Non-lockability of quantum max-entropy

Question

Lockability and non-lockability are explained in this paper. A real valued function of a quantum state is called non-lockable if its value does not change by too much after discarding a subsystem. The max-entropy of a quantum state is defined as

$$H_{\max }(A)_{\rho}= \log \operatorname{tr}(\rho_{A}^{1 / 2})$$

For a bipartite quantum state $\rho_{AB}$, I would like to know if the max-entropy is non-lockable i.e. is there any relationship of the form

$$\text{tr}(\rho_{AB}^{1/2}) \leq \text{tr}(\rho_{A}^{1/2})\cdot|B|$$

which, after taking logs on both sides, would yield

$$H_{\max}(AB)\leq H_{\max}(A) + \log|B|$$

A couple of easy numerical examples suggests this might be true but I have not been able to prove it.

Answer 1

Let $D_{\alpha}(\rho\|\sigma):= \frac{1}{\alpha - 1} \log \mathrm{Tr}[\rho^\alpha \sigma^{1-\alpha}]$ be the Petz-Rényi divergence for $\alpha \in (0,1)\cup(1,\infty)$. Note that for $\alpha \in (0,1)\cup(1,2]$ this quantity satisfies the data processing inequality $$ D_{\alpha}(\rho\|\sigma) \geq D_{\alpha}(\mathcal{E}(\rho) \| \mathcal{E}(\sigma)), $$ where $\mathcal{E}$ is any CPTP map.

Now let $\rho_{AB}$ be any state and take $\mathcal{E}$ to be the partial trace over the $B$ system. Then for $\alpha = 1/2$ we have $$ \begin{aligned} -2 \log \mathrm{Tr}[\rho^{1/2}_{AB}] &= D_{1/2}(\rho_{AB}\|I_{AB}) \\ &\geq D_{1/2}(\mathrm{Tr}_{B}[\rho_{AB}]\|\mathrm{Tr}_{B}[I_{AB}]) \\ &= D_{1/2}(\rho_{A}\||B|I_{A}) \\ &= -2 \log \mathrm{Tr}[\rho^{1/2}_{A} |B|^{1/2}] \end{aligned} $$

Rearranging you get $$ \log \mathrm{Tr}[\rho^{1/2}_{AB}] \leq \log \mathrm{Tr}[\rho^{1/2}_{A}] + \log|B|^{1/2} $$ which is actually a slightly stronger inequality with the exponent of $|B|$ being only $1/2$.