# Classical correlations in bipartite entangled mixed state

I have recently asked somewhat related question and got very illuminating answer. After some thinking however I have realized that (at least) one more point is unclear to me:

How can we check whether given bipartite entangled mixed state has classical correlations between its subsystems?

This is what I know about the subject so far. Suppose we are given some mixed state: $$\rho = \sum_{i} p_i |\psi_i\rangle\langle\psi_i|$$ here $$\{p_i \}$$ defines probabilistic mixture of pure states $$\{|\psi_i\rangle \}$$. This state could be one of three:

• simply separable(product state) $$\rho = \rho_A \otimes \rho_B$$. In this case there are no correlations(either classical or quantum) between subsystems.
• separable state $$\rho = \sum_k p_k \rho_A ^k \otimes \rho_B ^k \quad \sum_k p_k =1 \quad (1)$$ Firstly by definition this state has no quantum correlations between subsystems A and B. Secondly this state can be obtained from product state by LOCC. This means that given state has classical correlations between subsystems.
• Entangled state In this case given state can't be written in form (1). It will obviously have quantum correlations between subsystems. Yet what can we say about classical correlations between parts A and B? My guess would be to check whether this state can only be constructed out of product state by combination of both nonlocal operators $$U_A \otimes U_B$$ and LOCC. If LOCC is necessary then our state has classical correlations between subsystems. However this criterion(if correct at all) seems to be almost impossible to apply in real situation. Is there any other way to resolve the question?
• I don't want to mess too much with your formatting, but I would strongly discourage you from using MathJax for emphasis. If it's just text that you want to emphasize, it doesn't belong in a MathJax block. Use italics or bold as required, instead. Commented Sep 21, 2018 at 13:43
• You haven't said how you want to quantify correlations, so I don't think it's really meaningful to talk about separating classical from quantum correlations. Two examples to keep in mind: Commented Sep 21, 2018 at 17:06
• (1) Alice and Bob have two qubits each. They perform a joint unitary on qubits A1 and B1 and put them on a maximally entangled state, and they use LOCC to put qubits A2 and B2 on a state with maximal classical correlations, $\rho = \frac12 (|00⟩⟨00| + |11⟩⟨11|)$; within each side, qubits 1 and 2 are described by a joint state but they are otherwise kept separate. How should one quantify the correlations in this state? Commented Sep 21, 2018 at 17:12
• (2) Alice and Bob have one qubit each; they use a joint unitary to put them on the maximally entangled state $|\psi⟩ = \frac{1}{\sqrt{2}}(|00⟩ + |11⟩)$, and then each of them applies a local dephasing quantum channel such that $\Lambda (|0⟩⟨0|) = |0⟩⟨0|$, $\Lambda (|1⟩⟨1|) = |1⟩⟨1|$, $\Lambda (|0⟩⟨1|) = \Lambda (|1⟩⟨0|) = 0$, which reduces the global state to the maximally-classically-correlated state $\rho = \frac12(|00⟩⟨00| + |11⟩⟨11|)$. Should the correlations be considered "quantum" in some way? What if the dephasing is only partial (so $\Lambda (|0⟩⟨1|) =r|0⟩⟨1|$, $0<r<1$)? Commented Sep 21, 2018 at 17:16
• this seems like the kind of question that should be answered via a resource theory of entanglement, in which LOCC operations are taken to be the free resources. See for example chapter IV in arxiv.org/abs/1806.06107. I'm not versed enough in this topic to give a meaningful answer though
– glS
Commented Sep 22, 2018 at 16:07

Take my answer with a pinch of salt - I do not really know if this works (I guess we'll find out with the upvotes/downvotes).

It appears that the quantity $$L(\rho) = \inf_{CSS}S(\rho || CSS)$$ might be useful, where $$CSS$$ is the closest separable state and $$S$$ is the relative entropy. This is numerical, but it is a convex optimization problem so it is possible to do this on standard solvers.

The CSS state you find through such an optimization procedure can be diagonalized and it's diagonal entries should capture all possible the classical correlations. $$L(\rho)$$ represents the purely quantum correlations.

It seems that you would like to discern wether your state is in the form 1) or 2) of your bullet point list. That is, wether the state is a product state (1) or classically correlated (2).

Note that in (1) $$\rho_{A,B}$$ are the marginal of the state, i.e.

$$\rho_A = \mathrm{Tr}_B \rho_{AB}$$

And similarly for the other marginal.

So you need to check if

$$\rho_{AB} \stackrel{?}{=} \mathrm{Tr}_B \rho_{AB} \otimes \mathrm{Tr}_A \rho_{AB}.$$

Any distance function would do, for example you can compute

$$\left \Vert \rho_{AB} -\mathrm{Tr}_B \rho_{AB} \otimes \mathrm{Tr}_A \rho_{AB} \right \Vert_1$$

The above is zero if and only if the state is a product state.