Relativistic quantum reference frames: the operational meaning of spin

TL;DR

The paper addresses the challenge of defining and manipulating spin as a qubit in a relativistic setting, where spin and momentum become entangled under Lorentz boosts. It introduces a quantum reference frame transformation that implements a superposition of Lorentz boosts to move to the particle’s rest frame even when momentum is in a quantum superposition, enabling an operational spin definition via Stern–Gerlach measurements. By constructing observables 8_i that satisfy SU(2) and relate to rest-frame spin through a covariant lab-frame interaction, the authors provide a relativistic Stern–Gerlach procedure with probability conservation across QRFs. This framework yields a robust relativistic qubit description and opens avenues for spin-based quantum information protocols in the special-relativistic regime.

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 set-up in the non-relativistic (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 $\mathfrak{su}(2)$ 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.

Figures (2)

• Figure 1: (a) The state of a Dirac particle A with spin $\tilde{A}$ as seen from the laboratory perspective (C). When the state is in a superposition of relativistic velocities $-v_1$ and $-v_2$, the spin degree of freedom and the momentum degree of freedom are no longer separable. (b) The state of the spin $\tilde{A}$ and of the laboratory C as seen in the rest frame of the quantum particle A. In this quantum reference frame, the spin is operationally defined by means of the Stern-Gerlach experiment.
• Figure 2: The relativistic Stern-Gerlach experiment as seen from the QRF A (above) and from the QRF C (below). In the rest frame of particle A, the spin is operationally defined via the Stern-Gerlach experiment. To measure spin along direction $\vec{n}$ the spin (Pauli operator) $\vec{\sigma}$ is coupled to an inhomogeneous magnetic field oriented along $\vec{n}$. The particle is then deflected towards the direction $\vec{n}$ and $-\vec{n}$ corresponding to outcome "spin up" and "spin down" respectively. When transforming to the laboratory frame C, the magnetic field and the spin transform with a superposition of Lorentz boosts for $v_1$ and $v_2$. The interaction Hamiltonian is also transformed, giving rise to a coupling between the transformed vector $\vec{\mathcal{S}}_\Lambda(\vec{B}^{(A)}) = \hat{\gamma}_A \left[ \vec{B}^{(C)} - \frac{\hat{\gamma}_A}{\hat{\gamma}_A+1}\left(\vec{\hat{\beta}}_A \cdot \vec{B}^{(C)}\right)\vec{\hat{\beta}}_A + \left(\vec{\hat{\beta}}_A \times \vec{E}^{(C)}\right) \right]$ aligned in the same direction $\vec{n}$ as the magnetic field in the rest frame, and the transformed spin operator $\vec{\Xi}$. The particle is again deflected either to $\vec{n}$ or $-\vec{n}$ corresponding to the outcome "spin up" and "spin down" respectively. The probability of detecting the outcomes "spin up" and "spin down" is preserved under change of QRF.