  A formula for the norm of a bilinear Schur multiplier acting from the
Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the
Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in
terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of
all compact operators. Using this formula, we resolve Peller's problem on
Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice
continuously differentiable function $f$ with a bounded second derivative, a
self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in
\mathcal S^2$ such that $$
f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $$
