Pseudorandom states (PRS) are an important primitive in quantum cryptography. In this paper, we show that subset states can be used to construct PRSs. A subset state with respect to $S$, a subset of the computational basis, is \[ \frac{1}{\sqrt{|S|}}\sum_{i\in S} |i\rangle. \] As a technical centerpiece, we show that for any fixed subset size $|S|=s$ such that $s = 2^n/\omega(\mathrm{poly}(n))$ and $s=\omega(\mathrm{poly}(n))$, where $n$ is the number of qubits, a random subset state is information-theoretically indistinguishable from a Haar random state even provided with polynomially many copies. This range of parameter is tight. Our work resolves a conjecture by Ji, Liu and Song. Since subset states of small size have small entanglement across all cuts, this construction also illustrates a pseudoentanglement phenomenon.

翻译：伪随机态（PRS）是量子密码学中的重要原语。本文证明，子集态可用于构建伪随机态。相对于计算基的子集$S$，子集态定义为\[ \frac{1}{\sqrt{|S|}}\sum_{i\in S} |i\rangle \]。作为技术核心，我们证明：对于任意固定子集大小$|S|=s$，其中$s = 2^n/\omega(\mathrm{poly}(n))$且$s=\omega(\mathrm{poly}(n))$（$n$为量子比特数），即使提供多项式数量的副本，随机子集态在信息论意义上与哈达玛随机态不可区分。该参数范围具有紧致性。这一结果解决了季、刘和宋提出的猜想。由于小尺寸子集态在所有切割上的纠缠度均较小，该构造还揭示了伪纠缠现象。