Reference Frames Which Separately Store Noncommuting Conserved Quantities

Even in the presence of conservation laws, one can perform arbitrary transformations on a system if given access to a suitable reference frame, since conserved quantities may be exchanged between the system and the frame. Here we explore whether these quantities can be separated into different parts of the reference frame, with each part acting as a “battery” for a distinct quantity. For systems composed of spin-$\frac{1}{2}$ particles, we show that the components of angular momentum $S_x$, $S_y$, and $S_z$ (noncommuting conserved quantities) may be separated in this way, and also provide several extensions of this result. These results also play a key role in the quantum thermodynamics of noncommuting conserved quantities.

Figure 1

A spin-$\frac{1}{2}$ system, initially with spin $\mathbf{s}$ (color blue online) undergoes a rotation, after which its spin becomes ${\mathbf{s}}^{\prime }$ (color orange online). During this rotation, its spin components in the $x$, $y$, and $z$ direction change by $\mathrm{\Delta }{s}_x$, $\mathrm{\Delta }{s}_y$, and $\mathrm{\Delta }{s}_z$, respectively. This unitary evolution is implemented by acting on the system plus the particles of a reference frame (green box, online), composed of spin-$\frac{1}{2}$ particles that are divided into three groups: (i) the $x$ battery, composed of particles in the $+$ eigenstates of $s_y$ or $s_z$, (ii) the $y$ battery, composed of particles in the $+$ eigenstates of $s_x$ or $s_z$, and (iii) the $z$ battery, composed of particles in the $+$ eigenstates of $s_x$ or $s_y$. Up to arbitrary accuracy, the particles in the reference frame corresponding to the $k$ battery will absorb all and only the change $\mathrm{\Delta }{s}_k$ of the evolved spin system.