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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 antisymmetric 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. Antisymmetric 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

\displaystyle 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 {{n \choose 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}).


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

\displaystyle 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

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

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

\displaystyle 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

\displaystyle 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.