[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657384-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-657384-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","exercise-sheet-4-diagrams-and-quantum-channels","Exercise Sheet 4 - Diagrams and Quantum Channels","","Exercise Sheet 4 introduces graphical calculus for tensor networks, explaining how tensors with indices correspond to diagram elements such as boxes with legs, and how line directions distinguish kets and bras. It guides readers through constructing tensor networks for inner products, expectation values, trace expressions, and mixed-state partial traces. The sheet also derives the snake equation and related transpose identities using diagrammatic contraction. A final section motivates quantum channels by relating reduced density matrices and purification to subsystem descriptions.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/exercise-sheet-4-diagrams-and-quantum-channels/657384/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/exercise-sheet-4-diagrams-and-quantum-channels/657384.png","ImageObject",300,407,{"name":42,"@type":43},"Cipher","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-27","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"How do tensor network diagrams represent kets and bras?","Question",{"text":62,"@type":63},"Line directions in the diagrams distinguish kets from bras for states, corresponding to vectors and their duals in tensor notation.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How is the matrix product AB represented using tensor network contraction?",{"text":67,"@type":63},"Connecting the appropriate legs contracts indices, yielding a diagram that corresponds to the analytic expression (AB)ij = Σk AikBkj.",{"name":69,"@type":60,"acceptedAnswer":70},"What does the tensor network in Exercise 1(c) represent?",{"text":71,"@type":63},"Contracting over the open ends produces a diagram equal to XAii = Tr[A], showing how contraction encodes trace.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},657384,1793073856,{"code":4,"msg":81,"data":82},"success",[83,87,91,94,99,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":92,"slug":93},70,"exam",{"id":95,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},5,"Comic",60,"comic",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},6,"Technology",50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":95,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":22,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145},687208528416,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Exercise Sheet 4: Diagrams and Quantum Channels  \nGraphical calculus with tensor networks    \nAs you might have noticed, already for a little number of tensor factors even simple calculations can become hard to follow quite easily. Hence, an alternative approach to visualize such calculations was developed. We will give a short introduction into the basics of this calculation technique in this exercise. However, we encourage you to have a look into [https://arxiv.org/abs/1912.10049](https://arxiv.org/abs/1912.10049) , which gives a nice and complete overview over tensor networks. For this course, however, you won’t need most of the content.  \nIn tensor network notation, a tensor is simply an object that has indices, usually a set of complex numbers Ai1 , ...,in . A tensor with one index is a vector, a tensor with two indices is a matrix. A tensor with n indices is denoted as a box with n legs.  \nWe have the following correspondences between the objects we already know and with diagrams. First, we will use the direction of the lines to distinguish kets and bras for states (which correspond to vectors):  \n~~ ~~ ψ ≃ |ψ⟩ ψ ~~ ~~ ≃ ⟨ψ|  \nLinear operations, i.e. matrices, transform vectors into vectors and hence have one incoming leg and one outgoing leg:  \n~~ ~~ ≃ I ~~ ~~ A ~~ ~~ ≃ A  \nTensor products can be expressed very easily by writing the same objects next to each other. This corresponds to the idea that tensor products represent all possible products between the entries of the objects, and as such we go from two objects with one index each to one object  \nwith two indices.  \n~~ ~~ ψ ~~ ~~ ϕ  \n≃ |ψ⟩ ⊗ |ϕ⟩  \n~~ ~~ A ~~ ~~  \n~~ ~~ B ~~ ~~  \n≃ A ⊗ B  \nOne can think of each unconnected leg carrying a (dual) Hilbert space. Connecting two legs denotes contraction of the indices, so that for example the matrix product [AB]ij = Pdk=1 AikBkjis denoted by  \n~~ ~~ A ~~  ~~ B ~~ ~~ ≃ AB.  \nAnother important primitive that we will use to reason in terms of diagrams is the fact that a bend of the wires is related to a maximally entangled state:  \nd  \n ≃ X |ii⟩  \ni=1  \n5 P. Exercise 1 .  \n1 P. (a) Draw the inner product between two states ⟨ψ|ϕ⟩ as a tensor product.  \nSolution  \n1 P. (b) Draw the expectation value ⟨ψ|A|ψ⟩ as a tensor network. Solution  \n2 P. (c) What does the following tensor network represent?  \nA  \nSolution  \nConnecting two open ends is contracting over the corresponding index, hence the tensor network represents  \nd  \nXAii = Tr[A] . (1)  \ni=1  \n1 P. (d) Draw the expectation value for a mixed state, Tr[ρA], as a tensor network.  \nSolution  \nBefore we can come to the next exercise, we have to clarify that in the context of tensor networks, we formally identify tensor products of kets and bras of computational basis states  \nas outer products:  \n|i⟩ ⊗ ⟨j| ≃ |i⟩⟨j| . (2)  \nThe following result is a very basic but important prerequisite for manipulating tensor diagrams called the snake equation which has already made it into a popular TV show you might have watched.  \n5 P. Exercise 2 .  \n2 P. (a) Prove  \n = ~~ ~~ . (3)  \nSolution  \nThe tensor network represents the following analytical expression  \nd d  \nX (I ⊗ ⟨j| ⊗ ⟨j|)(|i⟩ ⊗ |i⟩ ⊗ I) = X |i⟩ ⊗ ⟨j|i⟩ ⊗ ⟨j|  \ni,j=1 i,j=1  \nd = X |i⟩ ⊗ ⟨i|  \ni=1  \nd ≃ X |i⟩⟨i|  \ni=1  \n= I.  \n2 P. (b) Prove  \nA = ~~ ~~ AT ~~ ~~ . (4)  \nSolution  \nThe proof proceeds in a similar way as in the previous exercise. The tensor network represents  \nd d  \nX (I ⊗ ⟨j| ⊗ ⟨j|)(I ⊗ A ⊗ I)(|i⟩ ⊗ |i⟩ ⊗ I) = X |i⟩ ⊗ ⟨j|A|i⟩ ⊗ ⟨j| i,j=1 i,j=1  \nd = X Aji (|i⟩ ⊗ ⟨i|)  \ni,j=1  \nd  \n≃ X Aji|i⟩⟨j|  \ni,j=1  \nd  \n= X [AT]ij|i⟩⟨j|  \ni,j=1  \n= AT .  \n1 P. (c) We emphasize that the results of the two preceding exercises also hold if you flip the tensor networks either horizontally or vertically. Use them to show that  \nA  \n=  \nAT  \n(5)  \nSolution  \nLet us next come to an exercise that illustrates that concepts that are difficult to visualize through math are much more understandable when using tensor networks.  \n2 P. Ex","cbCaismdgdJXFDYO","https://ap.wps.com/l/cbCaismdgdJXFDYO","pdf",708051,13,"English","# Exercise 1\n## (a) Inner product\n## (b) Expectation value\n## (c) Identify the represented tensor network\n## (d) Mixed-state expectation Tr[ρA]\n# Exercise 2\n## (a) Prove the snake equation\n## (b) Prove a transpose identity\n## (c) Show A = AT by flipping diagrams\n# Exercise 3\n## (a) Partial trace over system B\n## (b) Prove Tr[ρAB (OA⊗ IB)] identity\n# Bonus Exercise 1\n## (a) Tensor network for the flip operator F\n## (b) Prove Tr[A2] = Tr[F(A ⊗ A)]\n# Quantum Channels","[{\"question\":\"How do tensor network diagrams represent kets and bras?\",\"answer\":\"Line directions in the diagrams distinguish kets from bras for states, corresponding to vectors and their duals in tensor notation.\"},{\"question\":\"How is the matrix product AB represented using tensor network contraction?\",\"answer\":\"Connecting the appropriate legs contracts indices, yielding a diagram that corresponds to the analytic expression (AB)ij = Σk AikBkj.\"},{\"question\":\"What does the tensor network in Exercise 1(c) represent?\",\"answer\":\"Contracting over the open ends produces a diagram equal to XAii = Tr[A], showing how contraction encodes trace.\"}]","Exercise Sheet 4 - Diagrams and Quantum Channels | PDF",1792978317,33]