Related papers: Bounds on $k$-Uniform Quantum States

Bounds on $k$-Uniform Quantum States

URL: http://arxiv.org/abs/2310.06378v2
Date: Wed, 29 Nov 2023 08:38:11 GMT
Title: Bounds on $k$-Uniform Quantum States
Authors: Fei Shi, Yu Ning, Qi Zhao and Xiande Zhang
Abstract summary: We provide new upper bounds on the parameter $k$ for the existence of $k$-uniform states in $(mathbbCd)otimes N$. Since a $k$-uniform state in $(mathbbCd)otimes N$ corresponds to a pure $(N,1,k+1)_d$ quantum error-correcting codes, we also give new upper bounds on the minimum distance $k+1$ of pure $(N,1,k+1))_d$ quantum error-correcting codes
Score: 22.266687858571363
License: http://creativecommons.org/licenses/by/4.0/
Abstract: Do $N$-partite $k$-uniform states always exist when $k\leq \lfloor\frac{N}{2}\rfloor-1$? In this work, we provide new upper bounds on the parameter $k$ for the existence of $k$-uniform states in $(\mathbb{C}^{d})^{\otimes N}$ when $d=3,4,5$, which extend Rains' bound in 1999 and improve Scott's bound in 2004. Since a $k$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ corresponds to a pure $((N,1,k+1))_{d}$ quantum error-correcting codes, we also give new upper bounds on the minimum distance $k+1$ of pure $((N,1,k+1))_d$ quantum error-correcting codes. Furthermore, we generalize Scott's bound to heterogeneous systems, and show some non-existence results of absolutely maximally entangled states in $\mathbb{C}^{d_1}\otimes(\mathbb{C}^{d_2})^{\otimes 2n}$.

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