Swiveled Rényi entropies

Abstract

This paper introduces “swiveled Rényi entropies” as an alternative to the Rényi entropic quantities put forward in Berta et al. (Phys Rev A 91(2):022333, 2015). What distinguishes the swiveled Rényi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A consequence of this extra degree of freedom is that the swiveled Rényi entropies are ordered, which is an important property of the Rényi family of entropies. The swiveled Rényi entropies are, however, generally discontinuous at $\alpha =1$ and do not converge to the von Neumann entropy-based measures in the limit as $\alpha \rightarrow 1$, instead bounding them from above and below. Particular variants reduce to known Rényi entropies, such as the Rényi relative entropy or the sandwiched Rényi relative entropy, but also lead to ordered Rényi conditional mutual information and ordered Rényi generalizations of a relative entropy difference. Refinements of entropy inequalities such as monotonicity of quantum relative entropy and strong subadditivity follow as a consequence of the aforementioned properties of the swiveled Rényi entropies. Due to the lack of convergence at $\alpha =1$, it is unclear whether the swiveled Rényi entropies would be useful in one-shot information theory, so that the present contribution represents partial progress toward this goal.

Appendices

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].