Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general $K\times M$ problem and characterize the set of effectively different states (which cannot be related by local transformations). Thus, we generalize earlier results obtained for the simplest $2\times 2$ system, which lead to a stratification of the six-dimensional set of $N=4\mathrm{}$ pure states. We define the concept of absolutely separable states, for which all globally equivalent states are separable.

• Received 21 June 2000

DOI:https://doi.org/10.1103/PhysRevA.63.032307