Appendix 1: Limit as \(\alpha \rightarrow 1\)
Definition 6
Let \(\rho \), \(\sigma \), and \(\mathcal {N}\) be as given in Definition 1. For \(\alpha \in \left( 0,1\right) \cup \left( 1,\infty \right) \), let
$$\begin{aligned} \Delta _{\alpha }(\rho ,\sigma ,\mathcal {N})=\frac{1}{\alpha -1}\log Q_{\alpha }(\rho ,\sigma ,\mathcal {N}), \end{aligned}$$
(208)
where
$$\begin{aligned} Q_{\alpha }(\rho ,\sigma ,\mathcal {N})\equiv \left\| \left( \mathcal {N} (\rho )^{\left( 1-\alpha \right) /2}\mathcal {N}(\sigma )^{\left( \alpha -1\right) /2}\otimes I_{E}\right) U\sigma ^{\left( 1-\alpha \right) /2}\rho ^{\alpha /2}\right\| _{2}^{2}. \end{aligned}$$
(209)
Theorem 5
Let \(\rho \), \(\sigma \), and \(\mathcal {N}\) be as given in Definition 1 and such that \({\text {supp}} (\rho )\subseteq {\text {supp}}(\sigma )\). The following limit holds
$$\begin{aligned} \lim _{\alpha \rightarrow 1}\Delta _{\alpha }(\rho ,\sigma ,\mathcal {N})=D(\rho \Vert \sigma )-D\left( \mathcal {N}(\rho )\Vert \mathcal {N}(\sigma )\right) . \end{aligned}$$
(210)
Proof
Let \(\varPi _{\omega }\) denote the projection onto the support of \(\omega \). From the condition \({\text {supp}}(\rho )\subseteq {\text {supp}}(\sigma )\), it follows that \({\text {supp}}\left( \mathcal {N}(\rho )\right) \subseteq {\text {supp}}\left( \mathcal {N}(\sigma )\right) \) [32, Appendix B.4]. We can then conclude that
$$\begin{aligned} \varPi _{\sigma }\varPi _{\rho }=\varPi _{\rho },\qquad \varPi _{\mathcal {N}(\rho )} \varPi _{\mathcal {N}(\sigma )}=\varPi _{\mathcal {N}(\rho )}. \end{aligned}$$
(211)
We also know that \({\text {supp}}\left( U\rho U^{\dag }\right) \subseteq {\text {supp}}\left( \mathcal {N}(\rho )\otimes I_{E}\right) \) [32, Appendix B.4], so that
$$\begin{aligned} \left( \varPi _{\mathcal {N}(\rho )}\otimes I_{E}\right) \varPi _{U\rho U^{\dag }} =\varPi _{U\rho U^{\dag }}. \end{aligned}$$
(212)
When \(\alpha =1\), we find from the above facts that
$$\begin{aligned} Q_{1}(\rho ,\sigma ,\mathcal {N})&=\left\| \left( \varPi _{\mathcal {N}(\rho )}\varPi _{\mathcal {N}(\sigma )}\otimes I_{E}\right) U\varPi _{\sigma }\rho ^{1/2}\right\| _{2}^{2} \end{aligned}$$
(213)
$$\begin{aligned}&=\left\| \left( \varPi _{\mathcal {N}(\rho )}\otimes I_{E}\right) U\varPi _{\rho }\rho ^{1/2}\right\| _{2}^{2}\end{aligned}$$
(214)
$$\begin{aligned}&=\left\| \left( \varPi _{\mathcal {N}(\rho )}\otimes I_{E}\right) \varPi _{U\rho U^{\dag }}U\rho ^{1/2}\right\| _{2}^{2}\end{aligned}$$
(215)
$$\begin{aligned}&=\left\| \varPi _{U\rho U^{\dag }}U\rho ^{1/2}\right\| _{2}^{2}\end{aligned}$$
(216)
$$\begin{aligned}&=\left\| \rho ^{1/2}\right\| _{2}^{2}\end{aligned}$$
(217)
$$\begin{aligned}&=1. \end{aligned}$$
(218)
So from the definition of the derivative, this means that
$$\begin{aligned} \lim _{\alpha \rightarrow 1}\Delta _{\alpha }(\rho ,\sigma ,\mathcal {N})&=\lim _{\alpha \rightarrow 1}\frac{\log Q_{\alpha }(\rho ,\sigma ,\mathcal {N})-\log Q_{1}(\rho ,\sigma ,\mathcal {N})}{\alpha -1}\end{aligned}$$
(219)
$$\begin{aligned}&=\left. \frac{\hbox {d}}{\hbox {d}\alpha }\left[ \log Q_{\alpha }(\rho ,\sigma ,\mathcal {N})\right] \right| _{\alpha =1}\end{aligned}$$
(220)
$$\begin{aligned}&=\frac{1}{Q_{1}(\rho ,\sigma ,\mathcal {N})}\left. \frac{\hbox {d}}{\hbox {d}\alpha }\left[ Q_{\alpha }(\rho ,\sigma ,\mathcal {N})\right] \right| _{\alpha =1}\end{aligned}$$
(221)
$$\begin{aligned}&=\left. \frac{\hbox {d}}{\hbox {d}\alpha }\left[ Q_{\alpha }(\rho ,\sigma ,\mathcal {N} )\right] \right| _{\alpha =1}. \end{aligned}$$
(222)
Let \(\alpha ^{\prime }\equiv \alpha -1\). Consider that
$$\begin{aligned} Q_{\alpha }(\rho ,\sigma ,\mathcal {N})={\text {Tr}}\left\{ \rho ^{\alpha }\sigma ^{-\alpha ^{\prime }/2}\mathcal {N}^{\dag }\left( \mathcal {N} (\sigma )^{\alpha ^{\prime }/2}\mathcal {N}(\rho )^{-\alpha ^{\prime }} \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime } /2}\right\} . \end{aligned}$$
(223)
Now we calculate \(\frac{\hbox {d}}{\hbox {d}\alpha }Q_{\alpha }(\rho ,\sigma ,\mathcal {N})\):
$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}\alpha }{\text {Tr}}\left\{ \rho ^{\alpha }\sigma ^{-\alpha ^{\prime }/2}\mathcal {N}^{\dag }\left( \mathcal {N}(\sigma )^{\alpha ^{\prime } /2}\mathcal {N}(\rho )^{-\alpha ^{\prime }}\mathcal {N}(\sigma )^{\alpha ^{\prime } /2}\right) \sigma ^{-\alpha ^{\prime }/2}\right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \left[ \frac{\hbox {d}}{\hbox {d}\alpha }\rho ^{\alpha }\right] \sigma ^{-\alpha ^{\prime }/2}\mathcal {N}^{\dag }\left( \mathcal {N} (\sigma )^{\alpha ^{\prime }/2}\mathcal {N}(\rho )^{-\alpha ^{\prime }} \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime } /2}\right\} \nonumber \\&\quad \quad +\,{\text {Tr}}\left\{ \rho ^{\alpha }\left[ \frac{\hbox {d}}{\hbox {d}\alpha } \sigma ^{-\alpha ^{\prime }/2}\right] \mathcal {N}^{\dag }\left( \mathcal {N} (\sigma )^{\alpha ^{\prime }/2}\mathcal {N}(\rho )^{-\alpha ^{\prime }} \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime } /2}\right\} \nonumber \\&\quad \quad +\,{\text {Tr}}\left\{ \rho ^{\alpha }\sigma ^{-\alpha ^{\prime }/2} \mathcal {N}^{\dag }\left( \left[ \frac{\hbox {d}}{\hbox {d}\alpha }\mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right] \mathcal {N}(\rho )^{-\alpha ^{\prime }} \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime } /2}\right\} \nonumber \\&\quad \quad +\,{\text {Tr}}\left\{ \rho ^{\alpha }\sigma ^{-\alpha ^{\prime }/2} \mathcal {N}^{\dag }\left( \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\left[ \frac{\hbox {d}}{\hbox {d}\alpha }\mathcal {N}(\rho )^{-\alpha ^{\prime }}\right] \mathcal {N} (\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime }/2}\right\} \nonumber \\&\quad \quad +\,{\text {Tr}}\left\{ \rho ^{\alpha }\sigma ^{-\alpha ^{\prime }/2} \mathcal {N}^{\dag }\left( \mathcal {N}(\sigma )^{\alpha ^{\prime }/2} \mathcal {N}(\rho )^{-\alpha ^{\prime }}\left[ \frac{\hbox {d}}{\hbox {d}\alpha }\mathcal {N} (\sigma )^{\alpha ^{\prime }/2}\right] \right) \sigma ^{-\alpha ^{\prime } /2}\right\} \nonumber \\&\quad \quad +\,{\text {Tr}}\left\{ \rho ^{\alpha }\sigma ^{-\alpha ^{\prime }/2} \mathcal {N}^{\dag }\left( \mathcal {N}(\sigma )^{\alpha ^{\prime }/2} \mathcal {N}(\rho )^{-\alpha ^{\prime }}\mathcal {N}(\sigma )^{\alpha ^{\prime } /2}\right) \left[ \frac{\hbox {d}}{\hbox {d}\alpha }\sigma ^{-\alpha ^{\prime }/2}\right] \right\} \end{aligned}$$
(224)
$$\begin{aligned}&\quad =\Bigg [{\text {Tr}}\left\{ \rho ^{\alpha }\left[ \log \rho \right] \sigma ^{-\alpha ^{\prime }/2}\mathcal {N}^{\dag }\left( \mathcal {N} (\sigma )^{\alpha ^{\prime }/2}\mathcal {N}(\rho )^{-\alpha ^{\prime }} \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime } /2}\right\} \nonumber \\&\quad \quad -\,\frac{1}{2}{\text {Tr}}\left\{ \rho \left[ \log \sigma \right] \sigma ^{-\alpha ^{\prime }/2}\mathcal {N}^{\dag }\left( \mathcal {N} (\sigma )^{\alpha ^{\prime }/2}\mathcal {N}(\rho )^{-\alpha ^{\prime }} \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime } /2}\right\} \nonumber \\&\quad \quad +\,\frac{1}{2}{\text {Tr}}\left\{ \rho \sigma ^{-\alpha ^{\prime } /2}\mathcal {N}^{\dag }\left( \left[ \log \mathcal {N}(\sigma )\right] \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\mathcal {N}(\rho )^{-\alpha ^{\prime } }\mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime } /2}\right\} \nonumber \\&\quad \quad -\,{\text {Tr}}\left\{ \rho \sigma ^{-\alpha ^{\prime }/2}\mathcal {N}^{\dag }\left( \mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\left[ \log \mathcal {N} (\rho )\right] \mathcal {N}(\rho )^{-\alpha ^{\prime }}\mathcal {N}(\sigma )^{\alpha ^{\prime }/2}\right) \sigma ^{-\alpha ^{\prime }/2}\right\} \nonumber \\&\quad \quad +\,\frac{1}{2}{\text {Tr}}\left\{ \rho \sigma ^{-\alpha ^{\prime } /2}\mathcal {N}^{\dag }\left( \mathcal {N}(\sigma )^{\alpha ^{\prime } /2}\mathcal {N}(\rho )^{-\alpha ^{\prime }}\mathcal {N}(\sigma )^{\alpha ^{\prime } /2}\left[ \log \mathcal {N}(\sigma )\right] \right) \sigma ^{-\alpha ^{\prime }/2}\right\} \nonumber \\&\quad \quad -\,\frac{1}{2}{\text {Tr}}\left\{ \rho \sigma ^{-\alpha ^{\prime } /2}\mathcal {N}^{\dag }\left( \mathcal {N}(\sigma )^{\alpha ^{\prime } /2}\mathcal {N}(\rho )^{-\alpha ^{\prime }}\mathcal {N}(\sigma )^{\alpha ^{\prime } /2}\right) \sigma ^{-\alpha ^{\prime }/2}\left[ \log \sigma \right] \right\} \Bigg ].\nonumber \\ \end{aligned}$$
(225)
Taking the limit as \(\alpha \rightarrow 1\) gives
$$\begin{aligned} \left. \frac{\hbox {d}}{\hbox {d}\alpha }Q_{\alpha }(\rho ,\sigma ,\mathcal {N})\right| _{\alpha =1}= & {} {\text {Tr}}\left\{ \rho \left[ \log \rho \right] \varPi _{\sigma }\mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\varPi _{\mathcal {N} (\rho )}\varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} \nonumber \\&-\,\frac{1}{2}{\text {Tr}}\left\{ \rho \left[ \log \sigma \right] \varPi _{\sigma }\mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\varPi _{\mathcal {N}(\rho )}\varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} \nonumber \\&+\,\frac{1}{2}{\text {Tr}}\left\{ \rho \varPi _{\sigma }\mathcal {N}^{\dag }\left( \left[ \log \mathcal {N}(\sigma )\right] \varPi _{\mathcal {N}(\sigma )} \varPi _{\mathcal {N}(\rho )}\varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} \nonumber \\&-\,{\text {Tr}}\left\{ \rho \varPi _{\sigma }\mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\left[ \log \mathcal {N}(\rho )\right] \varPi _{\mathcal {N}(\rho )}\varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} \nonumber \\&+\,\frac{1}{2}{\text {Tr}}\left\{ \rho \varPi _{\sigma }\mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\varPi _{\mathcal {N}(\rho )}\varPi _{\mathcal {N} (\sigma )}\left[ \log \mathcal {N}(\sigma )\right] \right) \varPi _{\sigma }\right\} \nonumber \\&-\,\frac{1}{2}{\text {Tr}}\left\{ \rho \varPi _{\sigma }\mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\varPi _{\mathcal {N}(\rho )}\varPi _{\mathcal {N} (\sigma )}\right) \left[ \log \sigma \right] \varPi _{\sigma }\right\} .\nonumber \\ \end{aligned}$$
(226)
We now simplify the first four terms and note that the last two are Hermitian conjugates of the second and third:
$$\begin{aligned}&{\text {Tr}}\left\{ \rho \left[ \log \rho \right] \varPi _{\sigma } \mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\varPi _{\mathcal {N}(\rho )} \varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} ={\text {Tr}} \left\{ \rho \left[ \log \rho \right] \mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\rho )}\right) \right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \mathcal {N}\left( \rho \left[ \log \rho \right] \right) \varPi _{\mathcal {N}(\rho )}\right\} ={\text {Tr}}\left\{ U\rho \left[ \log \rho \right] U^{\dag }\left( \varPi _{\mathcal {N}(\rho )}\otimes I_{E}\right) \right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \varPi _{U\rho U^{\dag }}U\rho \left[ \log \rho \right] U^{\dag }\left( \varPi _{\mathcal {N}(\rho )}\otimes I_{E}\right) \right\} ={\text {Tr}}\left\{ \varPi _{U\rho U^{\dag }}U\rho \left[ \log \rho \right] U^{\dag }\right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \rho \left[ \log \rho \right] \right\} , \end{aligned}$$
(227)
$$\begin{aligned}&{\text {Tr}}\left\{ \rho \left[ \log \sigma \right] \varPi _{\sigma }\mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\varPi _{\mathcal {N}(\rho )} \varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} ={\text {Tr}} \left\{ \rho \left[ \log \sigma \right] \mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\rho )}\right) \right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \mathcal {N}\left( \rho \left[ \log \sigma \right] \right) \left( \varPi _{\mathcal {N}(\rho )}\right) \right\} ={\text {Tr}} \left\{ U\rho \left[ \log \sigma \right] U^{\dag }\left( \varPi _{\mathcal {N} (\rho )}\otimes I_{E}\right) \right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \varPi _{U\rho U^{\dag }}U\rho U^{\dag }U\left[ \log \sigma \right] U^{\dag }\left( \varPi _{\mathcal {N}(\rho )}\otimes I_{E}\right) \right\} ={\text {Tr}}\left\{ U\rho U^{\dag }U\left[ \log \sigma \right] U^{\dag }\right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \rho \left[ \log \sigma \right] \right\} , \end{aligned}$$
(228)
$$\begin{aligned}&{Tr}\left\{ \rho \varPi _{\sigma }\mathcal {N}^{\dag }\left( \left[ \log \mathcal {N}(\sigma )\right] \varPi _{\mathcal {N}(\sigma )}\varPi _{\mathcal {N} (\rho )}\varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} \nonumber \\&\qquad ={\text {Tr}}\left\{ \rho \mathcal {N}^{\dag }\left( \left[ \log \mathcal {N}(\sigma )\right] \varPi _{\mathcal {N}(\rho )}\right) \right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \mathcal {N}(\rho )\left[ \log \mathcal {N} (\sigma )\right] \varPi _{\mathcal {N}(\rho )}\right\} ={\text {Tr}}\left\{ \mathcal {N}(\rho )\left[ \log \mathcal {N}(\sigma )\right] \right\} , \end{aligned}$$
(229)
$$\begin{aligned}&{Tr}\left\{ \rho \varPi _{\sigma }\mathcal {N}^{\dag }\left( \varPi _{\mathcal {N}(\sigma )}\left[ \log \mathcal {N}(\rho )\right] \varPi _{\mathcal {N}(\rho )}\varPi _{\mathcal {N}(\sigma )}\right) \varPi _{\sigma }\right\} \nonumber \\&\qquad ={\text {Tr}}\left\{ \rho \mathcal {N}^{\dag }\left( \left[ \log \mathcal {N}(\rho )\right] \varPi _{\mathcal {N}(\rho )}\right) \right\} \nonumber \\&\quad ={\text {Tr}}\left\{ \mathcal {N}(\rho )\left( \left[ \log \mathcal {N}(\rho )\right] \varPi _{\mathcal {N}(\rho )}\right) \right\} ={\text {Tr}}\left\{ \mathcal {N}(\rho )\left[ \log \mathcal {N} (\rho )\right] \right\} . \end{aligned}$$
(230)
This then implies that the following equality holds
$$\begin{aligned}&\left. \frac{\hbox {d}}{\hbox {d}\alpha }Q_{\alpha }(\rho ,\sigma ,\mathcal {N})\right| _{\alpha =1}={\text {Tr}}\left\{ \rho \left[ \log \rho \right] \right\} -{\text {Tr}}\left\{ \rho \left[ \log \sigma \right] \right\} \nonumber \\&\quad +{\text {Tr}}\left\{ \mathcal {N}(\rho )\left[ \log \mathcal {N} (\sigma )\right] \right\} -{\text {Tr}}\left\{ \mathcal {N}(\rho )\left[ \log \mathcal {N}(\rho )\right] \right\} . \end{aligned}$$
(231)
Putting together (222) and (231), we can then conclude the statement of the theorem.
Appendix 2: Auxiliary lemmas and proofs
Lemma 1
Let \(\mathcal {A}\) and \(\mathcal {T}\) be compact metric spaces, and let \(f:\mathcal {A}\times \mathcal {T}\rightarrow \mathbb {R}\) be a continuous function. Then, \(g,h:\mathcal {A}\rightarrow \mathbb {R}\), defined as \(g(\alpha )=\max _{t\in \mathcal {T}}f(\alpha ,t)\) and \(h(\alpha )=\min _{t\in \mathcal {T}}f(\alpha ,t)\) are continuous.
Proof
By the Heine–Cantor theorem, f is uniformly continuous. Hence, for every \(\varepsilon >0\), there exists a \(\delta >0\) such that \(\left| f(\alpha ,t)-f(\alpha ^{\prime },t^{\prime })\right| <\varepsilon \) whenever \(D_{\mathcal {A}}(\alpha ,\alpha ^{\prime })<\delta \) and \(D_{\mathcal {T} }(t,t^{\prime })<\delta \), where \(D_{\mathcal {A}}\) and \(D_{\mathcal {T}}\) are the distance functions on \(\mathcal {A}\) and \(\mathcal {T}\) respectively. Now, given \(\alpha \in \mathcal {A}\), let t be such that \(g(\alpha )=f(\alpha ,t)\). Then, for any \(\alpha ^{\prime }\in \mathcal {A}\) with \(D_{\mathcal {A}}(\alpha ,\alpha ^{\prime })<\delta \) we have that
$$\begin{aligned} g(\alpha )=f(\alpha ,t)<f(\alpha ^{\prime },t)+\varepsilon \leqslant \max _{t^{\prime }\in \mathcal {T}}f(\alpha ^{\prime },t^{\prime })+\varepsilon =g(\alpha ^{\prime })+\varepsilon . \end{aligned}$$
By symmetry, we then have that \(\left| g(\alpha )-g(\alpha ^{\prime })\right| <\varepsilon \), which proves the continuity of g. A similar argument establishes the continuity of h. \(\square \)
Proof of Theorem 4
Let \(\rho \), \(\sigma \), and \(\mathcal {N}\) be as given in Definition 4. Let
$$\begin{aligned} G\left( z\right) =\left( \mathcal {N}(\rho )^{-z/2}\mathcal {N}(\sigma )^{z/2}\otimes I_{E}\right) U\sigma ^{-z/2}\rho ^{\left( 1+z\right) /2}. \end{aligned}$$
(232)
In the equation
$$\begin{aligned} \frac{1}{p_{\theta }}=\frac{\theta }{p_{0}}+\frac{1-\theta }{p_{1}}, \end{aligned}$$
(233)
choose \(p_{0}=2\) and \(p_{1}=2\), so that \(p_{\theta }=2\). Recalling that
$$\begin{aligned} M_{k}=\sup _{t\in \mathbb {R}}\left\| G\left( k+it\right) \right\| _{p_{k}}, \end{aligned}$$
(234)
for \(k=0,1\), we find that
$$\begin{aligned} \left\| G\left( \theta \right) \right\| _{p_{\theta }}\le M_{0}^{1-\theta }M_{1}^{\theta }. \end{aligned}$$
(235)
For our choices, we find that
$$\begin{aligned} M_{0}&=\sup _{t\in \mathbb {R}}\left\| G\left( it\right) \right\| _{2} \end{aligned}$$
(236)
$$\begin{aligned}&=\sup _{t\in \mathbb {R}}\left\| \left( \mathcal {N}(\rho )^{-it/2} \mathcal {N}(\sigma )^{it/2}\otimes I_{E}\right) U\sigma ^{-it/2}\rho ^{\left( 1+it\right) /2}\right\| _{2}\end{aligned}$$
(237)
$$\begin{aligned}&=\left\| \rho ^{1/2}\right\| _{2}=1, \end{aligned}$$
(238)
$$\begin{aligned} M_{1}&=\sup _{t\in \mathbb {R}}\left\| G\left( 1+it\right) \right\| _{2}\end{aligned}$$
(239)
$$\begin{aligned}&=\sup _{t\in \mathbb {R}}\left\| \left( \mathcal {N}(\rho )^{-\left( 1+it\right) /2}\mathcal {N}(\sigma )^{\left( 1+it\right) /2}\otimes I_{E}\right) U\sigma ^{-\left( 1+it\right) /2}\rho ^{\left( 1+\left( 1+it\right) \right) /2}\right\| _{2}\end{aligned}$$
(240)
$$\begin{aligned}&=\sup _{t\in \mathbb {R}}\left\| \left( \mathcal {N}(\rho )^{-1/2} \mathcal {N}(\sigma )^{it/2}\mathcal {N}(\sigma )^{1/2}\otimes I_{E}\right) U\sigma ^{-1/2}\sigma ^{-it/2}\rho \right\| _{2}\end{aligned}$$
(241)
$$\begin{aligned}&=\left[ \exp \sup _{t\in \mathbb {R}}D_{2}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P}_{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N}(\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) \right] ^{1/2}. \end{aligned}$$
(242)
Applying the three-line theorem gives
$$\begin{aligned}&\left\| \left( \mathcal {N}(\rho )^{-\theta /2}\mathcal {N}(\sigma )^{\theta /2}\otimes I_{E}\right) U\sigma ^{-\theta /2}\rho ^{\left( 1+\theta \right) /2}\right\| _{2}\nonumber \\&\quad \le \left[ \exp \sup _{t\in \mathbb {R}}D_{2}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P}_{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N}(\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) \right] ^{\theta /2}, \end{aligned}$$
(243)
and after a logarithm gives
$$\begin{aligned}&\frac{2}{\theta }\log \left\| \left( \mathcal {N}(\rho )^{-\theta /2}\mathcal {N}(\sigma )^{\theta /2}\otimes I_{E}\right) U\sigma ^{-\theta /2} \rho ^{\left( 1+\theta \right) /2}\right\| _{2}\nonumber \\&\quad \le \sup _{t\in \mathbb {R} }D_{2}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P} _{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N}(\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) . \end{aligned}$$
(244)
Take the limit as \(\theta \searrow 0\) to get
$$\begin{aligned} D(\rho \Vert \sigma )-D\left( \mathcal {N(}\rho )\Vert \mathcal {N(}\sigma )\right) \le \sup _{t\in \mathbb {R}}D_{2}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P}_{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N} (\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) . \end{aligned}$$
(245)
Now we prove the other inequality. Let \(\rho \), \(\sigma \), and \(\mathcal {N}\) be as given in Definition 1 and such that \({\text {supp}} (\rho )\subseteq {\text {supp}}(\sigma )\). Take
$$\begin{aligned} G\left( z\right) =\left( \mathcal {N}(\rho )^{z/2}\mathcal {N}(\sigma )^{-z/2}\otimes I_{E}\right) U\sigma ^{z/2}\rho ^{\left( 1-z\right) /2}. \end{aligned}$$
(246)
Then \(M_{0}=1\) again and
$$\begin{aligned} M_{1}&=\sup _{t\in \mathbb {R}}\left\| G\left( 1+it\right) \right\| _{2} \end{aligned}$$
(247)
$$\begin{aligned}&=\sup _{t\in \mathbb {R}}\left\| \left( \mathcal {N}(\rho )^{\left( 1+it\right) /2}\mathcal {N}(\sigma )^{-\left( 1+it\right) /2}\otimes I_{E}\right) U\sigma ^{\left( 1+it\right) /2}\rho ^{\left( 1-\left( 1+it\right) \right) /2}\right\| _{2}\end{aligned}$$
(248)
$$\begin{aligned}&=\sup _{t\in \mathbb {R}}\left\| \left( \mathcal {N}(\rho )^{1/2} \mathcal {N}(\sigma )^{-it/2}\mathcal {N}(\sigma )^{-1/2}\otimes I_{E}\right) U\sigma ^{1/2}\sigma ^{it/2}\rho ^{0}\right\| _{2}\end{aligned}$$
(249)
$$\begin{aligned}&=\exp \left\{ -\inf _{t\in \mathbb {R}}D_{0}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P}_{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N}(\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) \right\} ^{1/2}. \end{aligned}$$
(250)
Applying the three-line theorem gives
$$\begin{aligned}&\left\| \left( \mathcal {N}(\rho )^{\theta /2}\mathcal {N}(\sigma )^{-\theta /2}\otimes I_{E}\right) U\sigma ^{\theta /2}\rho ^{\left( 1-\theta \right) /2}\right\| _{2}\nonumber \\&\quad \le \left[ \exp \left\{ -\inf _{t\in \mathbb {R}}D_{0}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P}_{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N}(\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) \right\} \right] ^{\theta /2}, \end{aligned}$$
(251)
which after taking a logarithm gives
$$\begin{aligned}&\frac{2}{-\theta }\log \left\| \left( \mathcal {N}(\rho )^{\theta /2}\mathcal {N}(\sigma )^{-\theta /2}\otimes I_{E}\right) U\sigma ^{\theta /2} \rho ^{\left( 1-\theta \right) /2}\right\| _{2}\nonumber \\&\quad \ge \inf _{t\in \mathbb {R} }D_{0}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P} _{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N}(\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) . \end{aligned}$$
(252)
Take the limit as \(\theta \searrow 0\) to get
$$\begin{aligned} D(\rho \Vert \sigma )-D\left( \mathcal {N(}\rho )\Vert \mathcal {N(}\sigma )\right) \ge \inf _{t\in \mathbb {R}}D_{0}\left( \rho \Vert \left( \mathcal {U}_{\sigma ,-t}\circ \mathcal {P}_{\sigma ,\mathcal {N}}\circ \mathcal {U}_{\mathcal {N} (\sigma ),t}\right) \left( \mathcal {N}(\rho )\right) \right) . \end{aligned}$$
(253)
\(\square \)
Appendix 3: Taylor expansions
Here we show the following limit:
$$\begin{aligned} \lim _{\alpha \rightarrow 1}f\left( \alpha ,V_{\mathcal {N}(\sigma )},V_{\sigma }\right) =f\left( 1,V_{\mathcal {N}(\sigma )},V_{\sigma }\right) , \end{aligned}$$
(254)
where \(f\left( \alpha ,V_{\mathcal {N}(\sigma )},V_{\sigma }\right) \) is defined as
$$\begin{aligned}&f\left( \alpha ,V_{\mathcal {N}(\sigma )},V_{\sigma }\right) =\frac{1}{\alpha -1}\nonumber \\&\quad \log \left\| \left( \left[ \mathcal {N}\left( \rho \right) \right] ^{\left( 1-\alpha \right) /2}V_{\mathcal {N}(\sigma )}\left[ \mathcal {N} (\sigma )\right] ^{\left( \alpha -1\right) /2}\otimes I_{E}\right) U\sigma ^{\left( 1-\alpha \right) /2}V_{\sigma }\rho ^{\alpha /2}\right\| _{2}^{2}\qquad \end{aligned}$$
(255)
and \(f\left( 1,V_{\mathcal {N}(\sigma )},V_{\sigma }\right) \) in (53). From the fact that
$$\begin{aligned} \left. \log \left\| \left( \left[ \mathcal {N}(\rho )\right] ^{\left( 1-\alpha \right) /2}V_{\mathcal {N}(\sigma )}\left[ \mathcal {N}(\sigma )\right] ^{\left( \alpha -1\right) /2}\otimes I_{E}\right) U\sigma ^{\left( 1-\alpha \right) /2}V_{\sigma }\rho ^{\alpha /2}\right\| _{2}^{2}\right| _{\alpha =1}=0, \end{aligned}$$
(256)
we know (from the definition of derivative) that \(\lim _{\alpha \rightarrow 1}f\left( \alpha ,V_{\mathcal {N}(\sigma )},V_{\sigma }\right) \) is equal to
$$\begin{aligned}&\left. \frac{\hbox {d}}{\hbox {d}\alpha }\log \left\| \left( \left[ \mathcal {N}\left( \rho \right) \right] ^{\left( 1-\alpha \right) /2}V_{\mathcal {N}(\sigma )}\left[ \mathcal {N}(\sigma )\right] ^{\left( \alpha -1\right) /2}\otimes I_{E}\right) U\sigma ^{\left( 1-\alpha \right) /2}V_{\sigma }\rho ^{\alpha /2}\right\| _{2}^{2}\right| _{\alpha =1}\nonumber \\&\quad \quad =\left. \frac{\hbox {d}}{\hbox {d}\alpha }\left\| \left( \left[ \mathcal {N}\left( \rho \right) \right] ^{\left( 1-\alpha \right) /2}V_{\mathcal {N}(\sigma )}\left[ \mathcal {N}(\sigma )\right] ^{\left( \alpha -1\right) /2}\otimes I_{E}\right) U\sigma ^{\left( 1-\alpha \right) /2}V_{\sigma }\rho ^{\alpha /2}\right\| _{2}^{2}\right| _{\alpha =1}.\nonumber \\ \end{aligned}$$
(257)
We evaluate the latter derivative by employing Taylor expansions. Substitute \(\alpha =1+\gamma \), so that the quantity inside the derivative operation is equal to
$$\begin{aligned} \left\| \left( \left[ \mathcal {N}(\rho )\right] ^{-\gamma /2} V_{\mathcal {N}(\sigma )}\left[ \mathcal {N}(\sigma )\right] ^{\gamma /2}\otimes I_{E}\right) U\sigma ^{-\gamma /2}V_{\sigma }\rho ^{\left( 1+\gamma \right) /2}\right\| _{2}^{2}, \end{aligned}$$
(258)
which we can rewrite as
$$\begin{aligned} \left\| \left( \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N} (\rho )V_{\mathcal {N}(\sigma )}\right] ^{-\gamma /2}\left[ \mathcal {N} (\sigma )\right] ^{\gamma /2}\otimes I_{E}\right) U\sigma ^{-\gamma /2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{\left( 1+\gamma \right) /2}\right\| _{2}^{2}, \end{aligned}$$
(259)
due to the unitary invariance of the norm. Now we use that
$$\begin{aligned} \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{\left( 1+\gamma \right) /2}&=\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}+\frac{\gamma }{2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\log \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] +O\left( \gamma ^{2}\right) , \end{aligned}$$
(260)
$$\begin{aligned} \sigma ^{-\gamma /2}&=I-\frac{\gamma }{2}\log \sigma +O\left( \gamma ^{2}\right) ,\end{aligned}$$
(261)
$$\begin{aligned} \left[ \mathcal {N}(\sigma )\right] ^{\gamma /2}&=I+\frac{\gamma }{2} \log \left[ \mathcal {N}(\sigma )\right] +O\left( \gamma ^{2}\right) ,\end{aligned}$$
(262)
$$\begin{aligned} \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}\left( \rho \right) V_{\mathcal {N}(\sigma )}\right] ^{-\gamma /2}&=I-\frac{\gamma }{2} \log \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}(\rho )V_{\mathcal {N} (\sigma )}\right] +O\left( \gamma ^{2}\right) . \end{aligned}$$
(263)
The above implies that
$$\begin{aligned}&\left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}\left( \rho \right) V_{\mathcal {N}(\sigma )}\right] ^{-\gamma /2}\left[ \mathcal {N}(\sigma )\right] ^{\gamma /2}U\sigma ^{-\gamma /2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{\left( 1+\gamma \right) /2}\nonumber \\&\quad \quad =\left( I-\frac{\gamma }{2}\log \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}(\rho )V_{\mathcal {N}(\sigma )}\right] \right) \left( I+\frac{\gamma }{2}\log \left[ \mathcal {N}(\sigma )\right] \right) \nonumber \\&\quad \quad \times U\left( I-\frac{\gamma }{2}\log \sigma \right) \left( \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}+\frac{\gamma }{2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\log \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] \right) +O\left( \gamma ^{2}\right) .\nonumber \\ \end{aligned}$$
(264)
By working out the right-hand side above and neglecting terms of second order in \(\gamma \) and higher, we find that
$$\begin{aligned}&\left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}\left( \rho \right) V_{\mathcal {N}(\sigma )}\right] ^{-\gamma /2}\left[ \mathcal {N}(\sigma )\right] ^{\gamma /2}U\sigma ^{-\gamma /2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{\left( 1+\gamma \right) /2} \nonumber \\&\quad =U\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}-\frac{\gamma }{2} \log \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}\left( \rho \right) V_{\mathcal {N}(\sigma )}\right] U\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\nonumber \\&\quad \quad +\frac{\gamma }{2}\log \left[ \mathcal {N}(\sigma )\right] U\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\nonumber \\&\quad \quad -\frac{\gamma }{2}U\left[ \log \sigma \right] \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}+\frac{\gamma }{2}U\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\log \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] +O\left( \gamma ^{2}\right) .\nonumber \\ \end{aligned}$$
(265)
The Hermitian conjugate is
$$\begin{aligned}&\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}U^{\dag }-\frac{\gamma }{2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}U^{\dag }\log \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}(\rho )V_{\mathcal {N}(\sigma )}\right] \nonumber \\&\quad +\frac{\gamma }{2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}U^{\dag }\log \left[ \mathcal {N}(\sigma )\right] \nonumber \\&\quad -\frac{\gamma }{2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\left[ \log \sigma \right] U^{\dag }+\frac{\gamma }{2}\left[ \log \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] \right] \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}U^{\dag }+O\left( \gamma ^{2}\right) .\nonumber \\ \end{aligned}$$
(266)
Combining (265) with its Hermitian conjugate and neglecting higher order terms gives
$$\begin{aligned}&\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] -\gamma \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\mathcal {N}^{\dag }\left( \log \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}(\rho )V_{\mathcal {N}(\sigma )}\right] \right) \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\nonumber \\&\quad +\gamma \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\mathcal {N} ^{\dag }\left( \log \left[ \mathcal {N}(\sigma )\right] \right) \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}-\gamma \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\left[ \log \sigma \right] \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{1/2}\nonumber \\&\quad +\frac{\gamma }{2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] \log \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] +\frac{\gamma }{2}\left( \log \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] \right) \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] +O\left( \gamma ^{2}\right) . \end{aligned}$$
(267)
Taking a trace gives
$$\begin{aligned}&\text {Tr}\left\{ \rho \right\} -\gamma \text {Tr}\left\{ \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] \mathcal {N}^{\dag }\left( \log \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}(\rho )V_{\mathcal {N}(\sigma )}\right] \right) \right\} \nonumber \\&\quad +\gamma \text {Tr}\left\{ \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] \mathcal {N}^{\dag }\left( \log \left[ \mathcal {N}(\sigma )\right] \right) \right\} -\gamma \text {Tr}\left\{ \rho \left[ \log \sigma \right] \right\} \nonumber \\&\quad +\gamma \text {Tr}\left\{ \rho \log \rho \right\} +O\left( \gamma ^{2}\right) . \end{aligned}$$
(268)
We can now finally use the above development to conclude that
$$\begin{aligned}&\left. \frac{\hbox {d}}{\hbox {d}\alpha }\left\| \left( \left[ \mathcal {N}\left( \rho \right) \right] ^{\left( 1-\alpha \right) /2}V_{\mathcal {N}(\sigma )}\left[ \mathcal {N}(\sigma )\right] ^{\left( \alpha -1\right) /2}\otimes I_{E}\right) U\sigma ^{\left( 1-\alpha \right) /2}V_{\sigma }\rho ^{\alpha /2}\right\| _{2}^{2}\right| _{\alpha =1}\nonumber \\&\quad =\left. \frac{\hbox {d}}{\hbox {d}\gamma }\left\| \left( \left[ V_{\mathcal {N} (\sigma )}^{\dag }\mathcal {N}(\rho )V_{\mathcal {N}(\sigma )}\right] ^{-\gamma /2}\left[ \mathcal {N}(\sigma )\right] ^{\gamma /2}\otimes I_{E}\right) U\sigma ^{-\gamma /2}\left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] ^{\left( 1+\gamma \right) /2}\right\| _{2}^{2}\right| _{\gamma =0} \end{aligned}$$
(269)
$$\begin{aligned}&\quad ={\text {Tr}}\left\{ \rho \left[ \log \rho -\log \sigma \right] \right\} \nonumber \\&\quad -{\text {Tr}}\left\{ \mathcal {N}\left( \left[ V_{\sigma }\rho V_{\sigma }^{\dag }\right] \right) \left[ \log \left[ V_{\mathcal {N}(\sigma )}^{\dag }\mathcal {N}(\rho )V_{\mathcal {N}(\sigma )}\right] -\log \left[ \mathcal {N}(\sigma )\right] \right] \right\} \end{aligned}$$
(270)
$$\begin{aligned}&\quad =f(1,V_{\mathcal {N}(\sigma )},V_{\sigma }). \end{aligned}$$
(271)
A similar development with Taylor expansions leads to the conclusion that (63) holds. However, here one should employ the method outlined in the proof of [46, Proposition11].