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:2007.15592 · Physical Review A

Quantifying resources for the Page-Wootters mechanism: Shared asymmetry as relative entropy of entanglement

Rafael Simões do Carmo, Diogo O. Soares-Pinto

Published May 18, 2021• 12 citations
arXivHTMLPDF

Abstract

Recently, some attention has been given to the so-called Page-Wootters mechanism of quantum clocks. Among the various proposals to explore the mechanism using more modern techniques, some have chosen to use a quantum information perspective, defining and using informational measures to quantify how well a quantum system can stand as a reference frame for other quantum system. In this work, we explore the proposal based on a resource theory of asymmetry, known as mutual or shared asymmetry, which actually is equivalent to the approach from coherence theory in the case of interest here: quantum reference frames described by the $\text{U}(1)$ compact group. We extend some previous results in the literature about shared asymmetry and the Page-Wootters mechanism to more general cases, culminating in the enunciation of a theorem relating shared asymmetry of a bipartite state ${\ensuremath{\rho}}_{SR}$ with the relative entropy of entanglement of internal states ${\ensuremath{\rho}}_{M}$ on the charge sectors of the Hilbert space ${\mathcal{H}}_{S}\ensuremath{\bigotimes}{\mathcal{H}}_{R}$. Using this result, we reinterpret the relation between the Page-Wootters mechanism and entanglement, and we also open some paths to further studies.

DOI
10.1103/physreva.103.052420

Related papers

Quantifying quantum reference frames in composed systems: Local, global, and mutual asymmetries

Tiago Martinelli, Diogo O. Soares-Pinto

The Page-Wootters mechanism questioned the fundamental nature of time in quantum physics. The mechanism explored the notion that a given physical quantity is always defined and measured relative to a reference frame, in general, not explained in the theoretical description of quantum physical experiments. Recently, the resource theory of asymmetry deals explicitly with the physical conditions necessary for a quantum system to serve as a good reference frame. Nonetheless, to quantify a quantum reference frame in relation to another one, it is important to establish an internal description of quantum theory, i.e., without the need for a classical reference frame. In this work, we address this issue with the concept of mutual asymmetry and use this machinery in the Page-Wootters mechanism by identifying the concept of mutual asymmetry as mutual or internal coherence. To do so, the notion of quantum coherence in relation to a quantum reference frame is revisited and a quantifier is proposed in this scenario. Also, this opens space to investigate the link of internal coherence and correlations, as proposed by Page and Wootters, under a resource theory approach.

arXiv:1808.04209•Physical Review A•Apr 26, 2019•18 citations

Entanglement-Asymmetry Correspondence for Internal Quantum Reference Frames

Anne-Catherine de la Hamette, Stefan L. Ludescher, Markus P. Müller

In the quantization of gauge theories and quantum gravity, it is crucial to treat reference frames such as rods or clocks not as idealized external classical relata, but as internal quantum subsystems. In the Page-Wootters formalism, for example, evolution of a quantum system S is described by a stationary joint state of S and a quantum clock, where time dependence of S arises from conditioning on the value of the clock. Here, we consider (possibly imperfect) internal quantum reference frames R for arbitrary compact symmetry groups, and show that there is an exact quantitative correspondence between the amount of entanglement in the invariant state on RS and the amount of asymmetry in the corresponding conditional state on S. Surprisingly, this duality holds exactly regardless of the choice of coherent state system used to condition on the reference frame. Averaging asymmetry over all conditional states, we obtain a simple representation-theoretic expression that admits the study of the quality of imperfect quantum reference frames, quantum speed limits for imperfect clocks, and typicality of asymmetry in a unified way. Our results shed light on the role of entanglement for establishing asymmetry in a fully symmetric quantum world.

arXiv:2112.00046•Physical Review Letters•Dec 22, 2022•15 citations

The resource theory of asymmetry and some of its applications

Mehdi Ahmadi

A useful and unifying concept in quantum information theory is the idea of a consumable "resource". The basic idea behind quantum resource theories is that whenever we face restrictions on the set of operations we can perform, some states (resource states) enable us to perform particular tasks that otherwise would have been impossible. In this thesis we first review a resource theory that recently has gained a lot of attention, namely the resource theory of asymmetry in which one is restricted to symmetric operations and therefore asymmetric states are the resource states. The asymmetry properties of a state relative to some symmetry group specify how and to what extent the given symmetry is broken by the state. In the remainder of the thesis we explain how the ideas developed in this resource theory are related to three different areas in quantum theory: theory of quantum reference frames, the Wigner-Araki-Yanase(WAY) theorem and the conditional probability interpretation of time in quantum mechanics. Theory of quantum reference frames treats reference frames like any other physical system within the formalism of quantum theory, which causes the measurements to be an approximation of the measurements against their classical counterparts. We consider the dynamics of a quantum directional reference frame undergoing repeated interactions. These interactions induce a back-action on the reference which is the central focus of our study. The effect of a precise sequence of measurement outcomes on the reference frame is studied by looking at both the case that the measurement record is averaged over and the case wherein it is retained. We find, in particular, that there is interesting dynamics in the latter situation, which cannot be revealed by considering the average case. We then consider in detail how a sequence of rotationally invariant unitary interactions affects the reference frame, a situation, which leads to quite different dynamics than the case of repeated measurements. Different strategies for correcting reference frame drift is considered given that we have access to a set of particles with polarization opposite to the direction of drift. In particular, we find that, by implementing a suitably chosen unitary interaction after every two measurements, we can eliminate the rotational drift of the reference frame. The WAY theorem establishes an important constraint that conservation laws impose on quantum mechanical measurements. We formulate the WAY theorem in the broader context of resource theories, where one is constrained to a subset of quantum mechanical operations described by a symmetry group. Establishing connections with the theory of quantum state discrimination we obtain optimal unitaries describing the measurement of arbitrary observables, explain how prior information can permit perfect measurements that circumvent the WAY constraint, and provide a framework that establishes a natural ordering on measurement apparatuses through a decomposition into asymmetry and charge subsystems. Finally we review two di erent schemes in the conditional probability interpretation (CPI) of time in quantum mechanics. In these schemes parameter time "t" as it appears in Schrodinder equation is considered to be unobservable, yet one can use an extension of conditional probabilities in order to study dynamics of a system relative to another system, namely the quantum clock. We phrase CPI as a quantum communication protocol which enables us to use the machinery developed in the resource theory of asymmetry. This provides us with a deeper understanding of the decoherence effect caused due to the inaccessibility of the parameter time "t" and that how this effect can be suppressed by choosing an optimal initial state for the quantum clock.

•Oct 1, 2012•2 citations

Mecanismo de Page-Wootters: o papel da coerência e relógios finitos

Mendes, Leandro Raffhael da Silva

Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, this mechanism explored a stationary pure bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge and an equation of motion can be obtained for a quantum particle conditioned to the measurement of the state of the clock. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page–Wootters model. Connecting it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence. In a second step we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the systems initial conditions.

•Oct 25, 2021•0 citations

Interacting quantum reference frames

Leonardo Chataignier, Alexander R. H. Smith

We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space $\mathcal{H}_{\rm kin}$ by group averaging. We assume the gauge group $G$ is unimodular, and we define a quantum reference frame as a system of coherent states associated with a group $G_R$ of frame orientations acting on a tensor factor of $\mathcal{H}_{\rm kin}$. In direct analogy with the Faddeev-Popov procedure in path integrals, we show that the physical inner product can be written in terms of a choice of frame orientation with an appropriate operator measure, but that the physical Hilbert space is independent of this choice of frame. This yields a generalization of the Page-Wootters reduction map used to define conditional states and conditional probabilities of measurement outcomes given a frame orientation. These probabilities correspond to the expectation values of relational Dirac observables obtained from generalizations of previously considered relativization maps. Our main result is that the map that implements a change of frame is unitary, even when $G$ does not act as a tensor-product representation across a factorization of $\mathcal{H}_{\rm kin}$. The formalism therefore accommodates interactions between reference frames and the systems described relative to them. We apply this formalism to temporal frames, showing that a unitary Schrödinger equation emerges for an interacting clock and system, and that a previously reported time-nonlocal Schrödinger equation is recovered in an appropriate limit. With this, problems of non-unitarity are avoided by the use of the gauge-fixing measure in the physical inner product. We also discuss the case of interacting spatial frames, where the known correspondence between coherence in one frame and entanglement in another is shown to depend on the coupling.

quant-pharXiv:2610.28732•Oct 22, 2026•0 citations

Quantum Time: a novel resource for quantum information

M. B. Altaie

Time in relativity theory has a status different from that adopted by standard quantum mechanics, where time is considered as a parameter measured with reference to an external absolute Newtonian frame. This status strongly restricts its role in the dynamics of systems and hinders any formulation to merge quantum mechanics with general relativity, specifically when considering quantum gravity. To overcome those limitations, several authors tried to construct an operator which is conjugate to the Hamiltonian of quantum systems implementing some essential features of the relativistic time. These formulations use the concept of internal or intrinsic time instead of the universal coordinate time used in textbooks. Furthermore, recently it is remarked that the consideration of time with relativistic features could enhance the analysis techniques in quantum information processing and have an impact on its status in causal orders and causal structures of quantum information. The role of clocks, their accuracy and stability has become an important issue in quantum information processing. This article present a substantiative review of recent works which reflect the possibility of utilizing quantum time, measured by quantum clock devised according to Page-Wootters scheme, to stand as a resource for quantum information processing.

arXiv:2310.16617•Oct 25, 2023•0 citations
Recently, some attention has been given to the so-called Page-Wootters mechanism of quantum clocks. Among the various proposals to explore the mechanism using more modern techniques, some have chosen to use a quantum information perspective, defining and using informational measures to quantify how well a quantum system can stand as a reference frame for other quantum system. In this work, we explore the proposal based on a resource theory of asymmetry, known as mutual or shared asymmetry, which actually is equivalent to the approach from coherence theory in the case of interest here: quantum reference frames described by the U(1)\text{U}(1)U(1) compact group. We extend some previous results in the literature about shared asymmetry and the Page-Wootters mechanism to more general cases, culminating in the enunciation of a theorem relating shared asymmetry of a bipartite state \ensuremathρSR{\ensuremath{\rho}}_{SR}\ensuremathρSR​ with the relative entropy of entanglement of internal states \ensuremathρM{\ensuremath{\rho}}_{M}\ensuremathρM​ on the charge sectors of the Hilbert space HS\ensuremath⨂HR{\mathcal{H}}_{S}\ensuremath{\bigotimes}{\mathcal{H}}_{R}HS​\ensuremath⨂HR​. Using this result, we reinterpret the relation between the Page-Wootters mechanism and entanglement, and we also open some paths to further studies.
The Page-Wootters mechanism questioned the fundamental nature of time in quantum physics. The mechanism explored the notion that a given physical quantity is always defined and measured relative to a reference frame, in general, not explained in the theoretical description of quantum physical experiments. Recently, the resource theory of asymmetry deals explicitly with the physical conditions necessary for a quantum system to serve as a good reference frame. Nonetheless, to quantify a quantum reference frame in relation to another one, it is important to establish an internal description of quantum theory, i.e., without the need for a classical reference frame. In this work, we address this issue with the concept of mutual asymmetry and use this machinery in the Page-Wootters mechanism by identifying the concept of mutual asymmetry as mutual or internal coherence. To do so, the notion of quantum coherence in relation to a quantum reference frame is revisited and a quantifier is proposed in this scenario. Also, this opens space to investigate the link of internal coherence and correlations, as proposed by Page and Wootters, under a resource theory approach.
In the quantization of gauge theories and quantum gravity, it is crucial to treat reference frames such as rods or clocks not as idealized external classical relata, but as internal quantum subsystems. In the Page-Wootters formalism, for example, evolution of a quantum system S is described by a stationary joint state of S and a quantum clock, where time dependence of S arises from conditioning on the value of the clock. Here, we consider (possibly imperfect) internal quantum reference frames R for arbitrary compact symmetry groups, and show that there is an exact quantitative correspondence between the amount of entanglement in the invariant state on RS and the amount of asymmetry in the corresponding conditional state on S. Surprisingly, this duality holds exactly regardless of the choice of coherent state system used to condition on the reference frame. Averaging asymmetry over all conditional states, we obtain a simple representation-theoretic expression that admits the study of the quality of imperfect quantum reference frames, quantum speed limits for imperfect clocks, and typicality of asymmetry in a unified way. Our results shed light on the role of entanglement for establishing asymmetry in a fully symmetric quantum world.
A useful and unifying concept in quantum information theory is the idea of a consumable "resource". The basic idea behind quantum resource theories is that whenever we face restrictions on the set of operations we can perform, some states (resource states) enable us to perform particular tasks that otherwise would have been impossible. In this thesis we first review a resource theory that recently has gained a lot of attention, namely the resource theory of asymmetry in which one is restricted to symmetric operations and therefore asymmetric states are the resource states. The asymmetry properties of a state relative to some symmetry group specify how and to what extent the given symmetry is broken by the state. In the remainder of the thesis we explain how the ideas developed in this resource theory are related to three different areas in quantum theory: theory of quantum reference frames, the Wigner-Araki-Yanase(WAY) theorem and the conditional probability interpretation of time in quantum mechanics. Theory of quantum reference frames treats reference frames like any other physical system within the formalism of quantum theory, which causes the measurements to be an approximation of the measurements against their classical counterparts. We consider the dynamics of a quantum directional reference frame undergoing repeated interactions. These interactions induce a back-action on the reference which is the central focus of our study. The effect of a precise sequence of measurement outcomes on the reference frame is studied by looking at both the case that the measurement record is averaged over and the case wherein it is retained. We find, in particular, that there is interesting dynamics in the latter situation, which cannot be revealed by considering the average case. We then consider in detail how a sequence of rotationally invariant unitary interactions affects the reference frame, a situation, which leads to quite different dynamics than the case of repeated measurements. Different strategies for correcting reference frame drift is considered given that we have access to a set of particles with polarization opposite to the direction of drift. In particular, we find that, by implementing a suitably chosen unitary interaction after every two measurements, we can eliminate the rotational drift of the reference frame. The WAY theorem establishes an important constraint that conservation laws impose on quantum mechanical measurements. We formulate the WAY theorem in the broader context of resource theories, where one is constrained to a subset of quantum mechanical operations described by a symmetry group. Establishing connections with the theory of quantum state discrimination we obtain optimal unitaries describing the measurement of arbitrary observables, explain how prior information can permit perfect measurements that circumvent the WAY constraint, and provide a framework that establishes a natural ordering on measurement apparatuses through a decomposition into asymmetry and charge subsystems. Finally we review two di erent schemes in the conditional probability interpretation (CPI) of time in quantum mechanics. In these schemes parameter time "t" as it appears in Schrodinder equation is considered to be unobservable, yet one can use an extension of conditional probabilities in order to study dynamics of a system relative to another system, namely the quantum clock. We phrase CPI as a quantum communication protocol which enables us to use the machinery developed in the resource theory of asymmetry. This provides us with a deeper understanding of the decoherence effect caused due to the inaccessibility of the parameter time "t" and that how this effect can be suppressed by choosing an optimal initial state for the quantum clock.
Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, this mechanism explored a stationary pure bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge and an equation of motion can be obtained for a quantum particle conditioned to the measurement of the state of the clock. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page–Wootters model. Connecting it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence. In a second step we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the systems initial conditions.
We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space Hkin\mathcal{H}_{\rm kin}Hkin​ by group averaging. We assume the gauge group GGG is unimodular, and we define a quantum reference frame as a system of coherent states associated with a group GRG_RGR​ of frame orientations acting on a tensor factor of Hkin\mathcal{H}_{\rm kin}Hkin​. In direct analogy with the Faddeev-Popov procedure in path integrals, we show that the physical inner product can be written in terms of a choice of frame orientation with an appropriate operator measure, but that the physical Hilbert space is independent of this choice of frame. This yields a generalization of the Page-Wootters reduction map used to define conditional states and conditional probabilities of measurement outcomes given a frame orientation. These probabilities correspond to the expectation values of relational Dirac observables obtained from generalizations of previously considered relativization maps. Our main result is that the map that implements a change of frame is unitary, even when GGG does not act as a tensor-product representation across a factorization of Hkin\mathcal{H}_{\rm kin}Hkin​. The formalism therefore accommodates interactions between reference frames and the systems described relative to them. We apply this formalism to temporal frames, showing that a unitary Schrödinger equation emerges for an interacting clock and system, and that a previously reported time-nonlocal Schrödinger equation is recovered in an appropriate limit. With this, problems of non-unitarity are avoided by the use of the gauge-fixing measure in the physical inner product. We also discuss the case of interacting spatial frames, where the known correspondence between coherence in one frame and entanglement in another is shown to depend on the coupling.
Time in relativity theory has a status different from that adopted by standard quantum mechanics, where time is considered as a parameter measured with reference to an external absolute Newtonian frame. This status strongly restricts its role in the dynamics of systems and hinders any formulation to merge quantum mechanics with general relativity, specifically when considering quantum gravity. To overcome those limitations, several authors tried to construct an operator which is conjugate to the Hamiltonian of quantum systems implementing some essential features of the relativistic time. These formulations use the concept of internal or intrinsic time instead of the universal coordinate time used in textbooks. Furthermore, recently it is remarked that the consideration of time with relativistic features could enhance the analysis techniques in quantum information processing and have an impact on its status in causal orders and causal structures of quantum information. The role of clocks, their accuracy and stability has become an important issue in quantum information processing. This article present a substantiative review of recent works which reflect the possibility of utilizing quantum time, measured by quantum clock devised according to Page-Wootters scheme, to stand as a resource for quantum information processing.