# Does a subsystem being mixed imply the state is entangled?

If a pure state, $$\rho_{AB}$$, has subsystems described by mixed density matrices, the overall state is entangled (as far as I understand).

Can you conclude the same with an initially mixed bipartite density matrix i.e. if $$\rho_{AB}$$ is a mixed density matrix, it is entangled if its subsystems are described by mixed density matrices $$\rho_A$$ and $$\rho_B$$? Is there a counterexample; a separable, but mixed, $$\rho_{AB}$$ that has mixed subsystems?

• This might be of further interest. Jan 15, 2022 at 23:00

Let $$\rho_1$$ and $$\rho_2$$ be mixed density matrices. Then $$\rho=\rho_1\otimes\rho_2$$ is mixed and separable.

In response to the comment: Let us study the case $$\rho_1=\rho_2 = \frac{1}{2} \,\mathbb I_2 \quad ,$$ where $$\mathbb I_2$$ is the identity matrix on $$H\cong \mathbb C^2$$, i.e. these matrices are the maximally mixed density matrices of a qubit system.

Then these are the reduced density matrices of both the maximally entangled two-qubit Bell state

$$\sigma = |\psi^-\rangle \langle \psi^-|$$ and the separable mixed state $$\rho=\rho_1 \otimes \rho_2 \quad .$$ Additionally, consider a Werner state of the form $$\omega = \alpha \,\sigma + (1-\alpha)\, \rho \quad,$$ with some restriction on $$\alpha$$ such that it is an entangled mixed density matrix. We again find that $$\rho_1$$ and $$\rho_2$$ are its reduced density matrices.

In conclusion, we see that the two (mixed) reduced density matrices $$\rho_1$$ and $$\rho_2$$ could arise from a pure entangled, mixed separable or mixed entangled state. However, they cannot arise from a pure separable (product) state, since every pure state with those reduced density matrices necessarily has a Schmidt rank greater than one and is thus entangled.

• Is there anything we can conclude about the separability of the $\rho$ if we know $\rho_1$ and $\rho_2$ are mixed? You've shown that $\rho$ can be separable, but can it also be entangled or can we only find out by using something like the Concurrence? Jan 13, 2022 at 22:17
• @Angus Does this help (especially point 3. in the answer)? Jan 13, 2022 at 22:27
• @Angus I've edited the answer and tried to give an example. Please double check everything. Does this help? Jan 13, 2022 at 23:14
• It does, thanks for the great answer. Jan 13, 2022 at 23:31