We investigates the joint compatibility problem of the three pairwise concurrences, the three-tangle, and the Kempe invariant in arbitrary pure three-qubit states, and gives necessary and sufficient conditions for these five quantities to be simultaneously realizable by the same pure state. This five-dimensional compatibility region can be viewed as a fiber structure over the previously derived four-dimensional reachable region. We further find that the residual continuous local-unitary freedom depends on the position of the state within this region. In the generic interior, one continuous degree of freedom remains. It is fixed if the three-tangle vanishes or a pairwise concurrence becomes zero, and also at the boundary of the four-dimensional tangle region. The Kempe invariant gives the simplest algebraic parametrization of the fifth coordinate, but the same degree of freedom may be described by a known pure-state entanglement monotone. The compatibility conditions can also be expressed in terms of concurrence-based bipartite measures and multipartite quantities fixed by the pairwise concurrences and the three-tangle.We investigates the joint compatibility problem of the three pairwise concurrences, the three-tangle, and the Kempe invariant in arbitrary pure three-qubit states, and gives necessary and sufficient conditions for these five quantities to be simultaneously realizable by the same pure state. This five-dimensional compatibility region can be viewed as a fiber structure over the previously derived four-dimensional reachable region. We further find that the residual continuous local-unitary freedom depends on the position of the state within this region. In the generic interior, one continuous degree of freedom remains. It is fixed if the three-tangle vanishes or a pairwise concurrence becomes zero, and also at the boundary of the four-dimensional tangle region. The Kempe invariant gives the simplest algebraic parametrization of the fifth coordinate, but the same degree of freedom may be described by a known pure-state entanglement monotone. The compatibility conditions can also be expressed in terms of concurrence-based bipartite measures and multipartite quantities fixed by the pairwise concurrences and the three-tangle.

quant-pharXiv:2609.03864Sep 4, 20260 citations

Entanglement of four-qubit mixed states quantified with polynomial invariants

Christopher Eltschka, Oliver Viehmann, Jens Siewert

Verhandlungen der Deutschen Physikalischen GesellschaftJan 1, 20110 citations

Multipartite entanglement in four-qubit graph states

Mojtaba Jafarpour, Leila Assadi

The European Physical Journal DMar 1, 20163 citations

Three-tangle of assistance

Xi-Jun Ren, Zhen Wang, Xingxiang Zhou +2 more

In this report, we calculate the three tangle of assistance and give the corresponding best possible pure state decomposition of rank-two mixed states for the three-qubit Greenberger-Horne-Zeilinger (GHZ) state and $W$ state. As the three tangle can be transformed into bipartite concurrence with assistance, we achieve a lower bound of the assisted concurrence with local operations and classical communication (LOCC) constraints via these pure state decompositions. It is smaller than the total assisted concurrence, which shows that local assistance alone is not sufficient for maximizing bipartite entanglement. The increase of three tangle is attributed to the genuine multipartite entanglement in the purified state. A genuine four-qubit entanglement measure is calculated for the purified four-qubit state and compared to the three tangle of assistance.In this report, we calculate the three tangle of assistance and give the corresponding best possible pure state decomposition of rank-two mixed states for the three-qubit Greenberger-Horne-Zeilinger (GHZ) state and $W$ state. As the three tangle can be transformed into bipartite concurrence with assistance, we achieve a lower bound of the assisted concurrence with local operations and classical communication (LOCC) constraints via these pure state decompositions. It is smaller than the total assisted concurrence, which shows that local assistance alone is not sufficient for maximizing bipartite entanglement. The increase of three tangle is attributed to the genuine multipartite entanglement in the purified state. A genuine four-qubit entanglement measure is calculated for the purified four-qubit state and compared to the three tangle of assistance.

Physical Review AApr 13, 20103 citations

Relationship between the n-tangle and the residual entanglement of even n qubits

Xiangrong Li, Dafa Li

We show that for even $n$ qubits, $\ n$ -tangle is the square of the SLOCC polynomial invariant of degree 2. We find that the $n$ -tangle is not the residual entanglement for any even $n\geq 4$ \ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence $C_{1(2...n)}$ . The condition implies that the concurrence $C_{1(2...n)}$ is always positive for any entangled states while the $n$ -tangle vanishes for some entangled states. We argue that for even $n$ \ qubits, the concurrence $ C_{1(2...n)}$\ is equal to or greater than the $n$ -tangle.\ Further, we reveal that the residual entanglement is a partial measure for product states of any $n$ qubits while the $n$ -tangle is multiplicative for some product states.We show that for even $n$ qubits,$\ n$-tangle is the square of the SLOCC polynomial invariant of degree 2. We find that the $n$-tangle is not the residual entanglement for any even $n\geq 4$\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence $C_{1(2...n)}$. The condition implies that the concurrence $C_{1(2...n)}$ is always positive for any entangled states while the $n$-tangle vanishes for some entangled states. We argue that for even $n$\ qubits, the concurrence $ C_{1(2...n)}$\ is equal to or greater than the $n$-tangle.\ Further, we reveal that the residual entanglement is a partial measure for product states of any $n$ qubits while the $n$-tangle is multiplicative for some product states.

arXiv:1003.4774Quantum Information and ComputationNov 1, 20109 citations

Entanglement of four-qubit rank-2 mixed states

Eylee Jung, DaeKil Park

Quantum Information ProcessingJun 11, 20152 citations

W

state. As the three tangle can be transformed into bipartite concurrence with assistance, we achieve a lower bound of the assisted concurrence with local operations and classical communication (LOCC) constraints via these pure state decompositions. It is smaller than the total assisted concurrence, which shows that local assistance alone is not sufficient for maximizing bipartite entanglement. The increase of three tangle is attributed to the genuine multipartite entanglement in the purified state. A genuine four-qubit entanglement measure is calculated for the purified four-qubit state and compared to the three tangle of assistance.

We show that for even

n

qubits,

\ n

-tangle is the square of the SLOCC polynomial invariant of degree 2. We find that the

n

-tangle is not the residual entanglement for any even

n\geq 4

\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence

C_{1(2...n)}

. The condition implies that the concurrence

C_{1(2...n)}

is always positive for any entangled states while the

n

-tangle vanishes for some entangled states. We argue that for even

n

\ qubits, the concurrence $ C_{1(2...n)}$\ is equal to or greater than the

n

-tangle.\ Further, we reveal that the residual entanglement is a partial measure for product states of any

n

qubits while the

n

-tangle is multiplicative for some product states.