arXiv subjects

Jan Głowacki

Publications and source records attributed to Jan Głowacki.

9 recordsLinked to original sources

arXiv2026

Quantum Reference Frames on Homogeneous Spaces

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.

quant-ph↗

arXiv2026

W*-algebraic Integration Theory

Given a pair of $\mathrm{W}^*$-algebras $(\mathcal{M}_\mathcal{S},\mathcal{M}_\mathcal{R})$ with $(\mathcal{M}_\mathcal{S})_*$ separable, a measurable space $(Σ, \mathcal{F})$ and a POVM $\mathsf{E}: \mathcal{F} \to \mathcal{E}(\mathcal{M}_\mathcal{R})$, the integral of a function $f: Σ\to \mathcal{M}_\mathcal{S}$ is defined as an element of the spatial tensor product $\int f \otimes d\mathsf{E} \in \mathcal{M}_\mathcal{S} \bar{\otimes} \mathcal{M}_\mathcal{R}$. The space $B_b(Σ,\mathcal{F},\mathcal{M}_\mathcal{S})$ of uniformly bounded ultraweakly measurable functions is the universal domain of integration; once $\mathsf{E}$ is fixed it refines to the quotient $L^\infty_\mathsf{E}(Σ,\mathcal{M}_\mathcal{S}) = B_b(Σ,\mathcal{F},\mathcal{M}_\mathcal{S})/\mathcal{N}_\mathsf{E}$ by $\mathsf{E}$-null functions. When $(\mathcal{M}_\mathcal{R})_*$ is also separable, $L^\infty_\mathsf{E}(Σ,\mathcal{M}_\mathcal{S}) \cong \mathcal{M}_\mathcal{S} \bar{\otimes} L^\infty_\mathsf{E}(Σ)$ is a $\mathrm{W}^*$-algebra. The integration map is a faithful normal unital completely positive (CP) map, a $*$-homomorphism for PVMs and an isometry for localizable POVMs. It can be identified with the spatial tensor product $\boldsymbol{1}_{\mathcal{M}_\mathcal{S}} \hat{\otimes} Φ_\mathsf{E}$ where $Φ_\mathsf{E}: L^\infty_\mathsf{E}(Σ) \to \mathcal{M}_\mathcal{R}$ is the faithful normal positive map corresponding to $\mathsf{E}$. Complete positivity of integration maps is derived from Stinespring factorization through Naimark dilation. We establish an operator-valued Leibniz rule and Fubini theorem.

math-ph↗

arXiv2026

Foundations of Relational Quantum Field Theory I: Scalars

We develop foundations for a relational approach to quantum field theory (RQFT) based on the operational quantum reference frames (QRFs) framework considered in a relativistic setting. Unlike other efforts in combining QFT with QRFs, we use the latter to provide novel mathematical and conceptual foundations for the former. We focus on scalar fields in Minkowski spacetime and discuss the emergence of relational local (bounded) observables and (pointwise) fields from the consideration of Poincaré-covariant (quantum) frame observables defined over the space of (classical) inertial reference frames. We recover a relational notion of Poincaré covariance, with transformations on the system directly linked to the state preparations of the QRF. We introduce and analyse various causality conditions, and construct an explicit example of a covariant scalar relational quantum field which is causal relative to operationally meaningful preparations of a relativistic QRF. The theory makes direct contact with established foundational approaches to QFT. We demonstrate that the vacuum expectation values derived within our framework reproduce many of the essential properties of Wightman functions and carry out a detailed comparison of the proposed formalism with Wightman QFT with the frame smearing functions describing the QRF's localisation uncertainty playing the role of the Wightmanian test functions. We also show how the properties of algebras generated by relational local observables suitably extend the core axioms of Algebraic QFT. This work is an early step in revisiting the mathematical foundations of QFT from a relational and operational perspective.

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arXiv2025

Operational Quantum Reference Frame Transformations

Quantum reference frames are needed in quantum theory for much the same reasons that reference frames are in classical theories: to manifest invariance in line with fundamental relativity principles and to provide a basis for the definition of observable quantities. Though around since the 1960s, and used in a wide range of applications, only recently has the means for transforming descriptions between different quantum reference frames been tackled in detail. In this work, we provide a general, operationally motivated framework for quantum reference frames and their transformations, holding for locally compact groups. The work is built around the notion of operational equivalence, in which quantum states that cannot be physically distinguished are identified. For example, we describe the collection of relative observables as a subspace of the algebra of invariants on the composite of system and frame, and from here the set of relative states is constructed through the identification of states which cannot be distinguished by relative observables. Through the notion of framed observables -- the formation of joint observables of system and frame -- of which the relative observables can be understood as examples, quantum reference frame transformations are then maps between equivalence classes of relative states which respect the framing. We give an explicit realisation in the setting that the initial frame admits a highly localized state with respect to the frame observable. The transformations are invertible exactly when the final frame also has such a localizability property. The procedure we present is in operational agreement with other recent inequivalent constructions on the domain of common applicability, but extends them in a number of ways, and weakens claims of entanglement generation through frame changes.

quant-ph↗

arXiv2024

Towards Relational Quantum Field Theory

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.

quant-ph↗

arXiv2024

Relativization is naturally functorial

In this note, we provide some categorical perspectives on the relativization construction arising from quantum measurement theory in the presence of symmetries and occupying a central place in the operational approach to quantum reference frames. This construction provides, for any quantum system, a quantum channel from the system's algebra to the invariant algebra on the composite system also encompassing the chosen reference, contingent upon a choice of the pointer observable. These maps are understood as relativizing observables on systems upon the specification of a quantum reference frame. We begin by extending the construction to systems modelled on subspaces of algebras of operators to then define a functor taking a pair consisting of a reference frame and a system and assigning to them a subspace of relative operators defined in terms of an image of the corresponding relativization map. When a single frame and equivariant channels are considered, the relativization maps can be understood as a natural transformation. Upon fixing a system, the functor provides a novel kind of frame transformation that we call external. Results achieved provide a deeper structural understanding of the framework of interest and point towards its categorification and potential application to local systems of algebraic quantum field theories.

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arXiv2023

Operational Quantum Frames: An operational approach to quantum reference frames

The quantum reference frames program is based on the idea that reference frames should be treated as quantum physical systems. In this work, we combine these insights with the emphasis on operationality, understood as refraining from introducing into the framework objects not directly related to in principle verifiable probabilities of measurement outcomes, and identifying the setups indistinguishable as such. Based on intuitions from special relativity and gauge theory, we introduce an operational notion of a quantum reference frame -- which is defined as a quantum system equipped with a covariant positive operator-valued measure (POVM) -- and build a framework on the concept of operational equivalence that allows us to enforce operationality by quotienting the quantum state spaces with equivalence relation of indistinguishability by the available effects, assumed to be invariant under gauge transformations, and framed in the sense of respecting the choice of the frame's POVM. Such effects are accessed via the yen construction, which maps effects on the system to those on the composite system, satisfying gauge invariance and framing. They are called relative, and the classes of states indistinguishable by them are referred to as relative states. We show that when the frame is localizable, meaning that it allows for states that give rise to a highly localized probability distribution of the frame's observable, by restricting the relative description upon such localized frame preparation we recover the usual, non-relational formalism of quantum mechanics. We provide a consistent way of translating between different relative descriptions by means of frame-change maps and compare these with the corresponding notions in other approaches to QRF, establishing an operational agreement in the domain of common applicability.

quant-ph↗

arXiv2023

Quantum Reference Frames on Finite Homogeneous Spaces

We present an operationally motivated treatment of quantum reference frames in the setting that the frame is a covariant positive operator valued measure (POVM) on a finite homogeneous space, generalising the principal homogeneous spaces studied in previous work. We focus on the case that the reference observable is the canonical covariant projection valued measure on the given space, and show that this gives rise to a rank-one covariant POVM on the group, which can be seen as a system of coherent states, thereby making contact with recent work in the perspective-neutral approach to quantum reference frames.

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arXiv2022

Fact-nets: towards a mathematical framework for relational quantum mechanics

The relational interpretation of quantum mechanics (RQM) has received a growing interest since its first formulation in 1996. Usually presented as an interpretational layer over the usual quantum mechanics formalism, it appears as a philosophical perspective without proper mathematical counterparts. This state of affairs has direct consequences on the scientific debate on RQM which still suffers from misunderstandings and imprecise statements. In an attempt to clarify those debates, the present paper proposes a radical reformulation of the mathematical framework of quantum mechanics which is relational from the start: fact-nets. The core idea is that all statements about the world, facts, are binary entities involving two systems that can be symmetrically thought of as observed and observer. We initiate a study of the fact-nets formalism and outline how it can shed new relational light on some familiar quantum features.

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