Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2303.14002 · Quantum

Operational Quantum Reference Frame Transformations

Titouan Carette, Jan Glowacki, Leon Loveridge

Published Mar 27, 2025• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2025-03-27-1680

Related papers

Operational Quantum Frames: An operational approach to quantum reference frames

Jan Głowacki

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.

arXiv:2304.07021•Apr 14, 2023•2 citations

Quantum Reference Frames, Measurement Schemes and the Type of Local Algebras in Quantum Field Theory

Christopher J. Fewster, Daan W. Janssen, Leon Loveridge +2 more

© The Author(s) 2024. We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.

•Dec 18, 2024•0 citations

Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory

CJ Fewster, DW Janssen, LD Loveridge +2 more

We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed by the existence of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type $\textnormal{II}_1$ factor. Our results build upon recent work of Chandrasekaran, Longo, Penington and Witten [JHEP 2023, 82 (2023)], providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.

•Jan 1, 2025•0 citations

Quantum Reference Frames and The Poincaré Symmetry

Oleg Kabernik

The concept of a reference frame has been a part of the physical formalism since the early days of physics. The reference frame was always understood as a physical system with respect to which states of other physical systems are described. With the adoption of quantum theory, our understanding of physical systems has fundamentally changed, yet we still rely on classical systems for the definition of reference frames. If we acknowledge the fact that all systems are quantum mechanical, including reference frames, then we have to be able to describe quantum systems relative to other quantum systems. \n \nIn this thesis we develop a framework for integrating quantum reference frames into the formalism of quantum mechanics. We show how the description of states and measurements has to be refined in order to account for the quantum nature of the reference frame. Initially, we study the implications of the new description with a quantum reference frame associated with a generic compact group. Then, we apply the same approach to the study of relativistic quantum reference frames associated with the Poincaré group, which is not compact. \n \nOur findings for the generic compact group include an analysis of how well a quantum system defines a frame of reference for other systems. Also in the generic case we analyze the effects of a measurement on a quantum reference frame. In the relativistic case we first find how the massive representations of the Poincaré group decompose into irreducible representations. This is a key problem in the analysis of quantum reference frames. Finally, we analyze a relativistic quantum reference frame that is used for measuring the total momentum. We will show how the relativistic measurements of total momentum should be described with respect to a quantum reference frame. We will also show how these measurements affect the quantum reference frame.

•Sep 19, 2014•2 citations

Quantum Reference Frames, Measurement Schemes and the Type of Local Algebras in Quantum Field Theory

Christopher J. Fewster, Daan W. Janssen, Leon Loveridge +2 more

Abstract We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type $$\text {II}_1$$ II 1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780 ), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.

arXiv:2403.11973•Communications in Mathematical Physics•Dec 18, 2024•43 citations

Relativization is naturally functorial

Jan Głowacki

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.

arXiv:2403.03755•Mar 6, 2024•1 citation
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.
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.
© The Author(s) 2024. We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.
We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed by the existence of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1\textnormal{II}_1II1​ factor. Our results build upon recent work of Chandrasekaran, Longo, Penington and Witten [JHEP 2023, 82 (2023)], providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.
The concept of a reference frame has been a part of the physical formalism since the early days of physics. The reference frame was always understood as a physical system with respect to which states of other physical systems are described. With the adoption of quantum theory, our understanding of physical systems has fundamentally changed, yet we still rely on classical systems for the definition of reference frames. If we acknowledge the fact that all systems are quantum mechanical, including reference frames, then we have to be able to describe quantum systems relative to other quantum systems. \n \nIn this thesis we develop a framework for integrating quantum reference frames into the formalism of quantum mechanics. We show how the description of states and measurements has to be refined in order to account for the quantum nature of the reference frame. Initially, we study the implications of the new description with a quantum reference frame associated with a generic compact group. Then, we apply the same approach to the study of relativistic quantum reference frames associated with the Poincaré group, which is not compact. \n \nOur findings for the generic compact group include an analysis of how well a quantum system defines a frame of reference for other systems. Also in the generic case we analyze the effects of a measurement on a quantum reference frame. In the relativistic case we first find how the massive representations of the Poincaré group decompose into irreducible representations. This is a key problem in the analysis of quantum reference frames. Finally, we analyze a relativistic quantum reference frame that is used for measuring the total momentum. We will show how the relativistic measurements of total momentum should be described with respect to a quantum reference frame. We will also show how these measurements affect the quantum reference frame.
Abstract We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1\text {II}_1II1​ II 1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780 ), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.
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.