Reconstruction and Structural Separation of Heisenberg (1927)

Georg Karagounis

Antonios Valamontes

This work presents a complete reconstruction and structural analysis of Werner Heisenberg’s 1927 formulation of position–momentum uncertainty. The original presentation interwove operational measurement arguments, algebraic non-commutativity, Fourier reciprocity, and interpretive claims. The present study reconstructs the 1927 reasoning in full and separates these distinct logical layers within the formal structure of quantum mechanics. The canonical commutation relation [Q,P]=iℏ[Q,P]=i\hbar[Q,P]=iℏ is shown to imply the Robertson variance bound ΔQ ΔP≥ℏ/2\Delta Q\,\Delta P \ge \hbar/2ΔQΔP≥ℏ/2 independently of measurement modeling. Reciprocal localization arises from Fourier duality in L2(R)L^2(\mathbb{R})L2(R). Measurement–disturbance relations depend on explicit unitary system–apparatus couplings. Interpretive assertions concerning ontological indeterminacy are not formal consequences of operator algebra. A structural separation theorem is established distinguishing:(1) algebraic incompatibility,(2) analytic reciprocity,(3) dynamical disturbance, and(4) interpretive extension. The mathematical content of quantum mechanics remains unchanged. The contribution consists in clarifying the logical architecture underlying uncertainty relations and preventing conflation between algebraic structure and measurement dynamics.

Harry A. Mavromatis

Akseli Mäkinen, Joni Ikonen, Matti Partanen +1 more

We propose an approach to analytically solve the quantum dynamics of bosonic systems. The method is based on reconstructing the quantum state of the system from the moments of its annihilation operators, dynamics of which is solved in the Heisenberg picture. The proposed dynamical reconstruction method is general in the sense that it does not require assumptions on the initial conditions of the system such as separability, or the structure of the system such as linearity. It is an alternative to the standard master-equation approaches, which are analytically demanding especially for large multipartite quantum systems. To demonstrate the proposed technique, we apply it to a system consisting of two coupled damped quantum harmonic oscillators.

Nikita Romanov, Petr Ivashkov, Weiyuan Gong +4 more

We study ansatz-free Lindbladian learning, the problem of reconstructing the generator of an open quantum system without prior knowledge of its Hamiltonian or dissipator structures. This problem exhibits two fundamental precision limits. Hamiltonian components not obscured by dissipation are Heisenberg-limited, while full Lindbladian reconstruction is subject to the quadratically worse standard quantum limit. This creates an apparent algorithmic impasse: achieving the Heisenberg-limited Hamiltonian learning requires suppressing unknown dissipation, which seemingly demands prior reconstruction of the noise and thereby incurs the standard-quantum-limit cost. In this work, we resolve this obstruction by giving an algorithm that learns the Hamiltonian disjoint from dissipator (HDD) at the Heisenberg limit without prior knowledge of either the Hamiltonian or dissipator supports. Our main technical ingredient is a randomized recursive stabilizer-code construction that progressively identifies and suppresses the dominant dissipative terms without reconstructing the full dissipator. Building on this framework, we also introduce an efficient end-to-end algorithm that learns the entire sparse Lindbladian at the standard quantum limit. Finally, we prove that in the ansatz-free setting, the HDD terms constitute the maximal set of Hamiltonian coefficients uniformly learnable at the Heisenberg limit. We show that every term outside HDD is fundamentally standard-quantum limited, extending prior no-go results to the ansatz-free setting. Together, our protocols provide a scalable framework for characterizing open quantum systems, with quantum error correction serving as a key learning primitive.

Vlasie Catalin Mircea