Quantum reference frames provide a relational description of multipartite quantum systems in which physical states and observables are defined relative to quantum observers. Yet different observers can assign different entropies to the same system, raising the question of how such observer-dependence is constrained. We identify a family of frame-independent diagonal Rényi entropies for arbitrary subsystems, yielding a generalized multipartite coherence-entanglement tradeoff. For ideal frames, the observer-dependence of subsystem entropy admits an exact decomposition into a sum of single-frame coherences and inter-frame correlations; for non-ideal frames, it is instead bounded by the dimension of an effective relational Hilbert space determined by the representation structure of the frames. Our results place quantitative limits on how much quantum observers can disagree about subsystem entropy, with potential implications for observer-dependent entropy assignments in gravitational settings.
Quantum reference frames provide a relational description of multipartite quantum systems in which physical states and observables are defined relative to quantum observers. Yet different observers can assign different entropies to the same system, raising the question of how such observer-dependence is constrained. We identify a family of frame-independent diagonal Rényi entropies for arbitrary subsystems, yielding a generalized multipartite coherence-entanglement tradeoff. For ideal frames, the observer-dependence of subsystem entropy admits an exact decomposition into a sum of single-frame coherences and inter-frame correlations; for non-ideal frames, it is instead bounded by the dimension of an effective relational Hilbert space determined by the representation structure of the frames. Our results place quantitative limits on how much quantum observers can disagree about subsystem entropy, with potential implications for observer-dependent entropy assignments in gravitational settings.
One of the most basic notions in physics is the partitioning of a system into subsystems and the study of correlations among its parts. In this Letter, we explore this notion in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We demonstrate that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We further demonstrate that subalgebras which commute before imposing the symmetry constraint can translate into noncommuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame dependent.
In quantum information theory, standard notions typically presuppose a fixed partition of the world into subsystems, encoded in a tensor product structure of the Hilbert space. Yet, in some scenarios, the subsystem structure is not uniquely fixed. A case in point is the description of subsystems relative to different quantum reference frames (QRFs), where the tensor product structure, although fixed in a specific QRF, changes across different QRFs. Such a relativity of subsystems underlies many interesting features of QRFs, such as the relativity of entanglement, entropy, and coherence. It also raises a broader question: how do we compare the information across different subsystem perspectives, especially when the corresponding subsystem algebras overlap? Here, we propose a notion of quantum mutual information across different QRF perspectives, which captures the correlations between subsystems relative to different frames of reference. To do so, we focus on an algebraic definition of subsystems. Studying in detail the union and overlap between subsystem algebras relative to different QRFs, we find that the usual definition of mutual information in terms of the chain law overcounts the information associated to the overlap. Based on this insight, we propose a modified chain law and show that the resulting notion of mutual information satisfies subadditivity and connects naturally to other measures found in the literature [1]. Our investigations not only provide a way to quantify the relation between different QRF perspectives from an information-theoretic point of view but also sheds light on some important features of quantum information in absence of a fixed tensor product structure.
Viewing frames of reference as physical systems, subject to the same laws as the systems they describe, is central to the relational approach in physics. Under the assumption that quantum mechanics universally governs all physical entities, this perspective naturally leads to the concept of quantum reference frames (QRFs). We investigate the perspective-dependence of position and momentum uncertainties, correlations, covariance matrices, and entanglement within the QRF formalism. We show that the Robertson-Schrödinger uncertainty relations are frame-dependent, and so are correlations and variances, which satisfy various constraints described as inequalities. However, the determinant of the total covariance matrix, linked to the uncertainty volume in phase space, as well as variance-based entanglement criteria, remains invariant under changes of reference frame. Under specific conditions, the purities of subsystems are also invariant for different QRFs, but in general, they are perspective-dependent. These invariants suggest fundamental, robust measures of uncertainty and entanglement that persist despite changes in observational perspective, potentially inspiring dedicated quantum information protocols as well as further foundational studies.
We find a necessary condition for subsystems to become entangled after a quantum reference frame transformation. We then distinguish between subsystems that admit separable descriptions relative to a quantum reference frame, and those that do not. On the one hand, we show that separable descriptions maximize the entanglement internal to the subsystem, and relate our results with the conservation of entanglement and coherence under quantum reference frame transformations. On the other hand, systems that do not admit a separable description relative to any subsystem display a form of entanglement that is genuine to the quantum reference frame formalism.
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.
Multipartite quantum correlations are complex and intricate, posing challenges for their detection and quantification. Nevertheless, they are central to both fundamental quantum science and its numerous applications. We discuss several approaches for analyzing quantum correlations, with an emphasis on the Relativistic Independence framework which allows one to detect and differentiate classes of multipartite entangled states, while also yielding refined quantum bounds on multipartite systems. This framework is shown to rely on a recently demonstrated tradeoff between local and nonlocal correlations, signifying the enabling power of quantum uncertainty. Furthermore, we examine these notions from the broader perspective of quantum reference frames (QRFs), attempting to see which of the above concepts are perspective-dependent and which are not. Large parts of the current discussion, as well as many of the quantitative results presented, specifically rely on the structure of the covariance matrix, whose determinant remains invariant under QRF changes, thereby signaling its fundamental significance.