We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate $tr(\rho^2)$ of an unknown quantum state $\rho$ to additive error $\epsilon$. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate $tr(\rho\sigma)$ to additive error $\epsilon$ given copies of unknown quantum state $\rho$ and $\sigma$ 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 $O(median\{1/\epsilon^2,2^{n/2}/\epsilon,2^{n-k}/\epsilon^2\})$ copies of $\rho$ and $\sigma$. Our protocol can be modified to estimate the purity of an unknown quantum state $\rho$ using $k$-qubit quantum memory with the same complexity. We prove that arbitrary protocols with $k$-qubit quantum memory that estimate purity to error $\epsilon$ require $\Omega(median\{1/\epsilon^2,2^{n/2}/\sqrt{\epsilon},2^{n-k}/\epsilon^2\})$ copies of $\rho$. 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 $\Omega(\max\{1/\epsilon^2,2^{n/2}/\epsilon\})$ 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.

翻译：本文研究了量子纯度估计与量子内积估计这两类典型任务的样本复杂度。在纯度估计任务中，我们需要以加性误差ε估计未知量子态ρ的迹tr(ρ²)。而在量子内积估计任务中，Alice和Bob需要在仅使用经典通信与受限量子通信的条件下，利用未知量子态ρ和σ的副本以加性误差ε估计tr(ρσ)。本文揭示了有界量子内存下的纯度估计样本复杂度与有界量子通信及非纠缠测量下的内积估计样本复杂度之间的深刻联系。我们提出了一种协议，该协议利用k量子比特单向量子通信与非纠缠局域测量，以O(median{1/ε²,2^(n/2)/ε,2^(n-k)/ε²})个ρ和σ副本的代价解决量子内积估计问题。该协议可被修改为使用k量子比特量子内存以相同复杂度估计未知量子态ρ的纯度。我们证明了任何使用k量子比特量子内存、以误差ε估计纯度的协议都需要Ω(median{1/ε²,2^(n/2)/√ε,2^(n-k)/ε²})个ρ副本。这表明在单向k量子比特量子通信与经典通信及非纠缠局域测量下的量子内积估计具有相同下界。对于纯度估计，我们进一步将下界改进为Ω(max{1/ε²,2^(n/2)/ε})，该下界适用于任何采用相同单副本投影值测量的协议。此外，我们研究了混合态下无需量子通信的量子分布式内积估计判定变体，并给出了其样本复杂度的下界。