Exponential Separations between Quantum Learning with and without Purification

Understand

In quantum learning tasks, quantum memory can offer exponential reductions in statistical complexity compared to any single-copy strategies, but this typically necessitates at least doubling the system size.

• We show that such exponential reductions can also be achieved by having access to the purification of the target mixed state.
• Specifically, for a low-rank mixed state, only a constant number of ancilla qubits is needed for estimating properties related to its purity, cooled form, principal component and quantum Fisher information with constant sample complexity, which utilizes single-copy measurements on the purification.
• Without access to the purification, we prove that these tasks require exponentially many copies of the target mixed state for any strategies utilizing a bounded number of ancilla qubits, even with the knowledge of the target state's rank.