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The Robertson--Schrödinger Uncertainty Remainder as a Dirichlet--Neumann-Type Compatibility Deficit

Tamás Bódis

Published May 11, 2026• 0 citations
Publisher

Abstract

This preprint introduces an exact Hilbert-space decomposition of the Robertson--Schrödinger uncertainty relation into a canonical quantum minimum, a covariance contribution, and a non-negative orthogonal remainder. The remainder, denoted R_DN[ψ], is interpreted as a Dirichlet--Neumann-type compatibility deficit between the position and momentum fluctuation vectors. The work does not introduce a new uncertainty principle. Rather, it gives a geometric interpretation of the Robertson--Schrödinger saturation condition: the deficit vanishes precisely when the momentum fluctuation vector is a complex scalar multiple of the position fluctuation vector. This characterizes Robertson--Schrödinger-saturating Gaussian states, possibly with a quadratic phase. The approach is motivated by the Hamilton--Jacobi relation between coordinate-type and momentum-type boundary data. Examples involving an uncorrelated Gaussian state, a quadratic-phase Gaussian state, and a Fock state illustrate the role of the deficit.

DOI
10.5281/zenodo.20125970

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This preprint introduces an exact Hilbert-space decomposition of the Robertson--Schrödinger uncertainty relation into a canonical quantum minimum, a covariance contribution, and a non-negative orthogonal remainder. The remainder, denoted R_DN[ψ], is interpreted as a Dirichlet--Neumann-type compatibility deficit between the position and momentum fluctuation vectors. The work does not introduce a new uncertainty principle. Rather, it gives a geometric interpretation of the Robertson--Schrödinger saturation condition: the deficit vanishes precisely when the momentum fluctuation vector is a complex scalar multiple of the position fluctuation vector. This characterizes Robertson--Schrödinger-saturating Gaussian states, possibly with a quadratic phase. The approach is motivated by the Hamilton--Jacobi relation between coordinate-type and momentum-type boundary data. Examples involving an uncorrelated Gaussian state, a quadratic-phase Gaussian state, and a Fock state illustrate the role of the deficit.
We discuss recent findings relating the quantum Fisher information to convex roofs of variances. We present several improvements on the Robertson-Schrodinger uncertainty relation. In all these improvements, we consider a decomposition of the density matrix into a mixture of pure states, and use the fact that the Robertson-Schrodinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in F. Frowis, R. Schmied, and N. Gisin [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relations. We also gain new insights about the Cramer-Rao bound. By considering a concave roof of the bound in the Robertson-Schrodinger uncertainty relation, we obtain other type of improvements. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of canonical position and momentum operators for two-mode continuous variable systems and angular momentum operators for spin systems.
The uncertainty principle is traditionally viewed as a statement about the epistemic limits of observers or the intrinsic stochasticity of quantum states. We propose an operational reformulation in terms of experimental deficiency: the Robertson--Schr\"odinger inequality det⁡(Σ)≥ℏ2/4\det(\Sigma) \geq \hbar^2/4det(Σ)≥ℏ2/4 yields a lower bound on the Le Cam deficiency between incompatible experiments and any joint simulation. In the Gaussian (LAN) regime, this implies that every joint experiment attempting to simulate both position and momentum must incur strictly positive deficiency, with the Planck constant ℏ\hbarℏ setting the information-loss scale. We introduce the \textit{saturation ratio} η=(ℏ2/4)/det⁡(Σ)\eta = (\hbar^2/4)/\det(\Sigma)η=(ℏ2/4)/det(Σ) as a dimensionless diagnostic of experimental optimality, and interpret decoherence as deficiency inflation via increased effective covariance. The framework applies to covariant Gaussian joint measurements (e.g., Arthurs--Kelly-type schemes) and connects quantum uncertainty with classical statistical inference, identifying Gaussian measurements as deficiency-optimal. The resulting quantum--classical ``continuum'' claim is strictly operational: an ordering by saturation loss, not a metaphysical thesis of structural continuity, theory reduction, or ontological continuity.
We rederive the Schr\"odinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) ground state of the harmonic oscillator saturates the uncertainty relation.
Let ρ the density matrix of a mixed Gaussian state. Assuming that one of the Robertson–Schrödinger uncertainty inequalities is saturated by ρ, e.g. , we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of ρ and having the same variances and covariance Δ ρ X 1 , Δ ρ P 1 and Δ ρ ( X 1 , P 1 ) as ρ. This property can be viewed as an analytic version of Gromov's non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball. We conclude by giving a short geometric proof of the fact that pure Gaussian states are the only quantum states saturating the Robertson–Schrödinger uncertainty inequalities.
We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant $ \tfrac{1}{4} $ is replaced by the state-dependent coefficient (λmax⁡+λmin⁡)24(λmax⁡−λmin⁡)2, \frac{(λ_{\max} + λ_{\min})^2}{4(λ_{\max} - λ_{\min})^2}, 4(λmax​−λmin​)2(λmax​+λmin​)2​, where $ λ_{\max} $ and λmin⁡λ_{\min}λmin​ denote the largest and smallest eigenvalues of the density operator $ ρ$, respectively. We prove that this coefficient is optimal among all Robertson-type generalizations and does not admit further improvement. We also obtain a complete characterization of all nontrivial equality cases. Our relation becomes more pronounced as the quantum state becomes more mixed, capturing a trade-off in quantum uncertainty that the conventional Robertson's relation fails to detect. In addition, our result also provides a strict generalization of the Schrödinger's uncertainty relation, showing that the uncertainty trade-off is governed by the sum of the covariance term and a state-dependent improvement over the Robertson bound. As applications, we also refine error-disturbance trade-offs by incorporating spectral information of both the system and the measuring apparatus, thereby generalizing the Arthurs-Goodman and Ozawa inequalities.
Thermodynamic uncertainty principles make up one of the few rare anchors in the largely uncharted waters of nonequilibrium systems, the fluctuation theorems being the more familiar. In this work we aim to trace the uncertainties of thermodynamic quantities in nonequilibrium systems to their quantum origins, namely, to the quantum uncertainty principles. Our results enable us to make this categorical statement: For Gaussian systems, thermodynamic functions are functionals of the Robertson-Schrödinger uncertainty function, which is always non-negative for quantum systems, but not necessarily so for classical systems. Here, quantum refers to noncommutativity of the canonical operator pairs. From the nonequilibrium free energy, we succeeded in deriving several inequalities between certain thermodynamic quantities. They assume the same forms as those in conventional thermodynamics, but these are nonequilibrium in nature and they hold for all times and at strong coupling. In addition we show that a fluctuation-dissipation inequality exists at all times in the nonequilibrium dynamics of the system. For nonequilibrium systems which relax to an equilibrium state at late times, this fluctuation-dissipation inequality leads to the Robertson-Schrödinger uncertainty principle with the help of the Cauchy-Schwarz inequality. This work provides the microscopic quantum basis to certain important thermodynamic properties of macroscopic nonequilibrium systems.