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

Entanglement Classification via Roots of Polynomial Invariants

Adam Burchardt, Gonçalo M. Quinta, Rui André

Published Jun 1, 2021• 0 citations
OpenAlex

Abstract

We show that the invariance of an entanglement measure under Stochastic Local Operations with Classical Communication (SLOCC) expands to its convex roof extension. SLOCC may be represented by Mobius transformations on the roots of the entanglement measure on the Bloch sphere. We derive efficient conditions for n-qubit states to be SLOCC-equivalent. We apply our approach on 4-qubit states and show that roots of the 3-tangle measure classify 4-qubit generic states. Moreover, we propose a method to obtain the normal form of a 4-qubit state which bypasses the possibly infinite iterative procedure.

Related papers

Entanglement classification via a single entanglement measure

Adam Burchardt, Gonçalo M. Quinta, Rui André

We show that a single polynomial entanglement measure is enough to verify equivalence between generic $n$-qubit states under stochastic local operations with classical communication (SLOCC). SLOCC operations may be represented geometrically by M\"obius transformations on the roots of the entanglement measure on the Bloch sphere. Moreover, we show how the roots of the three-tangle measure classify four-qubit generic states, and propose a method to obtain the normal form of a four-qubit state, which bypasses the possibly infinite iterative procedure.

arXiv:2106.00850•Physical Review A•Mar 22, 2024•3 citations

Entanglement of four-qubit systems: A geometric atlas with polynomial compass II (the tame world)

Frédéric Holweck, Jean-Gabriel Luque, Jean‐Yves Thibon

We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by Stochastic Local Operations with Classical Communication (SLOCC) invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete et al. are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).

arXiv:1606.05569•Journal of Mathematical Physics•Feb 1, 2017•19 citations

INVARIANT POLYNOMIALS WITH APPLICATIONS TO QUANTUM COMPUTING

Alimzhan Amanov

In quantum information theory, understanding the complexity of entangled states within the context of SLOCC (stochastic local operations and classical communications) involving d qubits (or qudits) is essential for advancing our knowledge of quantum systems. This complexity is often analyzed by classifying the states via local symmetry groups. The resulting classes can be distinguished using invariant polynomials, which serve as a measure of entanglement. This paper introduces a novel method for obtaining invariant polynomials of the smallest degrees, which significantly enhances the efficiency of characterizing SLOCC classes of entangled quantum states. Our method not only simplifies the process of identifying these classes but also provides a robust tool for analyzing the entanglement properties of complex quantum systems. As a practical application, we demonstrate the derivation of minimal degree invariants in specific cases, illustrating the effectiveness of our approach in real-world scenarios. This advancement has the potential to streamline various processes in quantum information theory, making it easier to understand, classify, and utilize entangled states effectively.

Herald of Kazakh-British technical university•Jul 1, 2024•1 citation

Geometric approach to entanglement quantification with polynomial measures

Bartosz Regula, Gerardo Adesso

We show that the quantification of entanglement of any rank-2 state with any polynomial entanglement measure can be recast as a geometric problem on the corresponding Bloch sphere. This approach provides insight into the properties of entanglement and allows us to relate different polynomial measures to each other, simplifying their quantification. In particular, unveiling and exploiting the geometric structure of the concurrence for two qubits, we show that the convex roof of any polynomial measure of entanglement can be quantified exactly for all rank-2 states of an arbitrary number of qubits which have only one or two unentangled states in their range. We give explicit examples by quantifying the three-tangle exactly for several representative classes of three-qubit states. We further show how our methods can be used to obtain analytical results for entanglement of more complex states if one can exploit symmetries in their geometric representation.

arXiv:1606.06184•Physical Review A•Aug 19, 2016•10 citations

A new degree bound for local unitary and nnn-qubit SLOCC Invariants

Jacob Turner

Deep connections between invariant theory and entanglement have been known for some time and been the object of intense study. This includes the study of local unitary equivalence of density operators as well as entanglement that can be observed in stochastic local operations assisted by classical communication (SLOCC). An important aspect of both of these areas is the computation of complete sets of invariants polynomials. For local unitary equivalence as well as $n$-qubit SLOCC invariants, complete descriptions of these invariants exist. However, these descriptions give infinite sets; of great interest is finding generating sets of invariants. In this regard, degree bounds are highly sought after to limit the possible sizes of such generating sets. In this paper we give new upper bounds on the degrees of the invariants, both for a certain complete set of local unitary invariants as well as the $n$-qubit SLOCC invariants. We show that there exists a complete set of local unitary invariants of density operators in a Hilbert space $\mathcal{H}$, of dimension $d$, which are generated by invariants of degree at most $d^4$. This in turn allows us to show that the $n$-qubit SLOCC invariants are generated by invariants of degree at most $2^{4n}$.

arXiv:1706.00634•Jun 2, 2017•0 citations

Stochastic local operations and classical communication invariants via square matrix

Xin-Wei Zha, Irfan Ahmed, Da Zhang +2 more

Abstract In this paper, we raise a few corrections from the Li et al paper (2015 Phys. Rev . A 91 012302). We obtain some stochastic local operations and classical communication (SLOCC) invariants from second degree to sixth degree by defining square matrices. The relation of SLOCC invariants and the character polynomial of the square matrix are defined for three and four qubit states. This method comes as a useful resource in determining SLOCC invariants for n-qubit states.

Laser Physics•Jan 4, 2019•3 citations
We show that the invariance of an entanglement measure under Stochastic Local Operations with Classical Communication (SLOCC) expands to its convex roof extension. SLOCC may be represented by Mobius transformations on the roots of the entanglement measure on the Bloch sphere. We derive efficient conditions for n-qubit states to be SLOCC-equivalent. We apply our approach on 4-qubit states and show that roots of the 3-tangle measure classify 4-qubit generic states. Moreover, we propose a method to obtain the normal form of a 4-qubit state which bypasses the possibly infinite iterative procedure.
We show that a single polynomial entanglement measure is enough to verify equivalence between generic nnn-qubit states under stochastic local operations with classical communication (SLOCC). SLOCC operations may be represented geometrically by M\"obius transformations on the roots of the entanglement measure on the Bloch sphere. Moreover, we show how the roots of the three-tangle measure classify four-qubit generic states, and propose a method to obtain the normal form of a four-qubit state, which bypasses the possibly infinite iterative procedure.
We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by Stochastic Local Operations with Classical Communication (SLOCC) invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete et al. are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).
In quantum information theory, understanding the complexity of entangled states within the context of SLOCC (stochastic local operations and classical communications) involving d qubits (or qudits) is essential for advancing our knowledge of quantum systems. This complexity is often analyzed by classifying the states via local symmetry groups. The resulting classes can be distinguished using invariant polynomials, which serve as a measure of entanglement. This paper introduces a novel method for obtaining invariant polynomials of the smallest degrees, which significantly enhances the efficiency of characterizing SLOCC classes of entangled quantum states. Our method not only simplifies the process of identifying these classes but also provides a robust tool for analyzing the entanglement properties of complex quantum systems. As a practical application, we demonstrate the derivation of minimal degree invariants in specific cases, illustrating the effectiveness of our approach in real-world scenarios. This advancement has the potential to streamline various processes in quantum information theory, making it easier to understand, classify, and utilize entangled states effectively.
We show that the quantification of entanglement of any rank-2 state with any polynomial entanglement measure can be recast as a geometric problem on the corresponding Bloch sphere. This approach provides insight into the properties of entanglement and allows us to relate different polynomial measures to each other, simplifying their quantification. In particular, unveiling and exploiting the geometric structure of the concurrence for two qubits, we show that the convex roof of any polynomial measure of entanglement can be quantified exactly for all rank-2 states of an arbitrary number of qubits which have only one or two unentangled states in their range. We give explicit examples by quantifying the three-tangle exactly for several representative classes of three-qubit states. We further show how our methods can be used to obtain analytical results for entanglement of more complex states if one can exploit symmetries in their geometric representation.
Deep connections between invariant theory and entanglement have been known for some time and been the object of intense study. This includes the study of local unitary equivalence of density operators as well as entanglement that can be observed in stochastic local operations assisted by classical communication (SLOCC). An important aspect of both of these areas is the computation of complete sets of invariants polynomials. For local unitary equivalence as well as nnn-qubit SLOCC invariants, complete descriptions of these invariants exist. However, these descriptions give infinite sets; of great interest is finding generating sets of invariants. In this regard, degree bounds are highly sought after to limit the possible sizes of such generating sets. In this paper we give new upper bounds on the degrees of the invariants, both for a certain complete set of local unitary invariants as well as the nnn-qubit SLOCC invariants. We show that there exists a complete set of local unitary invariants of density operators in a Hilbert space H\mathcal{H}H, of dimension ddd, which are generated by invariants of degree at most d4d^4d4. This in turn allows us to show that the nnn-qubit SLOCC invariants are generated by invariants of degree at most 24n2^{4n}24n.