Environment and classical channels in categorical quantum mechanics

Categorical quantum mechanics [Abramsky and Coecke 04] provides a new perspective on quantum information processing. Particularly appealing is the fact that the symmetric monoidal categorical language comes with an intuitive graphical calculus [Penrose 71,Joyal and Street 91]. This approach has meanwhile led to new results in quantum information and quantum foundations. Graph states, a key resource for the measurement based quantum computational model, and translations thereof to the circuit model were studied in [Duncan and Perdrix 09, Duncan and Perdrix 10]. There is a compositional framework for studying the structure of general multipartite quantum entanglement [Coecke and Kissinger 10]. Quantum theory as well as Spekkens' toy theories have been casted within a single mathematical framework, enabling corresponding analysis of quantum non-locality in terms of 'phase groups' [Coecke,Edwards and Spekkens 11]. There exists a no-cloning theorem for a very general class of theories [Abramsky 09]. Meanwhile there also exists a software tool with not only a graphical interface but also a 'graphical internal logic', [quantomatic], which (semi-)automates graphical reasoning. It has been developed in a collaboration between Oxford, Edinburgh and since recently Google, and involved interpreting theories formulated in symmetric monoidal language in 2. Classicality vs. quantumness Let H be a Hilbert space. By a cloning map one means an operation which is such that for all |ψ ∈ H and some |0 ∈ H we have When fixing |0 within the argument we can instead consider ∆ := U (-⊗ |0 ) : H → H ⊗ H rather than U . It is well-known that there exists no cloning map, by the so-called no-cloning theorem [Dieks 82,Wootters and Zurek 82]. Now, if an operation clones certain pure states, say the basis vectors {|i } i , this does not imply that it clones mixtures of these too; setting ρ = i ρ i |i i| we have: However, what we do have is that where tr 1 and tr 2 respectively trace out the first and the second system. For arbitrary completely positive maps E Eqs. (2.1) becomes: tr 1 (E(ρ)) = tr 2 (E(ρ)) = ρ . (2.2) Universal validity of Eqs. (2.2) for a particular completely positive map E acting on the entire space of all density operators has been referred to as broadcasting. But the nobroadcasting theorem [Barnum et al. 96] states that only mixed states which share a basis in which they all are diagonal (i.e. mixed states that can be jointly simulated by classical probability distributions) can be broadcast by the same completely positive map. From the above discussion it follows that broadcastability is a strictly weaker requirement than cloneability. We have: pure classical mixed classical pure quantum mixed quantum broadcastable: yes no no no where by classical we refer to a set of density operators that are diagonal in the same basis. So (not non-cloneability but) non-broadcastability 'identifies' quantum relative to classical, in that classical states, both pure and mixed, can always be broadcast by a single quantum operation, while this is not possible for quantum states that cannot be jointly simulated classically. Now, taking the contrapositive, for us classicality will mean broadcastability. Equivalently, one can also conceive classicality as the result of decoherence [Zurek 91]. Concretely, 'total' decoherence is the completely positive map which erases all non-diagonal elements in the matrix representation of a given basis, that is, where δ ij is the Kronecker delta. Broadcasting and decoherence are indeed closely related: in the case of the first 'one copies into the environment', while in the case of the second 'one couples to the environment'. Decoherent then means invariance under this coupling. Formally, decoherent density operators relative to a fixed D {|i } i are exactly those collections of density operators that can be jointly broadcast; they all are diagonal in the basis {|i } i and hence can be simulated by the classical probability distributions that make up the diagonals. Summarizing the above: classical := broadcastable ≡ decoherent We will treat 'classicality' as a 'behavior' -i.e. behaves as if it is classical in the above discussed sense-rather than as the specification of the actual physical realization of a system. An important point in this context, already realized in [Coecke and Pavlovic 07,Coecke,Paquette and Pavlovic 09], is that by taking quantum to be the 'default behavior' within the mathematical universe of all operations, characterization of classical entities can be done in purely diagrammatic terms. In the concrete Hilbert space realization, this means that one only needs to rely on the multiplicative tensor product structure as a primitive connective, with no reference to the additive vector space structure. One could refer to this as 'classicization', in contrast to the standard notion of 'quantization' where one starts with a classical theory and then freely adjoins the additive vector space structure. As an example, consider a quantum measurement, which when applied to a quantum system changes the state of that quantum system and produces classical data. Since the resulting quantum state is an eigenstate for that measurement, hence broadcastable, it behaves precisely in the same manner as the classical data does. As a result, in the graphical calculus the classical data and the collapsed state won't be distinguishable once we omit explicit specification wether physically they are either classical or quantum. This ambiguity captures a feature of the particular manner in which quantum and classical data interact, namely, that the creation of classical data renders the quantum state in an eigenstate. Of course, if we later apply a non-classical unitary to the resulting quantum state, then we reassert its proper quantumness. Put in type-theoretic terms, there will be no such thing as a fixed 'classical type' and fixed 'quantum type' in our representation, since we can abstract away over these 'implementation details' without altering the essential structure. Of course, one can add those details in order to connect the graphical language to concrete physical protocols where the classical quantum distinction may be fundamental for the conceptual analysis, for example, in quantum teleportation it is crucial that the classical communication can indeed be realized by purely classical finitary means. In Section 7 we give several examples of protocols that come with specification of what is classical and what is quantum, and then pass to the abstract diagrammatic calculus where forgetting the physical realization is essential to perform the computation. In this paper we will work in the graphical representation of symmetric monoidal categories [Joyal and Street 91]. Mac Lane's strictification theorem [Mac Lane 00, p.257] allows us to take our symmetric monoidal categories to be strict, that is: Morphisms f : A 1 . . . A n → B 1 . . . B m , which we interpret as processes are respectively represented as boxes where the input wires represent the objects A 1 . . . A n and the output wires represent the objects B 1 . . . B m : Other shapes may be used to emphasize extra structure. Elements s : I → A 1 . . . A n , which in the graphical representation have no inputs, are interpret as 'states', and co-elements e : B 1 . . . B m → I with no outputs are interpreted as 'effects'. In standard quantum notation they would be kets |ψ and bras ψ| respectively. Composition and tensoring are respectively represented as: Theorem 3.1. [Joyal and Street 91, Selinger 11a] An equation follows from the axioms of symmetric monoidal categories if and only if it can be derived in the graphical language via diagram isomorphisms. A dagger functor on a symmetric monoidal category [Selinger 07] is an identity-onobjects contravariant involutive strict monoidal functor. It is graphically represented by flipping pictures upside-down, for instance: and it is unitary of both f and f † are isometries. A dagger compact category [Abramsky and Coecke 05] is a dagger symmetric monoidal category in which each object A comes with two morphisms η A : I → A * ⊗ A and ǫ A : A ⊗ A * → I which satisfy certain equations. In this paper we take all our objects to be self-dual, that is, A = A * .1 Graphically, we represent η A as: we take ǫ A to be its dagger, and the equations that govern η A and ǫ A are: The results in this paper can be extended to the case of non-self-dual compact structures, by relying on the results in [Coecke,Paquette and Perdrix 08]. This would, for example, be required when considering all three complementary measurements on a qubit. Theorem 3.3. [Kelly and Laplaza 80, Selinger 07] An equation follows from the axioms of (dagger) compact categories if and only if it can be derived in the corresponding graphical language via isotopy. The key difference between diagram isomorphism as in Theorem 3.1 and isotopy as in Theorem 3.3 is that diagram isomorphisms take specification of the boxes' inputs and outputs into account, while isotopy abstracts away these roles. Hence within the scope of Theorem 3.3 only the topology of the diagrams matters. By classical structures [Coecke and Pavlovic 07] we mean internal commutative special dagger Frobenius algebras in a dagger compact category for which we also require 'compatibility with the compact structure' (see below). We won't give an explicit definition here, but will rely on a remarkable normal form result that holds for morphisms build from this structure, namely, any morphism obtained by composing and tensoring the structural morphisms of a classical structure and the symmetric monoidal structure, and of which the diagrammatic representation is connected, only depends on n and m [Lack 04]. Graphically we represent this unique morphism as an n + m-legged spider : From the axioms of classical structures it follows that these spiders are invariant when one exchanges the roles of front-legs and back-legs, when one swaps two legs of either of these, and that the (1 + 1)-legged spider is the identity, that is, where σ A,A : A ⊗ A → A ⊗ A is the swap map, and, last but not least, that spiders which 'share' legs fuse together, i.e. spiders compose as follows: Conversely, the axioms of an internal commutative special dagger Frobenius algebra all follow from Eqs. (3.1) and (3.2). By compatibility of the classical structure with a given dagger compact structure we mean that for a spider on A we have η A = Ξ 0 2 , and consequently that ǫ A = Ξ 2 0 . In graphical terms, that is: = = Indeed, for spiders Ξ 0 2 and Ξ 2 0 we always have: = = and = that is, they form a dagger compact structure. In the graphical language we will depict elements (i.e. 'boxes without inputs') by triangles. By a pure classical element for a particular classical structure we mean an element which satisfies: i.e. it is 'copied'. Below, by e we will only denote such pure classical elements. In the dagger compact category FHilb which has finite dimensional Hilbert spaces as objects, linear maps as morphisms, the tensor product as the monoidal structure, and adjoints as the dagger, classical structures are in bijective correspondence orthonormal bases via this concept of pure classical elements [Coecke,Pavlovic and Vicary 12]. Concretely, the pure classical elements are exactly the basis vectors, and conversely, given an orthonormal basis {|i } i , the corresponding spiders are the linear maps with as only non-zero action on the basis vectors: i.e. arrays of identical basis vectors are mapped on arrays of identical basis vectors, and all other basis vectors are mapped on the zero vector. Important particular examples are which define the multiplication and its unit of the corresponding Frobenius algebra; their adjoints define the corresponding comultiplication and its counit. Remark 3.4. In any dagger symmetric monoidal category, the multiplication and its unit suffice to specify a classical structure; one can then construct any other spider by composing Ξ 1 2 , Ξ 1 0 , (Ξ 1 2 ) † and (Ξ 1 0 ) † to obtain a morphism with the required number of inputs n and outputs m, that is, the spider Ξ m n . Remark 3.5. Physically relevant, rather than FHilb, is the category WP (FHilb) which is obtained by subjecting FHilb to the congruence which identifies those linear maps of the same type that are equal up to a complex phase, i.e. The reason is that vectors which are equal up to a complex phase represent the same state in quantum theory. The precise connection between classical structures in FHilb and those in WP (FHilb) is studied in detail in [Coecke and Duncan 11]; roughly put -since this suffices for all practical purposes-classical structures are inherited. An example of such a complementary morphism is the familiar Hadamard matrix: We will use this 'topological trick' throughout this paper. We will consider two dagger compact categories, denoted C pure and C respectively, and we assume that C pure is a subcategory of C which inherits symmetric monoidal structure as well as dagger compact structure, and that |C| = |C pure |. (5.2) We will sometimes abbreviate environment structure to environment. Remark 5.2. Eqs. (5.1,5.2) had already been introduced in [Coecke 08], as part of an axiomatization of mixed states and completely positive maps, but it was never considered in relation to classicality, measurements, and complementarity thereof. Example 5.3. Let Dens be the category with C n×n for n ∈ N as objects, and with completely positive maps F : C n×n → C m×m as morphisms. A morphism f is pure, that is, f ∈ hom Dens pure (C n×n , C m×m ), if there exists a linear map L : C n → C m such that f :: ρ → LρL † . Then , the usual trace provides an environment structure for (Dens pure , Dens). Indeed, for any f :: ρ → LρL † and g :: ρ → M ρM † , equation 5.1 is satisfied: This example justifies the name 'environment': tracing a system out in quantum theory is interpreted as this system being part of the environment. Below we assume as given a pair (C pure , C) with an environment. We set (5.4) Below all elements depicted as triangles are normalized, diagrammatically: where 1 I is graphically represented by an empty picture. Proposition 5.5. A morphism f ∈ C pure is an isometry iff ⊤ B • f = ⊤ A , and hence, it is unitary iff we moreover have that f 6. Classical channels, measurements and classical control Definition 6.1. Let Ξ be a classical structure. The morphism: In the light of the discussion in Section 2, this picture can be interpreted as 'copying into the environment', that is, 'broadcasting', or in the decoherence view, 'being coupled to the environment'. Example 6.2. In Dens, for the classical structure of Eqs. (3.5), we have where D {|i } i was defined in Eq. (2.3). That is, a classical channel preserves the diagonal data relative to the basis that is specified by its basis structure. It will however destroy the non-diagonal data. In the light of the discussion of Section 2, it is a 'classicizing' operation, on the data it is applied to. Remark 6.3. While physically all classical channels are of course the same, our classical channels in addition carry specification of how the classical data it transmits has been obtained, in terms of a dependency on the classical structure Ξ which specifies a particular quantum measurement. In the light of the fact that by default we take all systems to be quantum, this specification of the classical structure relative to which classical data is classical is indeed unavoidable. It is for this reason that we choose to axiomatize the complementary morphism -cf. the discussion in Section 4-which enables us to restrict ourselves to a single classical structure. The following proposition shows that a classical channel leaves its pure classical elements invariant, and that it is idempotent. In fact, we could define more general classical elements p : I → A as those that satisfy C Ξ • p = p . (6.1) Physically, this means that a classical channel 'transmits' its classical elements. Equivalently, classical elements are invariant under decoherence. where the last equality holds due to the spider normal form theorem. We can now construct a measurement as follows: quantum output quantum input classical output (6.2) i.e. it copies the quantum data into a classical channel. A destructive measurement is obtained by 'feeding the quantum output itself into the environment'. Proposition 6.4 then yields: = . and the resulting shape of the destructive measurement is then: Note here in particular that destructive measurements and classical channels are 'semantically equivalent'. Similarly, by the spider normal form we have: = so the quantum output of a measurement is 'semantically equivalent' to its classical output, which captures change of the quantum state to an eigenstate. More generally, as a consequence of the structural power of the spider normal form theorem, classicality 'semantically spreads through a diagram'. Example 6.5. We now illustrate the above exposed diagrammatic analysis on the concrete example of measurement of a qubit. For |ψ = ψ 0 |0 + ψ 1 |1 to which we apply the morphism of of Eq. ( 6.2) which we assume to be in the computational basis, the first 'copying' operation yields ψ 0 |00 +ψ 1 |11 , the second one yields ψ 0 |000 +ψ 1 |111 and the effect of the environment yields, now necessarily in density matrix terms, Note that by idempotence of C Ξ it also follows that ⊥ A = ⊤ † A is a classical element, and in particular, that this does not depend on the choice of Ξ. We will call ⊥ A (unnormalized) maximal mixedness. Example 6.6. In Dens we indeed have that ⊥ H is diagonal in any basis. We call a morphism f : A → B disconnected if it factors along I, that is, if f = ψ • π for some ψ : I → B and π : A → I. In the graphical representation we indeed obtain a disconnected picture in this case: The topological disconnectedness physically stands for the fact that there is no information flowing from the input to the output. Remark 6.7. For non-trivial categories the morphisms ⊤ A cannot be pure; if they would be pure then setting f := ⊤ A and g := 1 A in Eq. 5.1, the righthandside becomes 1 I • ⊤ A = ⊤ A •1 A , which holds, and hence also the lefthandside holds: That is, the identity is 'disconnected'. This is obviously in conflict with the intuition that through a straight wire information flows without being modified, so one expects bad things to happen. Indeed, for any f : If we introduce H between C Ξ and itself, we obtain 'complementary behaviors'. The first equality of the following proposition implies that a measurement turns a pure classical element of a complementary measurement in maximal mixedness, i.e. any outcome is equally probable for that measurement (cf. 'unbiasedness'). The second one implies that there is no dataflow from the input to the output when we compose complementary measurements. Proposition 6.8. We now define what it means to have a family of unitaries of the same type, 'indexed' by a classical structure, that is, a controlled unitary. Definition 6.9. By a controlled unitary we mean an operation of the form: which 'for all classical input values is unitary', that is: Lemma 6.10. The following morphisms are controlled unitaries: (6.4) Proof. We have: = = and the remainder of the proof proceeds almost identical. 6.2. General non-degenerate measurements. We have identified an example of a nondegenerate measurement, namely the one of the shape (6.2), and an example of a nondegenerate destructive measurement, namely the one of the shape (6.3). Relative to a given classical structure we can define more general non-destructive measurements. The following Lemma shows how classical data can be composed in terms of classical structures, where we conceive classical structures as being specified by a multiplication and its unit, i.e. Ξ := (Ξ 1 2 , Ξ 1 0 ) -cf. Remark 3.4. Lemma 6.11. If (Ξ 1 2 , Ξ 1 0 ) and ( Ξ1 2 , Ξ1 0 ) are classical structures on A and à respectively, then the morphisms COECKE AND S. PERDRIX Lemma 6.14. The following morphism is unitary: CN OT := (6.5) Proof. Corollary 6.15. The following morphism is a non-degenerate destructive measurement on a pair of systems of the same type A: Bell-M eas := 6.3. Interpretation of graphical elements in Dens. The following tables translate the graphical language to Hilbert space quantum theory for the specific case of qubits. It is this translation which connects that diagrammatic presentation of the protocols in the following section to how one finds them usually described in textbooks. (pure) states & effects: Notation: Notation: (pure) gates: Notation: CP maps: Notation: In the statement of each proposition, we will specify protocols with explicit physical types, quantum channels being represented by full lines and classical channels being represented by dotted lines. We use the symbol ':≃' for the passage of this specification to the interpretation within the diagrammatic calculus. First we show that the teleportation protocol, by means of a Bell state and two classical channels, realizes a (perfect) quantum channel. As already mentioned in Example 5.4, the converse statement, that for any dagger compact category C the category CP M (C) provides an environment with purification also holds, up to a minor and physically justified assumption related to the fact that vectors which are equal up to a complex phase represent the same state in quantum theory. Concretely, this axiom states the for all pure elements ψ, ψ ′ : I → A we have: This equation follows from Eq. (5.1) when setting f := ψ † and g := ψ ′ † . Remark 8.3. The power of purification as an axiom for quantum theory has recently been exploited in [Chiribella,D'Ariano and Perinotti 09,Chiribella,D'Ariano and Perinotti 10], although there, the authors also require certain uniqueness properties. An axiomatization of the concept of environment resulted in a very simple comprehensive graphical calculus, which in particular enables one to reason about classical-quantum interaction in quantum informatic protocols. Several operationally distinct concepts turn out to have the same semantics within the graphical language (e.g. classical channel, measurement, preparation as in BB84). Consequently, all that one structurally truly needs are Propositions 6.4 and 6.8 on composition of classical channels and pure classical elements. The examples given here are simple but representative. This work and the earlier contributions on which we relied together successfully addresses a challenge for the categorical quantum mechanics research program which was set at the very beginning: to have a very simple graphical description of all basic quantum informatic protocols, in particular including classical-quantum interaction. The new graphical element 'environment' and the interaction rules for classical channels can now be integrated in the quantomatic software, so that it can now be used to (semi-)automate reasoning about full-blown quantum informatic protocols, including classicalquantum interaction. Here we only considered two complementary observables, and no phase data. We meanwhile also have several graphical calculi that are universal for quantum computing [Coecke and Duncan 11,Coecke and Kissinger 10]. The next step of this research strand would be to extend the graphical calculus presented here to these calculi, which include, for example, phases and W -states. This work could also be advanced in the direction of quantum information theory. In particular, one may want to study whether it would be possible to obtain a diagrammatic account on quantum informatic quantities. Some examples of diagrammatic quantum informatic quantities are in [Coecke 08]. A detailed study of the coherences for this situation is in[Selinger 10]. This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/2.0/ or send a letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or Eisenacher Strasse 2, 10777 Berlin, Germany