Entanglement depth

Entanglement depth appeared first in the context of cold gases, together with entanglement criteria that made it possible to bound it from below based on measured quantities.[2][3][4][5][6][7][8]

We will now present a general definition based on convex sets of quantum states. First, we will define ${\displaystyle k}$-producibility.[9] Let us consider a pure state that is the tensor product of multi-particle quantum states

${\displaystyle |\Psi \rangle =|\phi _{1}\rangle \otimes |\phi _{2}\rangle \otimes ...\otimes |\phi _{n}\rangle .}$

The pure state ${\displaystyle |\Psi \rangle }$ is told to be ${\displaystyle k}$-producible if all ${\displaystyle \phi _{i}}$ are states of at most ${\displaystyle k}$ particles. A mixed state is called ${\displaystyle k}$-producible, if it is a mixture of pure states that are all at most ${\displaystyle k}$-producible. The ${\displaystyle k}$-producible mixed states form a convex set.

A quantum state contains at least genuine multiparticle entanglement of ${\displaystyle k+1}$ particles, if it is not ${\displaystyle k}$-producible.

Finally, a quantum state has an entanglement depth ${\displaystyle k}$, if it is ${\displaystyle k}$-producible, but not ${\displaystyle (k-1)}$-producible.