This paper deals with quantum Uncertainty Relations (UR) subjected to the Standard Deviations (SD) of the relevant dynamical variables. It is important to note that these variables have to obey the two distinct periodicities of the knotted paths embedded on the torus. We compute generalized forms of the SDs and the subsequent URs (following the Kennard–Robertson formalism). These quantities explicitly involve the torus parameters and the knot parameters where restrictions on the latter have to be taken into account. These induce restrictions on the possible form of wave functions that are used to calculate the SDs and URs and in our simple example, two distinct SDs and URs are possible. In a certain limit (thin torus limit), our results will reduce to the results for a particle moving in a circle. An interesting fact emerges that in the case of the SDs and URs, the local geometry of the knots plays the decisive role and not their topological properties.
This paper deals with quantum Uncertainty Relations (UR) subjected to the Standard Deviations (SD) of the relevant dynamical variables. It is important to note that these variables have to obey the two distinct periodicities of the knotted paths embedded on the torus. We compute generalized forms of the SDs and the subsequent URs (following the Kennard–Robertson formalism). These quantities explicitly involve the torus parameters and the knot parameters where restrictions on the latter have to be taken into account. These induce restrictions on the possible form of wave functions that are used to calculate the SDs and URs and in our simple example, two distinct SDs and URs are possible. In a certain limit (thin torus limit), our results will reduce to the results for a particle moving in a circle. An interesting fact emerges that in the case of the SDs and URs, the local geometry of the knots plays the decisive role and not their topological properties.
Abstract We formulate the variance-based uncertainty relations (URs) via the Robertson’s inequality, for a 2-particle entangled system constrained on a torus and subject to a stationary magnetic field B B → . We explore the system’s parameter space and show that these new URs have field-tunable uncertainty bounds. Our analysis reveals that A A → (vector potential) induces a phase shift in the state, due to the Aharonov–Bohm effect, leading to a perturbed system dynamics which results in asymmetric product of variance ( POV POV ). Additionally, we give the critical range of A A → and B B → where the system acts as an entanglement amplifier; this amplification is also discussed under various geometric parameters. The possibility of reducing the POV POV of the conjugate pair [q, p] below the known benchmark value by the Generalized Uncertainty Relation (GUR) is also demonstrated. Finally, we check the susceptibility of state coherence to B B → by saturating the angular momentum uncertainty relation and identify the critical coherence value Bc B c such that when B=Bc B ≠ B c , the state decoheres.
Abstract In this paper, we study the subtle effect of constraints on the quantum dynamics of a point particle moving on a non-trivial torus knot. The particle is kept on the knot by the constraints, generated by curvature and torsion. In the Geometry-Induced Potential (GIP) approach, the Schrödinger equation for the system yields new results in particle energy eigenvalues and eigenfunctions, in contrast with existing results that ignored curvature and torsion effects. Our results depend on Γ, parameter that characterizes the global features of both the embedding torus and, more interestingly, the knottedness of the path.
Uncertainty relations (URs) such as the Heisenberg-Robertson or the time-energy UR are often considered to be hallmarks of quantum theory. Here, a simple derivation of these URs is presented based on a single classical inequality from estimation theory, a Cram\'er-Rao-like bound. The Heisenberg-Robertson UR is then obtained by using the Born rule and the Schr\"odinger equation. This allows a clear separation of the probabilistic nature of quantum mechanics from the Hilbert space structure and the dynamical law. It also simplifies the interpretation of the bound. In addition, the Heisenberg-Robertson UR is tightened for mixed states by replacing one variance by the quantum Fisher information. Thermal states of Hamiltonians with evenly gapped energy levels are shown to saturate the tighter bound for natural choices of the operators. This example is further extended to Gaussian states of a harmonic oscillator. For many-qubit systems, we illustrate the interplay between entanglement and the structure of the operators that saturate the UR with spin-squeezed states and Dicke states.
Abstract The uncertainty principle sets a limit to our capacity to predict the outcomes of two incompatible measurements. The Heisenberg uncertainty relation for two arbitrary observables A and B is usually discussed in textbooks. However, little or no attention is paid to the fact that Schrödinger generalised the Heisenberg relation taking into account the covariance between the observables A and B . This extended inequality is known as the Robertson–Schrödinger uncertainty relation. Here, we demonstrate the less known fact that two-level quantum states, i.e., qubits, satisfy the equality of the Robertson–Schrödinger uncertainty relation for two arbitrary observables A and B . Taking advantage of the homomorphism between SU(2) and SO(3) groups, it is possible to map the distributions of the expectation values and variances of the observables, and the Heisenberg and covariance terms on the Bloch sphere. The graphical visualisation of the relevant quantities involved in the uncertainty relations allows us to distinguish specific properties and symmetries that are not so evident in the algebraic formalism.