Pseudorandom and Pseudoentangled States from Subset States

• URL: http://arxiv.org/abs/2312.15285v2
• Date: Sat, 2 Mar 2024 13:34:39 GMT
• Title: Pseudorandom and Pseudoentangled States from Subset States
• Authors: Fernando Granha Jeronimo, Nir Magrafta, Pei Wu
• Abstract summary: A subset state with respect to $S$, a subset of the computational basis, is [ frac1sqrt|S|sum_iin S |irangle. We show that for any fixed subset size $|S|=s$ such that $s = 2n/omega(mathrmpoly(n))$ and $s=omega(mathrmpoly(n))$, a random subset state is information-theoretically indistinguishable from a Haar random state even provided
• Score: 49.74460522523316
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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