Uncertainty and Quantum Variance at the light of Quantum Information Geometry

Saptarshi Sahoo, Amit Kumar Mandal, Nirmalya Misra +1 more

Sk Sazim, Satyabrata Adhikari, Arun Kumar Pati +1 more

We introduce a concept, called the mutual uncertainty between two observables in a given quantum state, which enjoys features similar to those of the mutual information for two random variables. Further, we define conditional uncertainty as well as conditional variance and show that conditioning on more observables reduces the uncertainty. Given three observables, we prove a ``strong subadditivity'' relation for the conditional uncertainty under certain conditions. As an application, we show that by using the conditional variance one can detect bipartite higher dimensional entangled states. The efficacy of our detection method lies in the fact that it gives better detection criteria than most of the existing criteria based on geometry of the states. Interestingly, we find that for $N$-qubit product states, the mutual uncertainty is exactly equal to $N\ensuremath{-}\sqrt{N}$, and if it is other than this value, the state is entangled. We also show that using the mutual uncertainty between two observables, one can detect non-Gaussian steering where Reid's criterion fails to detect it. Our results may open a direction of exploration in quantum theory and quantum information using mutual uncertainty, conditional uncertainty, and the strong subadditivity for multiple observables.

Piero Chiarelli

This work shows that in the frame of the stochastic generalization of the quantum hydrodynamic analogy (QHA) the uncertainty principle is fully compatible with the postulate of finite transmission speed of light and information. The theory shows that the measurement process performed in the large scale classical limit in presence of background noise, cannot have a duration smaller than the time need to the light to travel the distance up to which the quantum non-local interaction extend itself. The product of the minimum measuring time multiplied by the variance of energy fluctuation due to presence of stochastic noise shows to lead to the minimum uncertainty principle. The paper also shows that the uncertainty relations can be also derived if applied to the indetermination of position and momentum of a particle of mass m in a quantum fluctuating environment.

Masahito Hayashi

Piero Chiarelli

This work shows that in the frame of the stochastic generalization of the quantum hydrodynamic analogy (QHA) the uncertainty principle can be derived by the postulate of finite transmission speed of light and information . The theory shows that the measurement process performed in the large scale classical limit of stochastic QHA (SQHA), cannot have a duration smaller than the time need to the light to travel the distance up to which the quantum non-local interaction extend itself. The product of the minimum measuring time multiplied by the variance of energy fluctuation due to presence of stochastic noise shows to lead to the minimum uncertainty principle. The paper also shows that the uncertainty relations can be also derived if applied to the indetermination of position and momentum of a particle of mass m in a quantum fluctuating environment.