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3

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 $$\widetilde{D}_{\alpha}(\rho\|\sigma) = \frac{1}{\... 3 Source of the problem The purported contradiction arises due to the use of incorrect assumptions for Klein equality$$ S(\rho||\sigma) \ge 0. $$The inequality does not require any particular relationship^1 between the support of \rho and the support of \sigma. However, it does require that \rho and \sigma be states, i.e. unit trace positive ... 2 There is a problem in the derivation you presented, since \rho \leq \lambda \sigma is only equivalent to \sigma^{-1/2} \rho \sigma^{-1/2} \leq \lambda I when \sigma is invertible (or at least when \mathrm{supp}(\rho) \subset \mathrm{supp}(\sigma), so that you can restrict the space to \mathrm{supp}(\sigma) instead of the whole Hilbert space). The ... 2 Can someone provide an example of a state \rho_{AB} for which \sigma^\star_B \neq \rho_B? Why not start very easily, with a separable state such as$$ \rho_{AB}=\left(p_0|0\rangle\langle 0|\otimes \tau_0+p_1|1\rangle\langle 1|\otimes \tau_1\right) $$where \tau_0 and \tau_1 are different (normalised) single-qubit density matrices. We have that$$ I=\...

2

I think I have an answer. The following should be the CVX code for one of the formulations found in this link. cvx_begin sdp variable X(2, 2) hermitian minimize(trace(id' * X)) % id is eye(2) subject to kron(id, X) >= rho_ab % the tensor product of two density matrices a, b X >= 0 cvx_end The optimal value found in this program is $$\text{optval} = ... 2 As you say,$$ \mathrm{Tr}[\rho_{XB} \log \rho_{XB}] = -S(X) + \sum_{x} p(x) \mathrm{Tr}[\rho_{B}^x \log \rho_B^x]. $$But if you can prove the above statement, then the exact same derivation gives you$$ \mathrm{Tr}[\rho_{XB} \log \sigma_{XB}] = - S(X) + \sum_{x} p(x) \mathrm{Tr}[\rho_B^x \log \sigma_B^x]. $$If you put both together then you get$$ \begin{...

2

Both limits are dealt with in a fair amount of detail in the work that originally defined the sandwiched entropies: On quantum Renyi entropies: a new generalization and some properties. In particular, you'll find the relevant results in section IV.C.

1

An easy (but perhaps cheating) answer: The relative entropy $S(A||B)$diverges when $A$ has support over the kernel of $B$ (e.g., wikipedia). Now, the kernel of $B=|0\rangle^{\otimes n}\langle 0|^{\otimes n}$ is everything other than $|0\rangle^{\otimes n}$, i.e., $\mathbb{I}-|0\rangle^{\otimes n}\langle 0|^{\otimes n}$! For the inequality to be meaningful, ...

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