We consider both the classical and quantum variations of $X$-secure, $E$-eavesdropped and $T$-colluding symmetric private information retrieval (SPIR). This is the first work to study SPIR with $X$-security in classical or quantum variations. We first develop a scheme for classical $X$-secure, $E$-eavesdropped and $T$-colluding SPIR (XSETSPIR) based on a modified version of cross subspace alignment (CSA), which achieves a rate of $R= 1 - \frac{X+\max(T,E)}{N}$. The modified scheme achieves the same rate as the scheme used for $X$-secure PIR with the extra benefit of symmetric privacy. Next, we extend this scheme to its quantum counterpart based on the $N$-sum box abstraction. This is the first work to consider the presence of eavesdroppers in quantum private information retrieval (QPIR). In the quantum variation, the eavesdroppers have better access to information over the quantum channel compared to the classical channel due to the over-the-air decodability. To that end, we develop another scheme specialized to combat eavesdroppers over quantum channels. The scheme proposed for $X$-secure, $E$-eavesdropped and $T$-colluding quantum SPIR (XSETQSPIR) in this work maintains the super-dense coding gain from the shared entanglement between the databases, i.e., achieves a rate of $R_Q = \min\left\{ 1, 2\left(1-\frac{X+\max(T,E)}{N}\right)\right\}$.

翻译：我们研究了经典和量子变体下的$X$-安全、$E$-窃听和$T$-合谋对称私密信息检索（SPIR）。这是首次在经典或量子变体中探讨具有$X$-安全性的SPIR工作。我们首先基于修正的交叉子空间对齐（CSA）方法，设计了一种经典$X$-安全、$E$-窃听和$T$-合谋SPIR（XSETSPIR）方案，其速率达到$R= 1 - \frac{X+\max(T,E)}{N}$。该修正方案在实现与$X$-安全PIR方案相同速率的基础上，额外提供了对称隐私保护。随后，我们将此方案扩展到基于$N$-求和盒抽象结构的量子对应版本。这是首次在量子私密信息检索（QPIR）中考虑窃听者存在的工作。在量子变体中，由于空中可解码性，窃听者通过量子信道比经典信道能更有效地获取信息。为此，我们专门设计了另一种对抗量子信道窃听者的方案。本文提出的$X$-安全、$E$-窃听和$T$-合谋量子SPIR（XSETQSPIR）方案，维持了数据库间共享纠缠的超级密集编码增益，即达到$R_Q = \min\left\{ 1, 2\left(1-\frac{X+\max(T,E)}{N}\right)\right\}$的速率。