On the sample complexity of purity and inner product estimation

We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate tr\(ρ²\) of an unknown quantum state ρ to additive error ε. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate tr(ρσ) to additive error ε given copies of unknown quantum state ρ and σ using classical communication and restricted quantum communication. In this paper, we show a strong connection between the sample complexity of purity estimation with bounded quantum memory and inner product estimation with bounded quantum communication and unentangled measurements. We propose a protocol that solves quantum inner product estimation with k-qubit one-way quantum communication and unentangled local measurements using O\(median\{1/ε²,2^n/2/ε,2^n-k/ε²\}\) copies of ρ and σ. Our protocol can be modified to estimate the purity of an unknown quantum state ρ using k-qubit quantum memory with the same complexity. We prove that arbitrary protocols with k-qubit quantum memory that estimate purity to error ε require Ω\(median\{1/ε²,2^n/2/sqrtε,2^n-k/ε²\}\) copies of ρ. This indicates the same lower bound for quantum inner product estimation with one-way k-qubit quantum communication and classical communication, and unentangled local measurements. For purity estimation, we further improve the lower bound to Ω\(max\{1/ε²,2^n/2/ε\}\) for any protocols using an identical single-copy projection-valued measurement. Additionally, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a lower bound on the sample complexity.