Extremal Maximal Entanglement
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913581001342976 |
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| author | Zhang, Wanchen Ning, Yu Shi, Fei Zhang, Xiande |
| author_facet | Zhang, Wanchen Ning, Yu Shi, Fei Zhang, Xiande |
| contents | A pure multipartite quantum state is called absolutely maximally entangled if all reductions of no more than half of the parties are maximally mixed. However, an $n$-qubit absolutely maximally entangled state only exists when $n$ equals $2$, $3$, $5$, and $6$. A natural question arises when it does not exist: which $n$-qubit pure state has the largest number of maximally mixed $\lfloor n/2 \rfloor$-party reductions? Denote this number by $Qex(n)$. It was shown that $Qex(4)=4$ in [Higuchi et al.Phys. Lett. A (2000)] and $Qex(7)=32$ in [Huber et al.Phys. Rev. Lett. (2017)]. In this paper, we give a general upper bound of $Qex(n)$ by linking the well-known Turán's problem in graph theory, and provide lower bounds by constructive and probabilistic methods. In particular, we show that $Qex(8)=56$, which is the third known value for this problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_12208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extremal Maximal Entanglement Zhang, Wanchen Ning, Yu Shi, Fei Zhang, Xiande Quantum Physics A pure multipartite quantum state is called absolutely maximally entangled if all reductions of no more than half of the parties are maximally mixed. However, an $n$-qubit absolutely maximally entangled state only exists when $n$ equals $2$, $3$, $5$, and $6$. A natural question arises when it does not exist: which $n$-qubit pure state has the largest number of maximally mixed $\lfloor n/2 \rfloor$-party reductions? Denote this number by $Qex(n)$. It was shown that $Qex(4)=4$ in [Higuchi et al.Phys. Lett. A (2000)] and $Qex(7)=32$ in [Huber et al.Phys. Rev. Lett. (2017)]. In this paper, we give a general upper bound of $Qex(n)$ by linking the well-known Turán's problem in graph theory, and provide lower bounds by constructive and probabilistic methods. In particular, we show that $Qex(8)=56$, which is the third known value for this problem. |
| title | Extremal Maximal Entanglement |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2411.12208 |