Quantum reference frames associated with noncompact groups: The case of translations and boosts and the role of mass

Quantum communication without a shared reference frame or the construction of a relational quantum theory requires the notion of a quantum reference frame. We analyze aspects of quantum reference frames associated with noncompact groups, specifically, the group of spatial translations and Galilean boosts. We begin by demonstrating how the usually employed group average, used to dispense of the notion of an external reference frame, leads to unphysical states when applied to reference frames associated with noncompact groups. However, we show that this average does lead naturally to a reduced state on the relative degrees of freedom of a system, which was previously considered by Angelo et al. [J. Phys. A: Math. Theor. 44, 145304 (2011)]. We then study in detail the informational properties of this reduced state for systems of two and three particles in Gaussian states.

Figure 1

The logarithmic negativity, as a measure of the entanglement between the center-of-mass and relation degrees of freedom, of the state associated with ${\mathbf{V}}_{CMR}$, when ${\mathbf{V}}_1={\mathbf{V}}_2$ and both ${\rho }_1$ and ${\rho }_2$ are pure, i.e., $det{\mathbf{V}}_1=det{\mathbf{V}}_2=1$, is plotted as a function of the squeezing parameter $r=r_1=r_2$ and the ratio of masses $m_1/(m_1+m_2)$ for different phase rotations $\theta ={\theta }_1={\theta }_2$: (a) $\theta =0$, (b) $\theta =\pi /32$, (c) $\theta =\pi /8$, and (d) $\theta =\pi /4$.

Figure 2

The logarithmic negativity, as a measure of the entanglement between the center-of-mass and relation degrees of freedom, of the state associated with ${\mathbf{V}}_{CMR}$, when $r=r_1=r_2, \theta ={\theta }_1={\theta }_2$, and particle 2 is a pure state, ${\mu }_2=1$, and particle 1 is not, for different purities of particle 1, ${\mu }_1$, and phase rotations $\theta$. In (a,b) $\theta =0$ and (c,d) $\theta =\pi /4$. In (a,c) ${\mu }_1=0.6$ and (b,d) $\mu =0.2$. Plots for $\theta =0$ and ${\mu }_1=1$ and $\theta =\pi /4$ and ${\mu }_1=1$ are shown in Figs. 1 and 1, respectively.

Figure 3

The logarithmic negativity, as a measure of the entanglement between the center-of-mass and relation degrees of freedom, of the state associated with ${\mathbf{V}}_{CMR}$ when $det{\mathbf{V}}_1=det{\mathbf{V}}_2=1$ for $r_2=\alpha r_1$, and for different phase rotations $\theta ={\theta }_1={\theta }_2$ and values $\alpha$. In (a,b) $\theta =0$ and (c,d) $\theta =\pi /4$. In (a,c) $\alpha =0$ and (b,d) $\alpha =0.5$. Plots for $\theta =0$ and $\alpha =1$ and $\theta =\pi /4$ and $\alpha =1$ are shown in Figs. 1 and 1, respectively.

Figure 4

The logarithmic negativity is plotted, as a measure of the entanglement between the center-of-mass and relation degrees of freedom, of the state associated with ${\mathbf{V}}_{CMR}$ describing three particles for different equal phase rotations ${\theta }_1={\theta }_2={\theta }_3=\theta$ with $det{\mathbf{V}}_1=det{\mathbf{V}}_2=det{\mathbf{V}}_3=1$. In (a) and (b) the logarithmic negativity is plotted for the case when $m_2=m_3$, as a function of the ratio $m_1/(m_1+m_2+m_3)$ and equal squeezing parameter $r_1=r_2=r_3=r$, with $\theta =0$ and $\theta =\pi /4$, respectively. In (c) and (d) the logarithmic negativity is plotted as a function of the two mass ratios $m_1/(m_1+m_2+m_3)$ and $m_2/(m_1+m_2+m_3)$ for $\theta =0$ and $\theta =\pi /4$, respectively, with the equal squeezing parameter fixed at $r=0.7$.

Figure 5

The logarithmic negativity of the relative state of particles 2 and 3 described by ${\mathbf{V}}_{23|1}$ is plotted, characterizing the entanglement among the relational degrees of freedom, for different equal phase rotations ${\theta }_1={\theta }_2={\theta }_3=\theta$ with $det{\mathbf{V}}_1=det{\mathbf{V}}_2=det{\mathbf{V}}_3=1$. In (a) and (b) the logarithmic negativity is plotted for the case $m_2=m_3$ as a function of the ratio $m_1/(m_1+m_2+m_3)$ and equal squeezing parameters $r_1=r_2=r_3=r$ for $\theta =0$ and $\theta =\pi /4$, respectively. In (c) and (d) logarithmic negativity is plotted as a function of the mass ratios $m_1/(m_1+m_2+m_3)$ and $m_2/(m_1+m_2+m_3)$ for equal squeezing parameter $r=0.7$ and $\theta =0$ and $\theta =\pi /4$, respectively.