{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "lieb-ruskai-strong-subadditivity-quantum-entropy-1973", "title": "Lieb and Ruskai: strong subadditivity of quantum entropy", "summary": "For a finite-dimensional tripartite quantum state, the von Neumann entropies satisfy S(ABC)+S(B) <= S(AB)+S(BC). The proof uses operator trace inequalities and is accompanied by convexity of negative conditional entropy; the original text develops extensions through controlled finite-dimensional approximation.", "boundary": "The displayed entropy combination is recorded where its terms are well-defined; it is not interpreted by subtracting divergent infinities in arbitrary infinite-dimensional states. This theorem is not a Shannon/Wehrl minimum-entropy conjecture, entropy-power inequality, or an equality/recovery characterization from later work.", "area": "Matrix analysis and quantum information", "topic": "Mathematics", "year": 1973, "lane": "HUMAN", "result": "THEOREM", "package_id": "lieb-ruskai-strong-subadditivity-quantum-entropy-1973", "target": { "record_id": "urn:uuid:b857695e-6e9e-4746-adc9-d927534406cb", "version": 1, "content_hash": "sha256:7871c4864a38b03c4754c5de493205d3c362631be5c26c39acc8f614650246d6" }, "humans": [ "Elliott H. Lieb", "Mary Beth Ruskai" ], "additional_contributors": [ { "name": "Barry Simon", "role": "Author of the appendix on entropy convergence used for infinite-dimensional extensions", "sources": [ { "title": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf" } ] } ], "authors_complete": true, "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.", "agents": [], "related": [], "subresults": [], "scientific_assessment": { "source_review": "SOURCE_CURATED", "scientific_review": "NOT_REVIEWED_BY_WRS", "formal_replay": "NOT_RUN", "provider_attestation": "NONE" }, "sources": [ { "id": "cur20261020-lieb-ruskai-numdam1973", "title": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf", "date": "1973" } ], "points": "650000", "published_grade": { "id": "lieb-ruskai-strong-subadditivity-quantum-entropy-1973", "entry_id": "lieb-ruskai-strong-subadditivity-quantum-entropy-1973", "band": "S", "modifier": null, "target": { "record_id": "urn:uuid:b857695e-6e9e-4746-adc9-d927534406cb", "version": 1, "content_hash": "sha256:7871c4864a38b03c4754c5de493205d3c362631be5c26c39acc8f614650246d6" }, "what_changed": "For a finite-dimensional tripartite quantum state, the von Neumann entropies satisfy S(ABC)+S(B) <= S(AB)+S(BC). The proof uses operator trace inequalities and is accompanied by convexity of negative conditional entropy; the original text develops extensions through controlled finite-dimensional approximation.", "scope": "For a finite-dimensional tripartite quantum state, the von Neumann entropies satisfy S(ABC)+S(B) <= S(AB)+S(BC). The proof uses operator trace inequalities and is accompanied by convexity of negative conditional entropy; the original text develops extensions through controlled finite-dimensional approximation. The displayed entropy combination is recorded where its terms are well-defined; it is not interpreted by subtracting divergent infinities in arbitrary infinite-dimensional states. 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Lieb", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf" } }, { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Mary Beth Ruskai", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "The displayed entropy combination is recorded where its terms are well-defined; it is not interpreted by subtracting divergent infinities in arbitrary infinite-dimensional states. This theorem is not a Shannon/Wehrl minimum-entropy conjecture, entropy-power inequality, or an equality/recovery characterization from later work.", "domain": "Matrix analysis and quantum information" }, "sources": [ { "citation": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf" } ], "statement": "# Lieb and Ruskai: strong subadditivity of quantum entropy\n\n## Information shared by three systems\n\nEntropy measures uncertainty in a specified state. For a quantum state, von Neumann entropy is calculated from the state's eigenvalues, rather than by assigning a definite hidden classical value to every measurement. Lieb and Ruskai proved a fundamental constraint for three parts A, B and C: the entropy of ABC plus that of B is no greater than the entropy of AB plus that of BC. This is strong subadditivity.\n\n## A classical example you can count\n\nUse two independent fair bits A and C. Define B to be zero when they agree and one when they differ. There are four equally likely triples. In this invented example, ABC has two bits of entropy, B has one, and each pair AB and BC has two. The inequality becomes 2 + 1 less than or equal to 2 + 2. All four entropies refer to the same joint distribution.\n\n## Why conditioning can reveal a connection\n\nBefore B is known, the bits A and C are independent. If B is zero, learning A tells you C; if B is one, learning A tells you the opposite of C. The difference between the two sides of the inequality is one bit here. This example helps visualize conditional correlation. Counting four classical outcomes does not prove the quantum theorem, whose states and operators require a different argument.\n\n## What the original proof supplies\n\nThe original uses operator trace inequalities and links the entropy result with a convexity statement. It also addresses infinite-dimensional extensions through approximation, including an appendix contributed by Barry Simon. Simon's appendix contribution remains distinct from the two-author main paper. Expressions involving infinite entropies require care; an undefined subtraction of infinities is not an application of the finite-dimensional formula.\n\n## A bounded way to continue\n\nRecalculate the classroom example from its four rows, then replace B with a fixed bit and compare the entropies. Keep the distribution and units visible. A source explanation can distinguish strong subadditivity from later equality and recovery results, entropy-power inequalities and other entropy conjectures. Similar terminology does not make these the same mathematical contribution.\n\n## Scientific scope\n\nThe displayed entropy combination is recorded where its terms are well-defined; it is not interpreted by subtracting divergent infinities in arbitrary infinite-dimensional states. This theorem is not a Shannon/Wehrl minimum-entropy conjecture, entropy-power inequality, or an equality/recovery characterization from later work.\n\n## Sources for this explanation\n\n- [Proof of the Strong Subadditivity of Quantum-Mechanical Entropy](https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n", "summary": "For a finite-dimensional tripartite quantum state, the von Neumann entropies satisfy S(ABC)+S(B) <= S(AB)+S(BC). The proof uses operator trace inequalities and is accompanied by convexity of negative conditional entropy; the original text develops extensions through controlled finite-dimensional approximation.", "title": "Lieb and Ruskai: strong subadditivity of quantum entropy" }, "record_id": "urn:uuid:b857695e-6e9e-4746-adc9-d927534406cb", "version": 1, "source_ref": { "record_id": "urn:uuid:b857695e-6e9e-4746-adc9-d927534406cb", "version": 1, "content_hash": "sha256:7871c4864a38b03c4754c5de493205d3c362631be5c26c39acc8f614650246d6" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:397335e3c7dc6d84d9f189053c9123bd079bbb052fafb48db0d03c00a0444346" }, "target": { "record_id": "urn:uuid:b857695e-6e9e-4746-adc9-d927534406cb", "version": 1, "content_hash": "sha256:7871c4864a38b03c4754c5de493205d3c362631be5c26c39acc8f614650246d6" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "lieb-ruskai-strong-subadditivity-quantum-entropy-1973", "entry_id": "lieb-ruskai-strong-subadditivity-quantum-entropy-1973", "revision": 1, "title": "Lieb and Ruskai: strong subadditivity of quantum entropy", "context": "Matrix analysis and quantum information", "track": "Exact theorem and reusable mathematical method", "band": "S", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "For a finite-dimensional tripartite quantum state, the von Neumann entropies satisfy S(ABC)+S(B) <= S(AB)+S(BC). The proof uses operator trace inequalities and is accompanied by convexity of negative conditional entropy; the original text develops extensions through controlled finite-dimensional approximation.", "scope": "For a finite-dimensional tripartite quantum state, the von Neumann entropies satisfy S(ABC)+S(B) <= S(AB)+S(BC). The proof uses operator trace inequalities and is accompanied by convexity of negative conditional entropy; the original text develops extensions through controlled finite-dimensional approximation. The displayed entropy combination is recorded where its terms are well-defined; it is not interpreted by subtracting divergent infinities in arbitrary infinite-dimensional states. This theorem is not a Shannon/Wehrl minimum-entropy conjecture, entropy-power inequality, or an equality/recovery characterization from later work.", "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The displayed entropy combination is recorded where its terms are well-defined; it is not interpreted by subtracting divergent infinities in arbitrary infinite-dimensional states. This theorem is not a Shannon/Wehrl minimum-entropy conjecture, entropy-power inequality, or an equality/recovery characterization from later work.", "references": [ { "title": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." } ], "roles": [ { "who": "The named authors in the research record", "credit": "SCIENCE", "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred." }, { "who": "Barry Simon", "credit": "SCIENCE", "did": "Author of the appendix on entropy convergence used for infinite-dimensional extensions" } ], "target": { "record_id": "urn:uuid:b857695e-6e9e-4746-adc9-d927534406cb", "version": 1, "content_hash": "sha256:7871c4864a38b03c4754c5de493205d3c362631be5c26c39acc8f614650246d6" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "650000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": { "id": "lieb-ruskai-strong-subadditivity-quantum-entropy-1973", "target": { "record_id": "urn:uuid:b857695e-6e9e-4746-adc9-d927534406cb", "version": 1, "content_hash": "sha256:7871c4864a38b03c4754c5de493205d3c362631be5c26c39acc8f614650246d6" }, "sections": [ { "title": "Information shared by three systems", "text": "Entropy measures uncertainty in a specified state. For a quantum state, von Neumann entropy is calculated from the state's eigenvalues, rather than by assigning a definite hidden classical value to every measurement. Lieb and Ruskai proved a fundamental constraint for three parts A, B and C: the entropy of ABC plus that of B is no greater than the entropy of AB plus that of BC. This is strong subadditivity." }, { "title": "A classical example you can count", "text": "Use two independent fair bits A and C. Define B to be zero when they agree and one when they differ. There are four equally likely triples. In this invented example, ABC has two bits of entropy, B has one, and each pair AB and BC has two. The inequality becomes 2 + 1 less than or equal to 2 + 2. All four entropies refer to the same joint distribution." }, { "title": "Why conditioning can reveal a connection", "text": "Before B is known, the bits A and C are independent. If B is zero, learning A tells you C; if B is one, learning A tells you the opposite of C. The difference between the two sides of the inequality is one bit here. This example helps visualize conditional correlation. Counting four classical outcomes does not prove the quantum theorem, whose states and operators require a different argument." }, { "title": "What the original proof supplies", "text": "The original uses operator trace inequalities and links the entropy result with a convexity statement. It also addresses infinite-dimensional extensions through approximation, including an appendix contributed by Barry Simon. Simon's appendix contribution remains distinct from the two-author main paper. Expressions involving infinite entropies require care; an undefined subtraction of infinities is not an application of the finite-dimensional formula." }, { "title": "A bounded way to continue", "text": "Recalculate the classroom example from its four rows, then replace B with a fixed bit and compare the entropies. Keep the distribution and units visible. A source explanation can distinguish strong subadditivity from later equality and recovery results, entropy-power inequalities and other entropy conjectures. Similar terminology does not make these the same mathematical contribution." } ], "references": [ { "title": "Proof of the Strong Subadditivity of Quantum-Mechanical Entropy", "url": "https://www.numdam.org/item/RCP25_1973__19__A5_0.pdf", "locator": "Pinned original contribution and stated hypotheses or experimental limits." } ] }, "reading_status": "EXPLANATORY_GUIDE" }