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Given a bipartite quantum system $ \mathcal{H}_A \otimes \mathcal{H}_B $, we define a quantum-classical (q-c) state to be any state that may be written in the form: $$ \rho = \sum_j q_j \rho_j^A \otimes | \psi_j \rangle \langle \psi_j | $$ where $ q_j \geq 0 $, $ \rho_j^A $ are density matrices acting on $ \mathcal{H}_A $, and $ \left\{ |\psi_j \rangle \right\}_j $ is an orthonormal set in $ \mathcal{H}_B $. Similarly, we define a classical-quantum (c-q) state as one that may be written as: $$ \rho = \sum_k p_k | \phi_k \rangle \langle \phi_k | \otimes \rho_k^B $$ with analogous conditions on $p_k, \rho_k^B$ and $ | \phi_k \rangle $. Finally, a classical-classical (c-c) state is a state that we may write as: $$ \rho = \sum_k p_k | \phi_k \rangle \langle \phi_k | \otimes | \psi_k \rangle \langle \psi_k | $$ where this time $ p_k \geq 0 $, $ \left\{ |\phi_k \rangle \right\}_k $ is an orthonormal set in $ \mathcal{H}_A $ and $ \left\{ |\psi_k \rangle \right\}_k $ is an orthonormal set in $ \mathcal{H}_B $. Here is my question:

Is there a state that is both c-q and q-c, but is not c-c ?

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The "classical" portion of the state should be unchanged by complete dephasing. This means that it will never have off-diagonal components in some particular basis. When a state is both q-c and c-q, there are a lot of off-diagonal components that must vanish! Such a state must satisfy $$(\langle \phi_k|\otimes \langle \psi_i |)\rho (| \phi_l\rangle\otimes | \psi_j \rangle)=0$$ when either $k\neq l$ or $i\neq j$. This immediately constrains the state to be c-c.

Take $k=l$. We find that $p_k\langle \psi_i|\rho_k^B|\psi_j\rangle=0$ for all $i\neq j$ by using the c-q expression and our above condition. This means $\rho_k^B$ must be diagonal in the $\{|\psi_j\rangle\}$ basis for all $k$, letting us write it as $$\rho_k^B=\sum_j\lambda_j^{(k)} |\psi_j\rangle\langle\psi_j|.$$ The overall state then becomes, again using the c-q expression: $$\rho=\sum_k\sum_j p_k\lambda_j^{(k)} |\phi_k\rangle\langle \phi_k|\otimes|\psi_j\rangle\langle \psi_j|.$$ This is a c-c state, just with each state $|\phi_k\rangle\langle \phi_k|$ and $|\psi_j\rangle\langle \psi_j|$ repeated multiple times in the expansion (actually every combination of the two is present, although the coefficients may vanish). Similar results can be found by starting from the q-c expression.

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