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.
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.
We have studied a relativistically covariant Stern-Gerlach (SG) experiment for a relativistic spin and a Dirac spin. We have obtained the relativistic spin in an arbitrary frame by using the classical spin dipole tensor, which gives the covariant spin dipole interactions, and the relation between a spin and a spin magnetic dipole moment. The relativistic spin is shown to have problems to become a proper spin operator for a massive relativistic particle because of two reasons. First, the relativistic spin three-vector operators cannot satisfy the spin algebra. Second, the SG experiment for the relativistic particle provides a paradox between two observers in the particle rest frame and the laboratory frame, in which the particle is moving. We have shown that the paradox in the SG experiment is resolved by the Dirac spin, which is covariantly defined by a Lorentz transformation in the Dirac spinor representation. The Dirac spin three-vector operators satisfy the spin algebra. It is shown that the SG experiment for the Dirac spin in the inertial frame, where there is only magnetic field, can determine the spin without the information of the momentum of the particle. This shows that the reduced spin density matrix for the Dirac particle can be well-defined by integrating out the momentum degrees of freedom.
Since their first introduction, Quantum Reference Frame (QRF) transformations have been extensively discussed, generalising the covariance of physical laws to the quantum domain. Despite important progress, a formulation of QRF transformations for Lorentz symmetry is still lacking. The present work aims to fill this gap. We first introduce a reformulation of relativistic quantum mechanics independent of any notion of preferred temporal slicing. Based on this, we define transformations that switch between the perspectives of different relativistic QRFs. We introduce a notion of ''quantum Lorentz transformations'' and ''superposition of Lorentz boosts'', acting on the external degrees of freedom of a quantum particle. We analyse two effects, superposition of time dilations and superposition of length contractions, that arise only if the reference frames exhibit both relativistic and quantum-mechanical features. Finally, we discuss how the effects could be observed by measuring the wave-packet extensions from relativistic QRFs.
Spin is an important quantum degree of freedom in relativistic quantum information theory. This paper provides a first-principles derivation of the observable corresponding to a Stern-Gerlach measurement with relativistic particle velocity. The specific mathematical form of the Stern-Gerlach operator is established using the transformation properties of the electromagnetic field. To confirm that this is indeed the correct operator we provide a detailed analysis of the Stern-Gerlach measurement process. We do this by applying a WKB approximation to the minimally coupled Dirac equation describing an interaction between a massive fermion and an electromagnetic field. Making use of the superposition principle we show that the +1 and -1 spin eigenstates of the proposed spin operator are split into separate packets due to the inhomogeneity of the Stern-Gerlach magnetic field. The operator we obtain is dependent on the momentum between particle and Stern-Gerlach apparatus, and is mathematically distinct from two other commonly used operators. The consequences for quantum tomography are considered.
Entangled states are often given as one of the most bizarre examples of “weirdness” described as inherent to quantum mechanics. The present work reinterprets entanglement as not being a property of states at all, but rather as a relationship between the reference frames in which the states reside, which proposes to reduce “weirdness” of interpretation. Using the geometric Algebra of Physical Space, it has been shown that a classical form of the Dirac equation can be satisfied by any eigenspinor, which is a Lorentz transformation operator describing the relative velocity and relative orientation of the rest frame of a system as seen from a particular lab frame from which it is described. The real linear nature of the Dirac equation means any real linear superposition of such eigenspinors are also solutions. Thus, with entanglement modelled as an operator consisting of a linear superposition of rotation operators describing the possible relative orientations of a particular particle frame and the frame from which it is observed, it too can satisfy a bipartite form of the Dirac equation. To investigate this model, the present work applies relativistic boost transformations to the entangling operator in various ways, including as an identical boost of both parts in the same direction, and also as equal and oppositely-directed boosts. The resulting “entangling eigenspinors” are then analyzed in various ways, including the application to specific spin states — only to discover that doing this results in a reduction of the information, which can be interpreted as a reduction in the amount of entanglement. By comparing this to the treatment of the Dirac equation in APS, it may be concluded that the application of the entangling eigenspinor to a state — which models the typical approach of simply boosting an entangled state — gives an incomplete account of what is happening. The full information is thus contained within the entangling eigenspinor, justifying the interpretation of the entanglement in terms of geometric information relating the reference frames, rather than as a property of the state.
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.
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.