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Chinese Physics Letters

Constructing a Maximally Entangled Seven-Qubit State via Orthogonal Arrays

Xin-Wei Zha, Minrui Wang, Ruo-Xu Jiang

Published Sep 1, 2020• 4 citations
Publisher

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 π 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

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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 π 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.
Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure ((7,1,4))_{2} quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.
A pure quantum state of NNN subsystems with ddd levels each is called kkk-multipartite maximally entangled state, which we call a kkk-uniform state, if all its reductions to kkk qubits are maximally mixed. Recently, Goyeneche and Zyczkowski proposed a method of constructing kkk-uniform states via orthogonal arrays, by which, we will also propose a new method to construct quantum states with orthogonal arrays in this paper, which is the method of increasing phase parameters. We can find the quantum states constructed via our new method by using the same orthogonal array, for four-qubit quantum states, not only are they 1-uniform states, also may become maximally entangled states; for five-qubit quantum states, we find that it can make the original 1-uniform states into 2-uniform states. In particular, for four-qubit quantum states, we can construct a series of new forms of maximally entangled states via our new method.
Quantum orthogonal arrays are combinatorial designs with a remarkable class of genuinely multipartite highly entangled states. In this paper, we propose the method for constructing maximally multi-qubit entangled state through phase parameters of orthogonal arrays. Using this method, we not only determine that maximally n-qubit entangled state, but also find the [Formula: see text] orthogonal array that gives maximally [Formula: see text] entangled states. Such states play an important role in quantum theory with a high degree of multipartite entanglement.
It is worth studying recurrence relation of multi-qubit states due to its benefits to the criterion and measurement of quantum entanglement. Recently, Huang et al. presented an expand recurrence relation of multi-qubit state. Using this expand recurrence relation and Brown state, we construct a new seven-qubit maximally entangled states of 32 and 3 balanced purities equal to 1/8 and 1/4, whose averaged subsystem purity is equal to 19/140. We found that the marginal density matrices for subsystems of 1 and 2 qubits all completely mixed in these states.