Translation between quantum reference frames

We first consider the case in which we change to the reference frame described by the position of the quantum system A at a particular instant of time \(\tau\), when the initial Hamiltonian for A and B is \(\hat H_{{\mathrm{AB}}}^{({\mathrm{C}})} = \frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}} + \frac{{\hat p_{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}}\). In this case, the operator \(\hat X_{\mathrm{A}}(t)\) generalises the function \(X(t) = X_0\), with X 0 being a constant, and takes the form \(\hat X_{\mathrm{A}}(t) = e^{\frac{{\mathrm{i}}}{\hbar }\frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}}\tau }\hat x_{\mathrm{A}}e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}}\tau }\), and the full operator \(\hat S\) is

$$\hat S_{\mathrm{T}} = {\mathrm{exp}}\left( { - \frac{{\mathrm{i}}}{\hbar }\frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}(t - \tau )} \right)\hat {\cal P}_{{\mathrm{AC}}}^{({\mathrm{x}})}{\mathrm{exp}}\left( {\frac{{\mathrm{i}}}{\hbar }\hat x_{\mathrm{A}}\hat p_{\mathrm{B}}} \right){\mathrm{exp}}\left( {\frac{{\mathrm{i}}}{\hbar }\frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}}(t - \tau )} \right),$$ (13)

where \(\hat {\cal P}_{{\mathrm{AC}}}^{{\mathrm{(x)}}} = \hat {\cal P}_{{\mathrm{AC}}}\) in Eq. (2) and we have introduced the term \({\mathrm{exp}}( { - \frac{{\mathrm{i}}}{\hbar }\frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}(t - \tau )} )\) to ensure that the position of the system A at time τ tranforms into the symmetric position of the system C, i.e. \(\hat S_{\mathrm{T}}( {\hat x_{\mathrm{A}} - \frac{{\hat p_{\mathrm{A}}}}{{m_{\mathrm{A}}}}(t - \tau )} )\hat S_{\mathrm{T}}^\dagger = - ( {\hat q_{\mathrm{C}} - \frac{{\hat \pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}(t - \tau )} )\). Notice that for t = τ the operator \(\hat S_{\mathrm{T}}\) in Eq. (13) is precisely the operator \(\hat S_{\mathrm{x}}\) in Eq. (2). Therefore, we can interpret \(\hat S_{\mathrm{x}}\) as the operator which performs the translation to a quantum reference frame when the dynamics is “frozen” at time τ. The transformation implemented by \(\hat S_{\mathrm{T}}\) is

$$\hat S_{\mathrm{T}}\hat x_{\mathrm{A}}\hat S_{\mathrm{T}}^\dagger = - \hat q_{\mathrm{C}} + \frac{{\hat \pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}(t - \tau ) - \frac{{\hat \pi _{\mathrm{B}} + \hat \pi _{\mathrm{C}}}}{{m_{\mathrm{A}}}}(t - \tau );\quad \quad \hat S_{\mathrm{T}}\hat p_{\mathrm{A}}\hat S_{\mathrm{T}}^\dagger = - (\hat \pi _{\mathrm{B}} + \hat \pi _{\mathrm{C}});$$ (14)

$$\hat S_{\mathrm{T}}\hat x_{\mathrm{B}}\hat S_{\mathrm{T}}^\dagger = \hat q_{\mathrm{B}} - \hat q_{\mathrm{C}} + \frac{{\hat \pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}(t - \tau );\quad \quad \hat S_{\mathrm{T}}\hat p_{\mathrm{B}}\hat S_{\mathrm{T}}^\dagger = \hat \pi _{\mathrm{B}}.$$ (15)

Equation (15) implies that the position at time t of system B from the point of view of C is mapped into the relative position between system B and the position of A at time \(\tau\), while the momentum of B remains unchanged. In addition, this transformation is a symmetry of the free particle according to the definition given in Eq. (6), because the Hamiltonian \(\hat H_{{\mathrm{AB}}}^{{\mathrm{(C)}}}\) is mapped through Eq. (5) to \(\hat H_{{\mathrm{BC}}}^{({\mathrm{A}})} = \frac{{\hat \pi _{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}} + \frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}\). Therefore, the transformation \(\hat S_{\mathrm{T}}\) in Eq. (13) constitutes a generalisation of the Galilean translations to quantum reference frames. The simplest example of dynamical conserved quantities, in this case, are the two momenta \(\hat C_1^{({\mathrm{C}})} = \hat p_{\mathrm{A}}\) and \(\hat C_2^{({\mathrm{C}})} = \hat p_{\mathrm{B}}\). It is immediate from Eqs. (14) and (15) to see that the choice \(\hat C_1^{({\mathrm{A}})} = \hat S_{\mathrm{T}}\hat C_2^{({\mathrm{C}})}\hat S_{\mathrm{T}}^\dagger = \hat \pi _{\mathrm{B}}\) and \(\hat C_2^{({\mathrm{A}})} = - \hat S_{\mathrm{T}}\hat C_1^{({\mathrm{C}})}\hat S_{\mathrm{T}}^\dagger - \hat S_{\mathrm{T}}\hat C_2^{({\mathrm{C}})}\hat S_{\mathrm{T}}^\dagger = \hat \pi _{\mathrm{C}}\) leads to the corresponding conserved quantities in the reference frame A. A similar procedure holds when we consider the extended set of conserved quantities composed of translations \(\hat p_i\) and Galilean boosts \(\hat G_{i} = \hat p_it - m_i\hat x_i\), \(i = {\mathrm{A,B}}\). Notice that this construction of the \(\hat S_{\mathrm{T}}\) operator satisfies the transitive property, meaning that changing the reference frame from C to A directly has the same effect as changing the reference frame first from C to B and then from B to A, i.e. \(\hat S_{\mathrm{T}}^{({\mathrm{C}} \to {\mathrm{A}})} = \hat S_{\mathrm{T}}^{({\mathrm{B}} \to {\mathrm{A}})}\hat S_{\mathrm{T}}^{({\mathrm{C}} \to {\mathrm{B}})}\).

Boosts between quantum reference frames

The second example we consider is the change to a reference frame moving with the velocity of a quantum system A, which is described as a free-particle from the point of view of the initial observer C. The total Hamiltonian for both systems A and B from C’s point of view is \(\hat H_{{\mathrm{AB}}}^{({\mathrm{C}})} = \frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}} + \frac{{\hat p_{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}}\). This section generalises the usual Galilean boost \(\hat U_{\mathrm{b}} = e^{\frac{{\mathrm{i}}}{\hbar }v\hat G_{\mathrm{B}}}\), with \(\hat G_{\mathrm{B}} = \hat p_{\mathrm{B}}t - m_{\mathrm{B}}\hat x_{\mathrm{B}}\) being the generator of the boost on system B, introduced in Results (Transformations between quantum reference frames), to situations in which the velocity of the reference frame is distributed according to its quantum state. With reference to Eq. (7), in this case we have \(\frac{d \hat{X}_A(t)}{dt} = \frac{{\hat p_{\mathrm{A}}}}{{m_{\mathrm{A}}}}\) generalising the parameter v in \(\hat U_{\mathrm{b}}\). The complete transformation, which we call \(\hat S_{\mathrm{b}}\), is

$$\hat S_{\mathrm{b}} = {\mathrm{exp}}\left( { - \frac{{\mathrm{i}}}{\hbar }\frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}t} \right)\hat {\cal P}_{{\mathrm{AC}}}^{({\mathrm{v}})}{\mathrm{exp}}\left( {\frac{{\mathrm{i}}}{\hbar }\frac{{\hat p_{\mathrm{A}}}}{{m_{\mathrm{A}}}}\hat G_{\mathrm{B}}} \right){\mathrm{exp}}\left( {\frac{{\mathrm{i}}}{\hbar }\frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}}t} \right),$$ (16)

where the ‘generalised parity operator’ \(\hat {\cal P}_{{\mathrm{AC}}}^{({\mathrm{v}})} = \hat {\cal P}_{{\mathrm{AC}}}{\mathrm{exp}}\left( {\frac{{\mathrm{i}}}{\hbar }{\mathrm{log}}\sqrt {\frac{{m_{\mathrm{C}}}}{{m_{\mathrm{A}}}}} (\hat x_{\mathrm{A}}\hat p_{\mathrm{A}} + \hat p_{\mathrm{A}}\hat x_{\mathrm{A}})} \right)\) maps the velocity of A to the opposite of the velocity of C via the standard parity-swap operator \(\hat {\cal P}_{{\mathrm{AC}}}\) and an operator scaling coordinates and momenta. Specifically, \(\hat {\cal P}_{{\mathrm{AC}}}^{({\mathrm{v}})}\hat x_{\mathrm{A}}\left( {\hat {\cal P}_{{\mathrm{AC}}}^{({\mathrm{v}})}} \right)^\dagger = - \frac{{m_{\mathrm{C}}}}{{m_{\mathrm{A}}}}\hat q_{\mathrm{C}}\), \(\hat {\cal P}_{{\mathrm{AC}}}^{({\mathrm{v}})}\hat p_{\mathrm{A}}\left( {\hat {\cal P}_{{\mathrm{AC}}}^{({\mathrm{v}})}} \right)^\dagger = - \frac{{m_{\mathrm{A}}}}{{m_{\mathrm{C}}}}\hat \pi _{\mathrm{C}}\). This choice of \(\hat S_{\mathrm{b}}\) ensures that the velocity of A in the reference frame of C is opposite to the velocity of C in the reference frame of A, i.e. \(\hat S_{\mathrm{b}}\frac{{\hat p_{\mathrm{A}}}}{{m_{\mathrm{A}}}}\hat S_{\mathrm{b}}^\dagger = - \frac{{\hat \pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}\). The coordinates and momenta transform as

$$\hat S_{\mathrm{b}}\hat x_{\mathrm{A}}\hat S_{\mathrm{b}}^\dagger = - \frac{{m_{\mathrm{C}}\hat q_{\mathrm{C}} + m_{\mathrm{B}}\hat q_{\mathrm{B}}}}{{m_{\mathrm{A}}}} + \frac{{\hat \pi _{\mathrm{C}} + \hat \pi _{\mathrm{B}}}}{{m_{\mathrm{A}}}}t - \frac{{\hat \pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}t,\hat S_{\mathrm{b}}\hat p_{\mathrm{A}}\hat S_{\mathrm{b}}^\dagger = - \frac{{m_{\mathrm{A}}}}{{m_{\mathrm{C}}}}\hat \pi _{\mathrm{C}},$$ (17)

$$\hat S_{\mathrm{b}}\hat x_{\mathrm{B}}\hat S_{\mathrm{b}}^\dagger = \hat q_{\mathrm{B}} - \frac{{\hat \pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}t,\quad \quad \hat S_{\mathrm{b}}\hat p_{\mathrm{B}}\hat S_{\mathrm{b}}^\dagger = \hat \pi _{\mathrm{B}} - \frac{{m_{\mathrm{B}}}}{{m_{\mathrm{C}}}}\hat \pi _{\mathrm{C}}.$$ (18)

This transformation, similarly to the transformation \(\hat S_{\mathrm{T}}\) discussed previously, is also a symmetry of the free-particle Hamiltonian, because it maps the initial Hamiltonian \(\hat H_{{\mathrm{AB}}}^{({\mathrm{C}})} = \frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}} + \frac{{\hat p_{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}}\) to the Hamiltonian in the new reference frame \(\hat H_{{\mathrm{BC}}}^{({\mathrm{A}})} = \frac{{\hat \pi _{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}} + \frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}\) through Eq. (5). Hence, this constitutes a Galilean boost transformation for quantum reference frames, which allows the system defining the reference frame to be in a superposition of velocities.

To illustrate this point, we consider the situation depicted in Fig. 5. We consider a state \(\left| {\Psi _t} \right\rangle _{{\mathrm{AB}}} = e^{ - \frac{{\mathrm{i}}}{\hbar }\hat H_{{\mathrm{AB}}}^{({\mathrm{C}})}t}\left| {\phi _0} \right\rangle _{\mathrm{A}}\left| {\psi _0} \right\rangle _{\mathrm{B}}\), where the initial state \(|\phi _0\rangle _{\mathrm{A}} = {\int} dp_{\mathrm{A}}\phi _0(p_{\mathrm{A}})|p_{\mathrm{A}}\rangle _{\mathrm{A}}\) of system A is in a superposition of momenta with respect to the initial reference frame C. We now change perspective to the reference frame A. No simple coordinate transformation of reference frames could capture this change. Our method gives, as a result of the transformation \(\hat S_{\mathrm{b}}\), the entangled state of B and C \(\hat S_{\mathrm{b}}\left| {\Psi _t} \right\rangle _{{\mathrm{AB}}} = {\int} d\pi _{\mathrm{C}}d\pi _{\mathrm{B}}e^{ - \frac{{\mathrm{i}}}{\hbar }\left( {\frac{{\pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}} + \frac{{\pi _{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}}} \right)t}\phi _0\left( { - \frac{{m_{\mathrm{A}}}}{{m_{\mathrm{C}}}}\pi _{\mathrm{C}}} \right)\psi _0(\pi _{\mathrm{B}})e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{\pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}\hat G_{\mathrm{B}}}\left| {\pi _{\mathrm{C}}} \right\rangle _{\mathrm{C}}\left| {\pi _{\mathrm{B}}} \right\rangle _{\mathrm{B}}\). The state of B is boosted by the velocity of A (which corresponds to the opposite of the velocity of C, given Eq. (18)) for each momentum in the superposition state of A, while the system C evolves as a free particle with opposite velocity to A.

Fig. 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 Full size image

In the special case of a free particle B in the general state |ψ 0 〉 B and the reference frame A having a state with a well-defined momentum (velocity) |ϕ 0 〉 A = |p A 〉 A , the transformed state \(\hat S_{\mathrm{b}}\left| {\Psi _t} \right\rangle _{{\mathrm{AB}}} = e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{\pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}t}\left| {\pi _{\mathrm{C}}} \right\rangle _{\mathrm{C}}e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{\pi _{\mathrm{C}}}}{{m_{\mathrm{C}}}}\hat G_{\mathrm{B}}}|\psi _t\rangle _{\mathrm{B}}\), where |ψ t 〉 B is the time evolved state and \(\pi _{\mathrm{C}} = - \frac{{m_{\mathrm{C}}}}{{m_{\mathrm{A}}}}p_{\mathrm{A}}\), reduces to the standard boost transformation \(\hat U_{\mathrm{b}}\) in the usual description of reference frames, with the difference that here C is a degree of freedom and hence evolved in time.

With a similar reasoning to the one presented in the previous section, it is possible to show that the set of the conserved quantities \(\hat p_{\mathrm{A}},\,\hat p_{\mathrm{B}},\,\hat G_{\mathrm{A}},\,\hat G_{\mathrm{B}}\) in the reference frame C is mapped to the set of the conserved quantities in the reference frame A \(\hat \pi _{\mathrm{B}},\hat \pi _{\mathrm{C}},\hat G_{\mathrm{B}},\hat G_{\mathrm{C}}\). Analogously to the generalised translations in the previous Subsection, this choice of the operator \(\hat S_{\mathrm{b}}\) also satisfies the transitive property, i.e. \(\hat S_{\mathrm{b}}^{({\mathrm{C}} \to {\mathrm{A}})} = \hat S_{\mathrm{b}}^{({\mathrm{B}} \to {\mathrm{A}})}\hat S_{\mathrm{b}}^{({\mathrm{C}} \to {\mathrm{B}})}\).

Notice that a time-independent version of the transformation \(\hat S_{\mathrm{b}}\), mapping to instantaneous relative velocities, would not preserve the invariance of the Hamiltonian. This example is discussed in Supplementary Note 4.

The weak equivalence principle in quantum reference frames

In this section we generalise the weak equivalence principle to quantum reference frames. By this, we mean that the physical effects as seen from a reference frame moving in a superposition of uniform gravitational fields are indistinguishable from those as seen from a system in superposition of accelerations. To achieve a superposition of accelerations, let us consider the situation depicted in Fig. 6, in which two particles A and B evolve in time according to the Hamiltonian

$$\hat H_{{\mathrm{AB}}}^{({\mathrm{C}})} = \hat H_{\mathrm{A}} + \hat H_{\mathrm{B}} = \frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}} + \frac{{\hat p_{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}} + V(\hat x_{\mathrm{A}})$$ (19)

in the reference frame of an observer C.

Fig. 6 Generalisation of the weak equivalence principle for quantum reference frames. The quantum system A is initially in a superposition of two localised wave amplitudes in a piecewise linear potential \(V(\hat x_{\mathrm{A}})\) from the point of view of another system C. The individual wave amplitudes are localised in spatial intervals corresponding to two different potential gradients. The system B evolves instead as a free particle. If we consider the motion for sufficiently short times, such that the two amplitudes remain localised within the corresponding intervals, the system A evolves as if it were in a superposition of the accelerations \(a_1\) and \(a_2\). We can then change perspective to the accelerated reference frame of A, by applying a transformation corresponding to a ‘superposition of accelerations’, and describe how the quantum system A sees the quantum systems B and C and their evolution. After the transformation, the system B evolves as if it was moving in a superposition of linear gravitational potentials, where the gravitational accelerations are such that \(\vec g_i = - \vec a_i\), where \(i = 1,2\). This means that the effects of a superposition of accelerations are indistinguishable from the effects of a superposition of gravitational fields Full size image

For the purpose of further analysis we will now consider the potential \(V(\hat x_{\mathrm{A}})\) to be piecewise linear and particle A to evolve in time t as a superposition of wave amplitudes, each localised in an interval that corresponds to a constant yet different potential gradient. For concreteness consider the superposition of two such amplitudes, \(\left| {\psi _0(t)} \right\rangle _{\mathrm{A}} = \frac{1}{{\sqrt 2 }}\left( {\left| {\psi _1(t)} \right\rangle _{\mathrm{A}} + \left| {\psi _2(t)} \right\rangle _{\mathrm{A}}} \right)\) (see Fig. 7). The state then is in a superposition of accelerations, i.e. the ‘acceleration operator’ applied on the state gives:

$$- \frac{1}{{m_{\mathrm{A}}}}\frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}\left| {\psi _0(t)} \right\rangle _{\mathrm{A}} \approx \frac{1}{{\sqrt 2 }}\left( {a_1\left| {\psi _1(t)} \right\rangle _{\mathrm{A}} +\, a_2\left| {\psi _2(t)} \right\rangle _{\mathrm{A}}} \right),$$ (20)

where \(a_1 = - \frac{1}{{m_{\mathrm{A}}}}\frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}|_{\begin{array}{c}x_1(t)\end{array}}\) and \(a_2 = - \frac{1}{{m_{\mathrm{A}}}}\frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}|_{\begin{array}{c}x_2(t)\end{array}}\), where x 1 (t) and x 2 (t) are the mean values of position operator for the individual localised amplitudes. Notice that the scalar accelerations a 1 and a 2 as well as the scalar positions x 1 (t) and x 2 (t) should be understood as multiplied by the identity operator.

Fig. 7 Representation of the conditions under which particle A effectively moves in a superposition of accelerations. The state of particle A and the piecewise linear potential \(V(\hat x_{\mathrm{A}})\) are represented for the initial time 0 and for the time t in two different yellow planes. At the initial time, in the background, the state is chosen to be a superposition of two coherent (Gaussian) states localised around x 1 , with width \(2\sigma _1\), and x 2 , with width \(2\sigma _2\). The individual states are localised within the spatial intervals that correspond to constant but different potential gradients. Under time evolution, the point where each of the two localised states is centred moves by \(\delta x_i\), \(i = 1,2\), and the wave-packet spreads. For each of them, it is possible to identify a maximal time such that the individual localised states in the superposition still remain in the region where the gradient of the potential is constant. Up to this time A evolves in a superposition of accelerations Full size image

In order to find the generalised version of the operator \(\hat U_{\mathrm{a}}\), discussed in Results (Transformations between quantum reference frames) and in detail in Supplementary Note 1, and get an expression analogous to the one in Eq. (7), we need the time derivative of the position operator \(\hat x_{\mathrm{A}}\) at time t. To calculate the evolved position operator, we write an explicit expression for \(\hat x_{\mathrm{A}}(t) = e^{\frac{{\mathrm{i}}}{\hbar }\hat H_{\mathrm{A}}t}\hat x_{\mathrm{A}}e^{ - \frac{{\mathrm{i}}}{\hbar }\hat H_{\mathrm{A}}t}\):

$$\hat x_{\mathrm{A}}(t) = \hat x_{\mathrm{A}} + \frac{{\hat p_{\mathrm{A}}}}{{m_{\mathrm{A}}}}t - \frac{1}{2}\frac{1}{{m_{\mathrm{A}}}}\frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}t^2.$$ (21)

From this expression we come to the generalised \(\hat X_{\mathrm{A}}(t) = \hat x_{\mathrm{A}}(t) - \hat x_{\mathrm{A}}\), which describes the change in time of the position of A as compared to the initial position and replaces the function X(t) in the extended Galilean transformation \(\hat U_{\mathrm{a}}\).

Following Eq. (7), the overall transformation \(\hat S_{{\mathrm{EP}}}\) reads

$$\hat S_{{\mathrm{EP}}} = e^{ - \frac{{\mathrm{i}}}{\hbar }\left( {\frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}} + \frac{{m_{\mathrm{C}}}}{{m_{\mathrm{A}}}}V( - \hat q_{\mathrm{C}})} \right)t}\hat {\cal P}_{{\mathrm{AC}}}^{{\mathrm{(v)}}}\hat Q_te^{\frac{{\mathrm{i}}}{\hbar }\left( {\frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}} + V(\hat x_{\mathrm{A}})} \right)t},$$ (22)

where the operator \(\hat {\cal P}_{{\mathrm{AC}}}^{{\mathrm{(v)}}}\) was defined below Eq. (16) and the operator \(\hat Q_t\) is defined as

$$\hat Q_t = e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{m_{\mathrm{B}}}}{{m_{\mathrm{A}}}}\left( {\hat p_{\mathrm{A}} - \frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}t} \right)\hat x_{\mathrm{B}}}e^{\frac{{\mathrm{i}}}{\hbar }\left( {\hat p_{\mathrm{A}} - \frac{1}{2}\frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}t} \right)\frac{{\hat p_{\mathrm{B}}}}{{m_{\mathrm{A}}}}t}e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{m_{\mathrm{B}}}}{{2m_{\mathrm{A}}^2}}{\int}_0^t ds\left( {\hat p_{\mathrm{A}} - \frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}s} \right)^2}$$ (23)

and represents the straightforward extension of the operator \(\hat U_{\mathrm{a}}\). Note that \(\left\langle {\frac{{d^2\hat x_{\mathrm{A}}}}{{dt^2}}} \right\rangle = - \left\langle {\frac{{d^2\hat q_{\mathrm{C}}}}{{dt^2}}} \right\rangle\). Using the transformation in Eq. (22), the new Hamiltonian from the point of view of A is

$$\hat H_{{\mathrm{BC}}}^{({\mathrm{A}})} = \frac{{\hat \pi _{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}} + \frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}} + \frac{{m_{\mathrm{C}}}}{{m_{\mathrm{A}}}}V( - \hat q_{\mathrm{C}}) - \frac{{m_{\mathrm{B}}}}{{m_{\mathrm{A}}}}\frac{{dV}}{{d\hat{x}_{\mathrm{A}}}} \Big|_{{ - \hat q_{\mathrm{C}}}}\, \hat q_{\mathrm{B}}.$$ (24)

From Eq. (24), we can see that B evolves in a potential which is determined by the first derivative of the potential at the position \(- \hat q_{\mathrm{C}}\), while C moves in a potential given by the sum of \(\frac{{m_{\mathrm{C}}}}{{m_{\mathrm{A}}}}V( - \hat q_{\mathrm{C}})\) and the interaction term involving its derivative. Hence, the quantum system B moves, in the reference frame of A, as if it were in a linear gravitational potential with a gravitational acceleration being an operator \(\hat g = 1/m_{\mathrm{A}}dV(\hat x_{\mathrm{A}})/d\hat x_{\mathrm{A}}|_{{ - \hat q_{\mathrm{C}}}}\)in the Hilbert space of C. This is a formulation of the weak equivalence principle in QRF.

As an example, we will now apply \(\hat S_{{\mathrm{EP}}}\) on an arbitrary state \(|\phi (t)\rangle _{\mathrm{B}}\) of B and on a state \(|\psi (t)\rangle _{\mathrm{A}}\) of A, for which we assume that the two localised wave amplitudes were initially prepared as non-overlapping coherent states (i.e. minimum uncertainty wave-packets) with well defined position \(x_i(0)\) and momenta \(p_i(0)\), \(i = 1,2\). We evolve the state of A for time t such that the two amplitudes still have well defined position \(x_i(t)\) and momenta \(p_i(t)\), where the momentum at time t is calculated analogously to \(\hat x_{\mathrm{A}}(t)\), i.e. \(\hat p_{\mathrm{A}}(t) = \hat p_{\mathrm{A}} - \frac{{dV(\hat x_{\mathrm{A}})}}{{d\hat x_{\mathrm{A}}}}t\). We denote the state of each amplitude as \(|\alpha _i(t)\rangle\), with \(i = 1,2\). Hence we obtain

$$\hat S_{{\mathrm{EP}}}\frac{1}{{\sqrt 2 }}(|\alpha _1(t)\rangle _{\mathrm{A}} + |\alpha _2(t)\rangle _{\mathrm{A}})|\phi \rangle _{\mathrm{B}} = \frac{1}{{\sqrt 2 }}(|\alpha _1{\!\! {\prime}} (t)\rangle _{\mathrm{C}}Q_t^1|\phi (t)\rangle _{\mathrm{B}} + |\alpha _2{\!\! {\prime}} (t)\rangle _{\mathrm{C}}Q_t^2|\phi (t)\rangle _{\mathrm{B}}),$$ (25)

where the transformed coherent state |α′ i (t)〉 C is centred in \(\left( { - \frac{{m_{\mathrm{A}}}}{{m_{\mathrm{C}}}}x_i(t), - \frac{{m_{\mathrm{C}}}}{{m_{\mathrm{A}}}}p_i(t)} \right)\) and \(Q_t^i = e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{m_{\mathrm{B}}}}{{m_{\mathrm{A}}}}\left( {p_i - m_{\mathrm{A}}a_it} \right)\hat x_{\mathrm{B}}}e^{\frac{{\mathrm{i}}}{\hbar }\left( {\frac{{p_it}}{{m_{\mathrm{A}}}} - \frac{{a_it^2}}{2}} \right)\hat p_{\mathrm{B}}}e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{m_{\mathrm{B}}}}{{2m_{\mathrm{A}}^2}}{\int}_0^t ds\left( {p_i - m_{\mathrm{A}}a_is} \right)^2}\). From A’s point of view, B evolves in a superposition of gravitational accelerations, which is controlled by the state of C.

We conclude that we have a generalised form of the weak equivalence principle which holds when the reference frame is a quantum particle in superposition of accelerations. This analysis can be extended to a general potential \(V(\hat x_{\mathrm{A}})\) acting for infinitesimal times, as we show in Supplementary Note 5.

The extension of the weak equivalence principle to quantum reference frames provides a good opportunity to test this framework in an experiment. In general terms, a suitable technique to verify the predictions of this section would be measuring relative degrees of freedom, for instance as done in refs. 36,37,38,39. However, a specific proposal on how to do so goes beyond the scope of this work.

It is possible to recover the usual notion of the weak equivalence principle if the potential is linear in the entire space, i.e. \(V(\hat x_{\mathrm{A}}) = m_{\mathrm{A}}a\hat x_{\mathrm{A}}\). The generalised form of the displacement of the reference frame reads \(\hat X_{\mathrm{B}}(t) = \frac{{\hat p_{\mathrm{B}}}}{{m_{\mathrm{A}}}}t - \frac{{at^2}}{2}\) and the operator \(\hat S_{{\mathrm{EP}}}\) is analogous to Eq. (22) with \({\hat{Q}}_t = e^{-\frac{\rm i}{{\hbar}}m_{\mathrm{B}} {\dot{\hat{X}}}_{\mathrm{A}}(t)\hat {x}_{\mathrm{B}}}e^{\,\frac{i}{\hbar}{\hat{X}}(t)\hat {p}_{\mathrm{B}}} e^{-\frac{1}{\hbar}\,\frac{m_{\mathrm{B}}}{2}} \int

olimits_{0}^{t} ds {\dot{\hat{X}}} {{2}\atop {\mathrm{A}}} (s)\), where \(\hat X_{\mathrm{A}}(t)\) has been previously defined and the dot indicates the time derivative. The initial Hamiltonian \(\hat H_{{\mathrm{AB}}}^{({\mathrm{C}})}\) in Eq. (19) is transformed to

$$\hat H_{{\mathrm{BC}}}^{({\mathrm{A}})} = \frac{{\hat \pi _{\mathrm{B}}^2}}{{2m_{\mathrm{B}}}} + \frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}} - m_{\mathrm{C}}a\hat q_{\mathrm{C}} - m_{\mathrm{B}}a\hat q_{\mathrm{B}}.$$ (26)

This result shows that the weak equivalence principle holds also if the reference frame is treated as a quantum system (and can therefore be delocalised) with its own dynamics.

Application: notion of rest frame of a quantum system

In this section, we show how our formalism enables us to define the notion of rest frame when the system is in a superposition of momenta from the point of view of the initial laboratory frame. The rest frame of a system is the frame of reference in which the system is at rest. Physical laws standardly take a simple form in the rest frame; for example, the rest frame Hamiltonian gives the dynamics of the internal degrees of freedom (e.g. spin). It is therefore useful to know how to map the descriptions in the rest and the laboratory frames of reference. As long as the system moves along a classical trajectory and is not treated as a dynamical degree of freedom the map can be achieved through a coordinate transformation between the two reference frames. However, in quantum mechanics, a system can evolve in a superposition of classical trajectories. How can we ‘move’ to the rest frame of a particle that is in superposition of momenta with respect to the laboratory reference frame? Here, by working out an explicit example, we show how our formalism can be used to recover the notion of the rest frame of a quantum system, when the semiclassical approximation fails.

We consider the situation illustrated in Fig. 8, in which an atom with its external (A) and internal (Ã) degrees of freedom interacts with a photon (B), as seen from the laboratory reference frame (C). We assume the internal degrees of freedom to be internal energy states of a two-level system. We want to find conditions under which the photon is in resonance with the internal energy levels of the atom from different frames of reference. More precisely, we want to find the state of the atom and photon such that the probability for the transition is maximised, in the case when the atom does not have a well-defined momentum in the laboratory reference frame. We know that when the source of a photon (where the source is at rest from the point of view of the laboratory reference frame) and the receiver (i.e. the atom) are in relative motion towards each other, and in the limit of small relative velocity between emitter and receiver, the frequency is Doppler-shifted according to \(\omega _{\mathrm{B}}^\prime = \omega _{\mathrm{B}}\left( {1 + \frac{v}{c}} \right)\), where ω B , \(\omega _{\mathrm{B}}^\prime\) are respectively the emitted and received frequency of the photon, v is the relative velocity between emitter and receiver, and c is the speed of light in the medium.

Fig. 8 Transformation to the rest frame of a quantum system. The interaction between a photon B and the internal degrees of freedom Ã of an atom as described from the point of view of a the rest frame of the atom A itself and b the laboratory C. We consider the situation when the atom does not have a sharp momentum in the laboratory reference frame and calculate which state of the photon and the atom we have to prepare to maximise the probability of absorbing the photon. The description of the situation is simplest in the rest reference frame (a). If the photon has spectral frequency \(\omega _{\mathrm{B}} = \frac{{{\mathrm{\Delta }}E}}{\hbar }\) corresponding to the atom’s energy gap, the probability is maximised. For simplicity of illustration, the state of the laboratory is described as a superposition of two amplitudes sharp around the velocities −v 1 and −v 2 . From the point of view of C, this situation is described as in b, in which the state of the photon and the external degrees of freedom of the atom are entangled. For each velocity \(v_i\), \(i = 1,2\) of atom A, the frequency of the photon is Doppler-shifted as: \(\omega _{{\mathrm{B}}i} = \omega _{\mathrm{B}}\left( {1 - \frac{{v_i}}{c}} \right)\) for \(i = 1,2\). The measurement of the frequency \(\hat \omega _{\mathrm{B}}\) in A’s reference frame, corresponds to the measurement of the observable \(\hat \omega _{\mathrm{B}}\left( {1 + \frac{{\hat p_{\mathrm{A}}}}{{cm_{\mathrm{A}}}}} \right)\) in the laboratory reference frame (see the next section). This ensures that if the photon is found absorbed (‘detected’) by the atom in its RF, so it will in the laboratory RF Full size image

The condition for the absorption of the photon is simplest in the rest frame of A. Suppose that in this frame the entire state is given as \(|\psi _t\rangle _{\mathrm{C}}|1,\omega _{\mathrm{B}}\rangle _{\mathrm{B}}|g\rangle _{{\tilde{\mathrm A}}}\). Here, the state of the laboratory at time t is \(|\psi _t\rangle _{\mathrm{C}} = {\int} d\pi _{\mathrm{C}}e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{\pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}t}\psi \left( { - \frac{{m_{\mathrm{A}}}}{{m_{\mathrm{C}}}}\pi _{\mathrm{C}}} \right)|\pi _{\mathrm{C}}\rangle _{\mathrm{C}}\) (this is related to the momentum distribution of atom A in laboratory reference frame C), \(|g\rangle _{{\tilde{\mathrm A}}}\) is the ground state of the internal degrees of freedom of the atom and \(|1,\omega _{\mathrm{B}}\rangle _{\mathrm{B}}\) is the 1-photon state of B with frequency \(\omega _{\mathrm{B}} = \frac{{\Delta E}}{\hbar }\), where \(\Delta E\) is the energy gap between the ground and the excited state of the internal energy. The Hamiltonian in the rest frame is taken to be \(\hat{H}^{(A)}_{{{{\tilde{\mathrm{A}}}\mathrm{BC}}}} = \frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}} + \hbar \hat \omega _{\mathrm{B}} + \hat H_{{\tilde{\mathrm A}}}\), where \(\hbar \hat \omega _{\mathrm{B}}\) is a simplified photon Hamiltonian in the one-particle sector. Here, we promote the frequency \(\omega _{\mathrm{B}}\) to an operator because the frequency shift due to the Doppler effect changes the mode of the photon state, but leaves the number of particles invariant. A more complete description of the Hamiltonian would involve the creation and annihilation operators, but we omit it here because it does not influence our results. The frequency operator \(\hat \omega _{\mathrm{B}}\) acts on the single-photon Hilbert space and is such that \(\hbar \hat \omega _{\mathrm{B}}|\omega _{\mathrm{B}}\rangle _{\mathrm{B}} = \hbar \omega _{\mathrm{B}}|\omega _{\mathrm{B}}\rangle _{\mathrm{B}}\), where the usual relation between momentum and frequency holds, i.e. \(\hbar \omega _{\mathrm{B}} = c|\pi _{\mathrm{B}}|\). Finally, \(\hat H_{{\tilde{\mathrm A}}} = E_g|g\rangle _{{\tilde{\mathrm A}}}\langle g| + E_e|e\rangle _{{\tilde{\mathrm A}}}\langle e|\) is the Hamiltonian of the internal degrees of freedom, with \(E_e - E_g = {\mathrm{\Delta }}E\).

To change the reference frame, we apply a boost transformation between A and C, and the transformation which gives the Doppler shift on the photon. Overall, we obtain

$$\hat S_{\mathrm{D}} = e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}}t}\hat {\cal P}_{{\mathrm{CA}}}^{{\mathrm{(v)}}}\hat R_{f_{\mathrm{C}}(\hat \pi _{\mathrm{C}})}^{\mathrm{B}}e^{\frac{{\mathrm{i}}}{\hbar }\frac{{\hat \pi _{\mathrm{C}}^2}}{{2m_{\mathrm{C}}}}t},$$ (27)

where the operator \(\hat {\cal P}_{{\mathrm{CA}}}^{{\mathrm{(v)}}}\) is the adjoint of \(\hat {\cal P}_{{\mathrm{AC}}}^{{\mathrm{(v)}}}\) defined after Eq. (16), i.e. \(\hat {\cal P}_{{\mathrm{CA}}}^{{\mathrm{(v)}}} = \left( {\hat {\cal P}_{{\mathrm{AC}}}^{{\mathrm{(v)}}}} \right)^\dagger\), and \(\hat R_{f_{\mathrm{C}}(\hat \pi _{\mathrm{C}})}^{\mathrm{B}} = {\mathrm{exp}}\left( {\frac{{\mathrm{i}}}{\hbar }{\mathrm{log}}\sqrt {f_{\mathrm{C}}(\hat \pi _{\mathrm{C}})} (\hat q_{\mathrm{B}}\hat \pi _{\mathrm{B}} + \hat \pi _{\mathrm{B}}\hat q_{\mathrm{B}})} \right)\), with \(f_{\mathrm{C}}(\hat \pi _{\mathrm{C}}) = 1 + \frac{{\hat \pi _{\mathrm{C}}}}{{cm_{\mathrm{C}}}}\). Specifically, the operator \(\hat R_{f_{\mathrm{C}}(\hat \pi _{\mathrm{C}})}^{\mathrm{B}}\) represents the Doppler shift of the photon. Finally, the transformation between the spatial degrees of freedom of A and C is the boost transformation in Eq. (16). We obtain

$$\hat S_{\mathrm{D}}\hat \pi _{\mathrm{C}}\hat S_{\mathrm{D}}^\dagger = - \frac{{m_{\mathrm{A}}}}{{m_{\mathrm{C}}}}\hat p_{\mathrm{A}};\quad \quad \hat S_{\mathrm{D}}\hat \pi _{\mathrm{B}}\hat S_{\mathrm{D}}^\dagger = \left( {1 + \frac{{\hat p_{\mathrm{A}}}}{{m_{\mathrm{A}}c}}} \right)\hat p_{\mathrm{B}}.$$ (28)

Applying this transformation to the Hamiltonian \({\hat{H}}^{(A)}_{{\tilde{\mathrm A}}{\mathrm{BC}}}\) yields

$$\hat H_{{\mathrm{A}} {{\tilde{\mathrm{A}}}}{\mathrm{B}}}^{{\mathrm{(C)}}} = \frac{{\hat p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}} + \hat H_{{\tilde{\mathrm A}}\prime } + \hbar \hat \omega _{\mathrm{B}}\left( {1 + \frac{{\hat p_{\mathrm{A}}}}{{m_{\mathrm{A}}c}}} \right),$$ (29)

where \(\hat \omega _{\mathrm{B}} = \frac{c}{\hbar }|\hat p_{\mathrm{B}}|\). From the perspective of the laboratory C, the Hamiltonian entangles the momentum of the atom A with the frequency of the photon, while the internal degrees of freedom are unchanged. The state of the joint system of the atom, with its internal and external degrees of freedom, and the photon is

$$|\Psi \rangle _{{\mathrm{A}} {{\tilde{\mathrm{A}}}}B}^{{\mathrm{(C)}}} \propto {\int} dp_{\mathrm{A}}e^{ - \frac{{\mathrm{i}}}{\hbar }\frac{{p_{\mathrm{A}}^2}}{{2m_{\mathrm{A}}}}t}\psi (p_{\mathrm{A}})|p_{\mathrm{A}}\rangle _{\mathrm{A}}|1,\omega _{\mathrm{B}}\left( {1 - \frac{{p_{\mathrm{A}}}}{{m_{\mathrm{A}}c}}} \right) \rangle _{\mathrm{B}}|g\rangle _{{\tilde{\mathrm A}}}.$$ (30)

The state (30) is the one which has to be prepared in the laboratory reference frame to maximise the absorption probability. We see that the frequency of the photon B is Doppler shifted by an amount that depends on the velocity of the atom A. In the next section we show that, by mapping the observables in reference frame A to those in reference frame C, the absorption of the photon is predicted consistently in both reference frames, i.e. if the photon is detected in A’s reference frame, so it will in C’s reference frame.

Measurements as seen from a quantum reference frame

In this section we analyse how a measurement procedure performed in one QRF looks like as seen from another QRF. We assume that an observer in reference frame C performs a measurement on the quantum systems A and B. How does an observer in the reference frame A describe this procedure? Note that the procedure in general includes also a measurement on the reference frame of A itself. This situation differs from the Wigner-friend scenario, in which one observer (friend) performs a measurement, while the other (Wigner) considers the process to be unitary24,41. In the present case both observers agree that a measurement is performed, though, as we will see, they might have a different view on which systems and which measurement is performed.

Consider that in C’s reference frame observable \(\hat O_{{\mathrm{AB}}}^{{\mathrm{(C)}}}\) is measured. The transformed observable in A’s reference frame is \(\hat O_{{\mathrm{BC}}}^{({\mathrm{A}})} = \hat S\hat O_{{\mathrm{AB}}}^{({\mathrm{C}})}\hat S^\dagger\), where \(\hat S\) is a general operator which implements the transformation from the reference frame of C to the reference frame of A. Using the cyclicity of the trace, it is immediate to verify that

$$\langle \hat O_{{\mathrm{AB}}}^{({\mathrm{C}})}\rangle = {\mathrm{Tr}}_{{\mathrm{AB}}}(\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})}\hat O_{{\mathrm{AB}}}^{({\mathrm{C}})}) = {\mathrm{Tr}}_{{\mathrm{BC}}}(\hat \rho _{{\mathrm{BC}}}^{{\mathrm{(A)}}}\hat O_{{\mathrm{BC}}}^{({\mathrm{A}})}) = \langle \hat O_{{\mathrm{BC}}}^{({\mathrm{A}})}\rangle ,$$ (31)

where \(\hat \rho _{{\mathrm{BC}}}^{({\mathrm{A}})}\) is the quantum state of B and C relative to A. An explicit example, using operator \(\hat S_{\mathrm{x}}\) in Eq. (2), is a measurement of the position operator \(\hat q_{\mathrm{B}}\) of the quantum system B in the reference frame of A, which is equivalent to the measurement of \(\hat x_{\mathrm{B}} - \hat x_{\mathrm{A}}\) in the reference frame of C.

To make these statements more concrete, we adopt a measurement scheme (see, for instance42) and check how the measurement procedure transforms when we change reference frame. The measurement procedure in C’s and A’s reference frames is depicted in Fig. 9. The measurement scheme consists in adding an ancillary system consisting of a pointer in the state \(\xi _{\mathrm{E}} \in {\cal H}_{\mathrm{E}}^{({\mathrm{C}})}\) and of external (position) degrees of freedom of the measurement apparatus in the state \(\sigma _{\mathrm{M}} \in {\cal H}_{\mathrm{M}}^{({\mathrm{C}})}\). The measurement of the observable \(\hat O_{{\mathrm{AB}}}^{({\mathrm{C}})}\) on the quantum system \(\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})}\) can be then described as an interaction between the pointer and the quantum system, followed by a projection in the Hilbert space of the pointer. The probability of measuring the outcome b* is

$$p(b^ \ast ) = {\mathrm{Tr}}_{{\mathrm{AB}}}\left[ {\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})}\hat O_{{\mathrm{AB}}}^{({\mathrm{C}})}(b^ \ast )} \right] = {\mathrm{Tr}}_{{\mathrm{ABEM}}}\left[ {{\cal C}(\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})} \otimes \xi _{\mathrm{E}}) \otimes \sigma _{\mathrm{M}}(1_{{\mathrm{ABM}}} \otimes \hat F_{\mathrm{E}}(b^ \ast ))} \right],$$ (32)

where \({\cal C}\) is a unitary channel entangling the states from the Hilbert spaces of A and B with those of the pointer E, and \(\hat F_{\mathrm{E}}\) is a projector on \({\cal H}_{\mathrm{E}}^{({\mathrm{C}})}\).

Fig. 9 Measurement in quantum reference frames. The measurement procedure as seen from the point of view of C, in a, and of A, in b. In a, C prepares an ancillary system, constituted by the position degrees of freedom of the apparatus M and a pointer E. The ancilla interacts with the quantum systems A and B. Subsequently, a projective measurement of the state of the pointer gives the outcome. In b, A describes the initial state as an entangled state of B, C, and M. The pointer E then interacts with both systems B and C. Finally, system E is measured. The measurement probability is the same as in the reference frame of C Full size image

We measure the outcome b* as \(\widehat O_{{\mathrm{AB}}}^{({\mathrm{C}})}\left( {b^ \ast } \right) = \left| {b^ \ast } \right\rangle _{{\mathrm{AB}}}\left\langle {b^ \ast } \right|\). Then, if we choose \(\hat F_{\mathrm{E}}\left( {b^ \ast } \right) = \left| {b^ \ast } \right\rangle _{\mathrm{E}}\left\langle {b^ \ast } \right|\), \(\xi _{\mathrm{E}}\) such that \(\left| {\xi _{\mathrm{E}}(x_{\mathrm{E}})} \right|^2 = \delta (x_{\mathrm{E}})\) and

$${\cal C}(\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})} \otimes |\xi _{\mathrm{E}}\rangle _{\mathrm{E}}\langle \xi _{\mathrm{E}}|) = C_{{\mathrm{ABE}}}(\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})} \otimes |\xi _{\mathrm{E}}\rangle _{\mathrm{E}}\langle \xi _{\mathrm{E}}|)C_{{\mathrm{ABE}}}^\dagger ,$$ (33)

where \(C_{{\mathrm{ABE}}} = {\mathrm{exp}}\left( { - \frac{{\mathrm{i}}}{\hbar }\hat O_{{\mathrm{AB}}}^{({\mathrm{C}})}\hat p_{\mathrm{E}}} \right)\), the condition (32) is satisfied for all \(\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})}\). Notice that by \(\hat x_{\mathrm{E}}\) and \(\hat p_{\mathrm{E}}\) we mean the position of the pointer in some abstract space of the internal degrees of freedom of the apparatus, not the real position (relative to C) of the measurement apparatus in space.

We now turn to the description from the point of view of A. For concreteness, we consider the transformation between two frames of reference C and A to be the map in Eq. (2); the formalism can be straightforwardly generalised to other maps. Considering the degrees of freedom of the ancilla the map is modified to include the external degrees of freedom of the measurement apparatus, i.e., \(\hat S_{{\mathrm{x,M}}} = \hat {\cal P}_{{\mathrm{AC}}}e^{\frac{{\mathrm{i}}}{\hbar }\hat x_{\mathrm{A}}(\hat p_{\mathrm{B}} + \hat p_{\mathrm{M}})}\), while the degrees of freedom of the pointer are considered as translational invariant, and therefore not transformed. From A’s reference frame the measurement process is

$$p(b^ \ast ) = {\mathrm{Tr}}_{{\mathrm{BCEM}}}\left[ {C_{{\mathrm{BCEM}}}^{({\mathrm{A}})}(\hat \rho _{{\mathrm{BCM}}}^{({\mathrm{A}})} \otimes \xi _{\mathrm{E}})(C_{{\mathrm{BCEM}}}^{({\mathrm{A}})})^\dagger (1_{{\mathrm{BCM}}} \otimes \hat F_{\mathrm{E}}(b^ \ast ))} \right],$$ (34)

where the state \(\hat \rho _{{\mathrm{BCM}}}^{({\mathrm{A}})} = \hat S_{{\mathrm{x,M}}}(\hat \rho _{{\mathrm{AB}}}^{({\mathrm{C}})} \otimes {\kern 1pt} \sigma _M)\hat S_{{\mathrm{x,M}}}^\dagger\) becomes entangled with the external degrees of freedom of the measurement apparatus, and \(C_{{\mathrm{BCEM}}}^{({\mathrm{A}})} = \hat S_{{\mathrm{x,M}}}C_{{\mathrm{ABE}}}\hat S_{{\mathrm{x,M}}}^\dagger = {\mathrm{exp}}\left( { - \frac{{\mathrm{i}}}{\hbar }\hat O_{{\mathrm{BC}}}^{({\mathrm{A}})}\hat p_{\mathrm{E}}} \right)\). We see that the two observers in the reference frames C and A disagree on which systems undergo the measurement and which observables are measured. For observer C, systems A and B are measured with the help of an ancilla E whose internal and external degrees of freedom are initially in a state that factorizes out. For observer A, systems B and C are measured via the ancilla whose external (but not internal) degrees of freedom are initially entangled with C. Therefore, a measurement model for C is transformed into a measurement model for A.

Notice that the measurement procedure just considered is different from A performing a measurement in her reference frame. In the previous paragraphs, when we changed the reference frame from C to A, we still described the measurement performed by C from the point of view of A. Clearly, A can apply the same measurement procedure as C, with the observables defined in her reference frame \(\hat O^{({\mathrm{A}})}\).

As a concrete example of this situation we consider the atom-photon interaction from the previous section. We make the following identification: the atom’s external degree of freedom is system A, the photon is system B, the atom’s internal degrees of freedom Ã are ancilla E, and finally the laboratory is system C. We consider that A ‘measures’ the frequency of the photon by transition of its internal level from the ground to the excited state. This transition can happen only if the frequency of the photon matches the energy gap of the atom. How is this condition written in the laboratory reference frame, in which the photon frequency is Doppler-shifted?

In the rest frame of the atom A, the channel \(C_{{\mathrm{B}}{\tilde{\mathrm{A}}}}^{{\mathrm{(A)}}}( \cdot ) = C_{{\mathrm{B}}{\tilde{\mathrm{A}}}}^{{\mathrm{(A)}}}( \cdot )C_{{\mathrm{B}}{\tilde{\mathrm{A}}}}^{{\mathrm{(A)}}\dagger} \) entangling the states of the Hilbert spaces of B with those of Ã is such that \(C_{{\mathrm{B}}{\tilde{\mathrm{A}}}}^{{\mathrm{(A)}}} = |e\rangle _{{\tilde{\mathrm A}}}\langle g| \otimes |0,\omega \rangle _{\mathrm{B}}\langle 1,\omega | + h.c.\), and the project or \(\hat F_{{\tilde{\mathrm A}}}^{{\mathrm{(A)}}} = |e\rangle _{{\tilde{\mathrm A}}}\langle e|\). Changing to the laboratory frame C via the application of the \(\hat S_{\mathrm{D}}\) operator in Eq. (27), the entangling channel becomes \(C_{{\mathrm{AB}}{\tilde{\mathrm{A}}}}^{{\mathrm{(C)}}} = 1_{\mathrm{A}} \otimes |e\rangle _{{\tilde{\mathrm A}}}\langle g| \otimes \hat R_{f_{\mathrm{A}}( - \hat p_{\mathrm{A}})}^{\mathrm{B}}|0,\omega \rangle _{\mathrm{B}}\langle 1,\omega |\hat R_{f_{\mathrm{A}}( - \hat p_{\mathrm{A}})}^{{\mathrm{B}}\dagger } + h.c.\), with \(f_{\mathrm{A}}( - \hat p_{\mathrm{A}}) = 1 - \frac{{\hat p_{\mathrm{A}}}}{{cm_{\mathrm{A}}}}\), while the projector is unchanged, i.e. \(\hat F_{{\tilde{\mathrm A}}}^{({\mathrm{A}})} = \hat F_{{\tilde{\mathrm A}}}^{({\mathrm{C}})}\). This ensures, by construction, that the probabilities of the photon being absorbed are the same in the reference frame A and C. What changes is the measured observable in the two reference frames: in the rest frame A one measures the frequency \(\hat \omega _{\mathrm{B}}\), while in the laboratory frame the measurement involves both the external degres of freedom of the atom and the photon. In this case, the observable is \(\left( {1 + \frac{{\hat p_{\mathrm{A}}}}{{cm_{\mathrm{A}}}}} \right)\hat \omega _{\mathrm{B}}\).

Comparison of previous approaches to quantum reference frames

In this Section, we compare different approaches to the topic of quantum reference frames, emphasising differences and similarities to our approach.

Much work has been done on the subject of QRFs starting from the seminal papers by Aharonov and Susskind1,2 and Aharonov and Kaufherr3. In refs. 1,2 the authors established a relation between superselection rules and the lack of a frame of reference. The authors then challenged the existence of superselection rules via some examples, where the superselection rule could be overcome by introducing a reference frame that was correlated with the system. The simplest example of this is described in ref. 4. There it is shown that, if two observers do not share information about their relative phase, this implies a superselection rule for photon number. This superselection rule can be overcome by introducing an appropriate quantum reference frame which is entangled with the system in such a way that the total photon number is conserved. In ref. 3 it was shown that it is possible to consistently formulate quantum theory without appealing to classical reference frames as well-localised laboratories of infinite mass.

QRFs have been considered as resources in quantum information protocols and quantum communication in refs. 4,5,6,7,8,9,10,11,12. These works mainly focus on (a) the consequences of the lack of a shared reference frame for quantum information tasks, on (b) the generalisation of the fact that superselection rules can be overcome by choosing an appropriate quantum system as a reference frame, and on (c) “bounded” reference frames. This means that, in a quantum communication protocol where a system is sent from A to B, the final reference frame only possesses limited or no information on the initial reference frame. In order to obviate this problem, most approaches resort to an encoding of quantum information in the relational degrees of freedom (see, e.g. ref. 5). The tool used to achieve this encoding is the \({\cal G}\)-twirl operation, which consists in an average over the group of symmetries of the external reference frame G. This operation consists in expressing the quantum state of the system under study, ρ, in a way that does not contain any information about the external reference frame. The \({\cal G}\)-twirl operation is mathematically expressed as \({\cal G}(\rho ) = {\int} d\mu (g)U(g)\rho U^\dagger (g)\), where \(U(g)\) is the unitary representation of the group element \(g \in G\), and \(\mu (g)\) is the group-invariant measure. This approach shares some methodological similarities with our work, such as the relevance of the relational degrees of freedom; however, there are some important differences. Firstly, we do not assume the existence of an external reference frame, whose degrees of freedom need to be averaged out. This means that, differently to the other approaches mentioned, we do not need to apply techniques, such as the \({\cal G}\)-twirling, in order to find the relational quantities, because our formalism is genuinely relational from the very start. Secondly, we do not address the problem of communication in absence of a shared reference frame. Finally, we assume that the relation between the initial and final reference frame is known and hence always given by a unitary transformation. Moreover, reference frames in our approach are not bounded; they allow to assign quantum states to all external systems to arbitrary precision.

Other authors focus on the possible role of QRFs in quantum gravity13, or point out how QRFs, together with a relational approach, can lead to intrinsic decoherence due to the finite size of the systems considered14,15. Considering reference frames quantum mechanically is a fundamental ingredient in formulating relational quantum theory, which makes no use of an external reference frame to specify its elements23. A relational approach to QRFs has been also considered in refs. 16,17,18, where the limit to an absolute reference frame was formalised, and in ref. 19, where a symmetry group for transformations of a spin system was reconstructed.

Among the works listed, special mention should be paid to ref. 8, which is the closest to our work. There, the \({\cal G}\)-twirl operation is introduced to average over all the external information to the joint system of two particles, A and S, one of which (A) serves as a reference frame. After this operation, the only quantities remaining are the relative variables of the system S with the quantum reference frame. This description that the QRF A gives of the system presents some similarities with ours. The main differences, at this level, are that in our method, after the transformation, we describe the initial reference frame in addition to the second particle, and we do not rely on any external frame. The rest of the paper, addressing the change to a different QRF, asks a different question to us, and thus arrives to a different result. In ref. 8, the authors switch to the description of a third quantum system B, whose relationship with the initial QRF is not known and will be acquired through a quantum measurement. To this end, they propose a protocol, consisting of multiple steps: (1) the quantum state of the QRF B, initially factored out from the system and the QRF A, undergoes a \({\cal G}\)-twirl operation, which removes the information about the external frame; (2) a quantum measurement on the joint state of the QRFs A and B is performed to establish the relation between the two QRFs; 3) The initial QRF A is discarded. The result of this series of operation is the relational description of the state of the system S with respect to B. As a result of the measurement, the group-averaging operation, and the discarding of the initial QRF the tranformation results in a decohered state of the system in general. The appearance of decoherence is a fundamental difference with our approach, where we assume that the relative description of the two QRFs (in our language, the state of the new QRF from the point of view of the old one) is known and hence the change of reference frame is unitary. In addition, in ref. 8 the system and the new QRF are never entangled at the beginning of the protocol, while our approach does not have this restriction.

Along the lines of1,2,3, the subject of quantum reference frames has been recently revised by Angelo and coworkers20,21,22, and fundamental contributions were provided to understand reference frames as quantum-mechanical systems. In these works, one methodologically begins by defining the state in an external frame, and then moves to the centre of mass and relative coordinates, tracing out the centre of mass coordinate as a degree of freedom that describes its position in this external frame. It was claimed that the equations of motion from the perspective of such relative degrees of freedom are compatible with Galilean relativity and the weak equivalence principle, but that the Hamilton formalism is not, as no Hamiltonian can be found that only depends on the coordinates accessible to the quantum frame of reference 22. This constitutes a fundamental difference with our work, where the Hamiltonian formalism can always be used. The main difference resides in the fact that our transformation is canonical, characteristics which automatically guarantees that a Hamiltonian system is transformed to another Hamiltonian system.

Differently to other approaches, our formalism is genuinely relational by construction. This means that, while the focus of the previous works cited has been to obtain the relational degrees of freedom, we consider from the very start physical degrees of freedom to be relational from the point of view of a chosen QRF. Moreover, similarly to ref. 8 and to refs. 20,21,22, we abandon the view that reference frames are abstract entities, which are useful to fix a set of coordinates, and instead treat the reference frame in the same way as any physical system, featuring its physical state and dynamics. Therefore, a QRF has a quantum state and a Hamiltonian relative to another QRF, and the latter QRF has a quantum state and a Hamiltonian relative to the former QRF. Our paper formalises the transformation of states, dynamics and measurements between these two QRFs, using exclusively relational quantities.

Every quantum state, as specified relative to a QRF, encodes the relational information in terms of probabilities for measuring all the degrees of freedom external to the QRF. As a consequence, our formalism does not appeal to an absolute reference frame and consequently does not require the existence of an ‘external’ perspective. Moreover, differently to other approaches in the literature, our work is not about the lack of a shared reference frame, and we do not consider our QRFs to be “bounded”, feature which usually leads to an imprecise state assigment or to a noisy measurement result. In contrast, our QRFs allow to make state assignments to external systems with arbitrary precision. We adopt an operational approach assuming that every QRF is equipped with hypothetical devices that allow for an operational justification of such state assignements. This operational view, indeed very useful, does not require to have laboratories (and possibly observers) in macroscopic superpositions. We will exemplify the relevance of our formalism for quantum particles by applying it to ‘move’ to the rest frame of a particle that is in a superposition of momenta with respect to the laboratory frame and has internal degrees of freedom that can serve as a ‘measurement device’. Possible tests of our framework would involve experimental techniques such as, for instance, those in refs. 36,37,38,39, which are able to probe the relative degrees of freedom.