The Peres-Horodecki separability criterion is studied for bipartite continuous variable states. The partial transpose operation in the continuous case is interpreted geometrically as mirror reflection in phase space. This leads to stronger uncertainty principles for separable states. For all bipartite Gaussian states, the Peres-Horodecki criterion is both necessary and sufficient for separability. Entanglement is central to quantum information and computation. The Peres-Horodecki criterion is necessary and sufficient for separability in 2x2 and 2x3 dimensional systems but not in higher dimensions. Continuous variable systems have gained interest due to experimental realizations like quantum teleportation of coherent states. The Peres-Horodecki criterion is shown to be easy to implement in the limit of infinite dimensions. The partial transpose operation is equivalent to mirror reflection in phase space, which imposes stronger uncertainty principles on separable states. This leads to a necessary and sufficient condition for separability in all bipartite Gaussian states. The Wigner distribution is used to describe the partial transpose operation, which transforms the Wigner distribution by inverting the p2 coordinate. The Peres-Horodecki criterion requires that the Wigner distribution under mirror reflection has the "Wigner quality" for separable states. The uncertainty principle for second moments is derived, and it is shown that the Peres-Horodecki criterion imposes additional restrictions on separable states. These restrictions are invariant under local transformations and mirror reflection. The criterion is shown to be equivalent to a condition on the variance matrix of the state. For Gaussian states, the Peres-Horodecki criterion is necessary and sufficient for separability. The criterion is shown to be invariant under local transformations and mirror reflection. The condition is applied to Gaussian states, and it is shown that states with det C ≥ 0 are separable. The main theorem is proved by considering cases where det C is positive or zero. The Peres-Horodecki criterion is shown to be easier to implement in practice compared to other criteria. It is invariant under local transformations and mirror reflection, making it a powerful tool for determining separability in continuous variable systems. The criterion is essential for understanding the separability of bipartite continuous variable states.The Peres-Horodecki separability criterion is studied for bipartite continuous variable states. The partial transpose operation in the continuous case is interpreted geometrically as mirror reflection in phase space. This leads to stronger uncertainty principles for separable states. For all bipartite Gaussian states, the Peres-Horodecki criterion is both necessary and sufficient for separability. Entanglement is central to quantum information and computation. The Peres-Horodecki criterion is necessary and sufficient for separability in 2x2 and 2x3 dimensional systems but not in higher dimensions. Continuous variable systems have gained interest due to experimental realizations like quantum teleportation of coherent states. The Peres-Horodecki criterion is shown to be easy to implement in the limit of infinite dimensions. The partial transpose operation is equivalent to mirror reflection in phase space, which imposes stronger uncertainty principles on separable states. This leads to a necessary and sufficient condition for separability in all bipartite Gaussian states. The Wigner distribution is used to describe the partial transpose operation, which transforms the Wigner distribution by inverting the p2 coordinate. The Peres-Horodecki criterion requires that the Wigner distribution under mirror reflection has the "Wigner quality" for separable states. The uncertainty principle for second moments is derived, and it is shown that the Peres-Horodecki criterion imposes additional restrictions on separable states. These restrictions are invariant under local transformations and mirror reflection. The criterion is shown to be equivalent to a condition on the variance matrix of the state. For Gaussian states, the Peres-Horodecki criterion is necessary and sufficient for separability. The criterion is shown to be invariant under local transformations and mirror reflection. The condition is applied to Gaussian states, and it is shown that states with det C ≥ 0 are separable. The main theorem is proved by considering cases where det C is positive or zero. The Peres-Horodecki criterion is shown to be easier to implement in practice compared to other criteria. It is invariant under local transformations and mirror reflection, making it a powerful tool for determining separability in continuous variable systems. The criterion is essential for understanding the separability of bipartite continuous variable states.