Treating reference frames fundamentally as quantum systems is inevitable in quantum gravity and also in quantum foundations once considering laboratories as physical systems. Both fields thereby face the question of how to describe physics relative to quantum reference systems and how the descriptions relative to different such choices are related. Here, we exploit a fruitful interplay of ideas from both fields to begin developing a unifying approach to transformations among quantum reference systems that ultimately aims at encompassing both quantum and gravitational physics. In particular, using a gravity inspired symmetry principle, which enforces physical observables to be relational and leads to an inherent redundancy in the description, we develop a perspectiveneutral structure, which contains all frame perspectives at once and via which they are changed. We show that taking the perspective of a specific frame amounts to a fixing of the symmetry related redundancies in both the classical and quantum theory and that changing perspective corresponds to a symmetry transformation. We implement this using the language of constrained systems, which naturally encodes symmetries. Within a simple one-dimensional model, we recover some of the quantum frame transformations of [1], embedding them in a perspective-neutral framework. Using them, we illustrate how entanglement and classicality of an observed system depend on the quantum frame perspective. Our operational language also inspires a new interpretation of Dirac and reduced quantized theories as perspective-neutral and perspectival quantum theories, respectively, and reveals the link between them. In this light, we suggest a new take on the relation between a 'quantum general covariance' and the diffeomorphism symmetry in quantum gravity. * p.hoehn@univie.ac.at arXiv:1809.00556v2 [quant-ph]

Every clock is a physical system and thereby ultimately quantum. A naturally arising question is thus how to describe time evolution relative to quantum clocks and, specifically, how the dynamics relative to different quantum clocks are related. This is a particularly pressing issue in view of the multiple choice facet of the problem of time in quantum gravity, which posits that there is no distinguished choice of internal clock in generic general relativistic systems and that different choices lead to inequivalent quantum theories. Exploiting a recent unifying approach to switching quantum reference systems [Vanrietvelde et al 2020 Quantum 4 225; Vanrietvelde et al 2018 arXiv:1809.05093[quant-ph])], we exhibit a systematic method for switching between different clock choices in the quantum theory. We illustrate it by means of the parametrized particle, which, like gravity, features a Hamiltonian constraint. We explicitly switch between the quantum evolution relative to the non-relativistic time variable and that relative to the particle’s position, which requires carefully regularizing the zero-modes in the so-called time-of-arrival observable. While this toy model is simple, our approach is general and, in particular, directly amenable to quantum cosmology. It proceeds by systematically linking the reduced quantum theories relative to different clock choices via the clock-choice-neutral Dirac quantized theory, in analogy to coordinate changes on a manifold. This method suggests a new perspective on the multiple choice problem, indicating that it is rather a multiple choice feature of the complete relational quantum theory, taken as the conjunction of Dirac quantized and quantum deparametrized theories. Precisely this conjunction permits one to consistently switch between different temporal reference systems, which is a prerequisite for a quantum notion of general covariance. Finally, we show that quantum uncertainties generically lead to a discontinuity in the relational dynamics when switching clocks, in contrast to the classical case.

Given the importance of quantum reference systems to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of quantum reference systems, which is valid in both fields. Here, we continue with such a unifying approach, begun in [1], whose key ingredients are an operational language and a gravity-inspired symmetry principle. The latter enforces physics to be relational and leads, thanks to gauge related redundancies, to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure turns out to be the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. Quantum reference frame switches thereby amount to a symmetry transformation. In the quantum theory, they require a transformation that takes one from the Dirac to a reduced quantum theory and we show that it amounts to a trivialization of the constraints and a subsequent projection onto the classical gauge fixing conditions. We illustrate this method in a general mechanical particle model, namely the relational N -body problem in three-dimensional space with rotational and translational symmetry. This model is particularly interesting because it features the generic Gribov problem so that globally valid gauge fixing conditions are impossible which, in turn, implies also that globally valid relational frame perspectives are absent in both the classical and quantum theory. We will show that the constraint surface is topologically non-trivial and foliated by three-, five-and six-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. In consequence, the N -body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can systematically construct the quantum reference frame transformations for the three-body problem.

We argue that the quantum-theoretical structures studied in several recent lines of research cannot be adequately described within the standard framework of quantum circuits. This is in particular the case whenever the combination of subsystems is described by a nontrivial blend of direct sums and tensor products of Hilbert spaces. We therefore propose an extension to the framework of quantum circuits, given by routed linear maps and routed quantum circuits. We prove that this new framework allows for a consistent and intuitive diagrammatic representation in terms of circuit diagrams, applicable to both pure and mixed quantum theory, and exemplify its use in several situations, including the superposition of quantum channels and the causal decompositions of unitaries. We show that our framework encompasses the `extended circuit diagrams' of Lorenz and Barrett [arXiv:2001.07774 (2020)], which we derive as a special case, endowing them with a sound semantics.

No quantum circuit can turn a completely unknown unitary gate into its coherently controlled version. Yet, coherent control of unknown gates has been realised in experiments, making use of a different type of initial resources. Here, we formalise the task achieved by these experiments, extending it to the control of arbitrary noisy channels, and to more general types of control involving higher dimensional control systems. For the standard notion of coherent control, we identify the information-theoretic resource for controlling an arbitrary quantum channel on a $d$-dimensional system: specifically, the resource is an extended quantum channel acting as the original channel on a $d$-dimensional sector of a $(d+1)$-dimensional system. Using this resource, arbitrary controlled channels can be built with a universal circuit architecture. We then extend the standard notion of control to more general notions, including control of multiple channels with possibly different input and output systems. Finally, we develop a theoretical framework, called supermaps on routed channels, which provides a compact representation of coherent control as an operation performed on the extended channels, and highlights the way the operation acts on different sectors.

Existing work on quantum causal structure assumes that one can perform arbitrary operations on the systems of interest. But this condition is often not met. Here, we extend the framework for quantum causal modelling to cases where a system can suffer sectorial contraints, that is, restrictions on the orthogonal subspaces of its Hilbert space that may be mapped to one another. Our framework (a) proves that a number of different intuitions about causal relations turn out to be equivalent; (b) shows that quantum causal structures in the presence of sectorial constraints can be represented with a directed graph; and (c) defines a fine-graining of the causal structure in which the individual sectors of a system bear causal relations, which provides a more detailed analysis than its coarse-grained counterpart. As an example, we apply our framework to purported photonic implementations of the quantum switch to show that while their coarse-grained causal structure is cyclic, their fine-grained causal structure is acyclic. We therefore conclude that these experiments realize indefinite causal order only in a weak sense. Notably, this is the first argument to this effect that is not rooted in the assumption that the causal relata must be localized in spacetime.

We introduce a notion of compatibility between constraint encoding and compositional structure. Phrased in the language of category theory, it is given by a composable constraint encoding. We show that every composable constraint encoding can be used to construct an equivalent notion of a constrained category in which morphisms are supplemented with the constraints they satisfy. We further describe how to express the compatibility of constraints with additional categorical structures of their targets, such as parallel composition, compactness, and time-symmetry. We present a variety of concrete examples. Some are familiar in the study of quantum protocols and quantum foundations, such as signalling and sectorial constraints; others arise by construction from basic categorical notions. We use the language developed to discuss the notion of intersectability of constraints and the simplifications it allows for when present, and to show that any time-symmetric theory of relational constraints admits a faithful notion of intersection.

Over the past decade, a number of quantum processes have been proposed which are logically consistent, yet feature a cyclic causal structure. However, there exists no general formal method to construct a process with an exotic causal structure in a way that ensures, and makes clear why, it is consistent.Here we provide such a method, given by an exended circuit formalism. This only requires directed graphs endowed with boolean matrices, which encode basic constraints on operations. Our framework (a) defines a set of elementary rules for checking the validity of any such graph, (b) provides a way of constructing consistent processes as a circuit from valid graphs, and (c) yields an intuitive interpretation of the causal relations within a process and an explanation of why they do not lead to inconsistencies. We display how several standard examples of exotic processes, including ones that violate causal inequalities, are among the class of processes that can be generated in this way; we conjecture that this class in fact includes all unitarily extendible processes.