We introduce a notion of quantum function, and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication, and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple. We analyse dualisable and invertible 1-morphisms in these 2-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.

We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertextransitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.

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We present a new scheme for teleporting a quantum state between two parties whose local reference frames are misaligned by the action of a finite symmetry group. Unlike other proposals, our scheme requires the same amount of classical communication and entangled resources as conventional teleportation, does not reveal any reference frame information, and is robust against changes in reference frame alignment while the protocol is underway. The mathematical foundation of our scheme is a unitary error basis which is permuted up to a phase by the conjugation action of the group. We completely classify such unitary error bases for qubits, exhibit constructions in higher dimension, and provide a method for proving nonexistence in some cases. IntroductionMotivation. It is now well recognized that a shared reference frame is an implicit assumption underlying the correct execution of many quantum protocols [4,18,13,34,14,31]. As quantum communication finds its way into handheld devices [35,9,10] and into space [26,38,2], it is increasingly important to develop protocols robust against reference frame error for situations where alignment is difficult [16,17,28] or undesired [3,15]. Considerable progress has already been made in this regard for quantum key distribution [8,39,36,20,29,19,30], and there is also a smaller body of work on quantum teleportation [6,21,22] without a shared reference frame, which our results extend.Main results. We consider the problem of quantum teleportation between two parties whose local reference frames are misaligned, where the set of possible local reference frame transformations forms a finite group G with a unitary representation ρ : G → U(d) on the d-dimensional system to be teleported. (This is the first paper in a series; the second paper [33] extends these results to the more common setting of infinite groups.) Success of the protocol is judged by a third-party observer who holds full reference frame information, and who must agree that the original state has been teleported perfectly up to a global phase. 1 We present a teleportation scheme for certain (G, ρ), where G is finite, which is guaranteed to succeed regardless of the parties' reference frame configurations and which additionally satisfies the following properties.• Tightness. The parties only require a d-dimensional maximally entangled resource state, and only 2 dits of classical information are communicated from Alice to Bob.• Dynamical robustness (DR). The scheme is not affected by changes in reference frame alignment during transmission of the classical message from Alice to Bob.• No reference frame leakage (NL). No information about either party's reference frame alignment is transmitted. 2Our scheme depends on the existence of a G-equivariant unitary error basis for the representation (G, ρ). We exhaustively classify these mathematical structures for two-dimensional representations, showing that they exist precisely when the image of the composite homomorphism G ρ → U(2) q → SO(3) is isomorphic to 1, Z 2 , Z 3 , Z 4 , ...

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