Entropy-Minimum POVM Router

Integrate this at the final classifier head or at the gating logits of a small mixture-of-experts model. Let the backbone output \(h\in\mathbb R^m\). Use a learned linear map \(L(h)\in\mathbb R^{d\times r}\) and construct the input density matrix \(\omega(h)=(L(h)L(h)^\top+\varepsilon I)/\operatorname{Tr}(L(h)L(h)^\top+\varepsilon I)\), with \(d=8\) or \(16\) initially. For each class or expert \(x\), maintain a learned matrix \(B_x\in\mathbb R^{d\times r}\), set \(R_x=B_xB_x^\top+\varepsilon I\), define \(\rho_x=R_x/\operatorname{Tr}(R_x)\), and set \(\sigma_x=p_x\rho_x\), where \(p_x\) is a learned prior normalized by softmax. Set \(\tau=I/d\) for the first experiment. Every 50--500 optimizer steps, solve for the Hermitian matrix \(A\) using damped L-BFGS or Newton iterations on the residual \(F(A)=\sum_x\exp(\log(\sigma_x)-A)-\tau\). Compute matrix logarithms and exponentials by eigendecomposition, and symmetrize every matrix as \((X+X^\top)/2\). Stop when \(\|F(A)\|_F/\|\tau\|_F<10^{-5}\), then cache \(M_x\) until the next refresh. During ordinary training, compute probabilities \(q_x=\operatorname{Tr}(M_x\omega(h))\), use cross-entropy \(-\log(q_y+10^{-8})\), and optionally backpropagate through cached operators or refresh them with implicit differentiation. The paper supplies the operator construction and completeness property; solver tolerance, refresh interval, rank \(r\), and feature parameterization are empirical choices. First test on CIFAR-10 with a small CNN or MLP, comparing a standard softmax head, cosine classifier, and POVM head at equal parameter count. Measure accuracy, expected calibration error, negative log-likelihood, and completeness error \(\|\sum_xM_x-I\|_F\). For routing, test a four-expert Transformer on a synthetic compositional task and measure expert load balance and accuracy at fixed capacity. Success means lower calibration error or higher accuracy without loss spikes, completeness error below \(10^{-4}\), and acceptable matrix-operation overhead.