Multipartite Entanglement from Local Bloch Vectors

We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for $n$ -qubit states, we express the condition that all $k$ -qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all $k$ vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group ${\mathcal{P}}_{n}$ that is isomorphic to ${\mathbb{Z}}_{2}^{n}$ . This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed $k$ for fixed $n$ . In particular, the ratio $k/n$ can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed $k$ -qubit reductions with $k>1$ . Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though $k>1$ is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for $n$-qubit states, we express the condition that all $k$-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all $k$ vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group ${\mathcal{P}}_{n}$ that is isomorphic to ${\mathbb{Z}}_{2}^{n}$. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed $k$ for fixed $n$. In particular, the ratio $k/n$ can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed $k$-qubit reductions with $k>1$. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though $k>1$ is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.

arXiv:1211.4118Physical Review AJan 18, 201383 citations

Geometry of two-body correlations in three-qubit states

Shravan Shravan, Simon Morelli, Otfried Gühne +1 more

We study restrictions of two-body correlations in three-qubit states, using three local-unitarily invariant coordinates based on the Bloch vector lengths of the marginal states. First, we find tight nonlinear bounds satisfied by all pure states and extend this result by including the three-body correlations. Second, we consider mixed states and conjecture a tight nonlinear bound for all three-qubit states. Finally, within the created framework, we give criteria to detect different types of multipartite entanglement as well as characterize the rank of the quantum state.We study restrictions of two-body correlations in three-qubit states, using three local-unitarily invariant coordinates based on the Bloch vector lengths of the marginal states. First, we find tight nonlinear bounds satisfied by all pure states and extend this result by including the three-body correlations. Second, we consider mixed states and conjecture a tight nonlinear bound for all three-qubit states. Finally, within the created framework, we give criteria to detect different types of multipartite entanglement as well as characterize the rank of the quantum state.

arXiv:2309.09549Physical Review ADec 16, 20242 citations

Bipartite Quantum Entanglement

Fabio Benatti

Lecture notes in physicsJan 1, 20101 citation

On sufficient and necessary criteria of multipartite quantum entanglement

Zhi-Bo Chen, Shao-Ming Fei

Abstract Based on the generalized Bloch representation, we study the separability and entanglement of arbitrary dimensional multipartite quantum states. Some sufficient and some necessary criteria are presented. For certain states, these criteria together are both sufficient and necessary. Detailed examples show that our criteria are better than some existing ones in identifying separability. Based on these criteria, the largest separable ball around the maximally mixed state for arbitrary multi-qubit systems is found, and it is proved that its radius defined via the l 1-norm of the Bloch vector is the constant 1. Furthermore, the criteria in this paper can be implemented experimentally.Abstract Based on the generalized Bloch representation, we study the separability and entanglement of arbitrary dimensional multipartite quantum states. Some sufficient and some necessary criteria are presented. For certain states, these criteria together are both sufficient and necessary. Detailed examples show that our criteria are better than some existing ones in identifying separability. Based on these criteria, the largest separable ball around the maximally mixed state for arbitrary multi-qubit systems is found, and it is proved that its radius defined via the l 1-norm of the Bloch vector is the constant 1. Furthermore, the criteria in this paper can be implemented experimentally.

arXiv:2503.10123Physica ScriptaAug 8, 20250 citations

Beyond the Entanglement of Qubit Pairs in Multipartite Systems

S. Shelly Sharma, Naresh Kumar Sharma

Jan 1, 20221 citation

We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for

n

-qubit states, we express the condition that all

k

-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all

k

vs $n\ensuremath{-}k$ bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group

{\mathcal{P}}_{n}

that is isomorphic to

{\mathbb{Z}}_{2}^{n}

. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed

k

for fixed

n

. In particular, the ratio

k/n

can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least $k=\ensuremath{\lfloor}0.189\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rfloor}$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed

k

-qubit reductions with

k>1

. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though

k>1

is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.