{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"9018a517d1aa1cef83ad623480042e3b3ba1f4b4ecbfa2d6946620d539156523","created":"2026-10-03T07:18:10Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011","ae4ecb85ec476ab2c7477e3c8cb53ccc9cdde05656075fc5faeac679dd015d50"],"salt":"6a1ad6c1d938882b52ef10bd694a95120bcbea79fba85e729b8a60b65edcc336","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"qi.quantum-thermo.clock-precision-entropy","kind":"well-posed","literature_status":"open","n":"1","parents":[{"note":"","parent_revision":"ae4ecb85ec476ab2c7477e3c8cb53ccc9cdde05656075fc5faeac679dd015d50","relation":"special_case"}],"plain":"A clock must waste energy to tick, and classical clocks become steadier only in proportion to the waste. A 2025 theory paper showed quantum clocks can become steadier exponentially faster, so the ultimate limit is open.","posed_since":"2017","precise":"For an autonomous clock driven by thermal baths, let $N = \\mu^2 / \\sigma^2$ be the precision of the waiting time between ticks and $\\Sigma$ the entropy produced per tick ($k_B = 1$); classical Markov clocks obey $N \\le \\Sigma / 2$. Meier et al. (Nature Physics 2025, arXiv:2407.07948) found a coherent many-body clock with $N$ growing exponentially in $\\Sigma$. Determine the optimal $N(\\Sigma, d)$ for clocks of Hilbert-space dimension $d$, and demonstrate $N > \\Sigma / 2$ experimentally.","problem_ref":null,"references":"","settled_by":"A proved upper bound on $N$ matched by a construction, plus an implementation exceeding $N = \\Sigma / 2$.","status_note":"Exponential scaling of $N$ with $\\Sigma$ was shown theoretically (Meier et al., Nature Physics 2025); a September 2026 superconducting experiment reached a current TUR ratio $1.71 \\pm 0.17$ below the classical value 2 (arXiv:2609.37149), but no clock with $N > \\Sigma/2$ has been reported; the optimal scaling is unknown.","title":"Maximal precision of a quantum clock per unit entropy","topic_ref":"742b38754a9f76dd32b7efe7d343f52ad880f5cd0826c1581b9a9198dd522611"},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"88f21f85a8e828b5563bc09df37a7b2ab3b835f1b3664f435695e0ee99844a17","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"5a2c927b08fabb8710b8418a9cc84266742c27538703516e017dc0c42412a587","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"t3xSIVUUZOu4TZMFrW7AaHCF-LbroIIvZlGfTJTwL-opQebDV8ijbLIQUr9Yqn7-Qm9OQg5ZuTkvse-tIcVQAQ"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIWiySewj6u4cQuEGKnMhCZnQsJ1OHA1FuAX3AxCQSpYfxIFoovlZsa0Mcl4cV6rzEQpbj1ymvAY8WbMZYB4VRuvOpCPEgTjQ11nDS3tx2eKi0_v-xa8mtHRQK-FmYC3_69lOh8ywI8SB6YLyJqq0RxJAt0oDYPWvH8SagJooL98Vaer4dzJvlSgjwILRJ64hGRXZZj_gRDwdAUmZte3GNBJ6j02GOahHSkawzCPAg9qu-Ww59HRqOQ27JV6hG-A3_slwIzTxN35UxRJTrPDcI8CDFkncYPaRj7AJppFaxX963BJFn6vfBnW-IA5zVcjJfewjxIPKV6XkIVxB6zTNwShcXvtclGhIqSpEaTf5Jioh6nDXPCPEgeXNVzcuD027sru55Y1PcT2z1CuWU2W4A3vqh452xOX0I8SCuet_E6kln8_ZRXavjld8BdUsVBp253AP3BX1K8IYKnwjwIK9SXvHBwgJE5CknXtp2lOqKcbHlJ-ncP9sNjnTHUJsmCPEgytIBB3x7TFDT-XqOV-9vpqw1NKoAu0dhtsCbAMql2L0I8CCtNZgmApuKULvPC9nsMCaM05arV-F3o5FTlnJCnRvdMgjwIBGYo5Vxy-d7U5zKv5vp3Ii7hJPH_BDLCzA13c47p9q7CP_wCHbghVgOFMuNCPAQPZK2lHbTMqcCYUotzOQcmQjwIJ7ruRIs6HnufKxCpmLUgeUl0aUu0ov4_3PqcppS-bddCPEgI530vHzdwz_3o9jYQ3WablSaV64Wfn7Pmby5WLyI-A0I8QRqwKw78AhITF7Tr3a1xACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AIfrBm5kAMH2cI8BAGKsbR52Yj2ZyRBj8fybyrCPEg15Kg4Ne2LGSKxM-WnEce5gVwiluZijYEWJ8P_lNnKQ4I8QRqwKw68AhHyujKJ6ONvACD3-MNLvkMji4taHR0cHM6Ly9hbGljZS5idGMuY2FsZW5kYXIub3BlbnRpbWVzdGFtcHMub3Jn"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEINTBAv2JJHYRShMRT4EeSj981Z2Upv7GrykKbfbDmfSvxIDUOFivyp7XO_ER1aQdB3RU4ezkwy7f2Mn_yV4XexhudCPEgpal9tDD_WFbnNX9F_LJBs5SPNkqJOXRdKhA2n2x1hKgI8SDWqWDpI3qwohOAIAvdQ53UScB0RzNSHhnvcNlKgdgP0AjwIKtMCm_Z2j3Hhrvec0O4aHzK7I5CSE2876y3QpJ-G9fuCPAg_bxz7TQ4Z0BaXZYjO8P7vQGg1Jj16lEx_81hkFQ_t5II8SDGYbdU2bTggfYnKkC8FLxQMlGRx9wcd7_DXmiddQsovQjxIJ_MNhWFPQ50jQqKiaasEy7dcYxiRqLqNafQ8_f7GFsLCPAgmMxSMgUf5aZiBNROkHFoikSpVjYtpC-itaBXF2xfRSkI8SC5_X58m0IpVwVSlvSNRol5kg0yrDlZeR8GcymoGQHlowjxINIT2A3d_DeKxwK521mnlCvK2wl2UdXYsWFx2YY9xgr-CPAg3v2QXfNkDO6Xzu9061umH0rhjRfHBKza3uG2nPQbpd8I8CCtNZgmApuKULvPC9nsMCaM05arV-F3o5FTlnJCnRvdMgjwIBGYo5Vxy-d7U5zKv5vp3Ii7hJPH_BDLCzA13c47p9q7CP_wCHbghVgOFMuNCPAQPZK2lHbTMqcCYUotzOQcmQjwIJ7ruRIs6HnufKxCpmLUgeUl0aUu0ov4_3PqcppS-bddCPEgI530vHzdwz_3o9jYQ3WablSaV64Wfn7Pmby5WLyI-A0I8QRqwKw78AhITF7Tr3a1xACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AIfrBm5kAMH2cI8BAGKsbR52Yj2ZyRBj8fybyrCPEg15Kg4Ne2LGSKxM-WnEce5gVwiluZijYEWJ8P_lNnKQ4I8QRqwKw68AhHyujKJ6ONvACD3-MNLvkMji4taHR0cHM6Ly9hbGljZS5idGMuY2FsZW5kYXIub3BlbnRpbWVzdGFtcHMub3Jn"]},"log":{"leaf_index":"2044","tree_size":"2123","proof":["270a2351320e599d28a610650285e92762f7efaf93cd15cc4f73108d05eb9575","732f4cb0ca8de3c57d31e93236887e9a7005ab3fabfe58d43e5a80de1c20884e","65e1713c7c6f1c8cdbab451a291c912cce7ada3520776ce11ee0f222e7513192","a4951ed4af3a33aae4098955cbbfa8e5c2fc3aa095cadcad23bcc4f05791519b","82b8faebbcdb67dcbc5128f325eaf4173682bb7ead485fcf3cc918b3091b6c02","084ae907d05713743322b23cb7b8fe3c224edce0deaedbfa06d556176d2850cb","fe2aec30c6f8f4d047d5b2e7db330580e8c0f364a6f4dc2a44c2a5ad4e208e48","d7d27dd04c95f43a1731a5e70153ff3ca2355d42fb83606a0239169940e818ff","89c88540ca484f1a3c3976a552c40aeb6520d68a5a5ecc248a08fd399e828af4","35f1d0a5e69da98f9992a5251e4e3df1933c2f26543c7a40b2681ecfa194443a","2b97406e15e4d451a5450b3b53b677749fe0fa759579146dbfa1544c1e61a0ed","addcf2e0ff7974e4aa99af5e30d409f90d9f4311a4d85f5baf3e125eeb45b4c1"],"checkpoint":"pubphys.com/log/v1\n2123\npXVcluz7BF4vgChQs6lShuU0386gCH0RxrC1sxyE3rI=\n\n— pubphys.com/log/v1 JFPwIOLzsUD22XoTVyTfk/FMVj/u3f3DxZk4V/Ws10KVSS+8WQSwhunQ//EAhyS9Td5CyM0E++85GMezmKd8240j9Q8=\n","promise":null},"orcid_key_evidence":[],"attesting":null}