{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:UVXVSAKHXTORCGY67BSZV6YRE4","short_pith_number":"pith:UVXVSAKH","canonical_record":{"source":{"id":"1609.01338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-09-05T21:55:39Z","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"title_canon_sha256":"4fdb370bef8cf0cf3f8f39eee5796a110307f1735e07589f4f52e086edca7e92","abstract_canon_sha256":"28ac988a1f974c0cd100f489915f0bfcecc8d03566c13bbfcaca38d47bdcda8f"},"schema_version":"1.0"},"canonical_sha256":"a56f590147bcdd111b1ef8659afb11273b69279d0323354c55412a6a29ef4ec3","source":{"kind":"arxiv","id":"1609.01338","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01338","created_at":"2026-05-18T01:05:41Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01338v1","created_at":"2026-05-18T01:05:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01338","created_at":"2026-05-18T01:05:41Z"},{"alias_kind":"pith_short_12","alias_value":"UVXVSAKHXTOR","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UVXVSAKHXTORCGY6","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UVXVSAKH","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:UVXVSAKHXTORCGY67BSZV6YRE4","target":"record","payload":{"canonical_record":{"source":{"id":"1609.01338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-09-05T21:55:39Z","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"title_canon_sha256":"4fdb370bef8cf0cf3f8f39eee5796a110307f1735e07589f4f52e086edca7e92","abstract_canon_sha256":"28ac988a1f974c0cd100f489915f0bfcecc8d03566c13bbfcaca38d47bdcda8f"},"schema_version":"1.0"},"canonical_sha256":"a56f590147bcdd111b1ef8659afb11273b69279d0323354c55412a6a29ef4ec3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:41.338632Z","signature_b64":"FlXFJDHQccoCmqdW8pQAcm4LVFG8lfmLS47pNZyhBEz1eXlounvbec53mdFIaZ5lesloHVBRujbsW6XL4p1dBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a56f590147bcdd111b1ef8659afb11273b69279d0323354c55412a6a29ef4ec3","last_reissued_at":"2026-05-18T01:05:41.338056Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:41.338056Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1609.01338","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fhZyB39il50Ww/P5jx6gXV6GrO8TjlnF7jTRP4lc94H4fQ30HWbx8u/GFM2Xap58WkFJ9JArkO1IE8kWs85iDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T12:59:22.346398Z"},"content_sha256":"d07504653a2c53269d25987a51ac9725b29cfaa003234851ca1d3ce01a0e0f6f","schema_version":"1.0","event_id":"sha256:d07504653a2c53269d25987a51ac9725b29cfaa003234851ca1d3ce01a0e0f6f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:UVXVSAKHXTORCGY67BSZV6YRE4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Perturbation Bounds for Williamson's Symplectic Normal Form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","quant-ph"],"primary_cat":"math.SP","authors_text":"Martin Idel, Michael M. Wolf, Sebatian Soto Gaona","submitted_at":"2016-09-05T21:55:39Z","abstract_excerpt":"Given a real-valued positive semidefinite matrix, Williamson proved that it can be diagonalised using symplectic matrices. The corresponding diagonal values are known as the symplectic spectrum. This paper is concerned with the stability of Williamson's decomposition under perturbations. We provide norm bounds for the stability of the symplectic eigenvalues and prove that if $S$ diagonalises a given matrix $M$ to Williamson form, then $S$ is stable if the symplectic spectrum is nondegenerate and $S^TS$ is always stable. Finally, we sketch a few applications of the results in quantum informatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01338","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+8Nv41E8iZXBTl9tzIWIfwxyXml2fwFw0jHmt7MPVRVfvc6BkMwywLkgf7HsNDC9thF8DHy6gT9QiV7Yc5vGCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T12:59:22.347073Z"},"content_sha256":"dca8a95aab8c9e96e7b43cc3f4f5cbe7de37e6b18f62620991599d4ef0ad964a","schema_version":"1.0","event_id":"sha256:dca8a95aab8c9e96e7b43cc3f4f5cbe7de37e6b18f62620991599d4ef0ad964a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UVXVSAKHXTORCGY67BSZV6YRE4/bundle.json","state_url":"https://pith.science/pith/UVXVSAKHXTORCGY67BSZV6YRE4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UVXVSAKHXTORCGY67BSZV6YRE4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T12:59:22Z","links":{"resolver":"https://pith.science/pith/UVXVSAKHXTORCGY67BSZV6YRE4","bundle":"https://pith.science/pith/UVXVSAKHXTORCGY67BSZV6YRE4/bundle.json","state":"https://pith.science/pith/UVXVSAKHXTORCGY67BSZV6YRE4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UVXVSAKHXTORCGY67BSZV6YRE4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UVXVSAKHXTORCGY67BSZV6YRE4","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"28ac988a1f974c0cd100f489915f0bfcecc8d03566c13bbfcaca38d47bdcda8f","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-09-05T21:55:39Z","title_canon_sha256":"4fdb370bef8cf0cf3f8f39eee5796a110307f1735e07589f4f52e086edca7e92"},"schema_version":"1.0","source":{"id":"1609.01338","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.01338","created_at":"2026-05-18T01:05:41Z"},{"alias_kind":"arxiv_version","alias_value":"1609.01338v1","created_at":"2026-05-18T01:05:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.01338","created_at":"2026-05-18T01:05:41Z"},{"alias_kind":"pith_short_12","alias_value":"UVXVSAKHXTOR","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UVXVSAKHXTORCGY6","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UVXVSAKH","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:dca8a95aab8c9e96e7b43cc3f4f5cbe7de37e6b18f62620991599d4ef0ad964a","target":"graph","created_at":"2026-05-18T01:05:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Given a real-valued positive semidefinite matrix, Williamson proved that it can be diagonalised using symplectic matrices. The corresponding diagonal values are known as the symplectic spectrum. This paper is concerned with the stability of Williamson's decomposition under perturbations. We provide norm bounds for the stability of the symplectic eigenvalues and prove that if $S$ diagonalises a given matrix $M$ to Williamson form, then $S$ is stable if the symplectic spectrum is nondegenerate and $S^TS$ is always stable. Finally, we sketch a few applications of the results in quantum informatio","authors_text":"Martin Idel, Michael M. Wolf, Sebatian Soto Gaona","cross_cats":["math-ph","math.MP","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-09-05T21:55:39Z","title":"Perturbation Bounds for Williamson's Symplectic Normal Form"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.01338","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d07504653a2c53269d25987a51ac9725b29cfaa003234851ca1d3ce01a0e0f6f","target":"record","created_at":"2026-05-18T01:05:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"28ac988a1f974c0cd100f489915f0bfcecc8d03566c13bbfcaca38d47bdcda8f","cross_cats_sorted":["math-ph","math.MP","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2016-09-05T21:55:39Z","title_canon_sha256":"4fdb370bef8cf0cf3f8f39eee5796a110307f1735e07589f4f52e086edca7e92"},"schema_version":"1.0","source":{"id":"1609.01338","kind":"arxiv","version":1}},"canonical_sha256":"a56f590147bcdd111b1ef8659afb11273b69279d0323354c55412a6a29ef4ec3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a56f590147bcdd111b1ef8659afb11273b69279d0323354c55412a6a29ef4ec3","first_computed_at":"2026-05-18T01:05:41.338056Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:41.338056Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FlXFJDHQccoCmqdW8pQAcm4LVFG8lfmLS47pNZyhBEz1eXlounvbec53mdFIaZ5lesloHVBRujbsW6XL4p1dBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:41.338632Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.01338","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d07504653a2c53269d25987a51ac9725b29cfaa003234851ca1d3ce01a0e0f6f","sha256:dca8a95aab8c9e96e7b43cc3f4f5cbe7de37e6b18f62620991599d4ef0ad964a"],"state_sha256":"fe884601250d306f46249182bea543a93c86b25a8d9178ab6f82612b0eaffc80"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IiClq2def5w0x/y59dOLj6C7v3MX6RvXjs+bpkAjy0jbRWr2JRxrR2XpOMF1i+EM0Kp7IzReyZQMked40CiIBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T12:59:22.350460Z","bundle_sha256":"9c9c68a6800652b6b581bc1ec0b9f3319ad635fe08f57f35ecaa6a0fe7baae23"}}