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:1106.4905

Local invariants for mixed qubit-qutrit states

Vladimir P. Gerdt, A. Khvedelidze, Dimitar M. Mladenov, Yuri Palii

Published Jun 24, 2011• 1 citation
arXivHTMLPDF

Abstract

In the present paper few steps are undertaken towards the description of the qubit-qutrit pair - quantum bipartite system composed of two and three level subsystems. The computational difficulties with the construction of the local unitary polynomial invariants are discussed. Calculations of the Molien functions and Poincare series for the qubit-qubit and qubit-qutrit local unitary invariants are outlined and compared with the known results. The requirement of positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the algebra su(6).

DOI
10.48550/arxiv.1106.4905

Related papers

A mixed two-qubit system and the structure of its ring of local invariants

Ronald C. King, Trevor A. Welsh, Peter Jarvis

The local invariants of a mixed two-qubit system are discussed. These invariants are polynomials in the elements of the corresponding density matrix. They are counted by means of group-theoretic branching rules which relate this problem to one arising in spin-isospin nuclear shell models. The corresponding Molien series and a refinement in the form of a 4-parameter generating function are determined. A graphical approach is then used to construct explicitly a fundamental set of 21 invariants. Relations between them are found in the form of syzygies. By using these, the structure of the ring of local invariants is determined, and complete sets of primary and secondary invariants are identified: there are 10 of the former and 15 of the latter.

Journal of Physics A Mathematical and Theoretical•Aug 1, 2007•22 citations

Revisiting the invariant ring of two-qubit mixed states

Bing Xie, Lin Zhang

Local unitary equivalence serves as the cornerstone for classifying entanglement in bipartite quantum systems. Mathematically, it reduces to the study of polynomial invariants of the density matrix under the action of local unitary groups. The collection of all such polynomial invariants forms a ring, known as the invariant ring. However, identifying the complete generators of the invariant ring is the central issue. In 2007, for the two-qubit system, King et al fully characterized the structure of the invariant ring and determined its Cohen--Macaulay decomposition. In this paper, we revisit their work, with a focus on the computation of the Molien series and the construction of invariants. On one hand, we rigorously derive the Molien series via explicit contour integration over the maximal torus, filling in all previously omitted computational steps. On the other hand, we systematically construct all invariants using a graphical method, and then reduce the candidate set by applying various identities and algebraic relations, obtaining a generating set consisting of 21 invariants. This paper aims to make this important result more widely accessible to researchers in quantum information and invariant theory through the above discussions.

quant-pharXiv:2607.23039•Jul 28, 2026•0 citations

Unitary invariants of qubit systems

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

We give an algorithm allowing the construction of bases of local unitary invariants of pure k -qubit states from a knowledge of the polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are made explicit and compared with various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

Mathematical Structures in Computer Science•Nov 23, 2007•26 citations

Separability of two-qubit state in terms of local invariants

Vladimir P. Gerdt, Yu. G. Palii, A. Khvedelidze

The separability condition for the two-qubit mixed state is formulated in terms of a system of inequalities in invariants of the adjoint SU (2) ⊗ SU (2) action on the space of density matrices, i.e., positive semidefinite 4 × 4 Hermitian matrices.

Physics of Particles and Nuclei Letters•Sep 1, 2011•1 citation

Bargmann Invariants Are Insufficient for Complete Local-Unitary Orbit Discrimination

Lin Zhang, Bing Xie

Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In $2\otimes2$ and $2\otimes3$ systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition

quant-pharXiv:2609.11090•Sep 11, 2026•0 citations

The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group $SL(3,{\mathbb {C}})$

Peter Jarvis

We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a `relativistic' transformation group analogous to that of the Lorentz group SL(2,C)_R ~ SO(3,1) for the qubit case. This is the group SL(3,C)_R, presented as a group of real 9x9 matrices acting linearly on the 9-dimensional space of projective coordinates for the qutrit density matrix. The counterpart, for qutrits, of the invariant 4x4 Minkowski metric of the qubit case, proves to be a certain 9x9x9 totally symmetric three-fold tensor generalizing the Gell-Mann d coefficient. Using this structure, we provide a count of the corresponding LSL polynomial invariants using group character methods. Finally, we give an explicit construction of the lowest degree quantity (the cubic invariant) and its expansion in terms of SU(3)xSU(3) invariants, and we indicate how to construct higher degree analogues. These quantities are proven to yield entanglement monotones. This work generalizes and partly extends the paper of King, Welsh and Jarvis (Journal of Physics A, 40(33):10083, 2007) on the mixed two qubit system, which is reviewed in an appendix.

arXiv:1312.7413•Journal of Physics A Mathematical and Theoretical•May 7, 2014•11 citations
In the present paper few steps are undertaken towards the description of the qubit-qutrit pair - quantum bipartite system composed of two and three level subsystems. The computational difficulties with the construction of the local unitary polynomial invariants are discussed. Calculations of the Molien functions and Poincare series for the qubit-qubit and qubit-qutrit local unitary invariants are outlined and compared with the known results. The requirement of positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the algebra su(6).
The local invariants of a mixed two-qubit system are discussed. These invariants are polynomials in the elements of the corresponding density matrix. They are counted by means of group-theoretic branching rules which relate this problem to one arising in spin-isospin nuclear shell models. The corresponding Molien series and a refinement in the form of a 4-parameter generating function are determined. A graphical approach is then used to construct explicitly a fundamental set of 21 invariants. Relations between them are found in the form of syzygies. By using these, the structure of the ring of local invariants is determined, and complete sets of primary and secondary invariants are identified: there are 10 of the former and 15 of the latter.
Local unitary equivalence serves as the cornerstone for classifying entanglement in bipartite quantum systems. Mathematically, it reduces to the study of polynomial invariants of the density matrix under the action of local unitary groups. The collection of all such polynomial invariants forms a ring, known as the invariant ring. However, identifying the complete generators of the invariant ring is the central issue. In 2007, for the two-qubit system, King et al fully characterized the structure of the invariant ring and determined its Cohen--Macaulay decomposition. In this paper, we revisit their work, with a focus on the computation of the Molien series and the construction of invariants. On one hand, we rigorously derive the Molien series via explicit contour integration over the maximal torus, filling in all previously omitted computational steps. On the other hand, we systematically construct all invariants using a graphical method, and then reduce the candidate set by applying various identities and algebraic relations, obtaining a generating set consisting of 21 invariants. This paper aims to make this important result more widely accessible to researchers in quantum information and invariant theory through the above discussions.
Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In 2⊗22\otimes22⊗2 and 2⊗32\otimes32⊗3 systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition
The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group SL(3,C)SL(3,{\mathbb {C}})SL(3,C)
We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a `relativistic' transformation group analogous to that of the Lorentz group SL(2,C)_R ~ SO(3,1) for the qubit case. This is the group SL(3,C)_R, presented as a group of real 9x9 matrices acting linearly on the 9-dimensional space of projective coordinates for the qutrit density matrix. The counterpart, for qutrits, of the invariant 4x4 Minkowski metric of the qubit case, proves to be a certain 9x9x9 totally symmetric three-fold tensor generalizing the Gell-Mann d coefficient. Using this structure, we provide a count of the corresponding LSL polynomial invariants using group character methods. Finally, we give an explicit construction of the lowest degree quantity (the cubic invariant) and its expansion in terms of SU(3)xSU(3) invariants, and we indicate how to construct higher degree analogues. These quantities are proven to yield entanglement monotones. This work generalizes and partly extends the paper of King, Welsh and Jarvis (Journal of Physics A, 40(33):10083, 2007) on the mixed two qubit system, which is reviewed in an appendix.