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MATRIX PERMANENT AND QUANTUM ENTANGLEMENT OF PERMUTATION INVARIANT STATES

Tzu-chieh Wei, Simone Severini B

Published Jan 1, 2009• 45 citations
OpenAlex

Abstract

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)].

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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 by Carlen et al. [Methods Appl. Anal. 13, 1 (2006)], we can prove explicit formulas of the geometric measure for permutation invariant basis states in a simple way.

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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
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)].
Given a complex n × n matrix A and an irreducible character χ of permutation group on n letters, generalized matrix function dχ(A) of A can be thought as combinatorial generalization of matrix permanent and determinant. Due to its combinatorial nature, it is usually a demanding task to assign the construction some geometrical meaning. In this presentation, we discuss how generalized matrix functions serve as essential tools in determining geometric measure of entanglement of certain quantum states. Along the way, we obtain some unexpected geometric interpretations and investigate some examples.
Quantum entanglement rapidly becomes unwieldy to calculate as the number of particlesand the dimension of the spaces associated to those particles increase. One meaningful approachwhich simplifies that analysis is the restriction to subsets of states which obey some physicallyrelevant symmetry. In this thesis, entanglement properties of totally permutation-symmetric,translationally invariant, and party-site symmetric states are examined, as well as those of smallbond-dimensional matrix product states.
We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical state space of a multi-particle system in which each particle has finitely many outcomes upon observation. Moreover, these invariant functions separate the entangled and unentangled states, and are therefore viewed as measurements of quantum entanglement. When the ranks of the unitary groups are large, we provide a graph theoretic interpretation for the dimension of the invariants of a fixed degree. We also exhibit a bijection between isomorphism classes of finite coverings of connected simple graphs and a basis for the space of invariants. The graph coverings are related to branched coverings of surfaces.
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching . We show that the new notion inherits many "classical" properties, but not all of them . This new notion goes somewhere beyound matroids . For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids . In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states . Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some "classical" complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.
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.