Abstract Physical Traces

arXiv:0910.3144v1 [quant-ph] 16 Oct 2009 ABSTRACT PHYSICAL TRACES SAMSON ABRAMSKY AND BOB COECKE ABSTRACT. We revise our ‘Physical Traces’ paper [Abramsky and Coecke CTCS‘02] in the light of the results in [Abramsky and Coecke LiCS‘04]. The key fact is that the notion of a strongly compact closed category allows abstract notions of adjoint, bipartite projector and inner product to be defined, and their key properties to be proved. In this paper we improve on the definition of strong compact closure as compared to the one presented in [Abramsky and Coecke LiCS‘04]. This modification enables an elegant characterization of strong compact closure in terms of adjoints and a Yanking axiom, and a better treatment of bipartite projectors. 1. Introduction In [Abramsky and Coecke CTCS‘02] we showed that vector space projectors P : V ⊗W →V ⊗W which have a one-dimensional subspace of V ⊗W as fixed-points, suffice to implement any linear map, and also the categorical trace [Joyal, Street and Verity 1996] of the category (FdVecK, ⊗) of finite-dimensional vector spaces and linear maps over a base field K. The interest of this is that projectors of this kind arise naturally in quantum mechanics (for K = C), and play a key role in information protocols such as [quantum teleportation 1993] and [entanglement swapping 1993], and also in measurement-based schemes for quantum computation. We showed how both the category (FdHilb, ⊗) of finite-dimensional com- plex Hilbert spaces and linear maps, and the category (Rel, ×) of relations with the cartesian product as tensor, can be physically realized in this sense. In [Abramsky and Coecke LiCS‘04] we showed that such projectors can be defined and their crucial properties proved at the abstract level of strongly compact closed cate- gories. This categorical structure is a major ingredient of the categorical axiomatization in [Abramsky and Coecke LiCS‘04] of quantum theory [von Neumann 1932]. It captures quantum entanglement and its behavioral properties [Coecke 2003]. In this paper we will improve on the definition of strong compact closure, enabling a characterization in terms of adjoints - in the linear algebra sense, suitably abstracted - and yanking, without ex- plicit reference to compact closure, and enabling a nicer treatment of bipartite projectors, coherent with the treatment of arbitrary projectors in [Abramsky and Coecke LiCS‘04]. Rick Blute, Sam Braunstein, Vincent Danos, Martin Hyland and Prakash Panangaden provided useful feed-back. 2000 Mathematics Subject Classification: 15A04,15A90,18B10,18C50,18D10,81P10,81P68. c⃝Samson Abramsky and Bob Coecke, 2004. Permission to copy for private use granted. 1 2 We are then able to show that the constructions in [Abramsky and Coecke CTCS‘02] for realizing arbitrary morphisms and the trace by projectors also carry over to the ab- stract level, and that these constructions admit an information-flow interpretation in the spirit of the one for additive traces [Abramsky 1996, Abramsky, Haghverdi and Scott 2002]. It is the information flow due to (strong) compact closure which is crucial for the abstract formulation, and for the proofs of correctness of protocols such as quantum teleportation [Abramsky and Coecke LiCS‘04]. A concise presentation of (very) basic quantum mechanics which supports the devel- opments in this paper can be found in [Abramsky and Coecke CTCS‘02, Coecke 2003]. However, the reader with a sufficient categorical background might find the abstract pre- sentation in [Abramsky and Coecke LiCS‘04] more enlightening. 2. Strongly compact closed categories As shown in [Kelly and Laplaza 1980], in any monoidal category C, the endomorphism monoid C(I, I) is commutative. Furthermore any s : I →I induces a natural transforma- tion sA : A ≃✲I ⊗A s ⊗1A✲I ⊗A ≃✲A . Hence, setting s • f for f ◦sA = sB ◦f for f : A →B, we have (s • g) ◦(r • f) = (s ◦r) • (g ◦f) for r : I →I and g : B →C. We call the morphisms s ∈C(I, I) scalars and s • −scalar multiplication. In (FdVecK, ⊗), linear maps s : K →K are uniquely determined by the image of 1, and hence correspond biuniquely to elements of K. In (Rel, ×), there are just two scalars, corresponding to the Booleans B. Recall from [Kelly and Laplaza 1980] that a compact closed category is a symmetric monoidal category C, in which, when C is viewed as a one-object bicategory, every one- cell A has a left adjoint A∗. Explicitly this means that for each object A of C there exists a dual object A∗, a unit ηA : I →A∗⊗A and a counit ǫA : A ⊗A∗→I, and that the diagrams A ≃✲A ⊗I 1A ⊗ηA ✲A ⊗(A∗⊗A) A 1A ❄ ✛ ≃ I ⊗A ✛ ǫA ⊗1A (A ⊗A∗) ⊗A ≃ ❄ (1) 3 and A∗ ≃✲I ⊗A∗ηA ⊗1A∗✲(A∗⊗A) ⊗A∗ A∗ 1A∗ ❄ ✛ ≃ A∗⊗I ✛ 1A∗⊗ǫA A∗⊗(A ⊗A∗) ≃ ❄ (2) both commute. Alternatively, a compact closed category may be defined as a ∗-autonomous category [Barr 1979] with a self-dual tensor, hence a model of ‘degenerate’ linear logic [Seely 1998]. For each morphism f : A →B in a compact closed category we can construct a dual f ∗, a name ⌜f⌝and a coname ⌞f

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