In the literature on quantum reference frames, the internal (relative) properties of a system are defined as those which are preserved under an arbitrary change of reference frame. For a system of quantum spins, these are all properties preserved by proper spatial rotations of the laboratory. However, this approach does not account for the hypothetical possibility of the laboratory becoming entangled to the system, as described by a second laboratory (the `Wigner's friend' scenario), in which case the relationship between the two laboratories is not a rotation, but is fundamentally quantum. To overcome this limitation, we re-define the reference frame transformations to be those that preserve the fidelities between subsystems. This enables us to derive U(2) as the correct symmetry group for transformations of a system of N spin-half particles. Next, we propose that systems having the same internal properties should be regarded as physically equivalent in the absence of an external frame. Remarkably, this implies that a single spin in a superposition relative to a spin magnet is equivalent to a macroscopic superposition of the magnet relative to the spin. We discuss the implications of this result for the Wigner's friend paradox.
In the literature on quantum reference frames, the internal (relative) properties of a system are defined as those which are preserved under an arbitrary change of reference frame. For a system of quantum spins, these are all properties preserved by proper spatial rotations of the laboratory. However, this approach does not account for the hypothetical possibility of the laboratory becoming entangled to the system, as described by a second laboratory (the `Wigner's friend' scenario), in which case the relationship between the two laboratories is not a rotation, but is fundamentally quantum. To overcome this limitation, we re-define the reference frame transformations to be those that preserve the fidelities between subsystems. This enables us to derive U(2) as the correct symmetry group for transformations of a system of N spin-half particles. Next, we propose that systems having the same internal properties should be regarded as physically equivalent in the absence of an external frame. Remarkably, this implies that a single spin in a superposition relative to a spin magnet is equivalent to a macroscopic superposition of the magnet relative to the spin. We discuss the implications of this result for the Wigner's friend paradox.
A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry group G , we define a general operator for reversibly changing between quantum reference frames associated to a group G . This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are `in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations of G . We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry group G is a semi-direct product G = N ⋊ P or a direct product G = N × P , providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.
In their strongest usage, quantum reference frames have been described as referring to "the measurements performed by a superposed lab", "the perspective of a quantum particle", "the point of view of a superposed observer", etc. While exciting, these operational proposals have remained brief and ambiguous, leading to misinterpretations and criticism. Here, we provide a detailed specification and defense of the notion of a position-superposed lab or observer. We argue that this requires no exotic claims about quantum physics and raises no greater interpretive difficulty than ordinary quantum measurements. We then derive several consequences of taking this operational meaning seriously. We stress that the position-superposed observers that define quantum references frames are different from, and considerably less problematic than, the outcome-superposed observers considered in Wigners' friend scenarios. In particular, we show that outcomes obtained by a position-superposed observer may (without decohering the superposition) be broadcast to a well-localised one, in contrast with Wigner's friend scenarios, which require the outcomes to remain internal to the system at hand. Finally, we defend the possibility to roleplay a quantum reference frame from a classical reference frame.
A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry group G, we define a general operator for reversibly changing between quantum reference frames associated to a group G. This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are 'in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations of G. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry group G is a semidirect product G=N (sic) P or a direct product G = N x P, providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.
This article suggests that thinking about the role of reference frames can provide new insight into Extended Wigner's Friend scenarios. This involves appealing to symmetries to make a principled distinction between properties of a system which are meaningful only relative to an external reference system and properties which are meaningful without further relativization. Thus we may propose that there are always well-defined facts about what observers have observed, but there are not necessarily well-defined facts about the relations between their reference frames, so there will not always exist a joint distribution over their outcomes which can meaningfully be compared to the predictions of quantum mechanics. In addition, this approach also offers a general argument against the idea that there should be a regress of relativization.
In physical experiments, reference frames are standardly modeled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e., spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum j and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large j, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonians (such as that of the Heisenberg model) are invariant under a larger group of quantum reference frame transformations. Our result is a development of the quantum reference frame formalism for a non-Abelian group.
In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other.Concretely, we consider a system of N relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the N particles appears "frozen", but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page-Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF.