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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).