Coherence and Asymmetry Cannot be Broadcast

In the presence of conservation laws, superpositions of eigenstates of the corresponding conserved quantities cannot be generated by quantum dynamics. Thus, any such coherence represents a potentially valuable resource of asymmetry, which can be used, for example, to enhance the precision of quantum metrology or to enable state transitions in quantum thermodynamics. Here we ask if such superpositions, already present in a reference system, can be broadcast to other systems, thereby distributing asymmetry indefinitely at the expense of creating correlations. We prove a no-go theorem showing that this is forbidden by quantum mechanics in every finite-dimensional system. In doing so, we also answer some open questions in the quantum information literature concerning the sharing of timing information of a clock and the possibility of catalysis in quantum thermodynamics. We also prove that even weaker forms of broadcasting, of which Åberg’s “catalytic coherence” is a particular example, can only occur in the presence of infinite-dimensional reference systems. Our results set fundamental limits on the creation and manipulation of quantum coherence and shed light on the possibilities and limitations of quantum reference frames to act catalytically without being degraded.

Figure 1

A weak broadcasting protocol (see Definition 3). A reference system $R$ distributes asymmetry or coherence to a system $S_n$, transforming it from an eigenstate to a superposition of eigenstates. The state of $R$ gets correlated to $S_n$ and is allowed to change, but it must be able to induce the same transition on a fresh copy of the system $S_{n+1}$ for all $n$. Åberg’s protocol [11] is a special case of weak broadcasting of coherence with $\dim R=\infty$.