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We show that for the $k$-fold product $G^k$ of the game $G$ (which represents the game $G$ played in parallel $k$ times independently),\n  $ \\omega^*(G^k) =\\left(1-(1-\\omega^*(G))^3\\right)^{\\Omega\\left(\\frac{k}{\\log(|\\mathcal{A}| \\cdot |\\mathcal{B}|)}\\right)} $, where $\\mathcal{A}$ and $\\mathcal{B}$ represent the sets from which the an"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1311.6309","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2013-11-25T14:07:52Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"e5028931ad00f99aa6399bf56ad537cfe0941fb73d0a7fd9005fbe422a78d2eb","abstract_canon_sha256":"08b9496c09b27ffcc6cc70a13ba4acb05296edae69e6a346c0c9bf09f71c980d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:47.092377Z","signature_b64":"yIFDJJUYrH7dyK8XQgssftYDDoflIt30n0weQYLW1AUsEsXFUyEbaQKMiR1Y514oWMrX+2jDSMTjvNpwI4ByCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a350226b920fb32eb9705001f60767d688197a2476846911a38ebbb6256091db","last_reissued_at":"2026-05-18T02:49:47.091823Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:47.091823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A parallel repetition theorem for entangled two-player one-round games under product distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"quant-ph","authors_text":"Attila Pereszl\\'enyi, Penghui Yao, Rahul Jain","submitted_at":"2013-11-25T14:07:52Z","abstract_excerpt":"We show a parallel repetition theorem for the entangled value $\\omega^*(G)$ of any two-player one-round game $G$ where the questions $(x,y) \\in \\mathcal{X}\\times\\mathcal{Y}$ to Alice and Bob are drawn from a product distribution on $\\mathcal{X}\\times\\mathcal{Y}$. We show that for the $k$-fold product $G^k$ of the game $G$ (which represents the game $G$ played in parallel $k$ times independently),\n  $ \\omega^*(G^k) =\\left(1-(1-\\omega^*(G))^3\\right)^{\\Omega\\left(\\frac{k}{\\log(|\\mathcal{A}| \\cdot |\\mathcal{B}|)}\\right)} $, where $\\mathcal{A}$ and $\\mathcal{B}$ represent the sets from which the an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6309","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1311.6309","created_at":"2026-05-18T02:49:47.091885+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.6309v2","created_at":"2026-05-18T02:49:47.091885+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6309","created_at":"2026-05-18T02:49:47.091885+00:00"},{"alias_kind":"pith_short_12","alias_value":"UNICE24SB6ZS","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"UNICE24SB6ZS5OLQ","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"UNICE24S","created_at":"2026-05-18T12:28:02.375192+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22","json":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22.json","graph_json":"https://pith.science/api/pith-number/UNICE24SB6ZS5OLQKAA7MB3H22/graph.json","events_json":"https://pith.science/api/pith-number/UNICE24SB6ZS5OLQKAA7MB3H22/events.json","paper":"https://pith.science/paper/UNICE24S"},"agent_actions":{"view_html":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22","download_json":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22.json","view_paper":"https://pith.science/paper/UNICE24S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.6309&json=true","fetch_graph":"https://pith.science/api/pith-number/UNICE24SB6ZS5OLQKAA7MB3H22/graph.json","fetch_events":"https://pith.science/api/pith-number/UNICE24SB6ZS5OLQKAA7MB3H22/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22/action/storage_attestation","attest_author":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22/action/author_attestation","sign_citation":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22/action/citation_signature","submit_replication":"https://pith.science/pith/UNICE24SB6ZS5OLQKAA7MB3H22/action/replication_record"}},"created_at":"2026-05-18T02:49:47.091885+00:00","updated_at":"2026-05-18T02:49:47.091885+00:00"}