When I encounter the definition of the mathematical definition of quantum entanglement. System composed by many parts $A$, $B$,.., $N$ can be described by a density matrix operator $\hat{\rho}$ acting on a Hilbert space of the tensorial product structure: $$H = H_A \otimes H_B \otimes \ldots \otimes H_N$$ The quantum state $\hat{\rho}$ is said to be at least $m$-separable if there exists splitting of the $N$ parties into $m$ parts $P_1, P_2, \ldots P_m$ such that [1] $$\hat{\rho}=\sum\limits_k p_k \hat{\rho}_k^{(1)} \otimes \hat{\rho}_k^{(2)} \otimes \ldots \hat{\rho}_k^{(m)}.$$ Otherwise the state is at least $m$-partite entangled.
All the quantum information science and quantum metrology (see Fisher information) is based on this definition and most importantly people apply this definition to indistinguishable particles (the parties are the collection of particles) which is not correct because the parties in the above definition are distinguishable (we are not dealing for example with the symmetrized Hilbert space). Maybe if the distinguishable particles are entangled according to the above definition they are also entangled in the indistinguishable case (some kind of lower bound)? How does this definition of entanglement applies to particle entanglement which are indistinguishable?
Here are some news phys.org.
UPDATE
Since my interest in the quantum metrology using atoms (e.g. atomic clocks) I wonder why the link between Quantum Fisher information (QFI) and multi-particle entanglement is correct [2]? Without doubt the QFI sets the achievable precision of interferometers, but the proof that it is connected to multi-particle entanglement is not convincing to me (as long as indistinguishable particles are concerned). It follows from standard definition of entanglement (above formula) where distinguishable particles are considered. Is it actually correct? In real experiments atoms are the same.
