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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.

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 ${p > 2}$. 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.

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).