Finite Dimensional Hilbert Spaces are Complete for Dagger Compact Closed Categories (Extended Abstract)

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1 Electronic Notes in Theoretical Computer Science 270 (1) (2011) 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 selinger@mathstat.dal.ca. Research supported by NSERC /$ see ront matter 2011 Elsevier.V. ll rights reserved. doi: /j.entcs

2 114 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) 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.

3 P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 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 m 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 :

4 116. P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) N = g 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.

5 P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) Take the same signature as above, and suppose M is the ollowing diagram: M = x y 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.

6 118 P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 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

7 P. Selinger / Electronic Notes in Theoretical Computer Science 270 (1) (2011) 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 [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 [3] Joyal,., R. Street and D. Verity, Traced monoidal categories, Mathematical Proceedings o the Cambridge Philosophical Society 119 (1996), pp [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

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