In this short post, I would like to discuss a special case of the construction introduced in the first part of the series, that is compute the set , where is the anti-symmetric subspace of the tensor product. This example plays an important role in the additivity problem for the minimum output entropy of quantum channels, as it was shown in [ghp].
1. Anti-symmetric vectors and matrices
For order two tensors , there are only two symmetry classes, the symmetric and the antisymmetric vectors. The antisymmetric vectors form a vector space
where the vectors in the span can be shown to form an orthonormal basis of the -dimensional subspace , whenever form a basis of . Let be the orthogonal projection on the subspace . It is easy to see that has entries in and it looks as below ().
Via the usual isomorphism , one can see antisymmetric tensors as antisymmetric (or skew-symmetric) matrices: one simply has to rearrange the complex coordinates of the tensor in an matrix, respecting the ordering of bases. Note that in this way we obtain antisymmetric () and not anti-Hermitian () elements.
2. Singular values of vectors in the antisymmetric subspace
It is well known that antisymmetric (complex) matrices can be 2-block diagonalized using orthogonal rotations
The matrix in the middle has either diagonal blocks of size 2, as shown, or null diagonal elements. Hence, the non-zero eigenvalues of an antisymmetric matrix come in pairs . Since antisymmetric matrices are normal, their singular values are just the moduli of the eigenvalues, so non-zero singular values have multiplicity at least 2. We conclude that
Actually, it is easy to see we have equality, since the vector
is a unit norm element of . We summarize everything in the following theorem, where denotes the majorization relation.
Theorem 1. The set of all possible singular values of antisymmetric vectors inside is given by
In particular, the set is convex and we have that for all . Hence, the minimum entropy of a vector inside is 1 bit.