⇦Chinese Physics Letters, 2020, Vol. 37, No. 9, Article code 090302 Constructing a Maximally Entangled Seven-Qubit State via Orthogonal Arrays Xin-Wei Zha (查新未), Min-Rui Wang (王敏锐)*, and Ruo-Xu Jiang (姜若虚) Affiliations School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China Received 15 March 2020; accepted 7 July 2020; published online 1 September 2020 *Corresponding author. Email: 503989460@qq.com Citation Text: Cha X W, Wang M R and Jiang R X 2020 Chin. Phys. Lett. 37 090302    Abstract Huber et al. [Phys. Rev. Lett. 118 (2017) 200502] have proved that a seven-qubit state whose three-body marginal states are all maximally mixed does not exist. Here, we propose a method to build a maximally entangled state based on orthogonal arrays to construct maximally entangled seven-qubit states. Using this method, we not only determine that a seven-qubit state whose three-body marginals are all maximally mixed does not exist, but also find the condition for maximally entangled seven-qubit states. We consider that $\pi_{\rm ME} =19/140$ is a condition for maximally entangled seven-qubit states. Furthermore, we derive three forms of maximally entangled seven-qubit states via orthogonal arrays. DOI:10.1088/0256-307X/37/9/090302 PACS:03.67.-a, 03.65.Ud, 03.67.Mn © 2020 Chinese Physics Society Article Text Entanglement is considered as a significant key resource for quantum information and computation.[1–4] In particular, the search for maximally entangled states has attracted great deal of attention.[5–14] With regard to an $n$-qubit pure state $|\psi\rangle$, if all $k$-qubit reduced states are maximally mixed, it is referred to as a $k$-uniform state.[15,16] It is particularly interesting that these $n$-qubit states are $[n/2]$-uniform. These states are also known as absolutely maximally entangled (AME) states. In fact, it has been shown that these states exist only for special values of $n$ ($n=2,3,5,6$). For instance, there is no AME state for four qubits,[17] seven qubits,[18] or for eight or more qubits.[13,19] It is therefore relevant to ask: which states are maximally entangled? Those states whose entanglement is maximal for every (balanced) bipartition are maximally multi-qubit entangled states (MMESs).[13] Therefore, we can determine whether a state is an MMES by determining whether the averaged purity over all $[n / 2]$-qubit subsystems reaches its minimum.[12] The averaged subsystem purity for a state is specified as: $$ \pi_{\rm ME} =\begin{pmatrix} n \\ n_{A} \end{pmatrix} ^{-1}\sum\limits_{| A |=n_A} {\pi_{A} },~~ \tag {1} $$ where $n_{A} =[n / 2]$ ($[n/2]$ takes the integer of $n/2$); $\pi_{A} ={\rm Tr}_{\!A}\, \rho_{A}^{2}$, and $\rho_{A} ={\rm Tr}_{\overline A } \,|\psi\rangle \langle \psi |$, used for calculating $\pi_{A}$, is the result after a partial trace operation over the complementary subsystem of $A$, i.e., $\overline{A}$, in the summation. It is obvious that for qubit cases, $\frac{1}{N_{A} }\leqslant \pi_{\rm ME} \leqslant 1$, where $N_{A} =2^{n_{A}}$. A maximally multi-qubit entangled state is a minimizer of $\pi_{\rm ME}$.[20] Although Huber et al.[18] have proved that no AME state exists for the seven-qubit case, they have not given a criterion for the identification of a maximally entangled seven-qubit state. In Ref. [21], we find a maximally entangled seven-qubit state. However, for seven-qubit entangled states, a general theoretical criterion for a maximally entangled seven-qubit state has not yet been found. In 2014, Goyeneche and Zyczkowski established a link between the combinatorial notions of orthogonal arrays.[15] In 2018, we proposed a method of adding phase parameters,[22] and were therefore able to construct four-qubit quantum states via an orthogonal array. In this study, using phase parameters and the minimum of $\pi_{\rm ME}$, we propose a new method for constructing a maximal seven-qubit entangled state. We find that $\pi_{\rm ME} =19/140$ is a condition for maximally entangled seven-qubit states.
Table 1. Orthogonal array of eight qubits.
A B C D E F G H
0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 0 1 1 1 1 0 0
0 1 0 1 0 1 0 1
0 1 0 1 1 0 1 0
0 1 1 0 0 1 1 0
0 1 1 0 1 0 0 1
1 0 0 1 0 1 1 0
1 0 0 1 1 0 0 1
1 0 1 0 0 1 0 1
1 0 1 0 1 0 1 0
1 1 0 0 0 0 1 1
1 1 0 0 1 1 0 0
1 1 1 1 0 0 0 0
1 1 1 1 1 1 1 1
A pure quantum state of $N$ subsystems with $d$ levels may be referred to as a $k$-maximally entangled state, expressed as $k$-uniform, if every reduction to $k$ qubits is maximally mixed. In addition, the connection between orthogonal arrays and $k$-uniform quantum states, as discussed in Ref. [15]. Goyeneche and Zyczkowski[15] gave the orthogonal array as listed in Table 1. This array leads to the following 3-uniform state of eight qubits: $$\begin{align} | \varphi_{8} \rangle ={}&\frac{1}{4}(| 00000000 \rangle +| 00001111 \rangle +| 00110011 \rangle \\ &+| 00111100 \rangle +| 01010101 \rangle +| 01011010 \rangle \\ &+| 01100110 \rangle +| 01101001 \rangle +| 10010110 \rangle \\ &+| 10011001 \rangle +| 10100101 \rangle +| 10101010 \rangle \\ &+| 11000011 \rangle +| 11001100 \rangle +| 11110000 \rangle\\ &+| 11111111 \rangle)_{12345678}.~~ \tag {2} \end{align} $$ It is easy to show $$ \pi_{ijk} ={\rm Tr}_{ijk}\, \rho^{2}_{ijk} =\frac{1}{8},~~ijk=123,\ldots,678.~~ \tag {3} $$ Furthermore, we can show that $$ \pi_{\rm ijkl} ={\rm Tr}_{\rm ijkl}\, \rho^{2}_{\rm ijkl} =\frac{1}{16},~~ijkl=1235,1236,\ldots~~ \tag {4} $$ except for $$\begin{align} &\pi_{1234} =\frac{1}{4},~ \pi_{1256} =\frac{1}{4},~ \pi_{1278} =\frac{1}{4},~ \pi_{1357} =\frac{1}{4},\\ &\pi_{1368} =\frac{1}{4},~ \pi_{1458} =\frac{1}{4},~ \pi_{1467} =\frac{1}{4}.~~ \tag {5} \end{align} $$ Now, we remove the last column from Table 1, thereby obtaining the orthogonal array in Table 2.
Table 2. Orthogonal array of seven qubits.
A B C D E F G
0 0 0 0 0 0 0
0 0 0 0 1 1 1
0 0 1 1 0 0 1
0 0 1 1 1 1 0
0 1 0 1 0 1 0
0 1 0 1 1 0 1
0 1 1 0 0 1 1
0 1 1 0 1 0 0
1 0 0 1 0 1 1
1 0 0 1 1 0 0
1 0 1 0 0 1 0
1 0 1 0 1 0 1
1 1 0 0 0 0 1
1 1 0 0 1 1 0
1 1 1 1 0 0 0
1 1 1 1 1 1 1
This array leads to the 2-uniform state of seven qubits: $$\begin{align} | \varphi_{7} \rangle ={}&\frac{1}{4}(| 0000000 \rangle +| 0000111 \rangle +| 0011001 \rangle \\ &+| 0011110 \rangle+| 0101010 \rangle +| 0101101 \rangle \\ &+| 0110011 \rangle +| 0110100 \rangle +| 1001011 \rangle \\ &+| 1001100 \rangle +| 1010010 \rangle +| 1010101 \rangle \\ &+| 1100001 \rangle +| 1100110 \rangle +| 1111000 \rangle \\ &+| 1111111 \rangle)_{1234567}.~~ \tag {6} \end{align} $$ It is easy to show that $$ \pi_{ijk} ={\rm Tr}_{ijk} \,\rho^{2}_{ijk} =\frac{1}{8},~~ijk=123,124,\ldots,467~~ \tag {7} $$ except for $$\begin{align} &\pi_{127} =\frac{1}{4},~ \pi_{136} =\frac{1}{4},~ \pi_{145} =\frac{1}{4},~ \pi_{235} =\frac{1}{4},\\ &\pi_{246} =\frac{1}{4},~ \pi_{347} =\frac{1}{4},~ \pi_{567} =\frac{1}{4}.~~ \tag {8} \end{align} $$ As a result of $$ \pi_{\rm ME} =\frac{1}{35}(\pi_{123} +\pi_{124} +\pi_{125} +\cdots+\pi_{456} +\pi_{457} +\pi_{467} +\pi_{567}),~~ \tag {9} $$ from Eqs. (7) and (8), we can obtain $$ \pi_{\rm ME} =\frac{1}{35}\Big({\frac{1}{8}\times 28+\frac{1}{4}\times 7} \Big)=\frac{3}{20}.~~ \tag {10} $$ Now, let us assume that $$\begin{align} | \varphi_{7} \rangle ={}&\frac{1}{4}(| 0000000 \rangle +e^{i\phi_{7} }| 0000111 \rangle +e^{i\phi_{25} }| 0011001 \rangle \\ &+e^{i\phi_{30} }| 0011110 \rangle +e^{i\phi_{42} }| 0101010 \rangle \\ &+e^{i\phi_{45} }| 0101101 \rangle +e^{i\phi_{51} }| 0110011 \rangle\\ & +e^{i\phi_{52} }| 0110100 \rangle+e^{i\phi_{75} }| 1001011 \rangle \\ &+e^{i\phi_{76} }| 1001100 \rangle +e^{i\phi_{82} }| 1010010 \rangle\\ &+e^{i\phi_{85} }| 1010101 \rangle +e^{i\phi_{97} }| 1100001 \rangle\\ & +e^{i\phi_{102} }| 1100110 \rangle +e^{i\phi_{120} }| 1111000 \rangle \\ &+e^{i\phi_{127} }| 1111111 \rangle)_{1234567},~~ \tag {11} \end{align} $$ we can then obtain $$ \pi_{ijk} ={\rm Tr}_{ijk} \,\rho_{ijk}^{2} =\frac{1}{8},~~ijk=123,124,\ldots,467~~ \tag {12} $$ except for $$\begin{align} \pi_{127} ={}&\frac{1}{64}[ 12+\cos ({\varphi_{127} -\varphi_{30} -\varphi_{97} })\\ &+\cos ({\varphi_{7} +\varphi_{120} -\varphi_{25} -\varphi_{102} })\\ &+\cos ({\varphi_{42} +\varphi_{85} -\varphi_{52} -\varphi_{75} }) \\ &+\cos ({\varphi_{45} +\varphi_{82} -\varphi_{51} -\varphi_{76} }) ], \end{align} $$ $$\begin{align} \pi_{136} =\,&\frac{1}{64}[ 12+\cos ({\varphi_{127} -\varphi_{45} -\varphi_{82} })\\ &+\cos ({\varphi_{7} +\varphi_{120} -\varphi_{42} -\varphi_{85} }) \\ &+\cos ({\varphi_{25} +\varphi_{102} -\varphi_{52} -\varphi_{75} })\\ &+\cos ({\varphi_{30} +\varphi_{97} -\varphi_{51} -\varphi_{76} }) ], \end{align} $$ $$\begin{align} \pi_{145} =\,&\frac{1}{64}[ 12+\cos ({\varphi_{127} -\varphi_{51} -\varphi_{76} })\\ &+\cos ({\varphi_{7} +\varphi_{120} -\varphi_{52} -\varphi_{75} }) \\ &+\cos ({\varphi_{25} +\varphi_{102} -\varphi_{42} -\varphi_{85} })\\ &+\cos ({\varphi_{30} +\varphi_{97} -\varphi_{45} -\varphi_{82} }) ], \end{align} $$ $$\begin{align} \pi_{235} =\,&\frac{1}{64}[ 12+\cos ({\varphi_{127} -\varphi_{52} -\varphi_{75} })\\ &+\cos ({\varphi_{7} +\varphi_{120} -\varphi_{51} -\varphi_{76} }) \\ &+\cos ({\varphi_{25} +\varphi_{102} -\varphi_{45} -\varphi_{82} })\\ &+\cos ({\varphi_{30} +\varphi_{97} -\varphi_{42} -\varphi_{85} }) ], \end{align} $$ $$\begin{align} \pi_{246} =\,&\frac{1}{64}[ 12+\cos ({\varphi_{127} -\varphi_{42} -\varphi_{85} })\\ &+\cos ({\varphi_{7} +\varphi_{120} -\varphi_{45} -\varphi_{82} }) \\ &+\cos ({\varphi_{25} +\varphi_{102} -\varphi_{51} -\varphi_{76} })\\ &+\cos ({\varphi_{30} +\varphi_{97} -\varphi_{52} -\varphi_{75} }) ], \end{align} $$ $$\begin{align} \pi_{347} =\,&\frac{1}{64}[ 12+\cos ({\varphi_{127} -\varphi_{25} -\varphi_{102} })\\ &+\cos ({\varphi_{7} +\varphi_{120} -\varphi_{30} -\varphi_{97} }) \\ &+\cos ({\varphi_{42} +\varphi_{85} -\varphi_{51} -\varphi_{76} })\\ &+\cos ({\varphi_{45} +\varphi_{82} -\varphi_{52} -\varphi_{75} }) ], \end{align} $$ $$\begin{alignat}{1} \pi_{567} =\,&\frac{1}{64}[ 12+\cos ({\varphi_{127} -\varphi_{7} -\varphi_{120} })\\ &+\cos ({\varphi_{25} +\varphi_{102} -\varphi_{30} -\varphi_{97} }) \\ &+\cos ({\varphi_{42} +\varphi_{85} -\varphi_{45} -\varphi_{82} })\\ &+\cos ({\varphi_{51} +\varphi_{76} -\varphi_{52} -\varphi_{75} }) ].~~ \tag {13} \end{alignat} $$ Given that $$\begin{align} \pi_{\rm ME} ={}&\frac{1}{35}(\pi_{123} +\pi_{124} +\pi_{125} +\cdots+\pi_{456}\\ & +\pi_{457} +\pi_{467} +\pi_{567}),~~ \tag {14} \end{align} $$ based on Eqs. (12) and (13), we then obtain $$ \pi_{\rm ME} =\frac{1}{35}\Big[\frac{1}{8}\times 28+\frac{1}{64}[84+f(\phi_{7,} \phi_{25,\ldots})]\Big],~~ \tag {15} $$ where $$\begin{align} f(\phi_{7,} \phi_{25,}\cdots)={}&\cos (\phi_{127} -\phi_{30} -\phi_{97})\\ &+\cos (\phi_{7} +\phi_{120} -\phi_{25} -\phi_{102})\\ &+\cos (\phi_{42} +\phi_{85} -\phi_{52} -\phi_{75})\\ &+\cos (\phi_{45} +\phi_{82} -\phi_{51} -\phi_{76}) \\ &+\cos (\phi_{127} -\phi_{45} -\phi_{82})\\ &+\cos (\phi_{7} +\phi_{120} -\phi_{42} -\phi_{85}) \\ &+\cos (\phi_{25} +\phi_{102} -\phi_{52} -\phi_{75})\\ &+\cos (\phi_{30} +\phi_{97} -\phi_{51} -\phi_{76}) \\ &+\cos (\phi_{127} -\phi_{51} -\phi_{76})\\ &+\cos (\phi_{7} +\phi_{120} -\phi_{52} -\phi_{75}) \\ &+\cos (\phi_{25} +\phi_{102} -\phi_{42} -\phi_{85})\\ &+\cos (\phi_{30} +\phi_{97} -\phi_{45} -\phi_{82}) \\ &+\cos (\phi_{127} -\phi_{52} -\phi_{75})\\ &+\cos (\phi_{7} +\phi_{120} -\phi_{51} -\phi_{76}) \\ &+\cos (\phi_{25} +\phi_{102} -\phi_{45} -\phi_{82})\\ &+\cos (\phi_{30} +\phi_{97} -\phi_{42} -\phi_{85}) \\ &+\cos (\phi_{127} -\phi_{42} -\phi_{85})\\ &+\cos (\phi_{7} +\phi_{120} -\phi_{45} -\phi_{82}) \\ &+\cos (\phi_{25} +\phi_{102} -\phi_{51} -\phi_{76})\\ &+\cos (\phi_{30} +\phi_{97} -\phi_{52} -\phi_{75}) \\ &+\cos (\phi_{127} -\phi_{25} -\phi_{102})\\ &+\cos (\phi_{7} +\phi_{120} -\phi_{30} -\phi_{97}) \\ &+\cos (\phi_{42} +\phi_{85} -\phi_{51} -\phi_{76})\\ &+\cos (\phi_{45} +\phi_{82} -\phi_{52} -\phi_{75})\\ &+\cos (\phi_{127} -\phi_{7} -\phi_{120})\\ &+\cos (\phi_{25} +\phi_{102} -\phi_{30} -\phi_{97}) \\ &+\cos (\phi_{42} +\phi_{85} -\phi_{45} -\phi_{82})\\ &+\cos (\phi_{51} +\phi_{76} -\phi_{52} -\phi_{75}).~~ \tag {16} \end{align} $$ For seven-qubit maximally entangled states, the function $f({\varphi_{7},\varphi_{25},\ldots })$ must reach its minimum. We can show that the minimum of $f({\varphi_{7},\varphi_{25},\ldots })$ is $-4$, thereby obtaining $\pi_{\rm ME} =19/140$. Therefore, we determine that $\pi_{\rm ME} =19/140$ is a condition for maximally entangled seven-qubit states. We now know that for maximally seven-qubit entangled states, it is definite that the function $f({\varphi_{7},\varphi_{25},\ldots })$ reaches its minimum, and the minimum of $f({\varphi_{7},\varphi_{25},\ldots })$ is $-4$. For example: (1) When $\varphi_{7} =\varphi_{25} =\varphi_{30} =\varphi_{42} =\varphi_{45} =\varphi_{51} =\varphi_{52} =\varphi_{97} =\varphi_{102} =\varphi_{120} =\varphi_{127} =0$ and $\varphi_{75} =\varphi_{76} =\varphi_{82} =\varphi_{85} =\pi, $ we obtain the following seven-qubit maximally entangled state: $$\begin{align} | \varphi_{m}^{1} \rangle ={}&\frac{1}{4}(| 0000000 \rangle +| 0000111 \rangle +| 0011001 \rangle \\ &+| 0011110 \rangle +| 0101010 \rangle +| 0101101 \rangle\\ &+| 0110011 \rangle +| 0110100 \rangle -| 1001011 \rangle\\ &-| 1001100 \rangle -| 1010010 \rangle -| 1010101 \rangle \\ &+| 1100001 \rangle +| 1100110 \rangle +| 1111000 \rangle\\ &+| 1111111 \rangle)_{1234567} \\ ={}&\frac{1}{4}[(| 000 \rangle +| 111 \rangle)_{127} (| 0000 \rangle +| 1111 \rangle)_{3456} \\ &+(| 001 \rangle +| 110 \rangle)_{127} (| 0011 \rangle +| 1100 \rangle)_{3456} \\ &+(| 010 \rangle -| 101 \rangle)_{127} (| 0101 \rangle +| 1010 \rangle)_{3456} \\ &+(| 011 \rangle -| 100 \rangle)_{127} (| 0110 \rangle +| 1001 \rangle)_{3456} ].~~ \tag {17} \end{align} $$ We can also obtain $$ \pi_{ijk} ={\rm Tr}_{ijk}\, \rho_{ijk}^{2} =\frac{1}{8},~~ijk=123,124,\ldots,467~~ \tag {18} $$ except for $$ \pi_{127} =\frac{1}{4},~~ \pi_{347} =\frac{1}{4},~~ \pi_{567} =\frac{1}{4}.~~ \tag {19} $$ (2) When $\varphi_{7 5} =\varphi_{7 6} =\varphi_{82} =\varphi_{85} =\varphi_{97} =\varphi_{102} =\varphi_{120} =\varphi_{127} =0$ and $\varphi_{7} =\frac{\pi }{4}$, $\varphi_{25} =\frac{2\pi }{4}$, $\varphi_{30} =\frac{3\pi }{4}$, $\varphi_{42} =\frac{4\pi }{4}$, $\varphi_{45} =\frac{5\pi }{4}$, $\varphi_{51} =\frac{6\pi }{4}$, $\varphi_{52} =\frac{7\pi }{4}$, we obtain the following seven-qubit maximally entangled state: $$\begin{align} | \varphi_{m}^{2} \rangle ={}&\frac{1}{4}\big(| 0000000 \rangle +e^{i\frac{\pi }{4}}| 0000111 \rangle\\ & +e^{i\frac{2\pi }{4}}| 0011001 \rangle +e^{i\frac{3\pi }{4}}| 0011110 \rangle \\ &+e^{i\frac{4\pi }{4}}| 0101010 \rangle +e^{i\frac{5\pi }{4}}| 0101101 \rangle\\ &+e^{i\frac{6\pi }{4}}| 0110011 \rangle +e^{i\frac{7\pi }{4}}| 0110100 \rangle \\ & +| 1001011 \rangle +| 1001100 \rangle+| 1010010 \rangle\\ & +| 1010101 \rangle +| 1100001 \rangle+| 1100110 \rangle \\ & +| 1111000 \rangle +| 1111111 \rangle \big)_{1234567}.~~ \tag {20} \end{align} $$ We can also obtain $$ \pi_{ijk} ={\rm Tr}_{ijk} \,\rho_{ijk}^{2} =\frac{1}{8},~~ijk=123,124,\ldots,467~~ \tag {21} $$ except for $$\begin{alignat}{1} &\pi_{127} =\frac{3}{16},~~ \pi_{136} =\frac{1}{32}({6-\sqrt 2 }),~~ \pi_{145} =\frac{3}{16},\\ &\pi_{235} =\frac{3}{16},~~ \pi_{347} =\frac{3}{16},~~ \pi_{567} =\frac{1}{32}({6+\sqrt 2 }).~~ \tag {22} \end{alignat} $$ (3) When $\varphi_{7} =\varphi_{25} =\varphi_{30} =\varphi_{42} =\varphi_{45} =\varphi_{51} =\varphi_{52} =0$ and $\varphi_{75} =\varphi_{120} =\frac{\pi }{2}$, $\varphi_{82} =\varphi_{97} =\pi$, $\varphi_{85} =\varphi_{102} =\frac{3\pi }{2}$, we obtain the following seven-qubit maximally entangled state: $$\begin{align} | \varphi_{m}^{3} \rangle ={}&\frac{1}{4}(| 0000000 \rangle +| 0000111 \rangle +| 0011001 \rangle \\ &+| 0011110 \rangle +| 0101010 \rangle +| 0101101 \rangle \\ &+| 0110011 \rangle +| 0110100 \rangle +i| 1001011 \rangle \\ &+| 1001100 \rangle -| 1010010 \rangle -i| 1010101 \rangle \\ &-| 1100001 \rangle -i| 1100110 \rangle +i| 1111000 \rangle \\ &+| 1111111 \rangle)_{1234567} \\ ={}&\frac{1}{4}[(| 000 \rangle +| 111 \rangle)_{145} (| 0000 \rangle +| 1111 \rangle)_{2367} \\ &+(| 001 \rangle +i| 110 \rangle)_{145} (| 0011 \rangle +| 1100 \rangle)_{2367} \\ &+(| 010 \rangle -i| 101 \rangle)_{145} (| 0101 \rangle +| 1010 \rangle)_{2367} \\ &+(| 011 \rangle -| 100 \rangle)_{145} (| 0110 \rangle +| 1001 \rangle)_{2367} ].~~ \tag {23} \end{align} $$ We can also obtain $$ \pi_{ijk} ={\rm Tr}_{ijk} \rho_{ijk}^{2} =\frac{1}{8},ijk=123,124,\ldots,467~~ \tag {24} $$ except for $$\begin{align} &\pi_{145} =\frac{1}{4},~~ \pi_{235} =\frac{3}{16},~~ \pi_{246} =\frac{3}{16},\\ &\pi_{347} =\frac{3}{16},~~ \pi_{567} =\frac{3}{16}.~~ \tag {25} \end{align} $$ In summary, based on previous study of $k$-uniform states and orthogonal arrays, we have proposed a new method for constructing seven-qubit maximally entangled states. For seven-qubit entangled states, a condition for a maximally entangled seven-qubit state has been found. We find that the maximally entangled seven-qubit states satisfy $\pi_{\rm ME} =19/140$. We also find some new forms of maximally entangled seven-qubit states. We believe that this new method can in theory be useful in the discovery of multi-qubit maximally entangled states. We thank Shengjun Wu for helpful discussion. References Quantum cryptography based on Bell’s theoremEntanglement and entropy rates in open quantum systemsPhase Transitions of Bipartite EntanglementHow entangled can two couples get?Bell inequality, Bell states and maximally entangled states for n qubitsMaximally entangled mixed states of two qubitsOptimal Bell Tests Do Not Require Maximally Entangled StatesMaximally entangled mixed states under nonlocal unitary operations in two qubitsFour Locally Indistinguishable Ququad-Ququad Orthogonal Maximally Entangled StatesMaximally Entangled Set of Multipartite Quantum StatesMultiqubit symmetric states with high geometric entanglementMaximally multipartite entangled statesGeneralized criterion for a maximally multi-qubit entangled stateGenuinely multipartite entangled states and orthogonal arraysExploring pure quantum states with maximally mixed reductionsAll maximally entangled four-qubit statesAbsolutely Maximally Entangled States of Seven Qubits Do Not ExistPurity distribution for bipartite random pure statesA Criterion to identify maximally entangled nine-qubit stateA maximally entangled seven-qubit stateA new method for constructing entangled states via orthogonal arrays
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