Separable states for the two-dimensional time-dependent anisotropic harmonic oscillator

What is it about?

Descartes was the pioneer to plant the root of the philosophy of modern science with his prescription to eliminate the plague of medieval sciences- "the influences from far away". Quantum entanglement seems to bring back the idea of "the influences from far away" in a different perspective. In particular, two entangled systems feel the existence of each other instantly, without violating the principles of special relativity. Despite the mental uneasyness, the experimental confirmation dictates that quantum entanglement is a key of nature. Theoretical study dictates that most of the quantum states of composite systems are entangled. Nonetheless, the measure of separable (not entangled) states are nonzero. Moreover, our experiment fits well with locality principle. That means, somehow the entanglement got balanced and allow us to treat a system in isolation locally. Therefore, it is important to find out the separable states and the limit at which the entangled states become separable. Peres and Horodecki found the criterion of separability of states for finite dimensional Hilbert space. The idea of Peres-Horodecki was generalized for infinite dimensional Hilbert space, in particular, for Gaussian states, by Simon. Subsequently, the formalism found to be fruitful to apply for a wide class of quantum systems. We show that the idea of Simon to find out the separability criterion for Gaussian states can be utilized for time-dependent system. Time-dependent anisotropic oscillator is studied in detail in our findings. The theoretical result of our study can be used for a wide class of systems. For instance, a charged harmonic oscillator in a magnetic field, Landau problems, an oscillator in a non-commutative space, Chern-Simon's model in long wavelength limit, and of similar type.

Featured Image

Why is it important?

Peres and Horodecki found the criterion of separability of states for finite dimensional Hilbert space. The idea of Peres-Horodecki was generalized for infinite dimensional Hilbert space, in particular, for Gaussian states, by Simon. Subsequently, the formalism found to be fruitful to apply for a wide class of quantum systems. We show that the idea of Simon to find out the separability criterion for Gaussian states can be utilized for time-dependent system. Time-dependent anisotropic oscillator is studied in detail in our findings. The theoretical result of our study can be used for a wide class of systems. For instance, a charged harmonic oscillator in a magnetic field, Landau problems, an oscillator in a non-commutative space, Chern-Simon's model in long wavelength limit, and of similar type.