1 No-Cloning In Categorical Quantum Mechanics

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3 1 No-Cloning In Categorical Quantum Mechanics Samson bramsky bstract The No-Cloning theorem is a basic limitative result or quantum mechanics, with particular signiicance or quantum inormation. It says that there is no unitary operation which makes perect copies o an unknown (pure) quantum state. stronger orm o this result is the No-Broadcasting theorem, which applies to mixed states. There is also a No-Deleting theorem. Recently, the author and Bob Coecke have introduced a categorical ormulation o Quantum Mechanics, as a basis or a more structural, high-level approach to quantum inormation and computation. This has been elaborated by ourselves, our colleagues, and other workers in the ield, and has been shown to yield an eective and illuminating treatment o a wide range o topics in quantum inormation. Diagrammatic calculi or tensor categories, suitably extended to incorporate the various additional structures which have been used to relect undamental eatures o quantum mechanics, play an important role, both as an intuitive and vivid visual presentation o the ormalism, and as an eective calculational device. It is clear that such a novel reormulation o the mathematical ormalism o quantum mechanics, a subject more or less set in stone since Von Neumann s classic treatise, has the potential to yield new insights into the oundations o quantum mechanics. In the present paper, we shall use it to open up a novel perspective on No-Cloning. What we shall ind, quite unexpectedly, is a link to some undamental issues in logic, computation, and the oundations o mathematics. striking eature o our results is that they are visibly in the same genre as a well-known result by Joyal in categorical logic showing that a Boolean cartesian closed category trivializes, which provides a major road-block to the computational interpretation 3

4 4 o classical logic. In act, they strengthen Joyal s result, insoar as the assumption o a ull categorical product (diagonals and projections) in the presence o a classical duality is weakened. This shows a heretoore unsuspected connection between limitative results in proo theory and No-Go theorems in quantum mechanics. 1.1 Introduction The No-Cloning theorem Dieks (1982); Wootters and Zurek (1982) is a basic limitative result or quantum mechanics, with particular signiicance or quantum inormation. It says that there is no unitary operation which makes perect copies o an unknown (pure) quantum state. stronger orm o this result is the No-Broadcasting theorem Barnum et al. (1996), which applies to mixed states. There is also a No-Deleting theorem Pati and Braunstein (2000). Recently, the author and Bob Coecke have introduced a categorical ormulation o Quantum Mechanics bramsky and Coecke (2004, 2005, 2008), as a basis or a more structural, high-level approach to quantum inormation and computation. This has been elaborated by ourselves, our colleagues, and other workers in the ield bramsky (2004, 2005, 2007); bramsky and Duncan (2006); Coecke and Pavlovic (2007); Coecke and Duncan (2008); Selinger (2007); Vicary (2007), and has been shown to yield an eective and illuminating treatment o a wide range o topics in quantum inormation. Diagrammatic calculi or tensor categories Joyal and Street (1991); Turaev (1994), suitably extended to incorporate the various additional structures which have been used to relect undamental eatures o quantum mechanics, play an important rôle, both as an intuitive and vivid visual presentation o the ormalism, and as an eective calculational device. It is clear that such a novel reormulation o the mathematical ormalism o quantum mechanics, a subject more or less set in stone since von Neumann s classic treatise von Neumann (1932), has the potential to yield new insights into the oundations o quantum mechanics. In the present paper, we shall use it to open up a novel perspective on No-Cloning. What we shall ind, quite unexpectedly, is a link to some undamental issues in logic, computation, and the oundations o mathematics. striking eature o our results is that they are visibly in the same genre as a well-known result by Joyal in categorical logic Lambek and Scott (1986) showing that a Boolean cartesian closed category trivializes, which provides a major road-block to the computational in-

5 No-Cloning In Categorical Quantum Mechanics 5 terpretation o classical logic. In act, they strengthen Joyal s result, insoar as the assumption o a ull categorical product (diagonals and projections) in the presence o a classical duality is weakened. This shows a heretoore unsuspected connection between limitative results in proo theory and No-Go theorems in quantum mechanics. The urther contents o the paper are as ollows: In the next section, we shall briely review the three-way link between logic, computation and categories, and recall Joyal s lemma. In section 3, we shall review the categorical approach to quantum mechanics. Our main results are in section 4, where we prove our limitative result, which shows the incompatibility o structural eatures corresponding to quantum entanglement (essentially, the existence o Bell states enabling teleportation) with the existence o a natural (in the categorical sense, corresponding essentially to basis-independent) copying operation. This result is mathematically robust, since it is proved in a very general context, and has a topological content which is clearly revealed by a diagrammatic proo. t the same time it is delicately poised, since non-natural, basis-dependent copying operations do in act play a key rôle in the categorical ormulation o quantum notions o measurement. We discuss this context, and the conceptual reading o the results. We conclude with some discussion o extensions o the results, urther directions, and open problems. 1.2 Categories, Logic and Computational Content: Joyal s Lemma Categorical logic Lambek and Scott (1986) and the Curry-Howard correspondence in Proo Theory Sørensen and Urzyczyn (2006) give us a beautiul three-way correspondence: Logic Categories Computation

6 6 More particularly, we have as a paradigmatic example: Intuitionistic Logic Cartesian Closed Categories λ-calculus Here we are ocussing on the ragment o intuitionistic logic containing conjunction and implication, and the simply-typed λ-calculus with product types. We shall assume amiliarity with basic notions o category theory MacLane (1998); Lawvere and Schanuel (1997). Recall that a cartesian closed category is a category with a terminal object, binary products and exponentials. The basic cartesian closed adjunction is C( B, C) = C(, B C). More explicitly, a category C with inite products has exponentials i or all objects and B o C there is a couniversal arrow rom to B, i.e. an object B o C and a morphism ev,b : ( B) B with the couniversal property: or every g : C B, there is a unique morphism Λ(g) : C B such that B Λ(g) ( B) ev,b B Λ(g) id g C C The correspondence between the intuitionistic logic o conjunction and implication and cartesian closed categories is summarized in the ollowing table:

7 No-Cloning In Categorical Quantum Mechanics 7 xiom Γ, Id π 2 : Γ Conjunction Γ Γ B I Γ B : Γ g : Γ B, g : Γ B Γ B Γ E1 : Γ B π 1 : Γ Γ B Γ B E2 : Γ B π 2 : Γ B Implication Γ, B Γ B I Γ B Γ E Γ B : Γ B Λ() : Γ ( B) : Γ ( B) g : Γ ev,b, g : Γ B Joyal s Lemma It is a very natural idea to seek to extend the correspondence shown above to the case o classical logic. Joyal s lemma shows that there is a undamental impediment to doing so. The natural extension o the notion o cartesian closed category, which corresponds to the intuitionistic logic o conjunction and implication, to the classical case is to introduce a suitable notion o classical negation. We recall that it is customary in intuitionistic logic to deine the negation by := where is the alsum. The characteristic property o the alsum is that it implies every proposition. In categorical terms, this translates into the notion o an initial object. Note that or any ixed object B in a cartesian closed category, there is a well-deined contravariant unctor C C op :: ( B). This will always satisy the properties corresponding to negation in min- It is customary to reer to this result as Joyal s lemma, although, apparently, he never published it. The usual reerence is Lambek and Scott (1986), who attribute the result to Joyal, but ollow the proo given by Freyd Freyd (1972). Our statement and proo are somewhat dierent to those in Lambek and Scott (1986).

8 8 imal logic, and i B = is the initial object in C, then it will satisy the laws o intuitionistic negation. In particular, there is a canonical arrow ( ) which is just the curried orm o the evaluation morphism. This corresponds to the valid intuitionistic principle. What else is needed in order to obtain classical logic? s is well known, the missing principle is that o proo by contradiction: the converse implication. This leads us to the ollowing notion. dualizing object in a closed category is one or which the canonical arrow ( ) is an isomorphism or all. We can now state Joyal s lemma: Proposition 1 (Joyal s Lemma) ny cartesian closed category with a dualizing object is a preorder (hence trivial as a semantics or proos or computational processes). Proo Note irstly that, i is dualizing, the induced negation unctor C C op is a contravariant equivalence C C op. Since ( ) = where is the terminal object, it ollows that is the dual o, and hence initial. So it suices to prove Joyal s lemma under the assumption that the dualizing object is initial. We assume that is a dualizing initial object in a cartesian closed category C. By cartesian closure, C(, B) = C(, B), which is a singleton by initiality o. It ollows that is initial. Now C(, B) = C(B, ) = C((B ), ). (1.1) Given any h, k : C, note that h = π 1 h, k, k = π 2 h, k. But =, hence by initiality π 1 = π 2, and so h = k, which by (1.1) implies that = g or, g : B. slicker proo simply notes that ( ) is a let adjoint by cartesian closure, and hence preserves all colimits, in particular initial objects.

9 No-Cloning In Categorical Quantum Mechanics Linearity and Classicality However, we know rom Linear Logic that there is no impediment to having a closed structure with a dualizing object, provided we weaken our assumption on the underlying context-building structure, rom cartesian to monoidal. Then we get a wealth o examples o -autonomous categories Barr (1979), which stand to Multiplicative Linear Logic as cartesian closed categories do to Intuitionistic Logic Seely (1998). Joyal s lemma can thus be stated in the ollowing equivalent orm. Proposition 2 -autonomous category in which the monoidal structure is cartesian is a preorder. Essentially, a cartesian structure is a monoidal structure plus natural diagonals, and with the tensor unit a terminal object, i.e. plus cloning and deleting! 1.3 Categorical Quantum Mechanics In this section, we shall provide a brie review o the structures used in categorical quantum mechanics, their graphical representation, and how these structures are used in ormalizing some key eatures o quantum mechanics. Further details can be ound elsewhere bramsky and Coecke (2008); bramsky (2005); Selinger (2007) Symmetric Monoidal Categories We recall that a monoidal category is a structure (C,, I, a, l, r) where: C is a category, : C C C is a unctor (tensor), I is a distinguished object o C (unit), a, l, r are natural isomorphisms (structural isos) with components: a,b,c : (B C) = ( B) C l : I = r : I = such that certain diagrams commute, which ensure coherence MacLane (1998), described by the slogan: ll diagrams only involving a, l and r must commute.

10 10 Examples: Both products and coproducts give rise to monoidal structures which are the common denominator between them. (But in addition, products have diagonals and projections, and coproducts have codiagonals and injections.) (N,, +, 0) is a monoidal category. Rel, the category o sets and relations, with cartesian product (which is not the categorical product). Vect k with the standard tensor product. Let us examine the example o Rel in some detail. We take to be the cartesian product, which is deined on relations R : X X and S : Y Y as ollows. (x, y) X Y, (x, y ) X Y. (x, y)r S(x, y ) xrx ysy. It is not diicult to show that this is indeed a unctor. Note that, in the case that R, S are unctions, R S is the same as R S in Set. Moreover, we take each a,b,c to be the associativity unction or products (in Set), which is an iso in Set and hence also in Rel. Finally, we take I to be the one-element set, and l, r to be the projection unctions: their relational converses are their inverses in Rel. The monoidal coherence diagrams commute simply because they commute in Set. Tensors and products s mentioned earlier, products are tensors with extra structure: natural diagonals and projections, corresponding to cloning and deleting operations. This act is expressed more precisely as ollows. Proposition 3 Let C be a monoidal category (C,, I, a, l, r). The tensor induces a product structure i there exist natural diagonals and projections, i.e. natural transormations :, p,b : B, q,b : B B,

11 No-Cloning In Categorical Quantum Mechanics 11 such that the ollowing diagrams commute. id p, id q, B,B ( B) ( B) p,b q,b id B B Symmetry symmetric monoidal category is a monoidal category (C,, I, a, l, r) with an additional natural isomorphism (symmetry), σ,b : B = B such that σ B, = σ 1,B, and some additional coherence diagrams commute Scalars Let (C,, I, l, a, l, r) be a monoidal category. We deine a scalar in C to be a morphism s : I I, i.e. an endomorphism o the tensor unit. Example 4 In FdVec K, linear maps K K are uniquely determined by the image o 1, and hence are in bijective correspondence with elements o K; composition corresponds to multiplication o scalars. In Rel, there are just two scalars, corresponding to the Boolean values 0, 1. The (multiplicative) monoid o scalars is then just the endomorphism monoid C(I, I). The irst key point is the elementary but beautiul observation by Kelly and Laplaza Kelly and Laplaza (1980) that this monoid is always commutative. Lemma 5 C(I, I) is a commutative monoid

12 12 Proo I s I t I r 1 I I I ====== I I l I I s 1 r 1 I I I l 1 I 1 t 1 t t s t li I I I I I ====== I I s 1 r I I s using the coherence equation l I = r I. The second point is that a good notion o scalar multiplication exists at this level o generality. That is, each scalar s : I I induces a natural transormation s : I s 1 I. with the naturality square s B B s B We write s or s = s B. Note that 1 = s (t ) = (s t) (s g) (t ) = (s t) (g ) (s ) (t g) = (s t) ( g) which exactly generalizes the multiplicative part o the usual properties o scalar multiplication. Thus scalars act globally on the whole category.

13 No-Cloning In Categorical Quantum Mechanics Compact Closed Categories category C is -autonomous Barr (1979) i it is symmetric monoidal, and comes equipped with a ull and aithul unctor such that a bijection ( ) : C op C C( B, C ) C(, (B C) ) exists which is natural in all variables. Hence a -autonomous category is closed, with B := ( B ). These -autonomous categories provide a categorical semantics or the multiplicative ragment o linear logic Seely (1998). compact closed category Kelly and Laplaza (1980) is a -autonomous category with a sel-dual tensor, i.e. with natural isomorphisms It ollows that u,b : ( B) B u I : I I. B B. n alternative deinition arises when one considers a symmetric monoidal category as a one-object bicategory. In this context, compact closure simply means that every object, qua 1-cell o the bicategory, has a speciied adjoint Kelly and Laplaza (1980). Deinition 6 (Kelly-Laplaza) compact closed category is a symmetric monoidal category in which to each object a dual object, a unit and a counit η : I ǫ : I are assigned, in such a way that the diagram r 1 I 1 η ( ) 1 l a,, I ( ) ǫ 1

14 14 and the dual one or both commute. Examples The symmetric monoidal categories (Rel, ) o sets, relations and cartesian product and (FdVec K, ) o inite-dimensional vector spaces over a ield K, linear maps and tensor product are both compact closed. In (Rel, ), we simply set X = X. Taking a one-point set { } as the unit or, and writing R or the converse o a relation R: η X = ǫ X = {(, (x, x)) x X}. For (FdVec K, ), we take V to be the dual space o linear unctionals on V. The unit and counit in (FdVec K, ) are n η V : K V V :: 1 ē i e i and i=1 ǫ V : V V K :: e i ē j ē j (e i ) where n is the dimension o V, {e i } n i=1 is a basis o V and ē i is the linear unctional in V determined by ē j (e i ) = δ ij. Deinition 7 The name and the coname o a morphism : B in a compact closed category are 1 B η I For R Rel(X, Y ) we have B 1 B I ǫ B B B R = {(, (x, y)) xry, x X, y Y } and R = {((x, y), ) xry, x X, y Y } and or FdVec K (V, W) with (m ij ) the matrix o in bases {e V i }n i=1 and {e W j }m j=1 o V and W respectively n,m : K V W :: 1 m ij ē V i e W j i,j=1

15 No-Cloning In Categorical Quantum Mechanics 15 and : V W K :: e V i ē W j m ij. Given : B in any compact closed category C we can deine : B as B l 1 B I B η 1 B B r I 1 ǫ B B B 1 1 B This operation ( ) is unctorial and makes Deinition 6 coincide with the one given at the beginning o this section. It then ollows by C( B, I) = C(, B) = C(I, B) that every morphism o type I B is the name o some morphism o type B and every morphism o type B I is the coname o some morphism o type B. In the case o the unit and the counit we have η = 1 and ǫ = 1. For R Rel(X, Y ) the dual is the converse, R = R Rel(Y, X), and or FdVec K (V, W), the dual is : W V :: φ φ Dagger Compact Categories In order to ully capture the salient structure o FdHilb, the category o inite-dimensional complex Hilbert spaces and linear maps, an important reinement o compact categories, to dagger- (or strongly-) compact categories, was introduced in bramsky and Coecke (2004, 2005). We shall not make any signiicant use o this reined deinition in this paper, since our results hold at the more general level o compact categories. Nevertheless, we give the deinition since we shall reer to this notion later. We shall adopt the most concise and elegant axiomatization o strongly We shall oten use the abbreviated orm compact categories instead o compact closed categories.

16 16 or dagger compact closed categories, which takes the adjoint as primitive, ollowing bramsky and Coecke (2005). It is convenient to build the deinition up in several stages, as in Selinger (2007). Deinition 8 dagger category is a category C equipped with a contravariant, identity-on-objects, strictly involutive unctor : 1 = 1, (g ) = g, =. We deine an arrow : B in a dagger category to be unitary i it is an isomorphism such that 1 =. n endomorphism : is sel-adjoint i =. Deinition 9 dagger symmetric monoidal category (C,, I, a, l, r, σ, ) combines dagger and symmetric monoidal structure, with the requirement that the natural isomorphisms a, l, r, σ are componentwise unitary, and moreover that is a strict monoidal unctor: ( g) = g. Finally we come to the main deinition. Deinition 10 dagger compact category is a dagger symmetric monoidal category which is compact closed, and such that the ollowing diagram commutes: I η ǫ σ, This implies that the counit is deinable rom the unit and the adjoint: ǫ = η σ, and similarly the unit can be deined rom the counit and the adjoint. Furthermore, it is in act possible to replace the two commuting diagrams required in the deinition o compact closure by one. We reer to bramsky and Coecke (2005) or the details.

17 No-Cloning In Categorical Quantum Mechanics Trace n essential mathematical instrument in quantum mechanics is the trace o a linear map. In quantum inormation, extensive use is made o the more general notion o partial trace, which is used to trace out a subsystem o a compound system. general categorical axiomatization o the notion o partial trace has been given by Joyal, Street and Verity Joyal et al. (1996). trace in a symmetric monoidal category C is a amily o unctions Tr U,B : C( U, B U) C(, B) or objects, B, U o C, satisying a number o axioms, or which we reer to Joyal et al. (1996) or?bramsky (2005). This specializes to yield the total trace or endomorphisms by taking = B = I. In this case, Tr() = Tr U I,I() : I I is a scalar. Expected properties such as the invariance o the trace under cyclic permutations Tr(g ) = Tr( g) ollow rom the general axioms. ny compact closed category carries a canonical (in act, a unique) trace. For an endomorphism :, the total trace is deined by Tr() = ǫ ( 1 ) σ, η. This deinition gives rise to the standard notion o trace in FdHilb Graphical Representation Complex algebraic expressions or morphisms in symmetric monoidal categories can rapidly become hard to read. Graphical representations exploit two-dimensionality, with the vertical dimension corresponding to composition and the horizontal to the monoidal tensor, and provide more intuitive presentations o morphisms. We depict objects by wires, morphisms by boxes with input and output wires, composition by connecting outputs to inputs, and the monoidal tensor by locating boxes side-by-side. B C g B B B B C C g B B h E g C B

18 18 lgebraically, these correspond to: 1 :, : B, g, 1 1 B, 1 C, g, ( g) h respectively. (The convention in these diagrams is that the upward vertical direction represents progress o time.) Kets, Bras and Scalars: special role is played by boxes with either no input or no output, i.e. arrows o the orm I or I respectively, where I is the unit o the tensor. In the setting o FdHilb and Quantum Mechanics, they correspond to states and costates respectively (c. Dirac s kets and bras Dirac (1947)), which we depict by triangles. Scalars then arise naturally by composing these elements (c. inner-product or Dirac s bra-ket): ψ π π oψ = π ψ Formally, scalars are arrows o the orm I I. In the physical context, they provide numbers ( probability amplitudes etc.). For example, in FdHilb, the tensor unit is C, the complex numbers, and a linear map s : C C is determined by a single number, s(1). In Rel, the scalars are the boolean semiring {0, 1}. This graphical notation can be seen as a substantial two-dimensional generalization o Dirac notation Dirac (1947): φ ψ φ ψ Note how the geometry o the plane (more precisely, the act that these diagrams are taken modulo planar isotopy) absorbs unctoriality and naturality conditions, e.g.:

19 No-Cloning In Categorical Quantum Mechanics 19 g = g ( 1) (1 g) = g = (1 g) ( 1) Cups and Caps We introduce a special diagrammatic notation or the unit and counit. ǫ : I η : I. The lines indicate the inormation low accomplished by these operations. Compact Closure The basic algebraic laws or units and counits become diagrammatically evident in terms o the inormation-low lines: = = (ǫ 1 ) (1 η ) = 1 (1 ǫ ) (η 1 ) = 1

20 20 Names and Conames in the Graphical Calculus The units and counits are powerul; they allow us to deine a closed structure on the category. In particular, we can orm the name o any arrow : B, as a special case o λ-abstraction, and dually the coname : : B I : I B This is the general orm o Map-State duality: C( B, I) = C(, B) = C(I, B) Formalizing Quantum Inormation Flow In this section, we give a brie glimpse o categorical quantum mechanics. While not needed or the results to ollow, it provides the motivating context or them. For urther details, see e.g. bramsky and Coecke (2008) Quantum Entanglement We consider or illustration two standard examples o two-qubit entangled states, the Bell state: and the EPR state:

21 No-Cloning In Categorical Quantum Mechanics 21 In quantum mechanics, compound systems are represented by the tensor product o Hilbert spaces: H 1 H 2. typical element o the tensor product has the orm: λ i φ i ψ i i where φ i, ψ i range over basis vectors, and the coeicients λ i are complex numbers. Superposition encodes correlation: in the Bell state, the o-diagonal elements have zero coeicients. This gives rise to Einstein s spooky action at a distance. Even i the particles are spatially separated, measuring one has an eect on the state o the other. In the Bell state, or example, when we measure one o the two qubits we may get either 0 or 1, but once this result has been obtained, it is certain that the result o measuring the other qubit will be the same. This leads to Bell s amous theorem Bell (1964): QM is essentially non-local, in the sense that the correlations it predicts exceed those o any local realistic theory. From paradox to eature : Teleportation φ x B 2 U x M Bell φ lice Bob In the teleportation protocol Bennet et al. (1993), lice sends an unknown qubit φ to Bob, using a shared Bell pair as a quantum channel. By perorming a measurement in the Bell basis on φ and her hal o the

22 22 entangled pair, a collapse is induced on Bob s qubit. Once the result x o lice s measurement is transmitted by classical communication to Bob (there are our possible measurement outcomes, hence this requires two classical bits), Bob can perorm a corresponding unitary correction U x on his qubit, ater which it will be in the state φ Categorical Quantum Mechanics and Diagrammatics We now outline the categorical approach to quantum mechanics developed in bramsky and Coecke (2004, 2005). The same graphical calculus and underlying algebraic structure which we have seen in the previous section has been applied to quantum inormation and computation, yielding an incisive analysis o quantum inormation low, and powerul and illuminating methods or reasoning about quantum inormatic processes and protocols bramsky and Coecke (2004). Bell States and Costates: The cups and caps we have already seen in the guise o deicit and cancellation operations, now take on the rôle o Bell states and costates (or preparation and test o Bell states), the undamental building blocks o quantum entanglement. (Mathematically, they arise as the transpose and co-transpose o the identity, which exist in any inite-dimensional Hilbert space by map-state duality ). * * The ormation o names and conames o arrows (i.e. map-state and map-costate duality) is conveniently depicted thus: =: =: (2) The key lemma in exposing the quantum inormation low in (bipartite) entangled quantum systems can be ormulated diagrammatically as ollows:

23 No-Cloning In Categorical Quantum Mechanics 23 g = g = g = g Note in particular the interesting phenomenon o apparent reversal o the causal order. While on the let, physically, we irst prepare the state labeled g and then apply the costate labeled, the global eect is as i we irst applied itsel irst, and only then g. Derivation o quantum teleportation. This is the most basic application o compositionality in action. We can read o the basic quantum mechanical potential or teleportation immediately rom the geometry o Bell states and costates: ψ = ψ = ψ lice Bob lice Bob lice Bob The Bell state orming the shared channel between lice and Bob appears as the downwards triangle in the diagram; the Bell costate orming one o the possible measurement branches is the upwards triangle. The inormation low o the input qubit rom lice to Bob is then immediately evident rom the diagrammatics. This is not quite the whole story, because o the non-deterministic nature o measurements. But in act, allowing or this shows the underlying design principle or the teleporation protocol. Namely, we ind a measurement basis such that each possible branch i through the measurement is labelled, under map-state duality, with a unitary map i. The corresponding correction is then just the inverse map i 1. Using our lemma, the ull description o teleportation becomes:

24 24 i i -1 i -1 = i = 1.4 No-Cloning Note that the proo o Joyal s lemma given in Section makes ull use o both diagonals and projections, i.e. o both cloning and deleting. Our aim is to examine cloning and deleting as separate principles, and to see how ar each in isolation is compatible with the strong orm o duality which, as we have seen, plays a basic structural rôle in the categorical axiomatization o quantum mechanics, and applies very directly to the analysis o entanglement xiomatizing Cloning Our irst task is to axiomatize cloning as a uniorm operation in the setting o a symmetric monoidal category. s a preliminary, we recall the notions o monoidal unctor and monoidal natural transormation. Let C and D be monoidal categories. (strong) monoidal unctor (F, e, m) : C D comprises: unctor F : C D n isomorphism e : I = FI natural isomorphism m,b : F FB F( B) subject to various coherence conditions. Let (F, e, m), (G, e, m ) : C D be monoidal unctors. monoidal natural transormation between them is a natural transormation t : F. G such that I e FI F FB m,b F( B) e t I GI t t B G GB m,b t B G( B)

25 No-Cloning In Categorical Quantum Mechanics 25 We say that a monoidal category has uniorm cloning i it has a diagonal, i.e. a monoidal natural transormation : which is moreover coassociative and cocommutative: 1 ( ) a,, ( ) 1 σ, Note that in the case when the monoidal structure is induced by a product, the standard diagonal : 1,1 automatically satisies all these properties. To simpliy the presentation, we shall henceorth make the assumption that the monoidal categories we consider are strictly associative. This is a standard manouevre, and by the coherence theorem or monoidal categories MacLane (1998) is harmless. Note that the unctor which is the codomain o the diagonal has as its monoidal structure maps m,b = B B 1 σ 1 B B e = I l 1 I I I. O course the identity unctor, which is the domain o the diagonal, has identity morphisms as its structure maps Compact categories with cloning (almost) collapse Theorem 11 Let C be a compact category with cloning. Then every endomorphism is a scalar multiple o the identity. More precisely, or :, = Tr() id. This means that or every object o C, C(, ) is a retract o C(I, I): α : C(, ) C(I, I) : β, α() = Tr(), β(s) = s id.

26 26 In a category enriched over vector spaces, this means that each endomorphism algebra is one-dimensional. In the cartesian case, there is a unique scalar, and we recover the relexive part o the posetal collapse o Joyal s lemma. But in general, the collapse given by our result is o a dierent nature to that o Joyal s lemma, as we shall see later. Note that our collapse result only reers to endomorphisms. In the dagger-compact case, every morphism : B has an associated endomorphism σ ( ) : B B. Moreover the passage to this associated endomorphism can be seen as a kind o projective quotient o the original category Coecke (2007). Thus in this case, the collapse given by our theorem can be read as saying that the projective quotient o the category is trivial Proving the Cloning Collapse Theorem We shall make some use o the graphical calculus in our proos. We shall use slightly dierent conventions rom those adopted in the previous section: Firstly, the diagrams to ollow are to be read downwards rather than upwards. Secondly, we shall depict the units and counits o a compact category simply as cups and caps, without any enclosing triangles. To illustrate these points, the units and counits will be depicted thus: η : I ǫ : I while the identities or the units and counits in compact categories will appear thus: = =

27 No-Cloning In Categorical Quantum Mechanics 27 The small nodes appearing in these diagrams indicate how the igures are built by composition rom basic igures such as cups, caps and identities. Finally, we shall depict diagonal morphisms diagrammatically by orking : or :. First step We shall begin by showing that Diagrammatically: parallel caps = nested caps = This amounts to a conusion o entanglements. In act, we shall ind it more convenient to prove this result in the ollowing orm: η η = ( ) (η η ) Here ( ) is the permutation acting on the tensor product o our actors which is built rom the symmetry isomorphisms in the obvious ashion. Diagrammatically: = Lemma 12 We have I = l 1 I : I I I. Proo This is an immediate application o the monoidality o, together with e = l 1 I or the codomain unctor.

28 28 Lemma 13 Let u : I B be a morphism in a symmetric monoidal category with cloning. Then Proo u u = ( ) (u u). Consider the ollowing diagram. I I I I u B B B B u u B B B B 1 σ 1 σ 1 B B The upper square commutes by naturality o. The upper triangle o the lower square commutes by monoidality o. The lower triangle commutes by cocommutativity o in the irst component, and then tensoring with the second component and using the biunctoriality o the tensor. Let = (u u) I, and g = ( B ) u. Then by the above diagram = (1 σ 1) (σ 1) g. simple computation with permutations shows that (1 σ 1) (σ 1) = ( ) ( ) = ( ) (1 σ 1). ppealing to the above diagram again, = (1 σ 1) g. Hence pplying the previous lemma: = (1 σ 1) (σ 1) g = ( ) (1 σ 1) g = ( ). u u = l I = ( ) l I = ( ) (u u). Diagrammatically, this can be presented as ollows:

29 No-Cloning In Categorical Quantum Mechanics 29 = = = and hence = Note that this lemma is proved in generality, or any morphism u o the required shape. However, we shall, as expected, apply it by taking u = η. It will be convenient to give the remainder o the proo in diagrammatic orm. Second step We use the irst step to show that the twist map = the identity in a compact category with cloning, by putting parallel and serial caps in a common context and simpliying using the triangular identities. The context is: We get:

30 30 = and: = We used the original picture o nested caps or clarity. I we use the picture directly corresponding to the statement o lemma 13, we obtain the same result: = The important point is that the let input is connected to the right output, and the right input to the let output. Third step Finally, we use the trace to show that any endomorphism : is a scalar multiple o the identity: = s 1 or s = Tr(). = =

31 No-Cloning In Categorical Quantum Mechanics 31 This completes the proo o the Cloning Collapse Theorem Examples We note another consequence o cloning. Proposition 14 In a monoidal category with cloning, the multiplication o scalars is idempotent. Proo This ollows immediately rom naturality I I I I together with lemma 12. s I I I I s s Thus the scalars orm a commutative, idempotent monoid, i.e. a semilattice. Given any semilattice S, we regard it qua monoid as a one-object category, say with object. We can deine a trivial strict monoidal structure on this category, with = = I. Biunctoriality ollows rom commutativity. natural diagonal is also given trivially by the identity element (which is the top element o the induced partial order, i we view the semilattice operation as meet). Units and counits are also given trivially by the identity. Note that the scalars in this category are o course just the elements o S. Thus any semilattice yields an example o a (trivial) compact category with cloning. Note the contrast with Joyal s lemma. While every boolean algebra is o course a semilattice, it orms a degenerate cartesian closed category as a poset, with many objects but at most one morphism in each homset. The degenerate categories we are considering are categories qua monoids, with arbitrarily large hom-sets, but only one object. Posets and monoids are opposite extremal examples o categories, which appear as contrasting degenerate examples allowed by these no-go results. Note that our result as it stands is not directly comparable with Joyal s, since our hypotheses are weaker insoar as we only assume a

32 32 monoidal diagonal rather than ull cartesian structure, but stronger insoar as we assume compact closure. boolean algebra which is compact closed qua category is necessarily the trivial, one-element poset, since meets and joins and in particular the top and bottom o the lattice are identiied Discussion The Cloning Collapse theorem can be read as a No-Go theorem. It says that it is not possible to combine basic structural eatures o quantum entanglement with a uniorm cloning operation without collapsing to degeneracy. It should be understood that the key point here is the uniormity o the cloning operation, which is ormalized as the monoidal naturality o the diagonal. suitable intuition is to think o this as corresponding to basis-independence. The distinction is between an operation that exists in a representation-independent orm, or logical reasons, as compared to operations which do intrinsically depend on speciic representations. In act, in turns out that much signiicant quantum structure can be captured in our categorical setting by non-uniorm copying operations Coecke and Pavlovic (2007). Given a choice o basis or a initedimensional Hilbert space H, one can deine a diagonal i ii. This is coassociative and cocommutative, and extends to a comonoid structure. pplying the dagger yields a commutative monoid structures, and the two structures interact by the Frobenius law. It can be shown that such dagger Frobenius structures on inite-dimensional Hilbert spaces correspond exactly to bases. Since bases correspond to choice o measurement context, these structures can be used to ormalize quantum measurements, and quantum protocols involving such measurements Coecke and Pavlovic (2007). It is o the essence o quantum mechanics that many such structures can coexist on the same system, leading to the idea o incompatible measurements. This too has been axiomatized in the categorical setting, enabling the eective description o many central eatures o quantum computation Coecke and Duncan (2008). In act, the original example which led Eilenberg and Mac Lane to deine naturality was the naturality o the isomorphism rom a inite-dimensional vector space to its second dual, as compared with the non-natural isomorphism to the irst dual.

33 No-Cloning In Categorical Quantum Mechanics 33 Thus the No-Go result is delicately poised on the issue o naturality. It seems possible that a rather sharp delineation between quantum and classical, and more generally a classiication o the space o possible theories incorporating various eatures, may be achieved by urther development o these ideas. 1.5 No-Deleting The issue o No-deleting is much simpler rom the current perspective. uniorm deleting operation is a monoidal natural transormation d : I. Note that the domain o this transormation is the identity unctor, while the codomain is the constant unctor valued at I. The ollowing result was originally observed by Bob Coecke in the dagger compact case: Proposition 15 I a compact category has uniorm deleting, then it is a preorder. Proo Given : B, consider the naturality square B I d B I I By monoidal naturality, d I = 1 I. So or all, g : B: d I = d B = g and hence = g. 1.6 Further Directions We conclude by discussing some urther developments and possible extensions o these ideas. In a orthcoming joint paper with Bob Coecke, the results are extended to cover No-Broadcasting by liting the Cloning Collapse theorem to the CPM category Selinger (2007), which provides a categorical treatment o mixed states.

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