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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.

This is the first post in a series about a problem inside RMT {\cap} 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 {k \leq n} are fixed constants. For a vector {y \in \mathbb R^k}, the symbol {y^\downarrow} denotes its ordered version, i.e. {y} and {y^\downarrow} are the same up to permutation of coordinates and {y^\downarrow_1 \geq \cdots \geq y^\downarrow_k}.

1. Singular values of vectors in a tensor product

Using the non-canonical isomorphism {\mathbb C^k \otimes \mathbb C^n \simeq \mathbb C^k \otimes (\mathbb C^n)^*}, one can see any vector

\displaystyle \mathbb C^k \otimes \mathbb C^n \ni x = \sum_{i=1}^k \sum_{j=1}^n x_{ij} e_i \otimes f_j

as a matrix

\displaystyle \mathcal M_{k \times n} \ni X = \sum_{i=1}^k \sum_{j=1}^n x_{ij} e_if_j^*.

In this way, by using the singular value decomposition of the matrix {X} (keep in mind that we assume {k \leq n}), one can write

\displaystyle x = \sum_{i=1}^k \sqrt{\lambda_i} e'_i \otimes f'_i,

where {(f'_i)}, resp. {(g'_i)} are orthonormal families in {\mathbb C^k}, resp. {\mathbb C^n}. The vector {\lambda = \lambda(x) \in \mathbb R_+^{k}} is the singular value vector of {x} and we shall always assume that it is ordered {\lambda(x) = \lambda(x)^\downarrow}. It satisfies the normalization condition

\displaystyle \sum_{i=1}^k \lambda_i(x)= \|x\|^2.

In particular, if {x} is a unit vector, then {\lambda(x) \in \Delta^\downarrow_k}, where {\Delta_k} is the probability simplex

\displaystyle \Delta_k = \left\{ y \in \mathbb R_+^k \, : \, \sum_{i=1}^k y_i = 1\right\}

and {\Delta^\downarrow_k} is its ordered version.

In QIT, the decomposition of x above is called the Schmidt decomposition and the numbers {\lambda_i(x)} are called the Schmidt coefficients of the pure state {|x\rangle}.

2. The singular value set of a vector subspace

Consider now a subspace {V \subset \mathbb C^k \otimes \mathbb C^n} of dimension {\mathrm{dim} V = d} and define the set

\displaystyle K_V = \{ \lambda(x) \, : \, x \in V \text{ and } \|x\| = 1 \} \subseteq \Delta^\downarrow_k,

called the singular value subset of the subspace {V}.

Below are some examples of sets {K_{V}}, in the case {k=3}, where the simplex {\Delta_{3}} is two-dimensional. In all the four cases, {k=n=3} and {d=2}. In the last two pictures, one of the vectors spanning the subspace {V} has singular values {(1/3,1/3,1/3)}.

3. Basic properties

Below is a list of very simple properties of the sets {K_{V}}.

Proposition 1. The set {K_V} is a compact subset of the ordered probability simplex {\Delta_k^\downarrow} having the following properties:

  1. Local invariance: {K_{(U_1 \otimes U_2)V} = K_V}, for unitary matrices {U_1 \in \mathcal U(k)} and {U_2 \in \mathcal U(n)}.
  2. Monotonicity: if {V_1 \subset V_2}, then {K_{V_1} \subset K_{V_2}}.
  3. If {d=1}, {V=\mathbb C x}, then {K_V=\{\lambda(x)\}}.
  4. If {d > (k-1)(n-1)}, then {(1,0,\ldots,0) \in K_V}.

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. \Box

4. So, what is the problem ?

The question one would like to answer is the following:

How does a typical K_V look 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.


[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).