[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-201989-105":59,"doc-detail-201989-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","quantum-causal-discovery-masters-thesis","Quantum Causal Discovery - Master’s Thesis","","Transfer entropy is a key tool in causal inference for time series, enabling detection of nonlinear directed relationships without assuming an underlying model. This work extends transfer entropy to time-parameterized quantum processes by defining quantum transfer entropy as a basis-dependent operational measure of one-way information flow through quantum channels. Nonzero values signal directed causal influence. The thesis further studies controlled-Z and Toffoli gates, including environment-conditioned cases, derives practical bounds, and applies the framework to causal discovery on a synthetic time-series dataset.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/quantum-causal-discovery-masters-thesis/201989/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/quantum-causal-discovery-masters-thesis/201989.png","ImageObject",300,407,{"name":92,"@type":93},"Maeve","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-19","2026-09-04",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does transfer entropy address in causal inference for time series?","Question",{"text":112,"@type":113},"Transfer entropy detects directed causal structures in time series as an information-theoretic measure, including nonlinear relationships, without requiring prior model assumptions.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is quantum transfer entropy defined and what does a nonzero value mean?",{"text":117,"@type":113},"Quantum transfer entropy generalizes transfer entropy to time-parameterized quantum processes as a basis-dependent one-way information-flow measure through quantum channels; a nonzero value indicates directed causal influence from source to target.",{"name":119,"@type":110,"acceptedAnswer":120},"What role do controlled-Z and coherence play in the thesis results?",{"text":121,"@type":113},"For controlled-Z channels, quantum transfer entropy becomes a linear function of the control qubit’s total information, incorporating both classical Shannon information and coherence in the chosen reference basis.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},201989,1788540711,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},5909877438554,"https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272","Politecnico di Torino  \nMaster’s Degree in Quantum Engineering A.a. 2025/2026  \nGraduation Session December 2025  \nQuantum Causal Discovery  \nSupervisor:  \nDavide Girolami  \nCandidate:  \nXuwei Ou  \nAbstract  \nTransfer entropy plays an important role in causal inference and causal discovery in time series analysis. As an information-theoretic measure, transfer entropy can detect nonlinear causal structures without any prior model assumptions. The transfer-entropy-based causal analysis scenario thus has a broader range of applications than the model-based methods like Granger causality tests. This study generalizes the definition of transfer entropy to time-parameterized quantum processes. We define a basis-dependent operational measure of one-way information flow through quantum channels, called quantum transfer entropy. Quantum transfer entropy provides a new characteristic of quantum channels. A nonzero quantum transfer entropy indicates the directed causal influence from the source to the target. In addition, we study controlled-Z channels and find that the quantum transfer entropy is a linear function of the total information of the control qubit—including both classical Shannon information and coherence in a given reference basis—thus providing an operational interpretation of information flow through quantum control channels. We further quantify the effect of a third party by studying the extreme case of the Toffoli gate, where the two control qubits are equivalent and symmetric with respect to the target qubit. This effect is characterized by the quantum transfer entropy conditioned on the third party (or the environment) . We illustrate several bounds on the conditional quantum transfer entropy for the Toffoli gate in terms of local systems. These bounds allow us to estimate the conditional transfer entropy without any knowledge of the environment qubit. Finally, we employ the proposed quantum transfer entropy for causal discovery in a synthetic time-series dataset.  \nTable of Contents  \nList of Tables iii  \nList of Figures iv  \n1 Causation in Classical Information Systems 1  \n1. 1 Measures of Information Flow . . . . . . . . . . . . . . . . . . . . . 1  \n1.2 Causality Modeling ........................... 2  \n2 Introduction to Quantum Information Theory 5  \n2. 1 State Space and Operators . . . . . . . . . . . . . . . . . . . . . . . 5  \n2.2 Quantum State Distance . . . . . . . . . . . . . . . . . . . . . . . . 7  \n2.3 Composite Systems ........................... 9  \n2.4 Quatnum Entropy ............................ 12  \n2.5 Holevo Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16  \n2.6 Entanglement and Quantum Discord ................. 17  \n2.7 Quantum Channels Theory ....................... 21  \n3 Measure of One-way Quantum Information 23  \n3.1 Quantum Transfer Entropy ....................... 23  \n3.2 Quantum-Classical Information Splitting of Controlled Rotation Channels ................................. 25  \n3.3 Bounds for Quantum Transfer Entropy ................ 27  \n3.4 Causality Interpretation ........................ 30  \n4 Quantum Causal Discovery Algorithm 32  \n4.1 Bipartite Systems ............................ 32  \n4.2 Tripartite Systems ........................... 37  \n5 Conclusion and Outlook 51  \nBibliography 54  \nList of Tables  \n2.1 Measures of entanglement for a bipartite state ρAB .......... 17  \n3.1 Linear Regression Results of Arbitrary input states of A ...... 29  \n3.2 Linear Regression Results of pure input states of A ......... 29  \n3.3 Linear Regression Results of classical input states of A ....... 30  \n4.1 The numerical results of the causal influence C (A → B), C (B → A) for several two-qubit channels. Both of the input states of A and Bare 2 (|0⟩ + |1⟩) ............................. 36  \n4.2 Truth Table of CCNOTzx→y ...................... 37  \n4.3 Probability Distribution for C(A → B|E) ............... 43  \n4.4 Probability Distribution for C(E → B|A) ............... 43  \n4.","cbCaikbjnVDoACwy","https://ap.wps.com/l/cbCaikbjnVDoACwy","pdf",3646946,64,"English","# Causation in Classical Information Systems\n## Measures of Information Flow\n## Causality Modeling\n# Introduction to Quantum Information Theory\n## State Space and Operators\n## Quantum State Distance\n## Composite Systems\n## Quatnum Entropy\n## Holevo Bound\n## Entanglement and Quantum Discord\n## Quantum Channels Theory\n# Measure of One-way Quantum Information\n## Quantum Transfer Entropy\n## Quantum-Classical Information Splitting of Controlled Rotation Channels\n## Bounds for Quantum Transfer Entropy\n## Causality Interpretation\n# Quantum Causal Discovery Algorithm\n## Bipartite Systems\n## Tripartite Systems\n# Conclusion and Outlook\n# Bibliography","[{\"question\":\"What problem does transfer entropy address in causal inference for time series?\",\"answer\":\"Transfer entropy detects directed causal structures in time series as an information-theoretic measure, including nonlinear relationships, without requiring prior model assumptions.\"},{\"question\":\"How is quantum transfer entropy defined and what does a nonzero value mean?\",\"answer\":\"Quantum transfer entropy generalizes transfer entropy to time-parameterized quantum processes as a basis-dependent one-way information-flow measure through quantum channels; a nonzero value indicates directed causal influence from source to target.\"},{\"question\":\"What role do controlled-Z and coherence play in the thesis results?\",\"answer\":\"For controlled-Z channels, quantum transfer entropy becomes a linear function of the control qubit’s total information, incorporating both classical Shannon information and coherence in the chosen reference basis.\"}]","Quantum Causal Discovery - Master’s Thesis | PDF",161]