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arXiv:quant-ph/0410035 · Physical Review A

Invariants of the local Clifford group

Maarten Van den Nest, Jeroen Dehaene, Bart De Moor

Published Feb 16, 2005• 14 citations
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Abstract

We study the algebra of complex polynomials which remain invariant under the action of the local Clifford group under conjugation. Within this algebra, we consider the linear spaces of homogeneous polynomials degree by degree and construct bases for these vector spaces for each degree, thereby obtaining a generating set of polynomial invariants. Our approach is based on the description of Clifford operators in terms of linear operations over GF(2). Such a study of polynomial invariants of the local Clifford group is mainly of importance in quantum coding theory, in particular in the classification of binary quantum codes. Some applications in entanglement theory and quantum computing are briefly discussed as well.

DOI
10.1103/physreva.71.022310

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We study the algebra of complex polynomials which remain invariant under the action of the local Clifford group under conjugation. Within this algebra, we consider the linear spaces of homogeneous polynomials degree by degree and construct bases for these vector spaces for each degree, thereby obtaining a generating set of polynomial invariants. Our approach is based on the description of Clifford operators in terms of linear operations over GF(2). Such a study of polynomial invariants of the local Clifford group is mainly of importance in quantum coding theory, in particular in the classification of binary quantum codes. Some applications in entanglement theory and quantum computing are briefly discussed as well.
In Phys. Rev. A 58, 1833 (1998), Grassl et al. presented a family of polynomial invariants which separate the orbits of multiqubit density operators $\ensuremath{\rho}$ under the action of the local unitary group. We consider this family of invariants for the class of those $\ensuremath{\rho}$ that are the projection operators describing stabilizer codes and give a complete translation of these invariants into the binary framework in which stabilizer codes are usually described. Such an investigation of local invariants of quantum codes is of natural importance in quantum coding theory, since locally equivalent codes have the same error-correcting capabilities and local invariants are powerful tools to explore their structure. Moreover, the present result is relevant in the context of multipartite entanglement and the development of the measurement-based model of quantum computation known as the one-way quantum computer.
The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary group---the tensor product of unitary group, and local general linear group---the tensor product of general linear group. For these two groups, we prove polynomial upper bounds on the degree of the generators of invariant polynomial rings. On the other hand, systematic methods are provided to to construct all homogenous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if rrr-copies of them are LU equivalent for some rrr.
We describe explicitly the algebra of polynomial functions on the Hilbert space of four-qubit states that are invariant under the group SL(2,C)4\mathrm{SL}(2,C{)}^{4}SL(2,C)4 of stochastic local quantum operations assisted by classical communication. From this description, we obtain a closed formula for the hyperdeterminant in terms of low degree invariants.
We describe stabilizer states and Clifford group operations using linear operations and quadratic forms over binary vector spaces. We show how the n-qubit Clifford group is isomorphic to a group with an operation that is defined in terms of a $(2n+1)\ifmmode\times\else\texttimes\fi{}(2n+1)$ binary matrix product and binary quadratic forms. As an application we give two schemes to efficiently decompose Clifford group operations into one- and two-qubit operations. We also show how the coefficients of stabilizer states and Clifford group operations in a standard basis expansion can be described by binary quadratic forms. Our results are useful for quantum error correction, entanglement distillation, and possibly quantum computing.