The least measurement uncertainty compatible with the fluctuations and correlations of quantum light

Variances and correlations are standard summaries of optical fluctuations. We determine the least uncertainty they enforce in measurements of the two conjugate field quadratures, quantified by the sum of their Shannon entropies. For any finite number \(m\) of modes and any prescribed finite physical covariance \(\Gamma\), we prove \[\inf_{\operatorname{Cov}\rho=\Gamma}\bigl[h(Q^m)+h(P^m)-m\ln(\pi e)\bigr]=\min_{\substack{G+i\Omega/2\succeq0\\G\preceq\Gamma}}\frac12\ln\det(4G_QG_P).\] Here \(\Omega\) is the canonical commutation matrix, \(G_Q,G_P\) are the two diagonal covariance blocks, and the vacuum quadrature variance is \(1/2\). A pure Gaussian core attains the matrix minimum. In one mode, let \(a=\operatorname{Var}Q\), \(b=\operatorname{Var}P\), \(c=\operatorname{Cov}(Q,P)\) and \(d=\sqrt{ab}-|c|\). The excess above \(\ln(\pi e)\) has infimum \[g(d)=\ln(d+1/(4d))\quad(0<d<1/2),\qquad g(d)=0\quad(d\ge1/2).\] Rare, widely displaced components carry the remaining covariance at vanishing entropy cost. Mixtures of Gaussian states and, separately, coherent pure-state constructions approach the same bound while preserving the target covariance exactly. Finite matrix certificates incorporate covariance errors, making the optimal uncertainty law usable with measured second moments.