Bounds on the Bures–Wasserstein distance

Question

The Bures–Wasserstein distance between $n\times n$ positive semidefinite matrices $A$ and $H$ is defined to be

$$ d(A,H) := \left[ \operatorname{tr} A + \operatorname{tr} H - 2\operatorname{tr} (A^{1/2}HA^{1/2})^{1/2} \right]^{1/2}. $$

This metric has several interpretations:

1. It is the Wasserstein 2-distance between centered normal distributions $\mathcal{N}(0,A)$ and $\mathcal{N}(0,H)$.
2. It is the optimal value of the unitary procrustes problem for $A^{1/2}$ and $H^{1/2}$, i.e., $$ d(A,H) = \min_{U \text{ unitary}} \|A^{1/2} - H^{1/2}U\|_{\rm F}. \tag{1} $$

More interpretations and much more about this metric are discussed in the article of Bhatia, Jain, and Lim.

I am interested in whether a bound of the form

$$ d^2(A,H) \stackrel{?}{\le} C(n) \frac{\| A - H \|_{\rm F}^2}{\min(\lambda_{\rm max}(A),\lambda_{\rm max}(H))} \tag{2} $$

holds, where $C(n)$ could be an arbitrary function of $n$.

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A weakened version of (2) holds where the maximum eigenvalue is replaced by the minimum inequality:

$$ d^2(A,H) \le \| A^{1/2} - H^{1/2} \|_{\rm F}^2 \le \frac{\|A - H \|_{\rm F}^2}{(\sqrt{\lambda_{\rm min}(A)} + \sqrt{\lambda_{\rm min}(H)})^2} \le \frac{\|A - H \|_{\rm F}^2}{\min(\lambda_{\rm min}(A),\lambda_{\rm min}(H))}. $$

The first inequality is (1) and the second inequality is a consequence of an inequality of van Hemmen and Ando (Prop. 3.2). This inequality seems quite loose because it uses the fixed unitary matrix $U = I$ in (1).

I feel like I might be missing something simple here, but despite my best efforts, I have been unable to prove an inequality of the form (2) or convince myself no such inequality holds.

Answer 1

A simple example shows the bound (2) cannot possibly hold. Consider

$$ A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad H = \begin{bmatrix} 1 & 0 \\ 0 & \varepsilon \end{bmatrix}. $$

Then $d(A,H) = \sqrt{\varepsilon}$, which is incompatible with the bound (2):

$$ d^2(A,H) = \varepsilon \le C(n)\varepsilon^2 =C(n) \frac{\|A-H\|_{\rm F}^2}{\min(\lambda_{\rm max}(A),\lambda_{\rm max}(H))} \quad \text{fails for sufficiently small $\varepsilon > 0$}. $$