Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Europhysics Letters (EPL)

Bloch vector norms of separable multi-partite quantum systems

Andrei Tănăsescu, Pantelimon George Popescu

Published Jul 25, 2019• 4 citations
Publisher

Abstract

In this letter, we find an upper bound for the norms of Bloch vectors for quantum systems comprised of an arbitrary number of qudits. We generalize a recent result of Li et al . for four-partite quantum systems. We apply our result to provide an upper bound on the entanglement measure given by the Bloch vector norm and we provide a necessary algebraic condition for separability of a general multi-partite quantum system under any arbitrary partition function. Finally we show that the finer the partitioning is, the smaller our upper bound becomes.

DOI
10.1209/0295-5075/126/60003

Related papers

The norms of Bloch vectors and classification of four-qudits quantum states

Li, M., Wang, Z., Wang, J. +2 more

We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. Theses upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four qudits quantum systems.

•Jan 1, 2019•13 citations

The norms of Bloch vectors and classification of four-qudits quantum states

Ming Li, Zong Wang, Jing Wang +2 more

We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measures defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. These upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four-qudits quantum systems.

arXiv:1903.04336•Europhysics Letters (EPL)•Feb 18, 2019•16 citations

Separability criterion for multipartite quantum states based on the Bloch representation of density matrices

Ali Saif M. Hassan, Pramod S. Joag

We give a new separability criterion, a necessary condition for separability of N-partite quantum states. The criterion is based on the Bloch representation of a N-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N-partite quantum states in $\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes \cdots \otimes \mathcal{H}^{d_N}.$ The criterion can test whether a N-partite state is entangled and can be applied to different partitions of the $N$-partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case N > 3, under certain condition. We also give a necessary and sufficient condition for separability of a class of N-qubit states which includes N-qubit PPT states.

arXiv:0704.3942•Quantum Information and Computation•Sep 1, 2008•39 citations

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.

arXiv:2503.10123•Physica Scripta•Aug 8, 2025•0 citations

Lower Bound of $l_{1}$ Norm of Coherence of Bipartite Qubit-Qudit System and its Application in the Detection of Entangled Tripartite Qudit-Qubit-Qudit System

Palash Garhwal, Pranav Chandhok, Satyabrata Adhikari

Quantum coherence and quantum entanglement are two strong pillars in quantum information theory. We study here for the possibility of any connection between these two important aspects of quantum mechanics while studying the entanglement detection problem for the detection of bipartite higher dimensional entangled states and multipartite entangled states. To achieve our goal, we derive the lower bound $L$ of $l_{1}$ norm of coherence of bipartite qubit-qudit system using the criterion that detect entanglement. Furthermore, we deduce the upper bound $U$ of $l_{1}$ norm of coherence of separable bipartite qubit-qudit system using the separability criterion. Thus, we find that if any $l_{1}$ norm of coherence of bipartite qubit-qudit system is greater than the upper bound $U$ then the given qubit-qudit state is entangled. Finally, we obtained the upper bound $U_{1}$ of $l_{1}$ norm of coherence of separable tripartite state lies either in $2 \otimes d \otimes d$ or $d \otimes 2 \otimes d$ or $d \otimes d \otimes 2$ dimensional Hilbert space using the upper bound $U$. We have shown that if the $l_{1}$ norm of coherence of any tripartite $qubit-qudit-qudit$ or $qudit-qubit-qudit$ or $qudit-qudit-qubit$ system is greater than the derived upper bound $U_{1}$ then the given tripartite system represent an entangled state.

arXiv:2203.12874•Mar 24, 2022•0 citations

Separability and classification of multipartite quantum states

Pan-Wen Ma, Hui Zhao, Naihuan Jing

Abstract We study separability in arbitrary multipartite quantum systems based on principal base matrices. Necessary conditions are presented for different kinds of separable states. These conditions can give a complete classification of multipartite quantum states. While the usual Bloch representation of a density matrix uses three types of generators, the representation with principal base matrices has one uniform type of generator which simplifies computation. In this paper, we take advantage of this simplicity to derive useful and operational criteria to detect multipartite separability. We first obtain criteria on detecting 1 − 3 separable, 2 − 2 separable, 1 − 1 − 2 separable and fully separable four-partite quantum states. We then study k-separability for multipartite quantum states in arbitrary dimensions. Detailed examples are given to show that our criteria are able to detect more entanglement states than some existing criteria.

Laser Physics Letters•Nov 8, 2023•1 citation
In this letter, we find an upper bound for the norms of Bloch vectors for quantum systems comprised of an arbitrary number of qudits. We generalize a recent result of Li et al . for four-partite quantum systems. We apply our result to provide an upper bound on the entanglement measure given by the Bloch vector norm and we provide a necessary algebraic condition for separability of a general multi-partite quantum system under any arbitrary partition function. Finally we show that the finer the partitioning is, the smaller our upper bound becomes.
We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measure defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. Theses upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four qudits quantum systems.
We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measures defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. These upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four-qudits quantum systems.
We give a new separability criterion, a necessary condition for separability of N-partite quantum states. The criterion is based on the Bloch representation of a N-partite quantum state and makes use of multilinear algebra, in particular, the matrization of tensors. Our criterion applies to arbitrary N-partite quantum states in H=Hd1⊗Hd2⊗⋯⊗HdN.\mathcal{H}=\mathcal{H}^{d_1}\otimes \mathcal{H}^{d_2} \otimes \cdots \otimes \mathcal{H}^{d_N}.H=Hd1​⊗Hd2​⊗⋯⊗HdN​. The criterion can test whether a N-partite state is entangled and can be applied to different partitions of the NNN-partite system. We provide examples that show the ability of this criterion to detect entanglement. We show that this criterion can detect bound entangled states. We prove a sufficiency condition for separability of a 3-partite state, straightforwardly generalizable to the case N > 3, under certain condition. We also give a necessary and sufficient condition for separability of a class of N-qubit states which includes N-qubit PPT states.
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.
Lower Bound of l1l_{1}l1​ Norm of Coherence of Bipartite Qubit-Qudit System and its Application in the Detection of Entangled Tripartite Qudit-Qubit-Qudit System
Quantum coherence and quantum entanglement are two strong pillars in quantum information theory. We study here for the possibility of any connection between these two important aspects of quantum mechanics while studying the entanglement detection problem for the detection of bipartite higher dimensional entangled states and multipartite entangled states. To achieve our goal, we derive the lower bound LLL of l1l_{1}l1​ norm of coherence of bipartite qubit-qudit system using the criterion that detect entanglement. Furthermore, we deduce the upper bound UUU of l1l_{1}l1​ norm of coherence of separable bipartite qubit-qudit system using the separability criterion. Thus, we find that if any l1l_{1}l1​ norm of coherence of bipartite qubit-qudit system is greater than the upper bound UUU then the given qubit-qudit state is entangled. Finally, we obtained the upper bound U1U_{1}U1​ of l1l_{1}l1​ norm of coherence of separable tripartite state lies either in 2⊗d⊗d2 \otimes d \otimes d2⊗d⊗d or d⊗2⊗dd \otimes 2 \otimes dd⊗2⊗d or d⊗d⊗2d \otimes d \otimes 2d⊗d⊗2 dimensional Hilbert space using the upper bound UUU. We have shown that if the l1l_{1}l1​ norm of coherence of any tripartite qubit−qudit−quditqubit-qudit-quditqubit−qudit−qudit or qudit−qubit−quditqudit-qubit-quditqudit−qubit−qudit or qudit−qudit−qubitqudit-qudit-qubitqudit−qudit−qubit system is greater than the derived upper bound U1U_{1}U1​ then the given tripartite system represent an entangled state.
Abstract We study separability in arbitrary multipartite quantum systems based on principal base matrices. Necessary conditions are presented for different kinds of separable states. These conditions can give a complete classification of multipartite quantum states. While the usual Bloch representation of a density matrix uses three types of generators, the representation with principal base matrices has one uniform type of generator which simplifies computation. In this paper, we take advantage of this simplicity to derive useful and operational criteria to detect multipartite separability. We first obtain criteria on detecting 1 − 3 separable, 2 − 2 separable, 1 − 1 − 2 separable and fully separable four-partite quantum states. We then study k-separability for multipartite quantum states in arbitrary dimensions. Detailed examples are given to show that our criteria are able to detect more entanglement states than some existing criteria.