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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 , where 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 , 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.

**References**

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

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This is the first post in a series about a problem inside RMT QIT that I have been working on for some time now [cn2,bcn]. Since I find it to be very simple and interesting, I will present it in a series of blog notes that should be accessible to a large audience. I will also use this material to prepare the talks I will be giving this summer on this topic ;).

In what follows, all vector spaces shall be assumed to be complex and are fixed constants. For a vector , the symbol denotes its ordered version, i.e. and are the same up to permutation of coordinates and .

**1. Singular values of vectors in a tensor product**

Using the non-canonical isomorphism , one can see any vector

as a matrix

In this way, by using the singular value decomposition of the matrix (keep in mind that we assume ), one can write

where , resp. are orthonormal families in , resp. . The vector is the singular value vector of and we shall always assume that it is ordered . It satisfies the normalization condition

In particular, if is a unit vector, then , where is the probability simplex

and is its ordered version.

In QIT, the decomposition of above is called the Schmidt decomposition and the numbers are called the Schmidt coefficients of the pure state .

**2. The singular value set of a vector subspace**

Consider now a subspace of dimension and define the set

called the singular value subset of the subspace .

Below are some examples of sets , in the case , where the simplex is two-dimensional. In all the four cases, and . In the last two pictures, one of the vectors spanning the subspace has singular values .

**3. Basic properties**

Below is a list of very simple properties of the sets .

**Proposition 1.** The set is a compact subset of the ordered probability simplex having the following properties:

- Local invariance: , for unitary matrices and .
- Monotonicity: if , then .
- If , , then .
- If , then .

**Proof:** The first three statements are trivial. The last one is contained in [cmw], Proposition 6 and follows from a standard result in algebraic geometry about the dimension of the intersection of projective varieties.

**4. So, what is the problem ?**

The question one would like to answer is the following:

How does a

typicallook like ?

In order to address this, I will introduce random subspaces in the ~~next post~~ future. In the next post, I look at the special case of antisymmetric tensors.

**References**

[bcn] S. Belinschi, B. Collins and I. Nechita, Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product, to appear in Invent. Math.

[cn2] B. Collins and I. Nechita, Random quantum channels II: Entanglement of random subspaces, Rényi entropy estimates and additivity problems, Adv. in Math. 226 (2011), 1181–1201.

[cmw] T. Cubitt, A. Montanaro and A. Winter, On the dimension of subspaces with bounded Schmidt rank, J. Math. Phys. 49, 022107 (2008).