Restricted numerical shadow and geometry of quantum entanglement

The restricted numerical range $W_R(A)$ of an operator $A$ acting on a $D$-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset $R$ of the of set of pure quantum states of dimension $D$. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of $A$ -- a normalized probability distribution on the complex plane supported in $W_R(A)$. Its value at point $z \in {\mathbbm C}$ is equal to the probability that the expectation value $< \psi|A|\psi>$ is equal to $z$, where $|\psi>$ represents a random quantum state in subset $R$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini--Study measure. Studying restricted shadows of operators of a fixed size $D=N_A N_B$ we analyse the geometry of sets of separable and maximally entangled states of the $N_A \times N_B$ composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on $D=2^3$ dimensional Hilbert space allows us to investigate the structure of the orbits of $GHZ$ and $W$ quantum states of a three--qubit system.