Two-Ievel (qubit) clock systems are often used to perform precise measurement of time. In this work, we propose a compression protocol for n identically prepared states of qubit clocks. The protocol faithfully encodes the states into (1/2) logn qubits and (1/2) logn classical bits and works even in the presence of noise. If the purity of the clock states is fixed, (1/2) logn qubits are sufficient. We also prove that this protocol requires the minimum amount of total memory among all protocols with vanishing error in the large n limit.

We study the compression of n quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with f independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size (f/2)\log n, including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of n identically prepared modes. In both cases, we show that (f/2)\log n is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.

We study the compression of arbitrary parametric families of n identically prepared finite-dimensional quantum states, in a setting that can be regarded as a quantum analogue of population coding. For a family with f free parameters, we propose an asymptotically faithful protocol that requires a memory of overall size (f/2) log n. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of the optimal compression of n identically prepared displaced thermal states. Our protocol achieves the ultimate bound predicted by quantum Shannon theory. In addition, we explore the minimum requirement for quantum memory: On the one hand, the amount of quantum memory used by our protocol can be made arbitrarily small compared to the overall memory cost; on the other hand, any protocol using only classical memory cannot be faithful.