Quantum mechanics and the covariance of physical laws in quantum reference frames

TL;DR

The paper tackles how to describe physics when reference frames are themselves quantum degrees of freedom, removing reliance on an external absolute frame. It introduces a general unitary QRF transformation and shows that the transformed dynamics follow $i\hbar \frac{d \rho^{(A)}_{BC}}{dt} = [H^{(A)}_{BC}, \rho^{(A)}_{BC}]$ with $H^{(A)}_{BC} = \hat{S} \hat{H}^{(C)}_{AB} \hat{S}^ ablager + i\hbar \frac{d \hat{S}}{dt} \hat{S}^ ablager$, thereby relating the evolution seen in different quantum frames. Key contributions include extending covariance to superpositions of spatial transformations, formulating a quantum weak equivalence principle, and providing a method to define the rest frame of a quantum system. The work reveals that entanglement and superposition are frame-dependent descriptors and offers a relational foundation with potential experimental tests and applications to clocks, gravity, and quantum information in moving frames.

Abstract

In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a "superposition of coordinate transformations". We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.

Figures (9)

• Figure 1: Illustration of the notion of quantum reference frames. Two quantum reference frames A and C and quantum system B. The reference frames A and C are pictorially represented as two laboratories equipped with their own instruments. In a realistic situation, however, the system A could be an atom, B a photon and C a laboratory (or another atom). The reference frame associated to A is in a superposition of positions as observed from the laboratory C (the superposition is illustrated by the fuzziness of laboratory A). Given the quantum states of A and B relative to the reference frame of C, what are the states of B and C as defined with respect to the reference frame of A?
• Figure 2: Transformation to relative coordinates. (a) Relative position coordinates of A and B from the point of view of C. (b) Relative position coordinates of B and C from the point of view of A. It is immediate to verify that $\hat{x}_\mathrm{B} \mapsto \hat{q}_\mathrm{B} - \hat{q}_\mathrm{C}$ and that $\hat{x}_\mathrm{A} \mapsto -\hat{q}_\mathrm{C}$.
• Figure 3: Examples of relative states in different quantum reference frames. The relative states are described from the reference frame of C (above in each subfigure) and A (below in each subfigure) in the position basis. Product states are represented as curves whose area is shaded, while entangled states as curves whose area is not shaded. In (a) A's state is well-localised from the point of view of C. In A's reference frame, B has the same state as seen from C, but translated, and C is well-localized. This case corresponds to the translation of a classical reference frame. In (b) A and B are in a product state, and A is in a superposition of two sharp-position states that do not overlap. From A's point of view B and C are entangled, but the relative distance between the states is unchanged. In (c) A and B are entangled and perfectly correlated, i.e. the relative distance between them is always $L$. In A's reference frame B is in a well-defined position and C is in a superposition of positions. Finally, in (d) A and B are entangled in an EPR state from C's point of view, i.e. $\left| \psi \right>_{\mathrm{AB}} = \int dx \left| x\right>_\mathrm{A}\left| x + X \right>_\mathrm{B}$. Changing to A, B appears in a fixed position, while C is spread over the whole space.
• Figure 4: Table summarising different quantum reference frame (QRF) transformations on system B. (The action on A is omitted) The time-independent transformation $\hat{S}_\mathrm{x}$ is the QRF generalisation of a standard reference frame (RF) transformation where the reference frame is moving along $X(t)$, and the relative coordinates describe the distance between systems A and B at time $t$. The three QRF transformations $\hat{S}_\mathrm{T}$, $\hat{S}_\mathrm{b}$, and $\hat{S}_\mathrm{EP}$ generalise the extended Galilean transformations to a reference frame which is respectively translated, moving with constant and uniform velocity, and moving with constant and uniform acceleration. In particular, $\hat{S}_\mathrm{T}$ transforms the old coordinate $x_\mathrm{B}$ into the relative position between system B at time $t$ and system A at time $\tau$, and reduces to $\hat{S}_\mathrm{x}$ for $t=\tau$. The transformation $\hat{S}_\mathrm{b}$ performs a Lorentz boost on system B, where the velocity is written in terms of dynamical variables of system A. Finally, the transformation $\hat{S}_\mathrm{EP}$, generalises the transformation to an accelerated reference frame to when system A moves in a superposition of accelerations. Importantly, the extension of the RF transformation to dynamical quantities of quantum systems makes it possible to introduce a generalised notion of symmetric transformation, which is exemplified, in this work, in the case of the $\hat{S}_\mathrm{T}$ and $\hat{S}_\mathrm{b}$ transformations.
• Figure 5: Schematic illustration of the descriptions in two quantum reference frames that are boosted with respect to each other: (a) the state of A and B as described from C, and (b) the state of B and C as described from A. In (a), the state of A and B is a product state, and the state of A is in a superposition of the two velocities $v_1$ and $v_2$. By applying a 'superposition of boosts' by the velocity of A, we find that, as seen from A, the state of B and C is entangled. In particular, the entanglement is such that if C moves with velocity $-v_i$, $i=1,2$, from A's point of view, B is boosted by $-v_i$.