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Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models

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Abstract

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete “operational” approach, in which the states and measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces–generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly self-dual, play an important role in reconstructions of various quantum-information theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol—hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the dagger-compactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism.

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Notes

  1. Bifunctoriality means that: (i) \(1_{A\otimes B} = 1_{A} \otimes 1_{B}\); and (ii) given morphisms \(f : A \rightarrow X\) and \(g : B \rightarrow Y\) in \({\mathcal C}\), there is a canonical product morphism \(f \otimes g : A \otimes B \rightarrow X \otimes Y\), such that \( (f \otimes g) \circ (f' \otimes g') = (f \circ f') \otimes (g \circ g')\).

  2. Sometimes just compact.

  3. We use the notation \(A'\), rather than the more standard \(A^{\ast }\), for the designated dual of an object in a compact closed category, because we wish to reserve the latter to denote, specifically, the dual space of a vector space.

  4. The self-dual setting requires some additional coherence conditions; see [40]

  5. Those new to categories should note that a functor from \({\mathcal C}^{op}\) to \({\mathcal C}\) is sometimes called a contravariant functor from \({\mathcal C} \rightarrow {\mathcal C}\); the description we have just given (minus the involutiveness condition) defines this notion without reference to \({\mathcal C}^{op}\).

  6. We can be more precise here: given a dagger-compact category of states and processes \({\mathcal C}\), any dagger-monoidal functor from \({\mathcal C}\) to FDHilb, the category of finite dimensional Hilbert spaces, will send the scalar \(\beta ^{\dag } \circ \alpha \) to the inner product \( \ensuremath {\langle \beta \mid \alpha \rangle }\).

  7. This is a straightforward extension of the definition in [13] to the context of possibly non-saturated models.

  8. This is closely related to the notion of regular composite introduced in [9].

  9. Technically we are relying on the isomorphisms between \(A \cong {\mathcal C} (I,A)\) and \(A^\# \cong {\mathcal C} (A,I)\)to guarantee that the internal representation of \(\hat {\omega } \circ \hat {f}\) defines the right linear map.

  10. When the state space is sufficiently symmetric, there may be a natural choice of state invariant under the symmetry group. For example, if the base-preserving automorphisms act transitively on the pure states, the state obtained by group-averaging is the natural choice.

  11. Strictly speaking, [12] deals with the case in which A and B are saturated, but the proof is easily extended to the general case.

  12. See [37, 46] for some recent progress on this question.

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Acknowledgments

HB and AW wish to thank Samson Abramsky and Bob Coecke for enabling them to visit the Oxford University Computing Laboratory in November 2009, where some of this work was done, and for helpful discussions during that time. The authors also wish to thank Peter Selinger for helpful discussions. RD was supported by EPSRC postdoctoral research fellowship EP/E04006/1. HB’s research was supported by Perimeter Institute for Theoretical Physics; work at Perimeter Institute is supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

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Author notes

  1. Howard Barnum

    Present address: Department of Physics and Astronomy, University of New Mexico, 1919 Lomas Blvd. NE, MSC07 4220, Albuquerque, NM, 87131-0001, USA

Authors and Affiliations

  1. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, Canada, N2L 2Y5

    Howard Barnum

  2. Laboratoire d’Information Quantique, Universit Libre de Bruxelles, Boulevard du Triomphe, Bruxelles, 1050, Belgium

    Ross Duncan

  3. Department of Mathematics, Susquehanna University, Selinsgrove, PA, 17837, USA

    Alexander Wilce

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Correspondence to Alexander Wilce.

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Barnum, H., Duncan, R. & Wilce, A. Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models. J Philos Logic 42, 501–523 (2013). https://doi.org/10.1007/s10992-013-9280-8

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