Local decoherence and entropy

Question

Consider a quantum system consisting of two subsystems, $A$ and $B$. Let $\rho$ be the density matrix of the whole system $A\cup B$. Let $|\alpha\rangle$, $\alpha = 1,2\cdots d_B$, be the states of the subsystem $B$. Then $\rho$ can be written as the following: $$ \rho = \sum^{d_B}_{\alpha=1}\sum^{d_B}_{\beta=1}\sigma_{\alpha\beta}\otimes|\alpha\rangle\langle\beta|, $$ where $\sigma_{\alpha\beta}$ are sub-density-matrices for subsystem $A$ of size $d_A\times d_A$. Here $d_A$ is the dimension of the Hilbert space of the subsystem $A$. The reduced density matrix of subsystem $A$ is given by $$ \rho_A = \sum^{d_B}_{\alpha=1}\sigma_{\alpha\alpha}, $$ and the reduced density matrix of subsystem $B$ is given by $$ \rho_B = \sum^{d_B}_{\alpha=1}\sum^{d_B}_{\beta=1}\mathrm{tr}(\sigma_{\alpha\beta})\otimes|\alpha\rangle\langle\beta|. $$

Let us consider a process after which the quantum coherence of subsystem $B$ is lost. The density matrix then becomes: $$ \rho' = \sum^{d_B}_{\alpha=1}\sigma_{\alpha\alpha}\otimes|\alpha\rangle\langle\alpha|. $$ I am interested to know whether it is possible to relate the Renyi entropy of the new density matrix $\rho'$, defined as $$ S_\alpha(\rho')=\frac{\ln\mathrm{tr}(\rho'^\alpha)}{1-\alpha}, $$ to the Renyi entropy of density matrices $\rho$, $\rho_A$, $\rho_B$, or similar quantities. If the quick answer is no, I hope someone could point me to useful references.

Answer 1

For Von Neumann entropies you would have:

$H(AB)_\rho\leq H(A)_\rho+H(B)_\rho$

$H(AB)_{\rho'}\leq H(A)_{\rho'} + H(B)_{\rho'}$

whatever happens in $B$ should not affect $A$ so:

$H(A)_\rho=H(A)_{\rho'}$

for $B$ itself:

$H(B)_\rho\leq H(B)_{\rho'}$

with equality if the decoherence process is in the basis that diagonalizes $B$. And:

$H(AB)_\rho\leq H(AB)_{\rho'}$

This is all textbook material, you can find it in Nielsen and Chuang or in Wilde's Quantum Shannon Theory. Now if you replace Von Neumann entropies by Renyi entropies, the third to fifth relations are still valid, but subadditivity doesn't hold and you would have something a little bit weaker for any bipartite system:

$S_\alpha(AB)\leq S_\alpha(A) + S_0(B)$

This relation was proved in (http://arxiv.org/abs/quant-ph/0204093)