AI 中文总结
研究在α-温和测量约束下估计高维量子态的问题，给出最优极小极大估计率，提出最优温和测量及实现方式，证明下界，揭示温和性损失与希尔伯特空间环境维度相关，区别于经典差分隐私。
AI 中文摘要
我们研究在测量为α-温和的约束下估计d维量子态ρ的任务。这种测量M不会使态坍缩，会产生包含统计信息的随机变量RM = ω和测量后态ρM→ω，使得||ρM→ω - ρ||Tr≤α。我们描述了温和测量及其与量子差分隐私的联系。结果表明，在Frobenius范数下的最优极小极大估计率为d³/(nα²)，而非一般测量的d²/n。对于秩为r（r≤d）的态，最优极小极大率为rd²/(nα²)，而非rd/n。令人惊讶的是，温和性损失d/α²与希尔伯特空间的环境维度相关，而非经典差分隐私中常见的参数数量rd。我们提出了最优温和测量，并指出如何使用辅助态和CNOT门通过物理方式实现，使其与初始态纠缠。我们注意到产生的随机变量的似然性满足局部差分隐私。通过应用于精心选择的（低秩）量子态流形中的态族的新量子信息理论不等式证明了下界。
英文摘要
We study the task of estimating a $d-$dimensional quantum state $ρ$ under the constraint that the measurement is $α-$gentle. Such measurements $M$ do not collapse the state; they issue both a random variable $R^M = ω$ containing statistical information and a post-measurement state $ρ_{M \to ω}$ such that $\|ρ_{M\to ω} - ρ\|_{Tr} \leq α$. We describe gentle measurements and their connection to quantum differential privacy. Our results show that the optimal minimax estimation rate in Frobenius norm is of order $d^3/(n α^2)$, instead of $d^2/n$ for general measurements. Moreover, for rank $r$ states with $r\leq d$ we prove that the optimal minimax rate is $rd^2/(n α^2)$, instead of $rd/n$. Very surprisingly, the loss for gentleness $d/α^2$ scales with the ambient dimension of the Hilbert space, rather than the number of parameters $rd$, typically seen in classical differential privacy. We propose optimal gentle measurements and indicate how they can be physically implemented using an ancillary state and a CNOT gate to entangle it with the initial state. We notice that the resulting random variable has a likelihood that satisfies local differential privacy. Lower bounds are proven through a new quantum information-theoretic inequality applied to well chosen families of states in the manifold of (small-rank) quantum states.
