Random subspaces of a tensor product (I)

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

\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

\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

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

\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

 \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 \dim V = d and define the set

 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.

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 anti-symmetric 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).