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

arXiv:1705.07160

Theoretical and computational aspects of entanglement

Harm Derksen, Shmuel Friedland, Lek‐Heng Lim, Li Wang

Published May 19, 2017• 13 citations
arXivHTMLPDF

Abstract

We show that the two notions of entanglement: the maximum of the geometric measure of entanglement and the maximum of the nuclear norm is attained for the same states. We affirm the conjecture of Higuchi-Sudberry on the maximum entangled state of four qubits. We introduce the notion of d-density tensor for mixed d-partite states. We show that d-density tensor is separable if and only if its nuclear norm is $1$. We suggest an alternating method for computing the nuclear norm of tensors. We apply the above results to symmetric tensors. We give many numerical examples.

DOI
10.48550/arxiv.1705.07160

Related papers

Entanglement and Quantumness - New Numerical Approaches -

Fabian Bohnet-Waldraff

The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.

•Sep 29, 2017•0 citations

Quantifying entanglement from the geometric perspective

Lisa T. Weinbrenner, Otfried Gühne

Abstract Quantum entanglement between several particles is essential for applications like quantum metrology or quantum cryptography, but it is also central for foundational phenomena like quantum non-locality. This leads to the problem of quantifying the amount of entanglement in a quantum state. We present a review on the geometric measure of entanglement, being a quantifier based on the distance of a state to the nearest separable state. We explain basic properties, existing methods to compute it, its operational interpretations, as well as scaling and complexity issues. We point out intimate relations to fundamental problems in mathematics concerning eigenvalues and norms of tensors. Consequently, the geometric measure of entanglement provides a playground where physical intuition and mathematical rigor benefit from each other.

arXiv:2505.01394•Europhysics Letters (EPL)•Aug 27, 2025•4 citations

Quantifying the dimensionality of multiparticle entanglement via partition rank

Sophia Denker, Ismaël Septembre, Robin Krebs +1 more

The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.

quant-pharXiv:2608.23399•Aug 25, 2026•0 citations

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

Guyan Ni, Liqun Qi, Minru Bai

We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rank-one approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.

SIAM Journal on Matrix Analysis and Applications•Jan 1, 2014•47 citations

Complete Hierarchies for the Geometric Measure of Entanglement

Lisa T. Weinbrenner, Albert Rico, Kenneth Goodenough +2 more

In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.

quant-pharXiv:2601.23243•Feb 2, 2026•0 citations

Numerical results for geometric measure of coherence and geometric measure of entanglement based on semidefinite programs

Zhou Zhang, Yue Dai, Yuli Dong +1 more

We present semidefinite programs for computing the geometric measure of coherence and the geometric measure of entanglement, respectively. For the geometric measure of coherence, the algorithm can be applied to any finite-dimensional mixed state. We test randomly generated single-qubit states, single-qutrit states, and a special kind of $d$-dimensional mixed states. For the geometric measure of entanglement, the algorithm is applied to all two-qubit and qubit-qutrit states, and some special kinds of higher dimensional mixed states. For other states, the algorithm can get a lower bound of the geometric measure of entanglement. Randomly generated two-qubit states, the isotropic states and the Werner states are tested. Furthermore, the maximum deviation between the analytical solution and the numerical solution is calculated.

•Mar 26, 2019•0 citations
We show that the two notions of entanglement: the maximum of the geometric measure of entanglement and the maximum of the nuclear norm is attained for the same states. We affirm the conjecture of Higuchi-Sudberry on the maximum entangled state of four qubits. We introduce the notion of d-density tensor for mixed d-partite states. We show that d-density tensor is separable if and only if its nuclear norm is 111. We suggest an alternating method for computing the nuclear norm of tensors. We apply the above results to symmetric tensors. We give many numerical examples.
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
Abstract Quantum entanglement between several particles is essential for applications like quantum metrology or quantum cryptography, but it is also central for foundational phenomena like quantum non-locality. This leads to the problem of quantifying the amount of entanglement in a quantum state. We present a review on the geometric measure of entanglement, being a quantifier based on the distance of a state to the nearest separable state. We explain basic properties, existing methods to compute it, its operational interpretations, as well as scaling and complexity issues. We point out intimate relations to fundamental problems in mathematics concerning eigenvalues and norms of tensors. Consequently, the geometric measure of entanglement provides a playground where physical intuition and mathematical rigor benefit from each other.
The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.
We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rank-one approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.
In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.
We present semidefinite programs for computing the geometric measure of coherence and the geometric measure of entanglement, respectively. For the geometric measure of coherence, the algorithm can be applied to any finite-dimensional mixed state. We test randomly generated single-qubit states, single-qutrit states, and a special kind of ddd-dimensional mixed states. For the geometric measure of entanglement, the algorithm is applied to all two-qubit and qubit-qutrit states, and some special kinds of higher dimensional mixed states. For other states, the algorithm can get a lower bound of the geometric measure of entanglement. Randomly generated two-qubit states, the isotropic states and the Werner states are tested. Furthermore, the maximum deviation between the analytical solution and the numerical solution is calculated.