# Inequalities involving entropy: quantum discord and mutual information

My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $$\rho$$ we define the entropy by the formula $$H(\rho):=-tr(\rho log \rho)$$ (by a state we mean in fact normal, possibly mixed state which we identify with the corresponding density matrix). Now consider a quantum system $$S$$ and an apparatus $$A$$: we describe both of them in the context of Hilbert spaces $$\mathcal{H}_S$$ and $$\mathcal{H}_A$$. Assume that the composite state $$\rho_{SA} \in \mathcal{L}^1(\mathcal{H}_S \otimes \mathcal{H}_A)$$ is given. Denote by $$\rho_S$$ the state $$tr_A(\rho_{SA})$$ obtained from $$\rho_{SA}$$ by taking the partial trace: similarly define $$\rho_A$$ and define the so called mutual information $$I(S:A):=H(\rho_S)+H(\rho_A)-H(\rho_{SA})$$. Quantum discord will be defined as the difference between mutual information $$I$$ and the other, quantum version of it, called $$J$$ which is defined as follows: choose a family of one dimensional mutually orthogonal projections $$\{\Pi_j^A\}_j$$ summing up to identity $$I_{\mathcal{H}_A}$$: put $$\rho_{S|\Pi_j^A}=\frac{1}{p_j}(I_{\mathcal{H}_S} \otimes \Pi_j^A) \rho_{SA} (I_{\mathcal{H}_S} \otimes \Pi_j^A)$$ where $$p_j=tr((I_{\mathcal{H}_S} \otimes \Pi_j^A)\rho_{SA})$$ and define $$H(\rho_S|\{\Pi_j^A\}_j):=\sum_j p_j H(\rho_{S|\Pi_j^A})$$. We define $$J(S:A)_{\{\Pi_j^A\}_j}:=H(\rho_S)-H(\rho_S|\{\Pi_j^A\}_j)$$. Finally we define the quantum discord as a difference $$\delta(S:A)_{\{\Pi_j^A\}_j}=I(S:A)-J(S:A)_{\{\Pi_j^A\}_j}.$$ Proposition 1 in the paper mentioned above gives another formula for $$H(\rho_S|\{\Pi_j^A\}_j)$$ as a difference $$H(\rho_{SA}^D)-H(\rho_A^D)$$ where $$\rho_{SA}^D=\sum_j (I \otimes \Pi_j^A)\rho_{SA}(I \otimes \Pi_j^A)$$ and $$\rho_A^D=\sum_j \Pi_j^A \rho_A \Pi_j^A$$. This allows us to express the quantum discord as follows: $$\delta(S:A)_{\{\Pi_j^A\}_j}=H(\rho_A)-H(\rho_A^D)+H(\rho_{SA}^D)-H(\rho_{SA})$$

Question 1 How to prove that $$\delta(S:A)_{\{\Pi_j^A\}_j} \geq 0$$?

In the paper authors claim that this follows somehow from concavity of $$H(\rho_{SA})-H(\rho_A)$$ but I don't see how (they cite some paper which I've browsed but didn't manage to find an answer)

Question 2 How to prove that $$\delta(S:A)_{\{\Pi_j^A\}_j}=0$$ implies $$\rho_{SA}=\rho_{SA}^D$$?

The authors give a proof of this fact but I'm not convinced by their argument: they consider another state $$\hat{\rho}_{SA}$$ obtained by removing off-diagonal terms and claim that $$H(\hat{\rho}_{SA}) >H(\rho_{SA})$$ while $$H(\hat{\rho}_A)=H(\rho_A)$$. I don't see why this is true: once we accept this we clearly obtain $$H(\rho_{SA})-H(\rho_A) and now they claim that also $$H(\hat{\rho}_{SA})-H(\hat{\rho}_A) \leq H(\rho_{SA}^D)-H(\rho_A^D)$$ which is also not clear for me (they again invoke concavity).

I will be very grateful for any help with this problem

EDIT: There is already one excellent answer below but unfortunately this answer adresses only the first of my questions: still I would like to see how to prove the second part. Maybe the following discussions will be relavant: see here and here but I'm affraid that somewhere we need strong inequality to finish the argument given by Olivier and Żurek.

The key point how to use concavity is the following: The map $$f\colon\rho_{SA}\mapsto H(\rho_{SA})-H(\mathrm{tr}_S(\rho_{SA}))$$ is invariant under conjugation with unitaries in $$\mathbb C 1\otimes \mathcal L(\mathcal H_A)$$. Moreover, if we denote by $$\mu$$ the normalized Haar measure on $$\mathbb T^m$$, we have $$\sum_{j=1}^m (I\otimes\Pi_j^A)x(I\otimes\Pi_j^A)=\int_{\mathbb T^m}\left(I\otimes\sum_{j=1}^m \lambda_j\Pi_j^A\right)x\left(I\otimes\sum_{k=1}^m\lambda_k\Pi_k^A\right)\,d\mu(\lambda)$$ since for $$j\neq k$$ one has $$\int_{\mathbb T^m}\lambda_j\lambda_k\,d\mu(\lambda)=0$$. Note that $$\sum_{j=1}^m \lambda_j\Pi_j^A$$ is a unitary in $$\mathcal L(\mathcal H_A)$$.
Then concavity and partial unitary invariance of $$f$$ imply \begin{align*} f(\rho_{SA}^{D})&=f\left(\int_{\mathbb T^m}\left(I\otimes\sum_{j=1}^m \lambda_j\Pi_j^A\right)\rho_{SA}\left(I\otimes\sum_{k=1}^m\lambda_k\Pi_k^A\right)\,d\mu(\lambda)\right)\\ &\geq \int_{\mathbb T^m}f\left(\left(I\otimes\sum_{j=1}^m \lambda_j\Pi_j^A\right)\rho_{SA}\left(I\otimes\sum_{k=1}^m\lambda_k\Pi_k^A\right)\right)\,d\mu(\lambda)\\ &=\int_{\mathbb{T}^m}f(\rho_{SA})\,d\mu(\lambda)\\ &=f(\rho_{SA}). \end{align*} This settles your first question (note that $$\mathrm{tr}_S(\rho^D_{SA})=\rho_A^D$$).