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Invariant Polynomial Functions on k qudits

Jean-Luc Brylinski·Ranee Brylinski
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Abstract

We study the polynomial functions on tensor states in $(C^n)^{\otimes k}$ which are invariant under $SU(n)^k$. We describe the space of invariant polynomials in terms of symmetric group representations. For $k$ even, the smallest degree for invariant polynomials is $n$ and in degree $n$ we find a natural generalization of the determinant. For $n,d$ fixed, we describe the asymptotic behavior of the dimension of the space of invariants as $k\to\infty$. We study in detail the space of homogeneous degree 4 invariant polynomial functions on $(C^2)^{\otimes k}$.

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Jean-Luc Brylinski, Ranee Brylinski. 2000-11-14. Invariant Polynomial Functions on k qudits. https://arxiv.org/abs/quant-ph/0010101

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