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arXiv · 2310.06378

Bounds on $k$-Uniform Quantum States

Fei Shi·Yu Ning·Qi Zhao·Xiande Zhang
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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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Fei Shi, Yu Ning, Qi Zhao, Xiande Zhang. 2023-10-10. Bounds on $k$-Uniform Quantum States. https://arxiv.org/abs/2310.06378

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