Part I: We present two new schemes for teleportation of a quantum state between parties whose local reference frames are misaligned by the action of a compact Lie group G. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference frame alignment during execution, suiting them to situations of rapid reference frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially ---including the case of qubit teleportation under arbitrary SU(2) reference frame uncertainty--- while communicating no information about either party's reference frame alignment at any time. Our perfect scheme performs perfect teleportation, but does communicate some reference frame information. The mathematical foundation of these schemes is a unitary error basis permuted up to a phase by the conjugation action of a finite subgroup of G; we completely classify such unitary error bases for qubits, exhibit constructions in higher dimension, and provide a method for proving nonexistence in some cases. Part II: Quantum pseudo-telepathy is a phenomenon whereby two non-communicating parties can use the non-signalling correlations from shared quantum entanglement to perform a task that would ordinarily be impossible without communication. Here we consider the graph isomorphism game: an instance of the game is defined by a pair of graphs, and a perfect deterministic classical strategy precisely corresponds to an isomorphism of those graphs. We call a pair of graphs for which a perfect strategy using shared entanglement (a quantum strategy) exists pseudo-telepathic, and call a quantum strategy a quantum isomorphism between the graphs. Using a deep connection between noncommutative mathematics and nonlocal games, we classify all graphs quantum isomorphic to a given graph in terms of the quantum automorphism group of the graph. This implies a construction of quantum isomorphisms from central type subgroups of the ordinary automorphism group; we give an explicit description of the resulting quantum strategy and the new graph. We consider generation of new examples of quantum contextuality from central type groups. For abelian central type groups we show that this reduces to construction of quantum solutions of a binary linear constraint system from the abelianisation of its homogeneous solution group.
Part I: We present two new schemes for teleportation of a quantum state between parties whose local reference frames are misaligned by the action of a compact Lie group G. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference frame alignment during execution, suiting them to situations of rapid reference frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially ---including the case of qubit teleportation under arbitrary SU(2) reference frame uncertainty--- while communicating no information about either party's reference frame alignment at any time. Our perfect scheme performs perfect teleportation, but does communicate some reference frame information. The mathematical foundation of these schemes is a unitary error basis permuted up to a phase by the conjugation action of a finite subgroup of G; we completely classify such unitary error bases for qubits, exhibit constructions in higher dimension, and provide a method for proving nonexistence in some cases. Part II: Quantum pseudo-telepathy is a phenomenon whereby two non-communicating parties can use the non-signalling correlations from shared quantum entanglement to perform a task that would ordinarily be impossible without communication. Here we consider the graph isomorphism game: an instance of the game is defined by a pair of graphs, and a perfect deterministic classical strategy precisely corresponds to an isomorphism of those graphs. We call a pair of graphs for which a perfect strategy using shared entanglement (a quantum strategy) exists pseudo-telepathic, and call a quantum strategy a quantum isomorphism between the graphs. Using a deep connection between noncommutative mathematics and nonlocal games, we classify all graphs quantum isomorphic to a given graph in terms of the quantum automorphism group of the graph. This implies a construction of quantum isomorphisms from central type subgroups of the ordinary automorphism group; we give an explicit description of the resulting quantum strategy and the new graph. We consider generation of new examples of quantum contextuality from central type groups. For abelian central type groups we show that this reduces to construction of quantum solutions of a binary linear constraint system from the abelianisation of its homogeneous solution group.
We present two schemes for quantum teleportation between parties whose local reference frames are misaligned by the action of a compact Lie group G. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference-frame alignment during execution, suiting them to situations of rapid reference-frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially---this includes the case of qubit teleportation under arbitrary SU(2) reference-frame uncertainty---while communicating no information about either party's reference-frame alignment at any time. Our perfect scheme performs perfect teleportation, but does communicate some reference-frame information. The mathematical foundation of these schemes is a unitary error basis permuted up to a phase by the conjugation action of a finite subgroup of G.
We present a 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 under way. 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.
Quantum permutations, or magic unitaries, have in recent years been explored in the context of identifying `genuinely quantum' isometries of graphs. Here, we import this tool in physics, showing that quantum permutations yield a generalisation of quantum reference frames in a discrete setting that is reminiscent of the passage from special to general relativity. We show that the typical quantum reference frames framework corresponds to quantum permutations classified as `classical' in the mathematical literature, and which we demonstrate are quantum controlled transformations (superpositions of classical coordinate maps). Genuinely quantum permutations (i) allow to construct \emph{non--commuting} quantum reference frames (ii) correspond to \emph{local}, as opposed to global, superpositions of transformations. Strikingly, we find that the non-commutativity of quantum fields, when used as reference systems, is exactly what implies that the change of frame is achieved through a genuine quantum permutation. We illustrate the above with several examples in both first and second quantization formalism, which demonstrate (a) simultaneous control on non--commuting variables, (b) the existence of bipartite states that can be localized with a genuine quantum permutation and cannot be localized with the usual quantum reference frame transformations (without introducing additional degrees of freedom), (c) extension of the Ising model symmetries to genuinely quantum permutations, and (d) extension of the symmetries of a scalar field action on curved spacetime to genuinely quantum permutations. While we have in mind applications in quantum gravity, we expect our formalism to be of interest in a wide range of topics in quantum information.
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 vertex-transitive 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.
We give a tight scheme for teleporting a quantum state between two parties whose reference frames are misaligned by an action of a finite symmetry group. Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the entangled resource is of the same size. In the terminology of Peres and Scudo, our protocol relies on classical communication of unspeakable information.
This work develops a process-based interpretation of quantum teleportation, quantum switching, and time, grounded in discrete process graphs rather than continuous space–time dynamics. We introduce a formal framework in which the fundamental unit of physical evolution is not the quantum state itself, but a discrete process step (process tact), represented as a unitary transformation acting on a structured process graph. Quantum teleportation is reinterpreted as a transition between isomorphic index layers of the same process graph, rather than as a transmission of a quantum state through space–time. In this framework, entanglement establishes a shared structural constraint, measurement performs index selection, and classical communication transfers index information only. The quantum state remains invariant under graph isomorphism, and teleportation is formalized as reindexation without a physical carrier. Time is shown to emerge as an ordering relation between process steps and index layers, not as a fundamental parameter. Changes in temporal ordering correspond to transitions between equivalent graph embeddings, enabling a natural interpretation of temporal displacement as index reassignment within a recursive process structure. Geometrically, these ideas are represented using recursive pentagon–pentagram embeddings, which serve as an exact structural analogy for quantum switching. The pentagram functions as a universal quantum switch, mediating transitions between distinct but isomorphic orderings. This construction provides an intuitive and mathematically consistent representation of indefinite causal order. The framework is directly connected to EPR correlations, quantum teleportation protocols, and the quantum switch, and is shown to be naturally compatible with the Everett (many-worlds) interpretation of quantum mechanics. Within this ontology, the wave function is treated as an objective process structure, and no collapse postulate is required. This work builds upon and extends previously published results on graph-based quantum key switching and emergent space–time structure, unifying them into a single process-theoretic description of quantum information transfer, causal order, and temporal indexing. Reference to previous foundational work: DOI 10.5281/zenodo.18498181. Reference to previous foundational work: DOI 10.5281/zenodo.18504244.