Pseudorandom and Pseudoentangled States from Subset States

TL;DR

This paper demonstrates that subset states can be used to construct pseudorandom states in quantum cryptography, showing their indistinguishability from Haar random states and illustrating pseudoentanglement phenomena.

Contribution

It introduces a method to generate pseudorandom states using subset states, resolving a conjecture and analyzing their indistinguishability and entanglement properties.

Findings

Subset states of certain sizes are indistinguishable from Haar random states.

The construction resolves a conjecture by Ji, Liu, and Song.

Small subset states exhibit pseudoentanglement across all cuts.

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 $s = 2^n/\omega(\mathrm{poly}(n))$ 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.