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Internal Structure of the Heisenberg and Robertson-Schrödinger Uncertainty Relations

Lubomír Skála

International Journal of Theoretical PhysicsMay 23, 20132 citations

Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

K. Mahesh Krishna

Jan 1, 20240 citations

Explorations into the Schrodinger Uncertainty Relation

Nathan Steiger, Jean-François S. Van Huele

Bulletin of the American Physical SocietyOct 17, 20080 citations

Generalized uncertainty relations and characteristic invariants for the multimode states

E. C. G. Sudarshan, Charles B. Chiu, G. Bhamathi

The close relationship between the zero-point energy, the uncertainty relation, coherent states, squeezed states, and correlated states for one mode is investigated. This group theoretic perspective of the problem enables the parametrization and identification of their multimode generalization. A simple and efficient method of determining the canonical structure of the generalized correlated states is presented. Implication of canonical commutation relations for correlations are not exhausted by the Heisenberg uncertainty relation, not even by the Schr\"odinger-Robertson uncertainty inequality, but there are relations in the multimode case that are the generalization of the Schr\"odinger-Robertson relation.The close relationship between the zero-point energy, the uncertainty relation, coherent states, squeezed states, and correlated states for one mode is investigated. This group theoretic perspective of the problem enables the parametrization and identification of their multimode generalization. A simple and efficient method of determining the canonical structure of the generalized correlated states is presented. Implication of canonical commutation relations for correlations are not exhausted by the Heisenberg uncertainty relation, not even by the Schr\"odinger-Robertson uncertainty inequality, but there are relations in the multimode case that are the generalization of the Schr\"odinger-Robertson relation.

Physical Review AJul 1, 199569 citations

Stronger Schrödinger-like uncertainty relations

Qiu-Cheng Song, Cong‐Feng Qiao

arXiv:1504.01137Physics Letters AJul 12, 201649 citations

Robertson–Schrödinger uncertainty relation for qubits: a visual approach

Julio C. Gutiérrez-Vega

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.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.

European Journal of PhysicsJan 8, 20214 citations

A group-theoretical approach to extending the Schrödinger-Robertson uncertainty inequality

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