Relational time in anyonic systems

In a seminal paper [Phys. Rev. D 27, 2885 (1983)], Page and Wootters suggest that time evolution could be described solely in terms of correlations between systems and clocks, as a means of dealing with the “problem of time” stemming from vanishing Hamiltonian dynamics in many theories of quantum gravity. Their approach seeks to identify relational dynamics given a Hamiltonian constraint on the physical states. Here we present a “state-centric” reformulation of the Page and Wootters model better suited to cases where the Hamiltonian constraint is satisfied, such as anyons emerging in Chern–Simons theories. We describe relational time by encoding logical “clock” qubits into topologically protected anyonic degrees of freedom. The minimum temporal increment of such anyonic clocks is determined by the universality of the anyonic braid group, with nonuniversal models naturally exhibiting discrete time. We exemplify this approach by using $\mathrm{SU}{\left(2\right)}_2$ anyons and discuss generalizations to other states and models.

Figure 1

Three $\sigma \mathrm{SU}{\left(2\right)}_2$ anyons with total charge $\sigma$ encode a logical qubit. Pauli measurements, $\mathrm{X}$, $\mathrm{Y}$, or $\mathrm{Z}$, on the qubit are implemented by fusing a pair of anyons (i.e., measuring their total charge), indicated by the colored ellipses. Fusion of two $\sigma$ yields $\mathbb{1}$ or $\psi$, corresponding to a projective measurement in one of the Pauli bases $\{\mathrm{X},\mathrm{Y},\mathrm{Z}\}$, depending on which pair is fused. Braiding among the three anyons effects $\frac{\pi }{2}$ rotations, $\sqrt{\mathrm{X}}$, $\sqrt{\mathrm{Y}}$, $\sqrt{\mathrm{Z}}$, around the three axes. $\sqrt{\mathrm{X}}$, $\sqrt{\mathrm{Y}}$, $\sqrt{\mathrm{Z}}$ are equivalent to swapping anyons (2,3), (1,3), and (1,2), respectively.

Figure 2

System qubit $\mathrm{s}$ and clock qubit $\mathrm{c}$ are prepared in a Bell state $|{\mathrm{\Psi }}_0{\rangle }_{\mathrm{cs}}$. To implement a POVM on the clock, the clock is coupled to a collection of $k$ ancilla via a unitary gate $\text{U}$. Depending on the universality class of the model, $\text{U}$ yields a POVM on the clock qubit with ${\mathcal{N}}_{\mathrm{c}}\le 2^{k+1}$ possible outcomes, which are in direct correspondence with the set of $\mathrm{Z}$-measurement outcomes, $\{{\mathfrak{z}}_{\mathrm{c}},{\mathfrak{z}}_1,\cdots ,{\mathfrak{z}}_k\}$, on the clock and ancilla qubits. An ordering of those outcomes gives the clock time $\tau \in \{0,1,...,{\mathcal{N}}_{\mathrm{c}}-1\}$.