Inequality involving tensor product of orthonormal unit vectors

Question

Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and define a subspace of $\mathbb{R}^{n\times n}$ as $$S = span\{e_1\otimes e_1,...,e_r\otimes e_r\}$$ $\mathbb{R}^{n\times n}$, as a Hilbert space, is also equipped with the usual matrix inner product $\langle,\rangle$ and the induced norm $\|\|$ (which is the Frobenius norm).

Take any $\Lambda\in S$ with $\|\Lambda\|=1$. Is the following conjecture true?

There exists a constant $C\geq1$, if $u_1,...,u_n$ are chosen such that $$1 -\delta\leq\sum_{i=1}^n\left|\langle\Lambda,u_i\otimes u_i\rangle\right|^2$$ for some very small $\delta>0$ (for example $0<\delta<\frac{1}{rn}$), then there exists a set $T\subset\{1,2,...,n\}$ with $|T| = r$ satisfying $$1-C\delta\leq\sum_{i\in T}\left|\langle\Lambda,u_i\otimes u_i\rangle\right|^2$$

The conjecture can be verified when $r=1$. Note that $C$ cannot depend on $\delta$, which means the conjecture states that there is a $C>0$ to make the assertion true for all $\delta$ small enough. For example, $C$ can be taken as 2 when $r=1$.

Answer 1

Without loss of generality, $u_1,\dots,u_n$ can be taken to be the standard basis of ${\bf R}^n$ (so $e_1,\dots,e_r$ is just some arbitrary orthonormal system). We can view $\Lambda$ as a real symmetric $n \times n$ matrix of Frobenius norm $1$ and rank at most $r$. (Conversely, every such matrix has a representation of the desired form for some $e_1,\dots,e_r$ by the spectral theorem, so this reformulation has not lost any information.) If $D = \sum_{i=1}^n \langle \Lambda, u_i \otimes u_i \rangle u_i \otimes u_i$ is the diagonal component of $\Lambda$, the hypothesis is then $\|D\|^2 \geq 1-\delta$.

Let $\lambda = \mathrm{diag}(\lambda_1,\dots,\lambda_r,0,\dots,0)$ be the diagonalisation of $\Lambda$ (the ordering of the eigenvalues is irrelevant), thus $\lambda$ is a unit vector supported on a set of $r$ indices in $\{1,\dots,n\}$. By the Schur-Horn theorem, $D$ is a convex combination of the permutations $\sigma(\lambda)$ of $\lambda$, $\sigma \in S_n$, where the symmetric group $S_n$ acts on diagonal matrices in the obvious manner, thus $$ D = \sum_{\sigma \in S_n} c_\sigma \sigma(\lambda)$$ for some non-negative coefficients $c_\sigma$ summing to $1$. (Again, the Schur-Horn theorem is an if-and-only-if statement, so we have still not lost any information so far.)

Now we exploit the uniform convexity of the Frobenius norm. We take the norm square $$ 1-\delta \leq \|D\|^2 = \sum_{\sigma,\sigma' \in S_n} c_\sigma c_{\sigma'} \langle \sigma(\lambda), \sigma'(\lambda) \rangle$$ and then apply the cosine rule to conclude $$ \sum_{\sigma,\sigma' \in S_n} c_\sigma c_{\sigma'} \| \sigma(\lambda) - \sigma'(\lambda)\|^2 \leq 2\delta$$ hence by pigeonholing there exists $\sigma_0 \in S_n$ such that $$ \sum_{\sigma \in S_n} c_\sigma \| \sigma(\lambda) - \sigma_0(\lambda)\|^2 \leq 2\delta$$ hence by Cauchy-Schwarz $$ \sum_{\sigma \in S_n} c_\sigma \| \sigma(\lambda) - \sigma_0(\lambda)\| \leq \sqrt{2\delta}$$ hence by the triangle inequality $$ \| \sum_{\sigma \in S_n} c_\sigma \sigma(\lambda) - \sigma_0(\lambda)\| \leq \sqrt{2\delta}$$ thus $$ \| D - \sigma_0(\lambda) \| \leq \sqrt{2\delta}.$$ The diagonal matrix $\sigma_0(\lambda)$ is supported on a set $T \subset \{1,\dots,n\}$ of cardinality $r$, and the above inequality implies that the Frobenius norm of $D$ outside of $T$ is at most $\sqrt{2\delta}$. Thus $$ \sum_{i \not \in T} |\langle \Lambda, u_i \otimes u_i \rangle|^2 \leq 2\delta$$ so that $$ \sum_{i \in T} |\langle \Lambda, u_i \otimes u_i \rangle|^2 \geq 1-3\delta$$ as required.