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

QUANTUM COVARIANCES AND SCHRÖDINGER TYPE OF UNCERTAINTY RELATION

Eunsang Kim, Tae Ryong Park

Far East Journal of Mathematical Sciences (FJMS)Mar 5, 20180 citations

Geometric uncertainty relation, the symplectic area, and the J-holomorphic maps for mixed quantum states

Hoshang Heydari

In this paper we will establish a relation between geometric uncertainty relation and the determinant of the quantum covariance matrix for mixed quantum states. We will show that determinant of the covariance matrix represents the squared metric area of a parallelogram. In this setting the geometric uncertainty relation compares a metric area to a symplectic area. Moreover, we will in details investigate relation between J-holomorphic maps and geometric uncertainty relation for mixed quantum states. We will argue that determinant of the quantum covariance matrix is equal to the harmonic energy of a holomorphic map that minimize the areas.In this paper we will establish a relation between geometric uncertainty relation and the determinant of the quantum covariance matrix for mixed quantum states. We will show that determinant of the covariance matrix represents the squared metric area of a parallelogram. In this setting the geometric uncertainty relation compares a metric area to a symplectic area. Moreover, we will in details investigate relation between J-holomorphic maps and geometric uncertainty relation for mixed quantum states. We will argue that determinant of the quantum covariance matrix is equal to the harmonic energy of a holomorphic map that minimize the areas.

arXiv:1612.08428Dec 26, 20160 citations

COVARIANCE APPROACH TO QUANTUM THEORY

Jan Naudts

International Symposium on Quantum Theory and Symmetries (QTS)Jun 1, 20022 citations

Invariant theoretic approach to uncertainty relations for quantum systems

J. Solomon Ivan, Krishna Kumar Sabapathy, Narasimhaiengar Mukunda +1 more

We present a general framework and procedure to derive uncertainty relations for observables of quantum systems in a covariant manner. All such relations are consequences of the positive semidefiniteness of the density matrix of a general quantum state. Particular emphasis is given to the action of unitary symmetry operations of the system on the chosen observables, and the covariance of the uncertainty relations under these operations. The general method is applied to the case of an $n$ -mode system to recover the $Sp(2n,\,R)$ -covariant multi mode generalization of the single mode Schrödinger-Robertson Uncertainty Principle; and to the set of all polynomials in canonical variables for a single mode system. In the latter situation, the case of the fourth order moments is analyzed in detail, exploiting covariance under the homogeneous Lorentz group $SO(2,\,1)$ of which the symplectic group $Sp(2,\,R)$ is the double cover.We present a general framework and procedure to derive uncertainty relations for observables of quantum systems in a covariant manner. All such relations are consequences of the positive semidefiniteness of the density matrix of a general quantum state. Particular emphasis is given to the action of unitary symmetry operations of the system on the chosen observables, and the covariance of the uncertainty relations under these operations. The general method is applied to the case of an $n$-mode system to recover the $Sp(2n,\,R)$-covariant multi mode generalization of the single mode Schrödinger-Robertson Uncertainty Principle; and to the set of all polynomials in canonical variables for a single mode system. In the latter situation, the case of the fourth order moments is analyzed in detail, exploiting covariance under the homogeneous Lorentz group $SO(2,\,1)$ of which the symplectic group $Sp(2,\,R)$ is the double cover.

arXiv:1205.5132May 23, 20124 citations

n

-mode system to recover the

Sp(2n,\,R)

-covariant multi mode generalization of the single mode Schrödinger-Robertson Uncertainty Principle; and to the set of all polynomials in canonical variables for a single mode system. In the latter situation, the case of the fourth order moments is analyzed in detail, exploiting covariance under the homogeneous Lorentz group

SO(2,\,1)

of which the symplectic group

Sp(2,\,R)

is the double cover.