Infinite-dimensionality in quantum foundations
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1 nfinite-dimensionality in quantum foundations W*-algebras as presheaves over matrix algebras Robert Furber (Aalborg University) Mathys Rennela (Radboud University) Sam Staton (Oxford University) QPL 16 Friday 10 June 2016 Page 1 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
2 Outline Key steps of the proof Related work Summary Page 2 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
3 Where we are, sofar Key steps of the proof Related work Summary
4 Overview of the talk W -Alg CP, category of W*-algebras together with completely positive maps. N CP, category of natural numbers n, seen as the algebra of n n complex matrices, and completely positive maps (CP-maps) between them. Set, category of sets and functions. Page 3 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
5 Overview of the talk W -Alg CP, category of W*-algebras together with completely positive maps. N CP, category of natural numbers n, seen as the algebra of n n complex matrices, and completely positive maps (CP-maps) between them. Set, category of sets and functions. Theorem: The hom-set functor W -Alg CP ( is full and faithful., =) : W -Alg op CP! [N CP, Set] Page 3 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
6 Overview of the talk W -Alg CP, category of W*-algebras together with completely positive maps. N CP, category of natural numbers n, seen as the algebra of n n complex matrices, and completely positive maps (CP-maps) between them. Set, category of sets and functions. Theorem: The hom-set functor W -Alg CP ( is full and faithful., =) : W -Alg op CP! [N CP, Set] What is the intuition behind this theorem? Page 3 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
7 Algebra of complex matrices C*-algebra: algebra of physical observables. ntuition: Measurable quantities of a physical system. Page 4 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
8 Algebra of complex matrices C*-algebra: algebra of physical observables. ntuition: Measurable quantities of a physical system. N CP, category of natural numbers n (seen as M n )andcp-maps. M n,c*-algebraofn n complex matrices. M k (M l ) = M k l ntuition: M k represents k possibly-entangled copies of C. Page 4 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
9 Algebra of complex matrices C*-algebra: algebra of physical observables. ntuition: Measurable quantities of a physical system. N CP, category of natural numbers n (seen as M n )andcp-maps. M n,c*-algebraofn n complex matrices. M k (M l ) = M k l ntuition: M k represents k possibly-entangled copies of C. M k (A), C*-algebraofk k matrices whose entries are in the C*-algebra A. ntuition: as a vector space, M k (A) is M k A. Page 4 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
10 Quantum channels = completely positive maps Alinearmapf : A! B is completely positive if it is n-positive for every n 2 N, i.e. the following map is positive for every n 2 N: M n (f ): M n (A)! M n (B) [x i,j ] i,japplen 7! [f (x i,j )] i,japplen Page 5 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
11 Quantum channels = completely positive maps Alinearmapf : A! B is completely positive if it is n-positive for every n 2 N, i.e. the following map is positive for every n 2 N: M n (f ): M n (A)! M n (B) [x i,j ] i,japplen 7! [f (x i,j )] i,japplen ntuition: Quantum channels, i.e. communication channels which transmit quantum information. Page 5 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
12 Quantum channels = completely positive maps Alinearmapf : A! B is completely positive if it is n-positive for every n 2 N, i.e. the following map is positive for every n 2 N: M n (f ): M n (A)! M n (B) [x i,j ] i,japplen 7! [f (x i,j )] i,japplen ntuition: Quantum channels, i.e. communication channels which transmit quantum information. Complete positivity is at the core of quantum computation Page 5 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
13 W*-algebras W*-algebras: C*-algebras with nice domain-theoretic properties. Example: the poset of positive elements below the unit forms a dcpo. Page 6 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
14 W*-algebras W*-algebras: C*-algebras with nice domain-theoretic properties. Example: the poset of positive elements below the unit forms a dcpo. Examples of W*-algebras All finite dimensional C*-algebras. Every algebra of bounded operators B(H) on any Hilbert space H. Every function space L 1 (X ) for any standard measure space X. The space `1(N) of bounded sequences. Page 6 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
15 W*-algebras W*-algebras: C*-algebras with nice domain-theoretic properties. Example: the poset of positive elements below the unit forms a dcpo. Examples of W*-algebras All finite dimensional C*-algebras. Every algebra of bounded operators B(H) on any Hilbert space H. Every function space L 1 (X ) for any standard measure space X. The space `1(N) of bounded sequences. Categories of W*-algebras W -Alg CP : category of W*-algebras and (normal) CP-maps W -Alg P : category of W*-algebras and (normal) positive maps Page 6 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
16 nfinite-dimensional structures: why should we care? Page 7 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
17 nfinite-dimensional structures: why should we care? Argument 1: Benefit from the full power of the theory of operator algebras. Page 7 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
18 nfinite-dimensional structures: why should we care? Argument 1: Benefit from the full power of the theory of operator algebras. Argument 2: nfinite dimensionality arise naturally in quantum field theory [Zeidler, 2008]. Page 7 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
19 nfinite-dimensional structures: why should we care? Argument 1: Benefit from the full power of the theory of operator algebras. Argument 2: nfinite dimensionality arise naturally in quantum field theory [Zeidler, 2008]. Argument 3: The register space in a scalable quantum computer arguably has an infinite dimensional aspect [Ralph et al., 2003]. Page 7 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
20 nfinite-dimensional structures: why should we care? Argument 1: Benefit from the full power of the theory of operator algebras. Argument 2: nfinite dimensionality arise naturally in quantum field theory [Zeidler, 2008]. Argument 3: The register space in a scalable quantum computer arguably has an infinite dimensional aspect [Ralph et al., 2003]. Argument 4: nfinite dimensionality comes into play in Quantum PL (e.g. [Gielerak and Sawerwain, 2010, Rennela and Staton, 2015]). Page 7 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
21 W -Alg CP, category of W*-algebras and CP-maps. N CP, category of natural numbers n (seen as M n )andcp-maps. Set, category of sets and functions. Theorem: The hom-set functor H = W -Alg CP (, =) : W -Alg op CP! [N CP, Set] is full and faithful, i.e. for A, B W*-algebras. W -Alg op CP (A, B) = [N CP, Set](H(A), H(B)) Page 8 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
22 Rewording The hom-set functor W -Alg CP (, =) : W -Alg op CP! [N CP, Set] is full and faithful. Page 9 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
23 Rewording The hom-set functor W -Alg CP (, =) : W -Alg op CP! [N CP, Set] is full and faithful. Matrix algebras are dense in W*-algebras. Page 9 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
24 Rewording The hom-set functor W -Alg CP (, =) : W -Alg op CP! [N CP, Set] is full and faithful. Matrix algebras are dense in W*-algebras. More precisely: All W*-algebras are canonical colimits of diagrams of matrix algebras and completely positive maps. Page 9 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
25 Rewording The hom-set functor W -Alg CP (, =) : W -Alg op CP! [N CP, Set] is full and faithful. Matrix algebras are dense in W*-algebras. More precisely: All W*-algebras are canonical colimits of diagrams of matrix algebras and completely positive maps. ntuition: n operator-theoretic categorical quantum foundations, finite-dimensional quantum structures can approximate their infinite-dimensional counterparts. Page 9 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations
26 Where we are, sofar Key steps of the proof Related work Summary
27 Key steps Page 10 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
28 Key steps (1) W -Alg CP! [N CP, W -Alg P ] is a full and faithful functor. Page 10 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
29 Key steps (1) W -Alg CP! [N CP, W -Alg P ] is a full and faithful functor. (2) W -Alg P (, C) :W -Alg op P! Cone is a full and faithful functor. Page 10 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
30 Key steps (1) W -Alg CP! [N CP, W -Alg P ] is a full and faithful functor. (2) W -Alg P (, C) :W -Alg op P (3) Natural maps in! Cone is a full and faithful functor. [N CP, Set] = [FdC -Alg CP, Set] are cone homomorphisms since FdCC -Alg CP, category of commutative C*-algebras, is equivalent to the Lawvere theory of abstract cones. Page 10 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
31 Key steps (1) W -Alg CP! [N CP, W -Alg P ] is a full and faithful functor. (2) W -Alg P (, C) :W -Alg op P (3) Natural maps in! Cone is a full and faithful functor. [N CP, Set] = [FdC -Alg CP, Set] are cone homomorphisms since FdCC -Alg CP, category of commutative C*-algebras, is equivalent to the Lawvere theory of abstract cones. Result: W -Alg CP (, =) : W -Alg op CP! [N CP, Set] is full and faithful. Page 10 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
32 Variation on [Rennela and Staton, 2015] Full and faithful functor M : W -Alg CP! [N CP, W -Alg P ] (C*-algebra) 7! (indexed family of C*-algebras) M(A) ={M n (A)} n (CP-map) 7! (natural family of positive maps) M(A f! B) ={M n (A) Mn(f )! M n (B)} n (Faithfulness is obvious. Fullness is more involved and requires the use of stabilizer states.) Page 11 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
33 W*-algebras as cones The homset W -Alg P (A, C) of positive linear functionals of a W*-algebra A is a cone, i.e. a module for the semiring of positive reals. The normal positive linear functional functor W -Alg P (, C) :W -Alg op P! Cone is full and faithful, where Cone is the category of cones and structure preserving functions between them. (Fullness essentially comes from the fact that the closedness and completeness of the positive cone imply that every positive linear map A! C is bounded (see e.g. [Namioka, 1957] or[schaefer, 1966, Th. V.5.5(ii)]).) Page 12 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
34 Lawvere and the cones FdC -Alg CP, category of all finite dimensional C*-algebras and completely positive maps. [N CP, Set] ' [FdC -Alg CP, Set] Page 13 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
35 Lawvere and the cones FdC -Alg CP, category of all finite dimensional C*-algebras and completely positive maps. [N CP, Set] ' [FdC -Alg CP, Set] FdCC -Alg CP, category of commutative finite-dimensional C*-algebras and completely positive maps. FdCC -Alg CP is equivalent to the Lawvere theory for abstract cones. [Furber and Jacobs, 2013, Prop. 4.3] Page 13 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
36 Lawvere and the cones FdC -Alg CP, category of all finite dimensional C*-algebras and completely positive maps. [N CP, Set] ' [FdC -Alg CP, Set] FdCC -Alg CP, category of commutative finite-dimensional C*-algebras and completely positive maps. FdCC -Alg CP is equivalent to the Lawvere theory for abstract cones. [Furber and Jacobs, 2013, Prop. 4.3] Consequence: Cone, category of cones, is a full subcategory of the category [FdCC -Alg CP, Set]. Page 13 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
37 Future work Study precisely the construction of W*-algebras as colimits. Page 14 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
38 Future work Study precisely the construction of W*-algebras as colimits. Characterize which presheaves on N CP correspond to W*-algebras. Page 14 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
39 Future work Study precisely the construction of W*-algebras as colimits. Characterize which presheaves on N CP correspond to W*-algebras. Link with Tobias Fritz perspective on infinite-dimensionality? [Fritz, 2013] Page 14 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
40 Future work Study precisely the construction of W*-algebras as colimits. Characterize which presheaves on N CP correspond to W*-algebras. Link with Tobias Fritz perspective on infinite-dimensionality? [Fritz, 2013] Tensor product as the unique extension of the standard tensor product of matrix algebras that preserves colimits of CP-maps in each argument. [Day, 1970] Page 14 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Key steps of the proof
41 Where we are, sofar Key steps of the proof Related work Summary
42 Colimits in operator theory AF C*-algebras are limits of directed diagrams of finite-dimensional C*-algebras and *-homomorphisms [Bratteli, 1972]. n a dual direction, C*-algebras and *-homomorphisms form a locally presentable category [Pelletier and Rosický, 1993]. Page 15 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
43 (Quantum) PL Defining computational constructs on dense subcategories is increasingly common (e.g. [Melliès, 2010]). Page 16 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
44 (Quantum) PL Defining computational constructs on dense subcategories is increasingly common (e.g. [Melliès, 2010]). [Malherbe et al., 2013] [Q op, Set] with Q related to N CP : Q is the category of finite sequences of finite-dimensional Hilbert spaces and trace non-increasing completely positive maps. Our result is a link between their proposal and operator theory. Page 16 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
45 (Quantum) PL Defining computational constructs on dense subcategories is increasingly common (e.g. [Melliès, 2010]). [Malherbe et al., 2013] [Q op, Set] with Q related to N CP : Q is the category of finite sequences of finite-dimensional Hilbert spaces and trace non-increasing completely positive maps. Our result is a link between their proposal and operator theory. [Pagani et al., 2014] N CP (biproduct completion of N CP ). Can we think about objects of this category as W*-algebras? Page 16 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
46 Contextuality Density theorems also appear in contextuality. Page 17 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
47 Contextuality Density theorems also appear in contextuality. Boolean algebras are dense in effect algebras. [Staton and Uijlen, 2015] Compact Hausdorff spaces are dense in piecewise C*-algebras. [Flori and Fritz, 2016, Thm. 4.5] ntuition: base category as a classical perspective. Page 17 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
48 Contextuality Density theorems also appear in contextuality. Boolean algebras are dense in effect algebras. [Staton and Uijlen, 2015] Compact Hausdorff spaces are dense in piecewise C*-algebras. [Flori and Fritz, 2016, Thm. 4.5] ntuition: base category as a classical perspective. How to study tensor products of C*/W*-algebras in this way? Page 17 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
49 Topological vector spaces TVect C, category of topological vector spaces over C and continuous C-linear maps. Consider a subcategory V of TVect C such that (1) C 2 V and A 2 V =) M n (A) 2 V (closed under matrices) (2) M : V C N Mat! TVect C factors through V V C the closure of the category V under matrices of morphisms. Theorem: the functor M : V C! [N Mat, V] is full and faithful. Page 18 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
50 Topological vector spaces TVect C, category of topological vector spaces over C and continuous C-linear maps. Consider a subcategory V of TVect C such that (1) C 2 V and A 2 V =) M n (A) 2 V (closed under matrices) (2) M : V C N Mat! TVect C factors through V V C the closure of the category V under matrices of morphisms. Theorem: the functor M : V C! [N Mat, V] is full and faithful. M : OpSpace! [N Mat, Banach] and M : OpSystem! [N Mat, OUS] full and faithful functors. Page 18 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Related work
51 Where we are, sofar Key steps of the proof Related work Summary
52 Summary The hom-set functor W -Alg CP (, =) : W -Alg op CP! [N CP, Set] is full and faithful. Matrix algebras are dense in W*-algebras. ntuition: n operator-theoretic categorical quantum foundations, finite-dimensional quantum structures can approximate their infinite-dimensional counterparts. Concrete applications? Quantum PL, contextuality,... We re looking into it! Page 19 of 19 Rennela 6-10/6/16 nfinite-dimensionality in quantum foundations Summary
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