Quantum Computing Asked on August 20, 2021
In John Watrous’ lectures, he defines the quantum min-relative entropy as
$$D_{min}(rho|sigma) = -log(F(rho, sigma)^2),$$
where $F(rho,sigma) = tr(sqrt{rhosigma})$. Here, I use this question and answer to make the definition simpler although one should note that the linked question uses a different definition of fidelity (squared vs not squared).
On the other hand, one of the early papers introducing this quantity (see Definition 2 of this paper) defines it as
$$D_{min}(rho|sigma) = -log(tr(Pi_rhosigma)),$$
where $Pi_rho$ is the projector onto the support of $rho$. It’s not clear if these definitions are equivalent since I can change $rho$ without altering its support.
How are the two definitions related to each other, if at all?
As @rnva points out these are not the same quantities. To give some clarity as to why they are both referred to as $D_{min}$ it is best to look at the as limiting cases of $alpha$-R'enyi divergences.
First, we have the sandwiched divergences which for $alpha in (0, 1) cup (1, infty)$ are defined as $$ widetilde{D}_{alpha}(rho|sigma) = frac{1}{alpha - 1} log mathrm{Tr}left[ (sigma^{frac{1-alpha}{2alpha}} rho sigma^{frac{1-alpha}{2alpha}} )^alpha right]. $$ These divergences are monotonically increasing in $alpha$ and satisfy the data processing inequality (DPI) for all $alpha geq 1/2$. Thus the smallest divergence in this family satisfying the DPI is $$ widetilde{D}_{min}(rho | sigma) = widetilde{D}_{1/2}(rho |sigma) = - log mathrm{Tr}[sqrt{rho} sqrt{sigma}]^2. $$
Another well studied family of divergences are the so-called Petz divergences defined for $alpha in (0,1) cup (1, infty)$ to be $$ overline{D}_{alpha}(rho | sigma) = frac{1}{alpha - 1} log mathrm{Tr}[rho^{alpha} sigma^{1-alpha}]. $$ This family satisfies the DPI for $alpha in (0,1) cup(1,2]$ and they are also monotonically increasing in $alpha$. Thus, the smallest divergence satisfying the DPI in this family is $$ overline{D}_{min}(rho | sigma) = lim_{alpha to 0^+} overline{D}_{alpha}(rho |sigma) = -log mathrm{Tr}[Pi_rho sigma ]. $$
Answered by Rammus on August 20, 2021
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