April 2025

This paper initiates a systematic study of operators arising as integrals of operator-valued functions with respect to positive operator-valued measures and utilizes these tools to provide relativization maps (Yen) for quantum reference frames (QRFs) defined on general homogeneous spaces. Properties of operator-valued integration are first studied and then employed to define general relativization maps and show their properties. The relativization maps presented here are defined for QRFs (systems of covariance) based on arbitrary homogeneous spaces of locally compact second countable topological groups and are shown to be contracting quantum channels, injective for localizable (norm-1 property) frames and multiplicative for the sharp ones (PVMs), extending the existing results.

This paper presents a research program aimed at establishing relational foundations for relativistic quantum physics. Although the formalism is still under development, we believe it has matured enough to be shared with the broader scientific community. Our approach seeks to integrate Quantum Field Theory on curved backgrounds and scenarios with indefinite causality. Building on concepts from the operational approach to Quantum Reference Frames, we extend these ideas significantly. Specifically, we initiate the development of a general integration theory for operator-valued functions (quantum fields) with respect to positive operator-valued measures (quantum frames). This allows us to define quantum frames within the context of arbitrary principal bundles, replacing group structures. By considering Lorentz principal bundles, we enable a relational treatment of quantum fields on arbitrarily curved spacetimes. A form of indefinite spatiotemporality arises from quantum states in the context of frame bundles. This offers novel perspectives on the problem of reconciling principles of generally relativistic and quantum physics and on modelling gravitational fields sourced by quantum systems.

We draw the current landscape of quantum algorithms, by classifying about 130 quantum algorithms, according to the fundamental mathematical problems they solve, their real-world applications, the main subroutines they employ, and several other relevant criteria. The primary objectives include revealing trends of algorithms, identifying promising fields for implementations in the NISQ era, and identifying the key algorithmic primitives that power quantum advantage.

We discuss a systematic way in which a relational dynamics can be established relative to periodic clocks both in the classical and quantum theories, emphasising the parallels between them. We show that: (1) classical and quantum relational observables that encode the value of a quantity relative to a periodic clock are only invariant along the gauge orbits generated by the Hamiltonian constraint if the quantity itself is periodic, and otherwise the observables are only transiently invariant per clock cycle (this implies, in particular, that counting winding numbers does not lead to invariant observables relative to the periodic clock); (2) the quantum relational observables can be obtained from a partial group averaging procedure over a single clock cycle; (3) there is an equivalence (‘trinity’) between the quantum theories based on the quantum relational observables of the clock-neutral picture of Dirac quantisation, the relational Schr”odinger picture of the Page-Wootters formalism, and the relational Heisenberg picture that follows from quantum deparametrisation, all three taken relative to periodic clocks (implying that the dynamics in all three is necessarily periodic); (4) in the context of periodic clocks, the original Page-Wootters definition of conditional probabilities fails for systems that have a continuous energy spectrum and, using the equivalence between the Page-Wootters and the clock-neutral, gauge-invariant formalism, must be suitably updated. Finally, we show how a system evolving periodically with respect to a periodic clock can evolve monotonically with respect to an aperiodic clock, without inconsistency. The presentation is illustrated by several examples, and we conclude with a brief comparison to other approaches in the literature that also deal with relational descriptions of periodic clocks.

Our fundamental theories, i.e., the quantum theory and general relativity, are invariant under time reversal. Only when we treat system from the point of view of thermodynamics, i.e., averaging between many subsystem components, an arrow of time emerges. The relation between thermodynamic and the quantum theory has been fertile, deeply explored and still a source of new investigations. The relation between the quantum theory and gravity, while it has not yet brought an established theory of quantum gravity, has certainly sparkled in depth analysis and tentative new theories. On the other hand, the connection between gravity and thermodynamics is less investigated and more puzzling. I review a selection of results in covariant thermodynamics, such as the construction of a covariant notion of thermal equilibrium by considering tripartite systems. I discuss how such construction requires a relational take on thermodynamics, similarly of what happens in the quantum theory and in gravity

Gaussian unitary transformations are generated by quadratic Hamiltonians, i.e., Hamiltonians containing quadratic terms in creations and annihilation operators, and are heavily used in many areas of quantum physics, ranging from quantum optics and condensed matter theory to quantum information and quantum field theory in curved spacetime. They are known to form a representation of the metaplectic and spin group for bosons and fermions, respectively. These groups are the double covers of the symplectic and special orthogonal group, respectively, and our goal is to analyze the behavior of the sign ambiguity that one needs to deal with when moving between these groups and their double cover. We relate this sign ambiguity to expectation values of the form $langle 0|exp{(-ihat{H})}|0rangle$, where $|0rangle$ is a Gaussian state and $hat{H}$ an arbitrary quadratic Hamiltonian. We provide closed formulas for $langle 0|exp{(-ihat{H})}|0rangle$ and show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space. Our construction relies on an explicit parametrization of these two groups (metaplectic, spin) in terms of symplectic and orthogonal group elements together with a twisted U(1) group.

In this paper, we study gravitational symmetry algebras that live on 2-dimensional cuts $S$ of asymptotic infinity. We define a notion of wedge algebra $mathcal{W}(S)$ which depends on the topology of $S$. For the cylinder $S=mathbb{C}^*$ we recover the celebrated $Lw_{1+infty}$ algebra. For the 2-sphere $S^2$, the wedge algebra reduces to a central extension of the anti-self-dual projection of the Poincar’e algebra. We then extend $mathcal{W}(S)$ outside of the wedge space and build a new Lie algebra $mathcal{W}_sigma(S)$, which can be viewed as a deformation of the wedge algebra by a spin two field $sigma$ playing the role of the shear at a cut of $mathscr{I}$. This algebra represents the gravitational symmetry algebra in the presence of a non trivial shear and is characterized by a covariantized version of the wedge condition. Finally, we construct a dressing map that provides a Lie algebra isomorphism between the covariant and regular wedge algebras.

We explore indefinite causal order between events in the context of quasiclassical spacetimes in superposition. We introduce several new quantifiers to measure the degree of indefiniteness of the causal order for an arbitrary finite number of events and spacetime configurations in superposition. By constructing diagrammatic and knot-theoretic representations of the causal order between events, we find that the definiteness or maximal indefiniteness of the causal order is topologically invariant. This reveals an intriguing connection between the field of quantum causality and knot theory. Furthermore, we provide an operational encoding of indefinite causal order and discuss how to incorporate a measure of quantum coherence into our classification.

We construct the spectral decomposition of field operators in bosonic quantum field theory as a limit of a strongly continuous family of POVM decompositions. The latter arise from integrals over families of bounded positive operators. Crucially, these operators have the same locality properties as the underlying field operators. We use the decompositions to construct families of quantum operations implementing measurements of the field observables. Again, the quantum operations have the same locality properties as the field operators. What is more, we show that these quantum operations do not lead to superluminal signaling and are possible measurements on quantum fields in the sense of Sorkin.

We present thought experiments to measure the Arnowitt-Deser-Misner and Bondi-Sachs energy of isolated systems in general relativity. The expression of the Bondi-Sachs energy used in the protocol is likely to have other applications. In particular, it is well-suited to to be promoted to an operator in non-perturbative loop quantum gravity.