Is purity an entanglement witness?

Question

Suppose an n-qubit mixed state $\rho$ with subsystems $a_1, a_2,..., a_n$

If purity of an arbitrary qubit $a_i$ is 1, then can i conclude its separatable from the entire system?

Answer 1

It is not an entanglement witness for two reasons:

1. Entanglement witnesses are linear maps in $\rho$. $\mathrm{tr}\,\rho^2$ is not a linear map.

2. Entanglement witnesses, by definition, detect entanglement in general (i.e., mixed) quantum states. This is not the case for the purity of a subsystem, as e.g. the maximally mixed and maximally entangled state have the same purity on each subsystem.

Answer 2

It is not an entanglement witness per se. A subsystem having purity = 1 does mean it is separable, i.e., it is a product state with the rest of the system. However, a subsystem having purity $\neq$ 1 does not mean it is always entangled. See a counter-example here. The exception for this is when your initial system of $n$ qubits is a pure state. See more here.