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arXiv:1907.05612 · Laser Physics Letters

A Criterion to identify maximally entangled nine-qubit state

Xinwei Zha, Irfan Ahmed, Da Zhang, Yanpeng Zhang

Published Jan 28, 2020• 9 citations
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Abstract

Abstract We present a generalized criterion for maximally entangled nine-qubit states, whose minimum averaged subsystem purity should be equal to 1/14. In this note, we prove that the absolutely maximally entangled state for nine qubits does not exist. In other words, we pronounce that the pure (9, 1, 5) 2 quantum error-correcting code does not exist. Further, we construct a new genuine nine-qubit maximally entangled states of 110,15 and 1 balanced purities equal to 1/16, 1/8 and 1/4, respectively. We found that the marginal density matrices for subsystems of 1,2,3- qubits all completely mixed in these states; therefore, maximally entangled nine-qubit state is a 3-uniform state.

DOI
10.1088/1612-202x/ab6a16

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Abstract We present a generalized criterion for maximally entangled nine-qubit states, whose minimum averaged subsystem purity should be equal to 1/14. In this note, we prove that the absolutely maximally entangled state for nine qubits does not exist. In other words, we pronounce that the pure (9, 1, 5) 2 quantum error-correcting code does not exist. Further, we construct a new genuine nine-qubit maximally entangled states of 110,15 and 1 balanced purities equal to 1/16, 1/8 and 1/4, respectively. We found that the marginal density matrices for subsystems of 1,2,3- qubits all completely mixed in these states; therefore, maximally entangled nine-qubit state is a 3-uniform state.
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.
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.
We first present a generalized criterion for maximally entangled states of 2–8 and in theory to arbitrary-number qubits. By this criterion, some known highly entangled multi-qubit states are examined and a new genuine eight-qubit maximally entangled state is obtained. For the four-, seven- and eight-qubit system in which no perfect maximally multi-qubit entangled state (MMES) was thought to exist before, we find that the proven four- and eight-qubit MMESs and the suspected seven-qubit MMES, are not completely mixed in subsystems with two, four and three qubits, respectively, but are completely mixed in subsystems with fewer qubits. The new criterion and MMES can play important roles in quantum information technology, such as teleportation and dense coding.
Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols and the construction of quantum error correcting codes. To our knowledge, the only known systematic construction of these states is based on a class of classical error correction codes known as maximum distance separable. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from maximum distance separable codes. Furthermore, we used our method to construct several examples of absolutely maximally entangled states whose existence was open so far.
A pure quantum state is called k-uniform if all its reductions to k-qudit are maximally mixed. We investigate the general constructions of k-uniform pure quantum states of n subsystems with d levels. We provide one construction via symmetric matrices and the second one through the classical error-correcting codes. There are three main results arising from our constructions. First, we show that for any given even n ≥ 2, there always exists an n/2-uniform n-qudit quantum state of level p for sufficiently large prime p. Second, both constructions show that there exist k-uniform n-qudit pure quantum states such that k is proportional to n, i.e., k = Ω(n) although the construction from symmetric matrices in general outperforms the one by error-correcting codes. Third, our symmetric matrix construction provides a positive answer to the open question on whether there exists a 3-uniform n-qudit pure quantum state for all n ≥ 8. In fact, we can further prove that, for every k, there exists a constant Mksuch that there exists a k-uniform n-qudit quantum state for all n ≥ Mk. In addition, by using the concatenation of algebraic geometry codes, we give an explicit construction of k-uniform quantum state when k tends to infinity.