Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 www.elsevier.com/locate/entcs Finite Dimensional Hilbert Spaces are Complete or Dagger Compact Closed Categories (Extended bstract) Peter Selinger 1 Department o Mathematics and Statistics Dalhousie University, Haliax, Nova Scotia, Canada bstract We show that an equation ollows rom the axioms o dagger compact closed categories i and only i it holds in inite dimensional Hilbert spaces. Keywords: Dagger compact closed categories, Hilbert spaces, completeness. 1 Introduction Hasegawa, Homann, and Plotkin recently showed that the category o inite dimensional vector spaces over any ixed ield k o characteristic 0 is complete or traced symmetric monoidal categories [2]. What this means is that an equation holds in all traced symmetric monoidal categories i and only i it holds in inite dimensional vector spaces. Via Joyal, Street, and Verity s Int -construction [3], it is a direct corollary that inite dimensional vector spaces are also complete or compact closed categories. The present paper makes two contributions: (1) we simpliy the proo o Hasegawa, Homann, and Plotkin s result, and (2) we extend it to show that inite dimensional Hilbert spaces are complete or dagger traced symmetric monoidal categories (and hence or dagger compact closed categories). 1 Email: selinger@mathstat.dal.ca. Research supported by NSERC. 1571-0661/$ see ront matter 2011 Elsevier.V. ll rights reserved. doi:10.1016/j.entcs.2011.01.010
114 2 Statement o the main result For a deinition o dagger compact closed categories, their term language, and their graphical language, see [1,4]. We also use the concept o a dagger traced monoidal category, which is a dagger symmetric monoidal category [4] with a trace operation [3] satisying Tr X U,V () =Tr X V,U ( ). We note that every dagger compact closed category is also dagger traced monoidal; conversely, by Joyal, Street, and Verity s Int construction, every dagger traced monoidal category can be ully embedded in a dagger compact closed category. We will make use o the soundness and completeness o the graphical representation, speciically o the ollowing result: Theorem 2.1 ([4]) well-typed equation between morphisms in the language o dagger compact closed categories ollows rom the axioms o dagger compact closed categories i and only i it holds, up to graph isomorphism, in the graphical language. n analogous result also holds or dagger traced monoidal categories. The goal o this paper is to prove the ollowing: Theorem 2.2 Let M,N : be two terms in the language o dagger compact closed categories. Suppose that [M ] = [N ] or every possible interpretation (o object variables as spaces and morphism variables as linear maps) in inite dimensional Hilbert spaces. Then M = N holds in the graphical language (and thereore, holds in all dagger compact closed categories). 3 Reductions P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 eore attempting to prove Theorem 2.2, we reduce the statement to something simpler. y arguments analogous to those o Hasegawa, Homann, and Plotkin [2], it suices without loss o generality to consider terms M,N that satisy some additional conditions. The additional conditions are: We may assume that M,N : I I, i.e., that both the domain and codomain o M and N are the tensor unit. Such terms are called closed. The restriction to closed terms is without loss o generality, because given general M,N :, we can extend the language with two new morphism variables : I and g : I, and apply the theorem to the terms M = g M and N = g N. Since g, are new symbols, g M = g N in the graphical language implies that M = N in the graphical language. It suices to consider terms M,N in the language o dagger traced monoidal categories. Namely, by Joyal, Street, and Verity s Int -construction [3], every statement about dagger compact closed categories can be translated to an equivalent statement about dagger traced monoidal categories. This is done by eliminating occurrences o the -operation: one replaces every morphism variable such as : C D E by an equivalent new morphism variable such as : D C E that does not use the -operation.
P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 115 It suices to consider terms whose graphical representation does not contain any trivial cycles. Trivial cycles are connected components o a diagram that do not contain any morphism variables, such as the ones obtained rom the trace o an identity morphism. The restriction is without loss o generality because i M, N have dierent numbers or types o trivial cycles, they can be easily separated in Hilbert spaces [2]. We say that a diagram is simple i it contains no trivial cycles. 4 Inormal outline o the result The ormal statement and proo o Theorem 2.2 requires a air amount o notation, and will be given elsewhere. Nevertheless, the main idea is simple, and we inormally illustrate it here. 4.1 Signatures, diagrams, and interpretations We assume given a set o object variables, denoted, etc., and a set o morphism variables, denoted,g etc. sort is a inite sequence o object variables. We usually write 1... n or an n-element sequence, and I or the empty sequence. We assume that each morphism variable is assigned two ixed sorts, called its domain and codomain respectively, and we write :. We urther require a ixpoint-ree involution ( ) on the set o morphism variables, such that : when :. The collection o object variables and morphism variables, together with the domain and codomain inormation and the dagger operation is called a signature Σ o dagger monoidal categories. Graphically, we represent a morphism variable : 1... n 1... m as a box n... 2 1 m... 2 1. The wires on the let are called the inputs o, and the wires on the right are called its outputs. Note that each box is labeled by a morphism variable, and each wire is labeled by an object variable. (closed simple dagger symmetric traced monoidal) diagram over a signature Σ consists o zero or more boxes o the above type, all o whose wires have been connected in pairs, such that each connection is between the output wire o some box and the input wire o some (possibly the same, possible another) box. Here is an example o a diagram N over the signature given by :, g :
116. P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 N = g 1 3 4 5 + 2 In the illustration, we have numbered the wires 1 to 5 to aid the exposition below; note that this numbering is not ormally part o the diagram. n interpretation o a signature in inite-dimensional Hilbert spaces consists o the ollowing data: or each object variable, a chosen inite-dimensional Hilbert space [], and or each morphism variable : 1... n 1... m,a chosen linear map [ ] : [ 1 ]... [ n ] [ 1 ]... [ m ], such that [ ] = [ ]. The denotation o a diagram M under a given interpretation is a scalar that is deined by the usual summation over internal indices ormula. For example, the denotation o the above diagram N is: [N ] = [g ] a1,b2 b 3,a 1 [ ] b 3 a5,a 4 [ ] a 5,a 4 b 2. (4.1) a 1,b 2,b 3,a 4,a 5 Here a 1,a 4,a 5 range over some orthonormal basis o [], b 2,b 3 range over some orthonormal basis o [ ], and [ ] b 3 a5,a 4 stands or the matrix entry a 5 a 4 [ ](b 3 ). s is well-known, this denotation is independent o the choice o orthonormal bases. 4.2 Proo sketch y the reductions in Section 3, Theorem2.2 is a consequence o the ollowing lemma: Lemma 4.1 (Relative completeness) Let M be a (closed simple dagger traced monoidal) diagram. Then there exists an interpretation [ ] M in inite dimensional Hilbert spaces, depending only on M, such that or all N, [N ] M =[M ] M holds i and only i N and M are isomorphic diagrams. Clearly, the right-to-let implication is trivial, or i N and M are isomorphic diagrams, then [N ] = [M ] holds under every interpretation; their corresponding summation ormulas dier at most by a reordering o summands and actors. It is thereore the let-to-right implication that must be proved. The general proo o this lemma requires quite a bit o notation, as well as more careul deinitions than we have given above. ull proo will appear elsewhere. Here, we illustrate the proo technique by means o an example.
P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 117 Take the same signature as above, and suppose M is the ollowing diagram: M = x 1 2 + y 3 5 4 g z gain, we have numbered the wires rom 1 to 5, and this time, we have also numbered the boxes x, y, andz. We must now construct the interpretation required by the Lemma. It is given as ollows. Deine [] M to be a 3-dimensional Hilbert space with orthonormal basis { 1, 2, 4 }. Deine [ ] M to be a 2-dimensional Hilbert space with orthonormal basis { 3, 5 }. Note that the names o the basis vectors have been chosen to suggest a correspondence between basis vectors o [] M and wires labeled in the diagram M, and similarly or [ ] M. Let x, y, andz be three algebraically independent transcendental complex numbers. This means that x, y, z do not satisy any polynomial equation p(x, y, z, x, ȳ, z) = 0 with rational coeicients, unless p 0. Deine three linear maps F x :[ ] M [] M [] M, F y :[] M [] M [ ] M, and F z :[] M [ ] M [ ] M [] M as ollows. We give each map by its matrix representation in the chosen basis. (F x ) i jk = x i i = 5, j = 2,andk = 1, 0 else, (F y ) ij k = y i i = 2, j = 1,andk = 3, 0 else, (F z ) ij kl = z i i = 4, j = 3, k = 5,andl = 4, 0 else. It is hopeully obvious how each o these linear unctions is derived rom the diagram M: each matrix contains precisely one non-zero entry, whose position is determined by the numbering o the input and output wires o the corresponding box in M. The interpretations o and g are then deined as ollows: [ ] M = F x + F y, [g ] M = F z. Note that we have taken the adjoint o the matrix F y, due to the act that the corresponding box was labeled. This inishes the deinition o the interpretation [ ] M. It can be done analogously or any diagram M.
118 P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 To prove the condition o the Lemma, we irst observe that the interpretation [N ] M o any diagram N is given by a summation ormula analogous to (4.1). Moreover, rom the deinition o the interpretation [ ] M, it immediately ollows that the scalar [N ] M can be (uniquely) expressed as a polynomial p(x, y, z, x, ȳ, z) withinteger coeicients in the variables x, y, z and their complex conjugates. We also note that this polynomial is homogeneous, and its degree is equal to the number o boxes in N. We claim that the coeicient o p at xyz is non-zero i and only i N is isomorphic to M. The proo is a direct calculation, using (4.1) and the deinition o [ ] M. Essentially, any non-zero contribution to xyz in the summation ormula must come rom a choice o a basis vector ψ(w) o [] M or each wire w labeled in N, and a choice o a basis vector ψ(w) o [ ] M or each wire w labeled in N, together with a bijection φ between the boxes o N and the set {x, y, z}; moreover, the contribution can only be non-zero i the choice o basis vectors is compatible with the bijection φ. Compatibility amounts precisely to the requirement that the maps ψ and φ determine a graph isomorphism rom N to M. For example, in the calculation o [N ] M according to equation (4.1), the only non-zero contribution to xyz in p comes rom the assignment a 1 4, b 2 3, b 3 5, a 4 1,and a 5 2, which corresponds exactly to the (in this case unique) isomorphism rom N to M. In act, we get a stronger result: the integer coeicient o p at xyz is equal to the number o dierent isomorphisms between N and M (usually 0 or 1, but it could be higher i M has non-trivial automorphisms). 5 Generalizations Other ields The result o this paper (Theorem 2.2) can be adapted to other ields besides the complex numbers. It is true or any ield k o characteristic 0 with a non-trivial involutive automorphism x x. (Non-trivial means that or some x, x x). The only special property o C thatwasusedintheproo,andwhichmaynot hold in a general ield k, was the existence o transcendentals. This problem is easily solved by irst considering the ield o ractions k(x 1,...,x n ), where the required transcendentals have been added reely. The proo o Lemma 4.1 then proceeds without change. Finally, once an interpretation over k(x 1,...,x n ) has been ound such that [M ] [N ], we use the act that in a ield o characteristic 0, any non-zero polynomial has a non-root. Thus we can instantiate x 1,...,x n to speciic elements o k while preserving the inequality [M ] [N ]. Note that thereore, Theorem 2.2 holds or k; however, Lemma 4.1 only holds or k(x 1,...,x n ). ounded dimension The interpretation [ ] M rom Section 4.2 uses Hilbert spaces o unbounded dimension. One may ask whether Theorem 2.2 remains true i the dimension o the Hilbert spaces is ixed to some n. This is known to be alse when n =2. Hereis
P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 113 119 119 a counterexample due to ob Paré: the equation tr() = tr() holds or all 2 2-matrices, but does not hold in the graphical language. Indeed, by the Cayley-Hamilton theorem, 2 = μ + νi or some scalars μ, ν. Thereore tr()=μ tr()+ν tr(), tr() =μ tr()+ν tr(), and the right-hand-sides are equal by cyclicity o trace. It is not currently known to the author whether Theorem 2.2 is true when restricted to spaces o dimension 3. 6 cknowledgments Thanks to Gordon Plotkin or telling me about this problem and or discussing its solution. Thanks to ob Paré or the counterexample in dimension 2. Reerences [1] bramsky, S. and. Coecke, categorical semantics o quantum protocols, in: Proceedings o the 19th nnual IEEE Symposium on Logic in Computer Science, LICS 2004, IEEE Computer Society Press, 2004, pp. 415 425. [2] Hasegawa, M., M. Homann and G. Plotkin, Finite dimensional vector spaces are complete or traced symmetric monoidal categories, in: Pillars o Computer Science: Essays Dedicated to oris (oaz) Trakhtenbrot on the Occasion o His 85th irthday, Lecture Notes in Computer Science 4800 (2008), pp. 367 385. [3] Joyal,., R. Street and D. Verity, Traced monoidal categories, Mathematical Proceedings o the Cambridge Philosophical Society 119 (1996), pp. 447 468. [4] Selinger, P., Dagger compact closed categories and completely positive maps, in: Proceedings o the 3rd International Workshop on Quantum Programming Languages, Electronic Notes in Theoretical Computer Science 170 (2007), pp. 139 163.