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:1105.3086 · Journal of Physics A Mathematical and Theoretical

All degree 6 local unitary invariants of k qudits

Szilárd Szalay

Published Jan 20, 2012• 9 citations
arXivHTMLPDF

Abstract

We give explicit index-free formulae for all the degree 6 (and also degree 4 and 2) algebraically independent local unitary invariant polynomials for finite-dimensional k -partite pure and mixed quantum states. We carry this out using graph-technical methods, which provide illustrations for this abstract topic.
DOI
10.1088/1751-8113/45/6/065302

Related papers

On local invariants of pure three-qubit states

Anthony Sudbery

We study invariants of three-qubit states under local unitary transformations, i.e. functions on the space of entanglement types, which is known to have dimension six. We show that there is no set of six algebraically independent polynomial invariants of degree ⩽ 6, and find such a set with maximum degree eight. We describe an intrinsic definition of a canonical state on each orbit, and discuss the (non-polynomial) invariants associated with it.

arXiv:quant-ph/0001116•Journal of Physics A Mathematical and General•Jan 12, 2001•139 citations

An algebraically independent generating set of the algebra of local unitary invariants

Péter Vrana

We show that the inverse limit of the graded algebras of local unitary invariant polynomials of finite dimensional k-partite quantum systems is free, and give an algebraically independent generating set. The number of degree 2d invariants in the generating set is equal to the number of conjugacy classes of index d subgroups of a free group on k-1 generators.

arXiv:1102.2861•Feb 14, 2011•0 citations

On the algebra of local unitary invariants of pure and mixed quantum states

Péter Vrana

We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we show that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2 m equals the number of conjugacy classes of index m subgroups in a free group on k − 1 generators. Similarly, we show that the inverse limit in the case of k -partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two statements are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m = 2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.

arXiv:1101.2514•Journal of Physics A Mathematical and Theoretical•May 4, 2011•11 citations

Unitary invariants of qubit systems

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

We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicited and compared to 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.

arXiv:quant-ph/0604202•Apr 27, 2006•5 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

Poincare series for local unitary invariants of mixed states of the qubit-qutrit system

Dragomir Ž. Djoković

We consider the mixed states of the bipartite quantum system with the first party a qubit and the second a qutrit. The group of local unitary transformations of the system, ignoring the overall phase factor, is the direct product G of SU(2) and SU(3). We compute the simply graded Poincare series of the algebra of G-invariant polynomial functions on the set of mixed states of the system, and construct several low degree invariants.

arXiv:quant-ph/0605018•May 1, 2006•1 citation
We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we show that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2 m equals the number of conjugacy classes of index m subgroups in a free group on k − 1 generators. Similarly, we show that the inverse limit in the case of k -partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two statements are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m = 2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.