At eleven parameters a frame is a body with a momentum. The quantum step of the theory makes that momentum an operator, and a frame body in a superposition of momenta is a quantum reference frame. This letter asks what using such a frame does to it, in a model with one assumption. An ideal meter comoving with the frame body reads the canonical spins of test bodies, and nothing else. Relative to the frame each spin is turned by the Wigner rotation of the composition law, by an angle that depends on the frame's own rapidity. So every spin the frame serves is a weak measurement of the frame's velocity. The quantum Fisher information per spin is the square of the derivative of the Wigner angle, in closed form. For a frame at rest it is tanh 2 (θt/2), θt the test body's rapidity, and it is bounded by 1/ cosh 2 θF . With polarised spins and the records kept, n spins sharpen the frame's rapidity, to the van Trees bound 1/(σ -2 + n F ) within one percent. With the records discarded, or with unpolarised spins, the frame's momentum coherence decays by the overlap of the two rotated spin states per spin. For a Gaussian frame its purity is 1/ 1 + N F σ 2 , equal to the purity of the spins relative to the frame. The origin of the frame is the conjugate reference. Each discarded spin kicks the frame's position by ±ℏφ ′ /2M , a random walk, and the kept records widen it by the Heisenberg bound. The velocity reference is kept or sharpened while the origin degrades, the two being the conjugate blocks of the eleven. The frame of an entangled pair with a sharp total momentum has nothing to learn and no origin to lose. The sixteen of a body being the operators of two qubits, a boost is a coupling that is not a gate, and its cost in a simulation is stated. The statements are checked numerically.
At eleven parameters a frame is a body with a momentum. The quantum step of the theory makes that momentum an operator, and a frame body in a superposition of momenta is a quantum reference frame. This letter asks what using such a frame does to it, in a model with one assumption. An ideal meter comoving with the frame body reads the canonical spins of test bodies, and nothing else. Relative to the frame each spin is turned by the Wigner rotation of the composition law, by an angle that depends on the frame's own rapidity. So every spin the frame serves is a weak measurement of the frame's velocity. The quantum Fisher information per spin is the square of the derivative of the Wigner angle, in closed form. For a frame at rest it is tanh 2 (θt/2), θt the test body's rapidity, and it is bounded by 1/ cosh 2 θF . With polarised spins and the records kept, n spins sharpen the frame's rapidity, to the van Trees bound 1/(σ -2 + n F ) within one percent. With the records discarded, or with unpolarised spins, the frame's momentum coherence decays by the overlap of the two rotated spin states per spin. For a Gaussian frame its purity is 1/ 1 + N F σ 2 , equal to the purity of the spins relative to the frame. The origin of the frame is the conjugate reference. Each discarded spin kicks the frame's position by ±ℏφ ′ /2M , a random walk, and the kept records widen it by the Heisenberg bound. The velocity reference is kept or sharpened while the origin degrades, the two being the conjugate blocks of the eleven. The frame of an entangled pair with a sharp total momentum has nothing to learn and no origin to lose. The sixteen of a body being the operators of two qubits, a boost is a coupling that is not a gate, and its cost in a simulation is stated. The statements are checked numerically.
Special relativity at eleven parameters (SR11) defines a frame by a body: a frame has a mass M , a momentum P , a spin and a position, the transformation to it is the boost L(P ) built from its momentum, and the composition law of the theory says how a second body looks from it. A frame is therefore a physical system with a quantum state, which is what the programme of quantum reference frames has been building; what the eleven add is a kinematics in which the frame body's mass appears explicitly as the scale of its rapidity and fixes its resolution. This paper takes the composition law of the eleven, with the frame body in a superposition of momenta, as the transformation law between quantum reference frames. The transformation is the composition law controlled by the frame's momentum; it carries the momentum of the body it describes linearly, and all its non-linearity sits in the Wigner rotation, which is the only place where the frame's momentum and the body's spin are correlated: the Wigner rotation is the entangler of spin with momentum, and the reduced spin of a body relative to a frame in a superposition of momenta has the purity , the result of Giacomini, Castro-Ruiz and Brukner with the frame's mass as the scale of its rapidity. The one number of the mixing is that mass. A frame body localised to ∆x has a transverse rapidity spread of at least ℏ/(2M c ∆x), so that the product of a frame's position and rapidity resolutions is bounded below by half its reduced Compton length, and the spin purity of a body of rapidity θ seen from it is 1 2 (1 + e -s 2 ) with s = (σ P /(M c)) tanh(θ/2), the small-angle Wigner rotation; the floor ∆x ≳ ℏ/(M c) below which the frame body is no longer one body is the length c 2 /a max on which the acceleration bound of SR11 is built, one scale read once as a resolution and once as a ceiling. Because the transformation is diagonal in the frame's momentum, the frame's pointer basis is its momentum, and it decoheres, when the bodies it frames are traced out, by the overlap of their relative momenta and spins; the thermal blur of a frame defined by a body in a Jüttner ensemble, computed in the thermal monograph, is that decoherence already done, 0.997, 0.992, 0.977, 0.964 at k B T /(mc 2 ) = 0.05, 0.135, 0.5, 1, and a frame defined by a conserved charge does not decohere at all. A clock is a body: its time, its energy and its dilatation charge close on an sl(2, R), the energy-time sector of the dilatation block, so that a body of the eleven is a Page-Wootters clock whose reading is a self-adjoint operator, Pauli's obstruction not applying to a commutator that is an operator; the price is that the reading is positive and its flow quadratic, x 0 -τ D + τ 2 p 0 when the clock is driven by its own energy, at the initial rate of its dilatation charge, which is Planck's constant in the Heisenberg contraction, so that the clock is covariant on neither the full line nor the full flow; and the compact conformal Hamiltonian, whose spectrum is E 0 + n, gives the clock a period, the time circle of the reverse paradigm; de Alfaro, Fubini and Furlan's conformal quantum mechanics is its model. The eleven are then placed among the kinematics that put the Planck scale into relativity, Snyder's and Yang's quantised spacetimes, doubly special relativity, κ-Poincaré, relative locality and triply special relativity, and among the quantum-reference-frame formalisms: they agree with the former on a bounded momentum, in the compact sector, and differ by the acceleration bound, the spin as one of the primary quantities, the reciprocity as an automorphism, and the linear frame composition and undeformed addition of momenta of the ordinary reading; they agree with 1 the latter that a frame is a quantum system and make explicit, in its mass, a parameter those formalisms carry and do not use. What the theory does not do is interpret: its quantum step is the Poisson bracket of the sixteen made a commutator, and it takes no position on measurement.
The spin is the prime example of a qubit. Encoding and decoding information in the spin qubit is operationally well defined through the Stern-Gerlach setup in the nonrelativistic (i.e., low velocity) limit. However, an operational definition of the spin in the relativistic regime is missing. The origin of this difficulty lies in the fact that, on the one hand, the spin gets entangled with the momentum in Lorentz-boosted reference frames, and on the other hand, for a particle moving in a superposition of velocities, it is impossible to "jump" to its rest frame, where spin is unambiguously defined. Here, we find a quantum reference frame transformation corresponding to a "superposition of Lorentz boosts," allowing us to transform to the rest frame of a particle that is in a superposition of relativistic momenta with respect to the laboratory frame. This enables us to first move to the particle's rest frame, define the spin measurements there (via the Stern-Gerlach experimental procedure), and then move back to the laboratory frame. In this way, we find a set of "relativistic Stern-Gerlach measurements" in the laboratory frame, and a set of observables satisfying the spin su(2) algebra. This operational procedure offers a concrete way of testing the relativistic features of the spin, and opens up the possibility of devising quantum information protocols for spin in the special-relativistic regime.
This paper is concerned with the spin–momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigenstates (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin and momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not an observer-dependent quantity in the inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and have shown that entanglement decreases with respect to the increase in observer velocity. Finally, we argue that Wigner rotation is induced by Lorentz transformations and can be realized as a controlling operator.
This paper was concerned with the spin-momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigen states (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin-momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not observer-dependent quantity in inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and has shown that entanglement decreases with respect to the increasing of observer velocity. Finally, we argue that, Wigner rotation is induced by Lorentz transformations can be realized as controlling operator.
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.
A still widely debated question in the field of relativistic quantum information is whether entanglement and the degree of violation of Bell's inequalities for massive relativistic particles are frame independent or not. At the core of this question is the effect that spin gets entangled with the momentum degree of freedom at relativistic velocities. Here, we show that Bell's inequalities for a pair of particles can be maximally violated in a special-relativistic regime, even without any postselection of the momentum of the particles. To this end, we use the methodology of quantum reference frames, which allows us to transform the problem to the rest frame of a particle, whose state can be in a superposition of relativistic momenta from the viewpoint of the laboratory frame. We show that, when the relative motion of two particles is noncollinear, the optimal measurements for violation of Bell's inequalities in the laboratory frame involve "coherent Wigner rotations." Moreover, the degree of violation of Bell's inequalities is independent of the choice of the quantum reference frame. Our results open up the possibility of extending entanglement-based quantum communication protocols to relativistic regimes.