Related question here - copying over the definitions.

The relative entropy between two quantum states is given by $D(\rho\|\sigma) = \operatorname{Tr}(\rho\log\rho -\rho\log\sigma)$. It is known that for any bipartite state $\rho_{AB}$ with reduced states $\rho_A$ and $\rho_B$, it holds that

$$D(\rho_{AB}\|\rho_A\otimes\rho_B)\leq D(\rho_{AB}\|\rho_A\otimes\sigma_B)$$

for all choices of $\sigma_B$. This can be seen by expanding both sides and noting that the relative entropy is nonnegative. Now define the relative entropy variance (see this reference, 2.16)

$$V(\rho \| \sigma):=\operatorname{Tr} [\rho(\log \rho-\log \sigma)^2]-D(\rho \| \sigma)^{2}$$

I would like to understand if this variance satisfies some type of continuity argument in the following sense. Let $\rho_{AB}$ be $d$ dimensional. I am given that $\sigma_B\approx_\varepsilon\rho_B$ although this property may not be very useful. Let

$$\Delta = D(\rho_{AB}\|\rho_A\otimes\sigma_B) - D(\rho_{AB}\|\rho_A\otimes\rho_B)$$

Is one able to then bound the difference in variances in the following way

$$V(\rho_{AB}\|\rho_A\otimes\rho_B) - V(\rho_{AB}\|\rho_A\otimes\sigma_B) \leq f(\Delta, d)?$$

EDIT: I ran following very simple MATLAB code (requires QETLAB and cvxquad for partial trace, random state generation and computing relative entropy) around 10000 times.

d = 10; %Dimension of rhoAB
rhoAB = RandomDensityMatrix(d^2,d^2);
rhoA = PartialTrace(rhoAB, 2, [d, d]);
rhoB = PartialTrace(rhoAB, 1, [d, d]);
sigmaB = RandomDensityMatrix(d,d);
v1 = trace(rhoAB*(logm(rhoAB) - logm(kron(rhoA, rhoB)))^2) -  quantum_rel_entr(rhoAB, kron(rhoA, rhoB))^2;
v2 = trace(rhoAB*(logm(rhoAB) - logm(kron(rhoA, sigmaB)))^2) - quantum_rel_entr(rhoAB, kron(rhoA, sigmaB))^2 ;
conjecture  = 1/sqrt(log(d))*(v1 - v2) <=  quantum_rel_entr(rhoAB, kron(rhoA, sigmaB)) - quantum_rel_entr(rhoAB, kron(rhoA, rhoB));

This suggests that the conjecture

$$V(\rho_{AB}\|\rho_A\otimes\rho_B) - V(\rho_{AB}\|\rho_A\otimes\sigma_B) \leq \Delta\sqrt{\log d}$$

holds for the randomly generated states I used without exception so far.



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