Abstract
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.
Highlights
The description of physical systems is standardly given in terms of coordinates as defined by reference frames
We have provided an operational description of the spin of a special-relativistic quantum particle
Such operational description is hard to obtain with standard methods due to the combined effect of special relativity, which makes the spin and momentum not separable, and quantum mechanics, which makes it impossible to jump to the rest frame with a standard reference frame transformation
Summary
The description of physical systems is standardly given in terms of coordinates as defined by reference frames. The set fulfills the su(2) algebra and is operationally defined through a ‘relativistic Stern-Gerlach experiment’: we construct the interaction and the measurement between the spin-momentum degrees of freedom and the electromagnetic field in the laboratory frame which gives the same probabilities as the Stern-Gerlach experiment in the rest frame. This set of observables in the laboratory frame allows us to partition the total Hilbert space into two (highly degenerate) subspaces corresponding to the two outcomes “spin up” and “spin down”. With QRFs techniques the relativistic spin can effectively be described as a qubit in an operationally well-defined way
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