Gaussian states as minimum uncertainty states

Pankaj Sharan

Ladislav Mišta, Klára Baksová

We develop a recently proposed theory of a quantifier of bipartite Gaussian entanglement called Gaussian intrinsic entanglement (GIE) [L. Mi\ifmmode \check{s}\else \v{s}\fi{}ta, Jr. and R. Tatham, Phys. Rev. Lett. 117, 240505 (2016)]. Gaussian intrinsic entanglement provides a compromise between computable and physically meaningful entanglement quantifiers and so far it has been calculated for two-mode Gaussian states including all symmetric partial minimum-uncertainty states, weakly mixed asymmetric squeezed thermal states with partial minimum uncertainty, and weakly mixed symmetric squeezed thermal states. We improve the method of derivation of GIE and show that all previously derived formulas for GIE of weakly mixed states in fact hold for states with higher mixedness. In addition, we derive analytical formulas for GIE for several other classes of two-mode Gaussian states with partial minimum uncertainty. Finally, we show that, like for all previously known states, also for all currently considered states the GIE is equal to Gaussian R\'enyi-2 entanglement of formation. This finding strengthens a conjecture about the equivalence of GIE and Gaussian R\'enyi-2 entanglement of formation for all bipartite Gaussian states.

David Stoler, S. Newman

Sang Pyo Kim

We show that the ground state of the well-known pseudo-stationary states for the Caldirola–Kanai Hamiltonian is a generalized minimum uncertainty state, which has the minimum allowed uncertainty Δ q Δ p = ℏσ 0 /2, where σ 0 (⩾1) is a constant depending on the damping factor and natural frequency. The most general symmetric Gaussian states are obtained as the one-parameter squeezed states of the pseudo-stationary ground state. It is further shown that the coherent states of the pseudo-stationary ground state constitute another class of the generalized minimum uncertainty states.

Jung Kon Kim, Sang Pyo Kim

By using the invariant method we find one-parameter squeezed Gaussian states for both time-independent and time-dependent oscillators. The squeezing parameter is expressed in terms of the energy expectation value for the time-independent case and represents the degree of mixing of positive and negative frequency solutions for the time-dependent case. A minimum uncertainty proposal is advanced to select, uniquely, vacuum states at each moment of time. We show that the Gaussian states with minimum uncertainty coincide with the true vacuum state for the time-independent oscillator and the Bunch-Davies vacuum for a massive scalar field in a de Sitter spacetime.

John H. Marburger, Edwin Albert Power