Random subspaces of a tensor product (II)

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 K_{A_n}, where A_n \subset \mathbb C^n \otimes \mathbb C^n 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 x \in \mathbb C^n \otimes \mathbb C^n, there are only two symmetry classes, the symmetric and the antisymmetric vectors. The antisymmetric vectors form a vector space

A_n = \mathrm{span}\left \{ \frac{1}{\sqrt 2} (e_i \otimes e_j - e_j \otimes e_i) \, : \, 1 \leq i < j \leq n \right \}

where the vectors in the span can be shown to form an orthonormal basis of the \binom n 2-dimensional subspace A_n, whenever e_i form a basis of \mathbb C^n. Let P_n \in \mathcal M_{n^2}(\mathbb C) be the orthogonal projection on the subspace A_n. It is easy to see that P_n has entries in {0,\pm 1/2} and it looks as below (n=10). antisymmetric-10

Via the usual isomorphism \mathbb C^n \otimes \mathbb C^n \simeq \mathbb C^n \otimes (\mathbb C^n)^* = \mathcal M_n(\mathbb C), one can see antisymmetric tensors as antisymmetric (or skew-symmetric) matrices: one simply has to rearrange the n^2 complex coordinates of the tensor in an n \times n matrix, respecting the ordering of bases. Note that in this way we obtain antisymmetric (X^t = -X) and not anti-Hermitian (X^* = -X) 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

X = O \begin{bmatrix} 0 & \lambda_1 & & & & \\ -\lambda_1 & 0 & & & & \\ & & 0 & \lambda_2 & & \\ & & -\lambda_2 & 0 & & \\ & & & & \ddots & \\ & & & & & 0 \end{bmatrix}O^t.

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  \{ \pm \lambda_i \}. 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

K_{A_n} \subset \left\{ (\lambda_1, \lambda_1, \lambda_2, \lambda_2, \ldots) \, : \, \lambda_i \geq 0, \, \sum_{i=1}^{\lfloor n/2 \rfloor} \lambda_i = 1/2\right \}.

Actually, it is easy to see we have equality, since the vector

 x_\lambda = \sum_{i=1}^{\lfloor n/2 \rfloor} \sqrt{ \lambda_i } (e_i \otimes e_{\lfloor n/2 \rfloor+i} - e_{\lfloor n/2 \rfloor+i} \otimes e_i)

is a unit norm element of A_n. We summarize everything in the following theorem, where \prec denotes the majorization relation.

Theorem 1. The set of all possible singular values of antisymmetric vectors inside \mathbb C^n \otimes \mathbb C^n is given by

K_{A_n} = \{ (\lambda_1, \lambda_1, \lambda_2, \lambda_2, \ldots) \in \Delta_n \, : \, \lambda_i \geq 0, \, \sum_{i=1}^{\lfloor n/2 \rfloor} \lambda_i = 1/2 \}.

In particular, the set K_{A_n} is convex and we have that \lambda \prec (1/2, 1/2, 0, \ldots, 0) for all \lambda \in K_{A_n}. Hence, the minimum entropy of a vector inside K_{A_n} is 1 bit.


[ghp] A. Grudka, M. Horodecki and L. Pankowski. Constructive counterexamples to the additivity of the minimum output Rényi entropy of quantum channels for all p>2. J. Phys. A: Math. Theor. 43 425304.