf-5d06145953b7 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 32
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Observation 2909be7b-8f24-421a-acad-74308bf9f2ed · outbound
Reasoning about Continuous-Variable Quantum Systems 2013.Perturbation theory for linear operators
Reference 33
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Observation 267a0502-4c6a-4caf-8d0d-986a015a4a55 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 34
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Observation 4f63f7c5-d50d-4c1b-a35a-829c0b4fefb0 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 35
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Observation 69c4ba7d-618d-42bf-b6ea-8707d9e4e6aa · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 36
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Observation b6cf3e8e-ee88-445b-b4d8-a57e65c180a6 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 37
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Observation b53ce3af-a533-491b-a9a7-8e9b920cc88d · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 38
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No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.
Observation 93870229-6eb3-4d62-8dd9-1218b217c922 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 39
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No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.
Observation 4d09749b-2812-4f79-93cd-ffd96eecf186 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 40
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Observation 6f26266c-a6d0-48f0-9dd2-d714e0f21625 · outbound
Reasoning about Continuous-Variable Quantum Systems Braunstein
Reference 41
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Observation eac11128-8e7b-4256-8902-2c165063cb94 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 42
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Observation 2d7b27d2-30cd-4f0c-872d-067bd03a91e5 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 43
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Observation 9defc0aa-2762-40da-9aae-21c2a2bb1e74 · outbound
Reasoning about Continuous-Variable Quantum Systems 2005.Abstraction, Refinement and Proof for Probabilistic Systems
Reference 44
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Observation f22d6969-7b67-4cdf-b983-c7248ca5dfa1 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 45
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Observation 8293bc3c-a9a3-475f-86da-ad9abc86b33b · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 46
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Observation f09a972c-2ab3-44a4-a64a-c5ff18a49f08 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 47
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Observation 5a1ef2ef-77d8-42fc-96a9-c868f09558ee · outbound
Reasoning about Continuous-Variable Quantum Systems 1980.Methods of modern mathematical physics: Functional analysis
Reference 48
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Observation 1f8e5e70-e4a3-49d1-b93c-e8cdefd68fda · outbound
Reasoning about Continuous-Variable Quantum Systems 1987.Real and complex analysis
Reference 49
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Observation 4b935485-54fc-4e76-b01f-1c86197cad34 · outbound
Reasoning about Continuous-Variable Quantum Systems 2012.Unbounded self-adjoint operators on Hilbert space
Reference 50
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Observation e441d4bd-419f-4862-a03c-bc92ffb0c881 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 51
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Observation 9cab4f38-82d3-49fc-8aec-8f98e8b9e896 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 52
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Observation 44adf98e-8eb7-4db1-afcd-ff1f018bbd40 · outbound
Reasoning about Continuous-Variable Quantum Systems Hoare meets Heisenberg: A Lightweight Logic for Quantum Programs
Reference 53
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Observation c45ad392-b6ab-4e3e-a942-3dc936273084 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 54
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Observation 6e228854-783a-4d36-a6e4-df7a48f7e766 · outbound
Reasoning about Continuous-Variable Quantum Systems Tse, Haocun Yu, N
Reference 55
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Observation 576501c9-397f-4b68-8d3d-cc16cca5b7e0 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 56
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Observation cf378017-2e60-4d5c-9b01-0ad7a5d0e258 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 57
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Observation 598d1591-5601-4023-9a57-21d612ee2fe8 · outbound
Reasoning about Continuous-Variable Quantum Systems Quantum references
Reference 58
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Observation 1a329be6-c29b-442a-9511-2653fafaea0c · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 59
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Observation 2eb4f01d-58c4-4673-b173-56eaae11a58f · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 60
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Observation e61204b9-6fe7-44fe-a04d-54a8dd71c0d0 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 61
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Observation b41cc290-2aca-41f0-a802-23120cc530ee · outbound
Reasoning about Continuous-Variable Quantum Systems QECV: Quantum Error Correction Verification
Reference 62
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Observation bde80235-a8c1-48e5-b48d-68b6eac70a8c · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 63
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Observation 23ad30d3-b1d4-456b-b8f3-4d7c32f7d91c · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 64
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Observation 87d345ad-85bc-4160-af1b-bfd0c55a2c75 · outbound
Reasoning about Continuous-Variable Quantum Systems Birkhoff-von Neumann Quantum Logic as an Assertion Language for Quantum Programs
Reference 65
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Observation 310a8191-281d-400e-8a3c-ceb8d985c1b0 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 66
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Observation e1bdcdf4-5117-4738-ace2-c93059798bb6 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 67
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Observation 147321c7-30c5-4ecd-9bfa-33c99445aa1d · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 68
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No event found in the named queried sources as of 2026-08-04T06:34:03.388597+00:00.
Observation 4761f284-f4f0-4038-a56c-58b5d04281b6 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 69
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Observation 81acf81d-dcf2-4bbc-bb85-c36850a37e5b · outbound
Reasoning about Continuous-Variable Quantum Systems Reasoning about Parallel Quantum Programs
Reference 70
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Observation f808191b-8c40-43b0-8163-c84f26cfbe7a · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 71
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Observation 224e5319-5746-4281-b783-6a182537a3b9 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 72
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Observation bdb14ada-54cf-4b17-b938-df7664a2b906 · outbound
Reasoning about Continuous-Variable Quantum Systems By the definition of closure, the domain Dom(𝑇) is dense inDom(𝔱 𝑇)with respect to the form norm
Reference 75
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Observation 8650cc3e-5ad6-4f87-959f-e89c9ec49966 · outbound
Reasoning about Continuous-Variable Quantum Systems continue
Reference 76
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Observation a7626f2b-1285-4716-9641-839b35c54c5d · outbound
Reasoning about Continuous-Variable Quantum Systems We show that Í 𝑖 𝔱𝑄[𝐾𝑖 u]= Í 𝑗 𝔱𝑄[𝐿𝑗 u]for any𝑢∈H
Reference 77
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Observation 4b7d3960-c7bc-4295-9b26-ff4fbe05784b · outbound
Reasoning about Continuous-Variable Quantum Systems By the spectral approximation𝑄= Ã 𝑛𝑄𝑛 used above, and the continuity of the trace over positive forms, we have: Tr(𝑄J𝑆K(𝜌))=sup 𝑛 Tr 𝑄𝑛J𝑆K(𝜌)
Reference 78
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Observation 84c17044-770a-4f2a-b9d7-ee5042928afd · outbound
Reasoning about Continuous-Variable Quantum Systems • Validity:By the Exact Duality established above,Tr(wlp(𝑆,𝑄)𝜌)=Tr(𝑄J𝑆K(𝜌))≥Tr(𝑄J𝑆K(𝜌))
Reference 79
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Observation 31cca163-a3cf-401d-aa49-0c3a09090579 · outbound
Reasoning about Continuous-Variable Quantum Systems By definition, Dom(𝔱𝑄)⊆Dom(𝔱 𝑃) and 𝔱𝑃[v]≤𝔱 𝑄[v] for all v∈Dom(𝔱 𝑄)
Reference 80
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Observation 8b3ac64e-1a4d-4d6b-bbef-311db79da02c · outbound
Reasoning about Continuous-Variable Quantum Systems By Theorem A.3, the form 𝔱𝑃 is defined by 𝔱𝑃[v]=Í 𝑗𝑐𝑗 𝔱𝑃 𝑗[v]and is a closed form
Reference 81
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Observation 53d9ca98-6360-4ae8-b3f2-f328d8f3efec · outbound
Reasoning about Continuous-Variable Quantum Systems By Theo- rem A.3,𝔱𝑃(v)=sup 𝑛 𝔱𝑃𝑛(v)
Reference 82
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Observation 3f347249-b6b3-4ece-a329-f83f10e7dcb7 · outbound
Reasoning about Continuous-Variable Quantum Systems Hence Tr(𝑃𝑠𝑦𝑛𝜌)=Tr(𝑄𝜌)=Tr(𝑄JskipK(𝜌))
Reference 83
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Observation 4bf704c4-cf11-46d7-9926-7666cdb4a3ee · outbound
Reasoning about Continuous-Variable Quantum Systems By the induction hypothesis for 𝑆2,𝑅∈P
Reference 84
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Observation 2fa1a916-e7b7-478d-8aed-f935da378a43 · outbound
Reasoning about Continuous-Variable Quantum Systems The measurement outcome space is denoted by(Ω𝑀,Σ𝑀,𝜇) , the Kraus-density operator at outcome𝜈 by𝑀𝜈, and the continuation with𝑥=𝜈 by𝑆 𝜈
Reference 85
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Observation 58ba0a12-c02a-41da-b741-4bdb3382d612 · outbound
Reasoning about Continuous-Variable Quantum Systems Define𝑋 0 =0, and 𝔱𝑋𝑛+1[u]= 𝔱 𝑄[𝑀0u]+𝔱 wlpsyn(𝑆𝑏𝑜𝑑 𝑦,𝑋𝑛)[𝑀1u]
Reference 86
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Observation e39d43ac-66af-4679-8cb8-4afee480074a · outbound
Reasoning about Continuous-Variable Quantum Systems The total cost is C(𝑆,𝜌)= ∞∑︁ 𝑘=0 𝜀Tr(𝜌 𝑘)+C(𝑆 𝑏𝑜𝑑𝑦,𝑀 1𝜌𝑘𝑀† 1) <∞
Reference 87
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Observation e5373e07-c39e-4645-aac2-bbfda9091f3d · outbound
Reasoning about Continuous-Variable Quantum Systems Their sum is 𝔱𝑄[u]+𝜀∥ u∥2, which exactly matches the syntactic rule 𝔱wp(skip,𝑄)[u]
Reference 88
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Observation d1daeaaf-4333-481b-8d58-9d1dcad2b09d · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 89
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Observation 84341996-d23a-49af-ab37-313baf6e6aa3 · outbound
Reasoning about Continuous-Variable Quantum Systems Unresolved cited work
Reference 90
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Observation f2158779-dd17-458d-ad68-3eaed26e0e28 · outbound
Reasoning about Continuous-Variable Quantum Systems Both components are defined as infinite limits of their 𝑁 -step truncations
Reference 91
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Observation 5905e428-a776-4f1c-83e0-2431673481df · outbound
Reasoning about Continuous-Variable Quantum Systems Thus, equality𝑃=wlp(𝑆,𝑄) holds trivially, which implies𝑃⊒wlp(𝑆,𝑄)
Reference 92
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Observation 28f08dc9-7fbf-4760-b4ba-3f9212b01dda · outbound
Reasoning about Continuous-Variable Quantum Systems By the monotonicity of the wlp transformer (Theorem C.2.1), wlp(𝑆 1,𝑅)⊒wlp(𝑆 1,wlp(𝑆 2,𝑄))
Reference 93
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Observation abf7e4d3-a4e1-4da6-a7ea-260535ef30e5 · outbound
Reasoning about Continuous-Variable Quantum Systems By induction hypothesis,𝑃𝜈⊒wlp(𝑆 𝜈,𝑄) , which means 𝔱𝑃𝜈[v]≥𝔱 wlp(𝑆 𝜈,𝑄)[v] for allv
Reference 94
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Observation 0ad88f5c-0ab3-46aa-a3b9-deea3bca392d · outbound
Reasoning about Continuous-Variable Quantum Systems By the induction hypothesis,𝑅⊒wlp(𝑆 𝑏𝑜𝑑𝑦,𝑃)
Reference 95
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Observation 51eceab7-ee70-4576-9ee5-77af61e8cf99 · outbound
Reasoning about Continuous-Variable Quantum Systems By the monotonicity of wlp,𝑄′⊒𝑄=⇒wlp(𝑆,𝑄 ′) ⊒wlp(𝑆,𝑄)
Reference 96
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Observation 4c62d466-e449-4d6b-beee-8852d2a26608 · outbound
Reasoning about Continuous-Variable Quantum Systems For abort, the semantic weakest liberal precondition is wlp(abort,𝑄)= 0(by Definition C.4)
Reference 97
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Observation 94012bae-7b12-4274-9d4f-ac8658601f67 · outbound
Reasoning about Continuous-Variable Quantum Systems By the induction hypothesis, ⊢ {𝑃𝜈}𝑆𝜈{𝑄} is derivable for all𝜈∈Ω 𝑀
Reference 98
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Observation a8f0e1b4-71e9-4600-88f0-03e28f2b65f4 · outbound
Reasoning about Continuous-Variable Quantum Systems We must derive⊢{wlp(𝑆,𝑄)}𝑆{𝑄}
Reference 99
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Observation 7b8c83af-34aa-46a5-ab2e-3397d9292ba8 · outbound
Reasoning about Continuous-Variable Quantum Systems Evaluating its quadratic form yields 𝔱𝑃𝑠 𝑦𝑛[u]= 𝔱 𝑄[u]+𝜀 ∥u∥2
Reference 100
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Observation 7aa68c1e-414e-4fa9-9194-b0f0eec1f580 · outbound
Reasoning about Continuous-Variable Quantum Systems By the induction hypothesis, we have𝑃⊒wp(𝑆 1,𝑅) and𝑅⊒wp(𝑆 2,𝑄)
Reference 101
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Observation 901021a3-9a14-4e39-abd1-90e52543764c · outbound
Reasoning about Continuous-Variable Quantum Systems The derived precondition is𝑃𝑝𝑟𝑒 =𝜀𝐼+ ∫ Ω𝑀 𝑀† 𝜈𝑃𝜈𝑀𝜈𝑑𝜇(𝜈)
Reference 102
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No inbound Pith citation observations are available.