Classical Reference Frame Research Articles

Considering a superposition of classical reference frames

A ubiquitous feature of quantum mechanical theories is the existence of states of superposition. This is expected to be no different for a quantum gravity theory. Guided by this consideration and others we consider a framework in which classical reference frames may be in superposition relative to one another. Mirroring standard quantum mechanics we introduce a complex-valued wavefunctional, which takes as input the transformations between the coordinates, Ψ[x(x′)], with the interpretation that an interaction between the reference frames may select a particular transformation with probability distribution given by the Born rule — P[x(x′)] = probability distribution functional ≡ |Ψ[x(x′)]|2. The cases of two and three reference frames in superposition are considered explicitly. It is shown that the set of transformations is closed. A rule for transforming wavefunctions from one system to another system in superposition is proposed and consistency with the Schrodinger equation is demonstrated.

The hands’ default location guides tactile spatial selectivity

Our skin is a two-dimensional sheet that can be folded into a multitude of configurations due to the mobility of our body parts. Parts of the human tactile system might account for this flexibility by being tuned to locations in the world rather than on the skin. Using adaptation, we scrutinized the spatial selectivity of two tactile perceptual mechanisms for which the visual equivalents have been reported to be selective in world coordinates: tactile motion and the duration of tactile events. Participants' hand position-uncrossed or crossed-as well as the stimulated hand varied independently across adaptation and test phases. This design distinguished among somatotopic selectivity for locations on the skin and spatiotopic selectivity for locations in the environment, but also tested spatial selectivity that fits neither of these classical reference frames and is based on the default position of the hands. For both features, adaptation consistently affected subsequent tactile perception at the adapted hand, reflecting skin-bound spatial selectivity. Yet, tactile motion and temporal adaptation also transferred across hands but only if the hands were crossed during the adaptation phase, that is, when one hand was placed at the other hand's typical location. Thus, selectivity for locations in the world was based on default rather than online sensory information about the location of the hands. These results challenge the prevalent dichotomy of somatotopic and spatiotopic selectivity and suggest that prior information about the hands' default position -right hand at the right side-is embedded deep in the tactile sensory system.

Algebraic Properties of Quantum Reference Frames: Does Time Fluctuate?

Quantum reference frames are expected to differ from classical reference frames because they have to implement typical quantum features such as fluctuations and correlations. Here, we show that fluctuations and correlations of reference variables, in particular of time, are restricted by their very nature of being used for reference. Mathematically, this property is implemented by imposing constraints on the system to make sure that reference variables are not physical degrees of freedom. These constraints not only relate physical degrees of freedom to reference variables in order to describe their behavior, they also restrict quantum fluctuations of reference variables and their correlations with system degrees of freedom. We introduce the notion of “almost-positive” states as a suitable mathematical method. An explicit application of their properties to examples of recent interest in quantum reference frames reveals previously unrecognized restrictions on possible frame–system interactions. While currently discussed clock models rely on assumptions that, as shown here, make them consistent as quantum reference frames, relaxing these assumptions will expose the models to new restrictions that appear to be rather strong. Almost-positive states also shed some light on a recent debate about the consistency of relational quantum mechanics.

Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system

Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently, a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However, these conditions are difficult to use in practice and their physical implications unclear. In the present theoretical work, we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed to make a given quantum state accessible under channels covariant with any given symmetry group.

Quantum superposition of spacetimes obeys Einstein's equivalence principle

We challenge the view that there is a basic conflict between the fundamental principles of Quantum Theory and General Relativity and, in particular, the fact that a superposition of massive bodies would lead to a violation of the Equivalence Principle. It has been argued that this violation implies that such a superposition must inevitably spontaneously collapse (like in the Diósi–Penrose model). We identify the origin of such an assertion in the impossibility of finding a local and classical reference frame in which Einstein's Equivalence Principle would hold. In contrast, we argue that the formulation of the Equivalence Principle can be generalized so that it holds for reference frames that are associated with quantum systems in a superposition of spacetimes. The core of this new formulation is the introduction of a quantum diffeomorphism to such Quantum Reference Frames. This procedure reconciles the principle of linear superposition in Quantum Theory with the principle of general covariance and the Equivalence Principle of General Relativity. Hence, it is not necessary to invoke a gravity-induced spontaneous state reduction when a massive body is prepared in a spatial superposition.

The group structure of dynamical transformations between quantum reference frames

Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.

Communicating without shared reference frames

We generalize a quantum communication protocol introduced by Bartlett et al. [New J. Phys. 11, 063013 (2009)], in which two parties communicating do not share a classical reference frame, to the case where changes of their reference frames form a one-dimensional noncompact Lie group. Alice sends to Bob the state ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$, where ${\ensuremath{\rho}}_{S}$ is the state of the system Alice wishes to communicate and ${\ensuremath{\rho}}_{R}$ is the state of an ancillary system serving as a token of her reference frame. Because Bob is ignorant of the relationship between his reference frame and Alice's, he will describe the state ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$ as an average over all possible reference frames. Bob measures the reference token and applies a correction to the system Alice wished to communicate conditioned on the outcome of the measurement. The recovered state ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ is decohered with respect to ${\ensuremath{\rho}}_{S}$, the amount of decoherence depending on the properties of the reference token ${\ensuremath{\rho}}_{R}$. We present an example of this protocol when Alice and Bob do not share a reference frame associated with the one-dimensional translation group and use the fidelity between ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ to quantify the success of the recovery operation.

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

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 what are the physical conditions 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 a important task to establish an internal description of quantum theory, i.e., without the need of a classical reference frame. In this work we address this issue by 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 of a quantum reference frame is revisited and a quantifier is proposed in this scenario. Also, this open space to investigate the link of internal coherence and correlations, as proposed by Page and Wootters, under a resource theory approach.

Characterizing Multipartite Entanglement with Moments of Random Correlations.

The experimental detection of multipartite entanglement usually requires a number of appropriately chosen local quantum measurements that are aligned with respect to a previously shared common reference frame. The latter, however, can be a challenging prerequisite, e.g., for satellite-based photonic quantum communication, making the development of alternative detection strategies desirable. One possibility for avoiding the distribution of classical reference frames is to perform a number of local measurements with settings distributed uniformly at random. In this Letter, we follow such a treatment and show that an improved detection and characterization of multipartite entanglement is possible by combining statistical moments of different order. To do so, we make use of designs that are pseudorandom processes allowing us to link the present entanglement criteria to ordinary reference frame independent ones. The strengths of our methods are illustrated in various cases, starting with two qubits and followed by more involved multipartite scenarios.

Macroscopic superpositions require tremendous measurement devices

We consider fundamental limits on the detectable size of macroscopic quantum superpositions. We argue that a full quantum mechanical treatment of system plus measurement device is required, and that a (classical) reference frame for phase or direction needs to be established to certify the quantum state. When taking the size of such a classical reference frame into account, we show that to reliably distinguish a quantum superposition state from an incoherent mixture requires a measurement device that is quadratically bigger than the superposition state. Whereas for moderate system sizes such as generated in previous experiments this is not a stringent restriction, for macroscopic superpositions of the size of a cat the required effort quickly becomes intractable, requiring measurement devices of the size of the Earth. We illustrate our results using macroscopic superposition states of photons, spins, and position. Finally, we also show how this limitation can be circumvented by dealing with superpositions in relative degrees of freedom.

Quantum parameter estimation with imperfect reference frames

Quantum metrology studies quantum strategies which enable us to outperform their classical counterparts. In this framework, the existence of perfect classical reference frames is usually assumed. However, such ideal reference frames might not always be available. The reference frames required in metrology strategies can either degrade or become misaligned during the estimation process. We investigate how the imperfectness of reference frames leads to noise which in general affects the ultimate precision limits in measurement of physical parameters. Moreover, since quantum parameter estimation can be phrased as a quantum communication protocol between two parties, our results provide deeper insight into quantum communication protocols with misaligned reference frames. Our framework allows for the study of noise on the efficiency of such schemes.

Entanglement detection with bounded reference frames

Quantum experiments usually assume the existence of perfect, classical reference frames (RFs), which allow for the specification of measurement settings (e.g. orientation of the Stern-Gerlach magnet in spin measurements) with arbitrary precision. If the RFs are ‘bounded’ (i.e. quantum systems themselves, having a finite number of degrees of freedom), only limited precision can be attained. Using spin coherent states as bounded RFs, we have found the minimum size needed for them to violate local realism for entangled spin systems. For composite systems of spin particles, RFs of very small size are sufficient for the violation; however, to see this violation for macroscopic entangled spins, the size of the RF must be at least quadratically larger than that of the spins. The unavailability of such RFs gives a possible explanation for the non-observance of violation of local realism in everyday experience.

Formulation and picture of quantum phase

Based on the concept of classical phase, we formulate a new explanation for the quantum phase from the quantum mechanical point of view. The quantum phase is the canonically conjugate variable of an angular momentum operator, which corresponds to the angular position φ in an actual physical space with a classical reference frame, but it takes a complex exponential form eiφ≡cosφ+i sinφ in the abstract Hilbert space of a quantum reference frame. This formulation is simply the famous Euler formula in a complex number field. In particular, when φ = π/2, the correlative quantum phase is a unitary pure imaginary number eiπ/2≡cos(π/2)+i sin(π/2) ≡ i. By using a photon state-vector function that is the general solution of photon Schrodinger equation and can completely describe a photon’s behavior, we discuss the relationship between the angular momentum of a photon and the phase of the photon; we also analyze the intrinsic relationship between the macroscopic light wave phase and the microscopic photon phase.

The resource theory of quantum reference frames: manipulations and monotones

Every restriction on quantum operations defines a resource theory, determining how quantum states that cannot be prepared under the restriction may be manipulated and used to circumvent the restriction. A superselection rule (SSR) is a restriction that arises through the lack of a classical reference frame and the states that circumvent it (the resource) are quantum reference frames. We consider the resource theories that arise from three types of SSRs, associated respectively with lacking: (i) a phase reference, (ii) a frame for chirality, and (iii) a frame for spatial orientation. Focusing on pure unipartite quantum states (and in some cases restricting our attention even further to subsets of these), we explore single-copy and asymptotic manipulations. In particular, we identify the necessary and sufficient conditions for a deterministic transformation between two resource states to be possible and, when these conditions are not met, the maximum probability with which the transformation can be achieved. We also determine when a particular transformation can be achieved reversibly in the limit of arbitrarily many copies and find the maximum rate of conversion. A comparison of the three resource theories demonstrates that the extent to which resources can be interconverted decreases as the strength of the restriction increases. Along the way, we introduce several measures of frameness and prove that these are monotonically non-increasing under various classes of operations that are permitted by the SSR.

QUANTUM ESTIMATION OF RELATIVE INFORMATION

We derive optimal schemes for preparation and estimation of relational degrees of freedom between two quantum systems. We specifically analyze the case of rotation parameters representing relative angles between elements of the SU(2) symmetry group. Our estimation procedure does not assume prior knowledge of the absolute spatial orientation of the systems and as such does not require information on the underlying classical reference frame in which the states are prepared.

Quantum chaos and order based on classically moving reference frames

We develop a mathematically consistent approach for treating the quantum systems based on moving classical reference frames. The classical and quantum exact solutions show excellently classical-quantum correspondence, in which the quantum chaotic coherent states correspond to the classically chaotic motions. Applying the approach to the periodically driven linear and nonlinear oscillators, the regular and chaotic quantum states and quantum levels, and the quantum chaotic regions are evidenced. The results indicate that chaos may cause the collapse of matter wave packets and suppress the quantum effect of energy.

Classical formulation of quantum mechanics

The concept of quantum state is given in terms of classical probability for position in squeezed and rotated classical reference frames in phase space. Stationary states and energy levels of the quantum system are obtained in a classical formulation of quantum mechanics. The positive probability density of the harmonic oscillator position is obtained by solving a new eigenvalue equation of standard quantum mechanics instead of the Schrodinger equation. The orthogonality and completeness relations are found for the eigendistributions.

Relativistic Reference Frames of Local Observer and Space Radiointerferometer

In a considerable number of works on relativistic astrometry (see, e.g. Kovalevsky and Brumberg 1986) the reference frames (RFs) are introduced either by means of coordinate representation of a space-time metric, such as using harmonicity conditions (Brumberg and Kopejkin 1989), or on the basis of invariant constructions like Fermi coordinates (Synge 1960; Ashby and Bertotti 1986; Boucher 1986). Both approaches must, probably, be combined in applications. We consider the local observer RFs (LORFs) based on the Fermi coordinates and on the optical ones (Synge 1960), which are rigorously defined for a general metric and are directly related to observable quantities. In particular, the optical RF operates with the observed direction of the light source, whereas the Fermi RF seems to be a natural generalization of the classical Cartesian RF.