Biunitary constructions in quantum information
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1 Biunitary constructions in quantum information David Reutter Jamie Vicary Department of Computer Science University of Oxford 14th International Conference on Quantum Physics and Logic Nijmegen, The Netherlands July 4, 2017 based on arxiv: David Reutter Biunitary constructions July 4, / 17
2 The plan Part 1. Quantum structures Part 2. igher algebra David Reutter Biunitary constructions July 4, / 17
3 Part 1 Quantum structures David Reutter Biunitary constructions July 4, / 17
4 Quantum structures A adamard matrix (ad) is a matrix Mat n (C) fulfilling i,j 2 = 1 = n1 David Reutter Biunitary constructions July 4, / 17
5 Quantum structures A adamard matrix (ad) is a matrix Mat n (C) fulfilling i,j 2 = 1 = n1 ( ) David Reutter Biunitary constructions July 4, / 17
6 Quantum structures A adamard matrix (ad) is a matrix Mat n (C) fulfilling i,j 2 = 1 = n1 ( ) A unitary error basis (UEB) is a collection of unitary matrices { Ui U(n) 1 i n 2} fulfilling Tr(U i U j) = nδ i,j David Reutter Biunitary constructions July 4, / 17
7 Quantum structures A adamard matrix (ad) is a matrix Mat n (C) fulfilling i,j 2 = 1 = n1 ( ) A unitary error basis (UEB) is a collection of unitary matrices { Ui U(n) 1 i n 2} fulfilling Tr(U i U j) = nδ i,j ( ), ( ), ( ) 0 -i i 0, ( ) David Reutter Biunitary constructions July 4, / 17
8 Quantum structures A adamard matrix (ad) is a matrix Mat n (C) fulfilling i,j 2 = 1 = n1 ( ) A unitary error basis (UEB) is a collection of unitary matrices { Ui U(n) 1 i n 2} fulfilling Tr(U i U j) = nδ i,j ( ), ( ), ( ) 0 -i i 0, ( ) They provide the mathematical foundation for: mutually unbiased bases, quantum key distribution, quantum teleportation, dense coding, quantum error correction David Reutter Biunitary constructions July 4, / 17
9 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard David Reutter Biunitary constructions July 4, / 17
10 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard family of adamards + family of adamards adamard David Reutter Biunitary constructions July 4, / 17
11 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard family of adamards + family of adamards adamard adamard + adamard + adamard UEB... David Reutter Biunitary constructions July 4, / 17
12 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard family of adamards + family of adamards adamard adamard + adamard + adamard UEB... (U ab ) c,d = 1 n A a,d B b,c C c,d David Reutter Biunitary constructions July 4, / 17
13 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard family of adamards + family of adamards adamard adamard + adamard + adamard UEB... These constructions are not obvious! (U ab ) c,d = 1 n A a,d B b,c C c,d David Reutter Biunitary constructions July 4, / 17
14 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard family of adamards + family of adamards adamard adamard + adamard + adamard UEB... These constructions are not obvious! Why do they work? (U ab ) c,d = 1 n A a,d B b,c C c,d David Reutter Biunitary constructions July 4, / 17
15 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard family of adamards + family of adamards adamard adamard + adamard + adamard UEB... These constructions are not obvious! Why do they work? Where do they come from? (U ab ) c,d = 1 n A a,d B b,c C c,d David Reutter Biunitary constructions July 4, / 17
16 Quantum constructions There are several known constructions of adamards and UEBs: adamard + adamard adamard family of adamards + family of adamards adamard adamard + adamard + adamard UEB... These constructions are not obvious! Why do they work? Where do they come from? ow can we find them? (U ab ) c,d = 1 n A a,d B b,c C c,d David Reutter Biunitary constructions July 4, / 17
17 Part 2 igher algebra David Reutter Biunitary constructions July 4, / 17
18 What is higher algebra? Ordinary algebra lets us compose along a line: xy 2 zyx 3 David Reutter Biunitary constructions July 4, / 17
19 What is higher algebra? Ordinary algebra lets us compose along a line: xy 2 zyx 3 igher algebra lets us compose in higher dimensions: L M N David Reutter Biunitary constructions July 4, / 17
20 Planar algebra = 2-category theory The language describing algebra in the plane is 2-category theory: f A A B A η B objects 1-morphism 2-morphism g f David Reutter Biunitary constructions July 4, / 17
21 Planar algebra = 2-category theory The language describing algebra in the plane is 2-category theory: f A A B A η B objects 1-morphism 2-morphism We can compose 2-morphisms like this: g f A ɛ η B A η B ɛ C vertical composition These are pasting diagrams. horizontal composition David Reutter Biunitary constructions July 4, / 17
22 Planar algebra = 2-category theory The language describing algebra in the plane is 2-category theory: g A A B A η B f f objects 1-morphism 2-morphism We can compose 2-morphisms like this: A ɛ η B A η B ɛ C vertical composition These are pasting diagrams. The dual diagrams are the graphical calculus. horizontal composition David Reutter Biunitary constructions July 4, / 17
23 Monoidal dagger pivotal 2-categories We use monoidal dagger pivotal 2-categories: David Reutter Biunitary constructions July 4, / 17
24 Monoidal dagger pivotal 2-categories We use monoidal dagger pivotal 2-categories: Dagger pivotal 2-categories have a very flexible graphical calculus. η η = David Reutter Biunitary constructions July 4, / 17
25 Monoidal dagger pivotal 2-categories We use monoidal dagger pivotal 2-categories: Dagger pivotal 2-categories have a very flexible graphical calculus. In a monoidal 2-category, we can layer surfaces on top of each other. η η = µ ν David Reutter Biunitary constructions July 4, / 17
26 Monoidal dagger pivotal 2-categories We use monoidal dagger pivotal 2-categories: Dagger pivotal 2-categories have a very flexible graphical calculus. In a monoidal 2-category, we can layer surfaces on top of each other. η η = µ ν Summary. The graphical calculus of monoidal dagger pivotal 2-categories is given by surfaces, wires and vertices in three-dimensional space. David Reutter Biunitary constructions July 4, / 17
27 A model for quantum computation: 2ilb The 2-category 2ilb is a model for quantum computation. David Reutter Biunitary constructions July 4, / 17
28 A model for quantum computation: 2ilb The 2-category 2ilb is a model for quantum computation. Objects are natural numbers n, m,... David Reutter Biunitary constructions July 4, / 17
29 A model for quantum computation: 2ilb The 2-category 2ilb is a model for quantum computation. Objects are natural numbers n, m,... 1-morphisms n m are matrices of ilbert spaces 11 1n..... m1 mn David Reutter Biunitary constructions July 4, / 17
30 A model for quantum computation: 2ilb The 2-category 2ilb is a model for quantum computation. Objects are natural numbers n, m,... 1-morphisms n m are matrices of ilbert spaces 2-morphisms = φ are matrices of linear maps φ n φ 1n n 1n m1 φ mn m1 m1 φ mn m1... mn mn David Reutter Biunitary constructions July 4, / 17
31 A model for quantum computation: 2ilb The 2-category 2ilb is a model for quantum computation. Objects are natural numbers n, m,... 1-morphisms n m are matrices of ilbert spaces 2-morphisms = φ are matrices of linear maps φ n φ 1n n 1n m1 φ mn m1 m1 φ mn m1... mn mn This well-studied structure plays a key role in higher representation theory. David Reutter Biunitary constructions July 4, / 17
32 A model for quantum computation: 2ilb The 2-category 2ilb is a model for quantum computation. Objects are natural numbers n, m,... 1-morphisms n m are matrices of ilbert spaces 2-morphisms = φ are matrices of linear maps φ n φ 1n n 1n m1 φ mn m1 m1 φ mn m1... mn mn This well-studied structure plays a key role in higher representation theory. Theorem. 2ilb is a monoidal dagger pivotal 2-category. David Reutter Biunitary constructions July 4, / 17
33 Biunitarity In a monoidal dagger pivotal 2-category, a 4-valent 2-morphism U is biunitary when the following holds: David Reutter Biunitary constructions July 4, / 17
34 Biunitarity In a monoidal dagger pivotal 2-category, a 4-valent 2-morphism U is biunitary when the following holds: It is (vertically) unitary: U = U U U = David Reutter Biunitary constructions July 4, / 17
35 Biunitarity In a monoidal dagger pivotal 2-category, a 4-valent 2-morphism U is biunitary when the following holds: It is (vertically) unitary: U = U U U = It is horizontally unitary: U U = λ U U = λ David Reutter Biunitary constructions July 4, / 17
36 Biunitarity In a monoidal dagger pivotal 2-category, a 4-valent 2-morphism U is biunitary when the following holds: It is (vertically) unitary: = = It is horizontally unitary: = λ = λ These look just like the second Reidemeister move. David Reutter Biunitary constructions July 4, / 17
37 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: David Reutter Biunitary constructions July 4, / 17
38 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: David Reutter Biunitary constructions July 4, / 17
39 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: UEBs are biunitaries of the following type: U David Reutter Biunitary constructions July 4, / 17
40 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: UEBs are biunitaries of the following type: U David Reutter Biunitary constructions July 4, / 17
41 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: UEBs are biunitaries of the following type: U David Reutter Biunitary constructions July 4, / 17
42 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: UEBs are biunitaries of the following type: U David Reutter Biunitary constructions July 4, / 17
43 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: UEBs are biunitaries of the following type: U Families of adamards (ad*) are biunitaries of the following type: David Reutter Biunitary constructions July 4, / 17
44 Quantum structures are biunitaries in 2ilb adamards are biunitaries of the following type: UEBs are biunitaries of the following type: U Families of adamards (ad*) are biunitaries of the following type: David Reutter Biunitary constructions July 4, / 17
45 Composing biunitaries Let U and V be biunitaries of the following types: U V David Reutter Biunitary constructions July 4, / 17
46 Composing biunitaries Let U and V be biunitaries of the following types: U V Then their diagonal composites are also biunitary: V U David Reutter Biunitary constructions July 4, / 17
47 Composing biunitaries Let U and V be biunitaries of the following types: U V Then their diagonal composites are also biunitary: U V David Reutter Biunitary constructions July 4, / 17
48 Composing biunitaries U ad UEB ad David Reutter Biunitary constructions July 4, / 17
49 Composing biunitaries U ad UEB ad B A David Reutter Biunitary constructions July 4, / 17
50 Composing biunitaries U ad UEB ad ad A B David Reutter Biunitary constructions July 4, / 17
51 Composing biunitaries U ad UEB ad ad + ad A B David Reutter Biunitary constructions July 4, / 17
52 Composing biunitaries U ad UEB ad ad + ad ad A B David Reutter Biunitary constructions July 4, / 17
53 Composing biunitaries U ad UEB ad ad + ad ad A B ab,cd = A b a,cb c b,d David Reutter Biunitary constructions July 4, / 17
54 Composing biunitaries U ad UEB ad ad + ad ad A B ab,cd = A b a,cb c b,d David Reutter Biunitary constructions July 4, / 17
55 Composing biunitaries U ad UEB ad B C A David Reutter Biunitary constructions July 4, / 17
56 Composing biunitaries U ad UEB ad ad B C A David Reutter Biunitary constructions July 4, / 17
57 Composing biunitaries U ad UEB ad ad + ad B C A David Reutter Biunitary constructions July 4, / 17
58 Composing biunitaries U ad UEB ad ad + ad + ad B A C David Reutter Biunitary constructions July 4, / 17
59 Composing biunitaries U ad UEB ad ad + ad + ad UEB B A C David Reutter Biunitary constructions July 4, / 17
60 Composing biunitaries U ad UEB ad ad + ad + ad UEB B A C (U ab ) c,d = 1 n A a,d B b,c C c,d David Reutter Biunitary constructions July 4, / 17
61 Composing biunitaries U ad UEB ad ad + ad + ad UEB B A C (U ab ) c,d = 1 n A a,d B b,c C c,d David Reutter Biunitary constructions July 4, / 17
62 Composing biunitaries U P U abc,de,fg = b,c a,eg P c,g e,b,f Q c,g,d Q 6 ad UEB ad V 11 Q W U abc,def,gh := r V b,c a,rf,g Qc b,r,d W rc,e,h 9 P Q B V D A Q K P C U abc,de,fg = r b,c a,r P c,r,d Q r,b,f V r,e,g U abcd,ef,gh = 1 n r,s A f,hb s,f C r,h D s,r d a,s K c b,r Q d,s,ep r,c,g David Reutter Biunitary constructions July 4, / 17
63 Composing biunitaries U P Q U abc,de,fg = b,c a,eg P c,g e,b,f Q c,g,d 6 ad UEB ad Q W V U abc,def,gh := r V b,c a,rf,g Qc b,r,d rc,e,h W P Q V U abc,de,fg = r b,c a,r P c,r,d Q r,b,f V r,e,g U abcd,ef,gh = 1 n r,s A f,hb s,f C r,h D s,r d a,s K c b,r Q d,s,ep r,c,g David Reutter Biunitary constructions July 4, / 17 K Q D P B C A
64 Summary In the paper, we have: David Reutter Biunitary constructions July 4, / 17
65 Summary In the paper, we have: Characterized quantum structures in terms of biunitaries in 2ilb. David Reutter Biunitary constructions July 4, / 17
66 Summary In the paper, we have: Characterized quantum structures in terms of biunitaries in 2ilb. Built new structures by composing biunitaries diagonally. David Reutter Biunitary constructions July 4, / 17
67 Summary In the paper, we have: Characterized quantum structures in terms of biunitaries in 2ilb. Built new structures by composing biunitaries diagonally. Unified many existing constructions, and found new ones. David Reutter Biunitary constructions July 4, / 17
68 Summary In the paper, we have: Characterized quantum structures in terms of biunitaries in 2ilb. Built new structures by composing biunitaries diagonally. Unified many existing constructions, and found new ones. Discovered new unitary error bases, which we prove could not have been build using existing techniques. David Reutter Biunitary constructions July 4, / 17
69 Summary In the paper, we have: Characterized quantum structures in terms of biunitaries in 2ilb. Built new structures by composing biunitaries diagonally. Unified many existing constructions, and found new ones. Discovered new unitary error bases, which we prove could not have been build using existing techniques. This shows how higher category theory can have a direct impact in an applied domain. David Reutter Biunitary constructions July 4, / 17
70 Summary In the paper, we have: Characterized quantum structures in terms of biunitaries in 2ilb. Built new structures by composing biunitaries diagonally. Unified many existing constructions, and found new ones. Discovered new unitary error bases, which we prove could not have been build using existing techniques. This shows how higher category theory can have a direct impact in an applied domain. Thanks for listening! David Reutter Biunitary constructions July 4, / 17
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