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

Journal of Physics Conference Series

Hubbard operators in multiqubit systems

Marco Enríquez

Published Jun 26, 2015• 3 citations
Publisher

Abstract

The properties of the Kronecker product of 2 x 2 matrices are reviewed in terms of Hubbard operators. This framework constitutes a shorthand notation to deal with the tensor algebra of operators acting on multiqubit states. As an application we derive some analytical expressions related to the geometric measure of entanglement of a certain class of multiqubit invariant permutation states. Our results can be straightaway extended for systems of larger dimensions.

DOI
10.1088/1742-6596/624/1/012019

Related papers

Some applications of the Kronecker product in Hubbard representation

Marco Enríquez, Óscar Rosas-Ortiz

The properties of the Kronecker product are revisited in terms of Hubbard operators. The simplest representation of a Hubbard operator X i,j n is a square matrix of size n with an entry equal to 1 and zero elsewhere. This framework simplifies the calculation of the Kronecker product of arbitrary matrices no matter the size or the number of the involved factors. Some applications are presented, these include the algebra of permutation matrices, the Hadamard matrix, the XXX Heisenberg model and the interaction of an atom with radiation fields.

Journal of Physics Conference Series•Oct 10, 2014•2 citations

THE TENSOR PRODUCT OF TENSOR OPERATORS OVER QUANTUM ALGEBRAS: SOME APPLICATIONS TO QUANTUM SPIN CHAINS

Manfred Scheunert

The present work is a direct sequel to a recent article by the author, in which he has analysed the tensor product of tensor operators over quantum algebras. Here the results obtained there are summarized and then specialized and extended to prepare possible applications to quantum spin chains. In particular, certain invariant two-point operators are introduced (whose expectation values yield the invariant two-point correlation functions) and their multiplicative properties are derived.

International Journal of Modern Physics B•Nov 1, 1994•3 citations

On the Relation between States and Maps in Infinite Dimensions

Janusz Grabowski, Marek Kuś, Giuseppe Marmo

Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.

arXiv:0706.2617•Open Systems & Information Dynamics•Dec 1, 2007•18 citations

Kronecker Products

Christian Bauckhage, Rafet Sifa

Abstract Tensor products or, more specifically, Kronecker products play a fundamental role in quantum computing and can also help with modeling certain combinatorial problems. In this chapter, we therefore discuss Kronecker products and their properties. To segue into this discussion, we first introduce the important notion of pseudo Boolean functions and look at their relations to multilinear algebra. Given this use case, we can easily understand the role of different kinds of tensor products and investigate algebraic properties especially of Kronecker products.

Cognitive technologies•Nov 17, 2025•0 citations

Permutation symmetry and entanglement in quantum states of heterogeneous systems

Gururaj Kadiri, S. Sivakumar

Permutation symmetries of multipartite quantum states are defined only when the constituent subsystems are of equal dimensions. In this work we extend this notion of permutation symmetry to heterogeneous systems, that is, systems composed of subsystems having unequal dimensions. Given a tensor product space of $k$ subsystems (of arbitrary dimensions) and a permutation operation $σ$ over $k$ symbols, these states are such that they have identical decompositions (up to an overall phase) in the given tensor product space and the tensor product space obtained by the permuting the subsystems by $σ$. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation $σ$. Eigenvectors of this matrix have the required symmetry. We then examine entanglement of states in the igenspaces of these matrices. It is found that all nonsymmetric eigenspaces of such matrices are completely entangled subspaces, with states being equally entangled in both the given tensor product space and the permuted tensor product space.

arXiv:1702.04948•Feb 16, 2017•0 citations

MATRIX PERMANENT AND QUANTUM ENTANGLEMENT OF PERMUTATION INVARIANT STATES

Tzu-chieh Wei, Simone Severini B

We point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states. This connection allows us to interpret the permanent as an angle between vectors. By employing a recently introduced permanent inequality [Carlen, Loss and Lieb, Meth. and Appl. of Analysis, 13 (2006), no. 1, 1–17], we can write combinatorial formulas for quantifying the entanglement of permutation invariant basis states. When applying the geometric measure to permutation invariant states with nonnegative coefficients, we show that the overlap with a product state is maximized by a tensor product of the same single-party state. This extends some observations in [Hayashi et al., Phys. Rev. A 77, 012104 (2008)].

•Jan 1, 2009•45 citations
The properties of the Kronecker product of 2 x 2 matrices are reviewed in terms of Hubbard operators. This framework constitutes a shorthand notation to deal with the tensor algebra of operators acting on multiqubit states. As an application we derive some analytical expressions related to the geometric measure of entanglement of a certain class of multiqubit invariant permutation states. Our results can be straightaway extended for systems of larger dimensions.
The properties of the Kronecker product are revisited in terms of Hubbard operators. The simplest representation of a Hubbard operator X i,j n is a square matrix of size n with an entry equal to 1 and zero elsewhere. This framework simplifies the calculation of the Kronecker product of arbitrary matrices no matter the size or the number of the involved factors. Some applications are presented, these include the algebra of permutation matrices, the Hadamard matrix, the XXX Heisenberg model and the interaction of an atom with radiation fields.
Relations between states and maps, which are known for quantum systems in finite-dimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dimensional Hilbert spaces and a simple proof of a generalized Choi Theorem. The natural framework is based on spaces of Hilbert-Schmidt operators [Formula: see text] and the corresponding tensor products [Formula: see text] of Hilbert spaces. It is proved that the corresponding isomorphisms cannot be naturally extended to compact (or bounded) operators, nor reduced to the trace-class operators. On the other hand, it is proven that there is a natural continuous map [Formula: see text] from trace-class operators on [Formula: see text] (with the nuclear norm) into compact operators mapping the space of all bounded operators on [Formula: see text] into trace class operators on [Formula: see text] (with the operator-norm). Also in the infinite-dimensional context, the Schmidt measure of entanglement and multipartite generalizations of state-maps relations are considered in the paper.
Abstract Tensor products or, more specifically, Kronecker products play a fundamental role in quantum computing and can also help with modeling certain combinatorial problems. In this chapter, we therefore discuss Kronecker products and their properties. To segue into this discussion, we first introduce the important notion of pseudo Boolean functions and look at their relations to multilinear algebra. Given this use case, we can easily understand the role of different kinds of tensor products and investigate algebraic properties especially of Kronecker products.
Permutation symmetries of multipartite quantum states are defined only when the constituent subsystems are of equal dimensions. In this work we extend this notion of permutation symmetry to heterogeneous systems, that is, systems composed of subsystems having unequal dimensions. Given a tensor product space of kkk subsystems (of arbitrary dimensions) and a permutation operation σσσ over kkk symbols, these states are such that they have identical decompositions (up to an overall phase) in the given tensor product space and the tensor product space obtained by the permuting the subsystems by σσσ. Towards this, we construct a matrix whose action is to simultaneously permute the subsystem label and subsystem dimension of a given state according to permutation σσσ. Eigenvectors of this matrix have the required symmetry. We then examine entanglement of states in the igenspaces of these matrices. It is found that all nonsymmetric eigenspaces of such matrices are completely entangled subspaces, with states being equally entangled in both the given tensor product space and the permuted tensor product space.
We point out that a geometric measure of quantum entanglement is related to the matrix permanent when restricted to permutation invariant states. This connection allows us to interpret the permanent as an angle between vectors. By employing a recently introduced permanent inequality [Carlen, Loss and Lieb, Meth. and Appl. of Analysis, 13 (2006), no. 1, 1–17], we can write combinatorial formulas for quantifying the entanglement of permutation invariant basis states. When applying the geometric measure to permutation invariant states with nonnegative coefficients, we show that the overlap with a product state is maximized by a tensor product of the same single-party state. This extends some observations in [Hayashi et al., Phys. Rev. A 77, 012104 (2008)].