Pseudorandom and Pseudoentangled States from Subset States
Fernando Granha Jeronimo, Nir Magrafta, Pei Wu
TL;DR
The paper resolves a conjecture that random subset states can generate pseudorandom quantum states (PRS) by proving information-theoretic indistinguishability from Haar randomness in a nontrivial subset-size regime: for $s$ with $t(n)=\omega(\mathrm{poly}(n))$ and $t(n)\le s \le 2^n/t(n)$, and any polynomially many copies $k$, the $k$-copy Haar mixture and the average over subset states are close in trace distance. The key technical contribution is a bound on $\left\|\int \psi^{\otimes k} d\mu(\psi) - \mathbb{E}_{S:|S|=s} \phi_S^{\otimes k}\right\|_1 \le O\left(\frac{k^2}{d} + \frac{k}{\sqrt{s}} + \frac{sk}{d}\right)$, derived via a decomposition linking the Haar state to a weighted sum of adjacency matrices of generalized Johnson graphs. This yields a PRS from a quantum-secure PRP family: the state $\frac{1}{\sqrt{s}}\sum_{x\in [s]} |\mathrm{PRP}_k(x)\rangle$ is a PRS on $n$ qubits for $t(n)\le s \le 2^n/t(n)$, and shows that subset states with small $s$ exhibit $h(n)$-pseudoentanglement with entanglement entropy $O(\log |S|)$ across cuts. The work advances the understanding of PRS by achieving a single-phase construction (no phase randomness) and connects to Johnson-graph spectra, resolving the conjecture while highlighting a pseudoentanglement phenomenon; concurrent independent work corroborates the core results.
Abstract
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 $s = 2^n/ω(\mathrm{poly}(n))$ and $s=ω(\mathrm{poly}(n))$, 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.