Quantum Rényi Divergences and the Strong Converse Exponent of State Discrimination in Operator Algebras

Abstract

The sandwiched Rényi $\alpha$-divergences of two finite-dimensional quantum states play a distinguished role among the many quantum versions of Rényi divergences as the tight quantifiers of the trade-off between the two error probabilities in the strong converse domain of state discrimination. In this paper, we show the same for the sandwiched Rényi divergences of two normal states on an injective von Neumann algebra, thereby establishing the operational significance of these quantities. Moreover, we show that in this setting, again similarly to the finite-dimensional case, the sandwiched Rényi divergences coincide with the regularized measured Rényi divergences, another distinctive feature of the former quantities. Our main tool is an approximation theorem (martingale convergence) for the sandwiched Rényi divergences, which may be used for the extension of various further results from the finite-dimensional to the von Neumann algebra setting. We also initiate the study of the sandwiched Rényi divergences of pairs of states on a $C^*$-algebra and show that the above operational interpretation, as well as the equality to the regularized measured Rényi divergence, holds more generally for pairs of states on a nuclear $C^*$-algebra.

Appendices

Relative Modular Operators

Let ${\mathcal {M}}$ be a general von Neumann algebra with the predual ${\mathcal {M}}_*$, and ${\mathcal {M}}_*^+$ be the positive part of ${\mathcal {M}}_*$ consisting of normal positive linear functionals on ${\mathcal {M}}$. We consider ${\mathcal {M}}$ in its standard form $({\mathcal {M}},{\mathcal {H}},J,{\mathcal {P}})$ [21], that is, ${\mathcal {M}}$ is represented on a Hilbert space ${\mathcal {H}}$ with the modular conjugation (a conjugate-linear involution) J and the natural cone (a self-dual cone) ${\mathcal {P}}$, satisfying the following properties:

• (1) $JMJ=M'$ ($M'$ being the commutant of ${\mathcal {M}}$),

• (2) $JxJ=x^*$, $x\in {\mathcal {M}}\cap {\mathcal {M}}'$ (the center of ${\mathcal {M}}$),

• (3) $J\xi =\xi$, $\xi \in {\mathcal {P}}$,

• (4) $xJxJ{\mathcal {P}}\subseteq {\mathcal {P}}$, $x\in {\mathcal {M}}$.

Any von Neumann algebra has a unique (up to unitary conjugation) standard form; see [21, Theorem 2.3]. Any $\sigma \in {\mathcal {M}}_*^+$ has a unique vector representative $\xi _\sigma$ in ${\mathcal {P}}$ so that $\sigma (x)=\langle \xi _\sigma ,x\xi _\sigma \rangle$, $x\in {\mathcal {M}}$. The support $s(\sigma )=s_{\mathcal {M}}(\sigma )\in {\mathcal {M}}$ of $\sigma$ is the orthogonal projection onto $\overline{{\mathcal {M}}'\xi _\sigma }$, while the ${\mathcal {M}}'$-support $s_{{\mathcal {M}}'}(\sigma )\in {\mathcal {M}}'$ is the orthogonal projection onto $\overline{{\mathcal {M}}\xi _\sigma }$ so that $s_{{\mathcal {M}}'}(\sigma )=Js_{\mathcal {M}}(\sigma )J$.

For any $\rho ,\sigma \in {\mathcal {M}}_*^+$, the closable conjugate-linear operators $S_{\rho ,\sigma }$ and $F_{\rho ,\sigma }$ are defined by

$$\begin{aligned} S_{\rho ,\sigma }(x\xi _\sigma +\eta )&:=s_{\mathcal {M}}(\sigma )x^*\xi _\sigma ,\ \ \qquad x\in {\mathcal {M}},\ \eta \in (\textbf{1}-s_{{\mathcal {M}}'}(\sigma )){\mathcal {H}}, \\ F_{\rho ,\sigma }(x'\xi _\sigma +\zeta )&:=s_{{\mathcal {M}}'}(\sigma )x'^*\xi _\sigma ,\qquad x'\in {\mathcal {M}}',\ \zeta \in (\textbf{1}-s_{\mathcal {M}}(\sigma )){\mathcal {H}}, \end{aligned}$$

for which $S_{\rho ,\sigma }^*={{\overline{F}}}_{\rho ,\sigma }$. The relative modular operator $\Delta _{\rho ,\sigma }$ [3] is

$$\begin{aligned} \Delta _{\rho ,\sigma }:=S_{\rho ,\sigma }^*{{\overline{S}}}_{\rho ,\sigma } \end{aligned}$$

and the polar decomposition of ${{\overline{S}}}_{\rho ,\sigma }$ is given as ${{\overline{S}}}_{\rho ,\sigma }=J\Delta _{\rho ,\sigma }^{1/2}$. When $\rho =\sigma$, $\Delta _{\sigma ,\sigma }$ is the modular operator $\Delta _\sigma$.

When ${\mathcal {M}}=\mathcal {B}({\mathcal {H}})$ on a Hilbert space ${\mathcal {H}}$, consider the Hilbert–Schmidt class ${\mathcal {C}}_2({\mathcal {H}})$ with the Hilbert–Schmidt inner product $\langle X,Y\rangle _\textrm{HS}:=\textrm{Tr}(X^*Y)$, $X,Y\in {\mathcal {C}}_2({\mathcal {H}})$; then, the standard form of $\mathcal {B}({\mathcal {H}})$ is given as

$$\begin{aligned} (\mathcal {B}({\mathcal {H}}),{\mathcal {C}}_2({\mathcal {H}}), J=\,^*,{\mathcal {C}}_2({\mathcal {H}})_+), \end{aligned}$$

where $\mathcal {B}({\mathcal {H}})$ is represented on ${\mathcal {C}}_2({\mathcal {H}})$ by the left multiplications $L_AX:=AX$ for $A\in \mathcal {B}({\mathcal {H}})$, $X\in {\mathcal {C}}_2({\mathcal {H}})$, and ${\mathcal {C}}_2({\mathcal {H}})_+:=\{X\in {\mathcal {C}}_2({\mathcal {H}}):X\ge 0\}$. Each $\rho \in \mathcal {B}({\mathcal {H}})_*^+$ is identified with a trace-class operator ${\hat{\rho }}\ge 0$ so that $\rho (X)=\textrm{Tr}({\hat{\rho }} X)=\langle {\hat{\rho }}^{1/2},X{\hat{\rho }}^{1/2}\rangle _\textrm{HS}$, $X\in \mathcal {B}({\mathcal {H}})$, and ${\hat{\rho }}^{1/2}\in {\mathcal {C}}_2({\mathcal {H}})_+$ is the vector representative of $\rho$. For $\rho ,\sigma \in \mathcal {B}({\mathcal {H}})_*^+$, the relative modular operator $\Delta _{\rho ,\sigma }$ is written as $\Delta _{\rho ,\sigma }=L_{{\hat{\rho }}}R_{{\hat{\sigma }}^{-1}}$, where ${\hat{\sigma }}^{-1}$ is the generalized inverse (i.e., the inverse with restriction to the support $s(\sigma ){\mathcal {H}}$) of ${\hat{\sigma }}$ and $R_{{\hat{\sigma }}^{-1}}$ is the right multiplication by ${\hat{\sigma }}^{-1}$. Of course, when $\dim {\mathcal {H}}<+\infty$, we have ${\mathcal {C}}_2({\mathcal {H}})=\mathcal {B}({\mathcal {H}})$.

Haagerup’s $L^p$-spaces

Assume that ${\mathcal {M}}$ is $\sigma$-finite, i.e., there exists a faithful $\omega \in {\mathcal {M}}_*^+$. Let us denote by ${\mathcal {N}}$ the crossed product ${\mathcal {M}}\rtimes _\omega {\mathbb {R}}$ of ${\mathcal {M}}$ by the modular automorphism group $\sigma _t^\omega =\Delta _\omega ^{it}(\cdot )\Delta _\omega ^{-it}$, $t\in {\mathbb {R}}$. Le $\theta _s$, $s\in {\mathbb {R}}$, be the dual action of ${\mathcal {N}}$ so that $\tau \circ \theta _s=e^{-s}\tau$, $s\in {\mathbb {R}}$, where $\tau$ is the canonical trace on ${\mathcal {N}}$; the crossed product construction was developed in the structure theory of von Neumann algebras [68]. Let ${\widetilde{{\mathcal {N}}}}$ denote the space of $\tau$-measurable operators [55, 72] affiliated with ${\mathcal {N}}$. For each $p\in (0,+\infty ]$, Haagerup’s $L^p$-space $L^p({\mathcal {M}})$ [72] is defined by

$$\begin{aligned} L^p({\mathcal {M}}):=\{x\in {\widetilde{{\mathcal {N}}}}:\theta _s(x)=e^{-s/p}x,\,s\in {\mathbb {R}}\} \end{aligned}$$

(in particular, $L^\infty ({\mathcal {M}})={\mathcal {M}}$), whose positive part is $L^p({\mathcal {M}})_+:=L^p({\mathcal {M}})\cap {\widetilde{{\mathcal {N}}}}_+$. There exists an order isomorphism ${\mathcal {M}}_*\cong L^1({\mathcal {M}})$, given as $\psi \in {\mathcal {M}}_*\mapsto h_\psi \in L^1({\mathcal {M}})$, so that $\textrm{tr}(h_\psi ):=\psi ({\textbf{1}})$, $\psi \in {\mathcal {M}}_*$, defines a positive linear functional $\textrm{tr}$ on $L^1({\mathcal {M}})$. For $1\le p<+\infty$, the $L^p$-norm $\Vert a\Vert _p$ of $a\in L^p({\mathcal {M}})$ is given by $\Vert a\Vert _p:=\textrm{tr}(|a|^p)^{1/p}$, and the $L^\infty$-norm $\left\| \cdot \right\| _\infty$ is the operator norm on ${\mathcal {M}}$. For $1\le p<+\infty$, $L^p({\mathcal {M}})$ is a Banach space with the norm $\left\| \cdot \right\| _p$, whose dual Banach space is $L^q({\mathcal {M}})$, where $1/p+1/q=1$, by the duality

$$\begin{aligned} (a,b)\in L^p({\mathcal {M}})\times L^q({\mathcal {M}})\,\longmapsto \,\textrm{tr}(ab)\ (=\textrm{tr}(ba)). \end{aligned}$$

In particular, $L^2({\mathcal {M}})$ is a Hilbert space with the inner product $\langle a,b\rangle =\textrm{tr}(a^*b)$ ($=\textrm{tr}(ba^*)$). Then,

$$\begin{aligned} ({\mathcal {M}},L^2({\mathcal {M}}),J=\,^*,L^2({\mathcal {M}})_+) \end{aligned}$$

becomes a standard form of ${\mathcal {M}}$, where ${\mathcal {M}}$ is represented on $L^2({\mathcal {M}})$ by the left multiplication. Each $\rho \in {\mathcal {M}}_*^+$ is represented as

$$\begin{aligned} \rho (x)=\textrm{tr}(xh_\rho )=\langle h_\rho ^{1/2},xh_\rho ^{1/2}\rangle ,\qquad x\in {\mathcal {M}}, \end{aligned}$$

with the vector representative $h_\rho ^{1/2}\in L^2({\mathcal {M}})_+$. Note that the support projection $s(\rho )$ ($\in {\mathcal {M}}$) of the functional $\rho$ coincides with that of the operator $h_\rho$. For any projection $e\in {\mathcal {M}}$, Haagerup’s $L^p$-space $L^p(e{\mathcal {M}}e)$ is identified with $eL^p({\mathcal {M}})e$ and the standard form of $e{\mathcal {M}}e$ is given by $(e{\mathcal {M}}e,eL^2({\mathcal {M}})e,J=\,^*,eL^2({\mathcal {M}})_+e)$.

Note that $L^p({\mathcal {M}})$ is independent (up to isometric isomorphism) of the choice of $\omega$ (where $\omega$ can be a faithful normal semifinite weight unless ${\mathcal {M}}$ is $\sigma$-finite), and that when ${\mathcal {M}}$ is semifinite with a faithful normal semifinite trace $\tau _0$, $L^p({\mathcal {M}})$ can be identified with the tracial $L^p$-space $L^p({\mathcal {M}},\tau _0)$ (see, e.g., [55]). In particular, when ${\mathcal {M}}=\mathcal {B}({\mathcal {H}})$ with $\omega =\textrm{Tr}$ (and so $\Delta _\omega ={\textbf{1}}$), note that ${\mathcal {N}}={\mathcal {M}}{\overline{\otimes }} L^\infty ({\mathbb {R}})$ on ${\mathcal {H}}\otimes L^2({\mathbb {R}})$ and the canonical trace on ${\mathcal {N}}$ is $\tau =\textrm{Tr}\otimes \int _{\mathbb {R}}(\cdot )e^t\,dt$, so that $L^p({\mathcal {M}})={\mathcal {C}}_p({\mathcal {H}})\otimes e^{-t/p}$ with $\Vert X\otimes e^{-t/p}\Vert _{L^p({\mathcal {M}})}=\Vert X\Vert _p$ for $X\in {\mathcal {C}}_p({\mathcal {H}})$. Here, the symbol $e^{-t/p}$ is used to denote the multiplication operator on $L^2({\mathbb {R}})$, and ${\mathcal {C}}_p({\mathcal {H}})$ is the Schatten–von Neumann p-class with $\Vert X\Vert _p:=(\textrm{Tr}\,|X|^p)^{1/p}$. Therefore, $L^p({\mathcal {M}})$ coincides with ${\mathcal {C}}_p({\mathcal {H}})$ by just neglecting the superfluous tensor factor $e^{-t/p}$; see [30, Remark 8.16, Example 9.11] for more details on this matter.

It might be instructive to note that Haagerup’s $L^p({\mathcal {M}})$ is different from the tracial $L^p$-space $L^p({\mathcal {N}},\tau )$ with the canonical trace $\tau$, even when ${\mathcal {M}}=\mathcal {B}({\mathcal {H}})$. In this case, $L^p({\mathcal {M}})={\mathcal {C}}_p({\mathcal {H}})\otimes e^{-t/p}$ as stated above, and for every $X\in {\mathcal {C}}_p({\mathcal {H}})$,

$$\begin{aligned} \Vert X\otimes e^{-t/p}\Vert _{L^p({\mathcal {N}},\tau )}=\Vert X\Vert _p\Bigl (\int _{\mathbb {R}}(e^{-t/p})^pe^t\,dt\Bigr )^{1/p} =\Vert X\Vert _p\Bigl (\int _{\mathbb {R}}dt\Bigr )^{1/p}=+\infty \end{aligned}$$

unless $X=0$. However, in the general case of ${\mathcal {M}}$, the exact relation of elements in $L^p({\mathcal {M}})$ with the canonical trace $\tau$ on ${\mathcal {N}}$ is expressed as follows: for every $a\in L^p({\mathcal {M}})$ and $p\in (0,+\infty )$,

$$\begin{aligned} \mu _t(a)=t^{-1/p}\Vert a\Vert _p,\qquad t>0, \end{aligned}$$

where $\mu _t(a)$ is the tth generalized s-number of a with respect to $\tau$; see [19, Lemma 4.8] and [30, Lemma 9.14]. The above expression is sometimes useful though it is not used in this paper.

Kosaki’s Interpolation $L^p$-spaces

Assume that ${\mathcal {M}}$ is $\sigma$-finite and let a faithful $\omega \in {\mathcal {M}}_*^+$ be given with $h_{\omega }\in L^1({\mathcal {M}})_+$. Consider an embedding ${\mathcal {M}}$ into $L^1({\mathcal {M}})$ by $x\mapsto h_\omega ^{1/2}xh_\omega ^{1/2}$. Defining $\Vert h_\omega ^{1/2}xh_\omega ^{1/2}\Vert _\infty :=\Vert x\Vert _\infty$ on $h_\omega ^{1/2}{\mathcal {M}}h_\omega ^{1/2}$, we have a pair $(h_\omega ^{1/2}{\mathcal {M}}h_\omega ^{1/2},L^1({\mathcal {M}}))$ of compatible Banach spaces (see, e.g., [7]). For $1<p<+\infty$ Kosaki’s (symmetric) $L^p$-space $L^p({\mathcal {M}},\omega )$ [42] with respect to $\omega$ is the complex interpolation Banach space

$$\begin{aligned} C_{1/p}(h_\omega ^{1/2}{\mathcal {M}}h_\omega ^{1/2},L^1({\mathcal {M}})) \end{aligned}$$

equipped with the interpolation norm $\left\| \cdot \right\| _{p,\omega }$ ($=\left\| \cdot \right\| _{C_{1/p}}$) [7]. Moreover, $L^1({\mathcal {M}},\omega ):=L^1({\mathcal {M}})$ with $\left\| \cdot \right\| _{1,\omega }=\left\| \cdot \right\| _1$ and $L^\infty ({\mathcal {M}},\omega ):=h_\omega ^{1/2}{\mathcal {M}}h_\omega ^{1/2}$ ($\cong {\mathcal {M}}$) with $\left\| \cdot \right\| _{\infty ,\omega }=\left\| \cdot \right\| _\infty$. Kosaki’s theorem [42, Theorem 9.1] says that for every $p\in [1,+\infty ]$ and $1/p+1/q=1$,

$$\begin{aligned}&L^p({\mathcal {M}},\omega )=h_\omega ^{1\over 2q}L^p({\mathcal {M}})h_\omega ^{1\over 2q}\ (\subseteq L^1({\mathcal {M}})), \end{aligned}$$

(C.1)

$$\begin{aligned}&\Vert h_\omega ^{1\over 2q}ah_\omega ^{1\over 2q}\Vert _{p,\omega }=\Vert a\Vert _p,\qquad a\in L^p({\mathcal {M}}), \end{aligned}$$

(C.2)

that is, $L^p({\mathcal {M}})\cong L^p({\mathcal {M}},\omega )$ by the isometry $a\mapsto h_\omega ^{1\over 2q}ah_\omega ^{1\over 2q}$. Interpolation $L^p$-spaces were introduced in [42] in terms of more general embeddings $x\in {\mathcal {M}}\mapsto h_\omega ^\eta xh_\omega ^{1-\eta }\in L^1({\mathcal {M}})$ with $0\le \eta \le 1$. (The $\eta =1/2$ case is the above symmetric $L^1({\mathcal {M}},\omega )$.) When ${\mathcal {M}}$ is general and $\omega \in {\mathcal {M}}_*^+$ is not faithful with the support projection $e:=s(\omega )\in {\mathcal {M}}$, Kosaki’s $L^p$-space $L^p({\mathcal {M}},\omega )$ with respect to $\omega$ is still defined over $e{\mathcal {M}}e$ so that (C.1) and (C.2) hold with $eL^p({\mathcal {M}})e$ in place of $L^p({\mathcal {M}})$.

Consider now the special case ${\mathcal {M}}=\mathcal {B}({\mathcal {H}})$, and let $\omega \in \mathcal {B}({\mathcal {H}})_*^+$ be given with $e:=s(\omega )$ and ${\hat{\omega }}\in {\mathcal {C}}_1({\mathcal {H}})_+$ representing $\omega$. When $1\le p\le +\infty$ and $1/p+1/q=1$, Kosaki’s $L^p$-space with respect to $\omega$ is $L^p(\mathcal {B}({\mathcal {H}}),\omega )={\hat{\omega }}^{1\over 2q}{\mathcal {C}}_p({\mathcal {H}}){\hat{\omega }}^{1\over 2q}$ with $\Vert {\hat{\omega }}^{1\over 2q}A{\hat{\omega }}^{1\over 2q}\Vert _{p,\omega }=\Vert A\Vert _p$ for $A\in e{\mathcal {C}}_p({\mathcal {H}})e$ (where ${\mathcal {C}}_\infty ({\mathcal {H}})=\mathcal {B}({\mathcal {H}})$). In particular, when $\dim {\mathcal {H}}<+\infty$, $L^p(\mathcal {B}({\mathcal {H}}),\omega )=e\mathcal {B}({\mathcal {H}})e=\mathcal {B}(e{\mathcal {H}})$ and the interpolation $L^p$-norm is $\Vert A\Vert _{p,\omega }=\Vert {\hat{\omega }}^{-{1\over 2q}}A{\hat{\omega }}^{-{1\over 2q}}\Vert _p$ for any $A\in \mathcal {B}(e{\mathcal {H}})$. The interpolation norm in the finite-dimensional case was used in [6] for instance.

Generalized Conditional Expectations

Let ${\mathcal {M}}$ and ${\mathcal {N}}$ be ($\sigma$-finite) von Neumann algebras, with standard forms $({\mathcal {M}},{\mathcal {H}},J,{\mathcal {P}})$ and $({\mathcal {N}},{\mathcal {H}}_0,J_0,{\mathcal {P}}_0)$, respectively (see Appendix A). Let $\Phi :{\mathcal {N}}\rightarrow {\mathcal {M}}$ be a unital positive map. Let a faithful $\omega \in {\mathcal {M}}_*^+$ be given, and assume that $\omega \circ \Phi$ is normal and faithful on ${\mathcal {N}}$. In this case, $\Phi$ is automatically normal and faithful (i.e., $\Phi (x^*x)=0$ $\implies$ $x=0$). Then, it was shown in [1] that there exists a unique unital normal positive map $\Phi _\omega ^*:{\mathcal {M}}\rightarrow {\mathcal {N}}$ such that

$$\begin{aligned} \langle Jx\Omega ,\Phi (y)\Omega \rangle =\langle J_0\Phi _\omega ^*(x)\Omega _0,y\Omega _0\rangle , \qquad x\in {\mathcal {M}},\ y\in {\mathcal {N}}, \end{aligned}$$

(D.1)

where $\Omega \in {\mathcal {P}}$ and $\Omega _0\in {\mathcal {P}}_0$ are the vector representatives of $\omega$ and $\omega \circ \Phi$, respectively. The map $\Phi _\omega ^*$ is also faithful. Moreover, we have

$$\begin{aligned} \omega \circ \Phi \circ \Phi _\omega ^*=\omega , \end{aligned}$$

(D.2)

and $\Phi _\omega ^*$ is completely positive if and only if so is $\Phi$. This map $\Phi _\omega ^*$ is called the $\omega$-dual map of $\Phi$, or the Petz recovery map (see [59]), whose definition by (D.1) is independent of the choice of the standard forms of ${\mathcal {M}},{\mathcal {N}}$. In terms of Haagerup’s $L^1$-elements $h_\omega$ and $h_{\omega \circ \Phi }$ (see Appendix B), we note [29, Lemma 8.3] that the map $\Phi _\omega ^*$ is determined by

$$\begin{aligned} \Phi _*(h_\omega ^{1/2}xh_\omega ^{1/2}) =h_{\omega \circ \Phi }^{1/2}\Phi _\omega ^*(x)h_{\omega \circ \Phi }^{1/2}, \qquad x\in {\mathcal {M}}, \end{aligned}$$

(D.3)

where $\Phi _*:L^1({\mathcal {M}})\rightarrow L^1({\mathcal {N}})$ is the predual map of $\Phi$ via ${\mathcal {M}}_*\cong L^1({\mathcal {M}})$ and ${\mathcal {N}}_*\cong L^1({\mathcal {N}})$, i.e., $\Phi _*(h_\psi )=h_{\psi \circ \Phi }$, $\psi \in {\mathcal {M}}_*$. Note that the construction of $\Phi _\omega ^*$ is possible even when $\omega$ and/or $\omega \circ \Phi$ are not faithful (see [29, Theorem 6.1 and Lemma 8.3]), though the above setting is sufficient for our present purpose.

In particular, let ${\mathcal {N}}$ be a von Neumann subalgebra of ${\mathcal {M}}$ containing the unit of ${\mathcal {M}}$, and $\omega \in {\mathcal {M}}_*^+$ be faithful. The $\omega$-dual map $\Phi _\omega ^*$ of the injection $\Phi :{\mathcal {N}}\hookrightarrow {\mathcal {M}}$ is called the generalized conditional expectation with respect to $\omega$ [1], which we denote by ${\mathcal {E}}_{{\mathcal {N}},\omega }:{\mathcal {M}}\rightarrow {\mathcal {N}}$. The map ${\mathcal {E}}_{{\mathcal {N}},\omega }$ is unital, normal, completely positive, and faithful. Property (D.2) becomes

$$\begin{aligned} \omega \circ {\mathcal {E}}_{{\mathcal {N}},\omega }=\omega . \end{aligned}$$

In the present case, the standard Hilbert space ${\mathcal {H}}_0$ for ${\mathcal {N}}$ is taken as ${\mathcal {H}}_0=\overline{{\mathcal {N}}\Omega }$, where the vector representative $\Omega _0$ of $\omega \circ \Phi =\omega |_{\mathcal {N}}$ is equal to $\Omega$. Let P be the orthogonal projection from ${\mathcal {H}}=\overline{{\mathcal {M}}\Omega }$ onto ${\mathcal {H}}_0=\overline{{\mathcal {N}}\Omega }$. In this situation, note [1] that ${\mathcal {E}}_{{\mathcal {N}},\omega }=\Phi _\omega ^*$ given in (D.1) and (D.3) can be written more explicitly as

$$\begin{aligned} {\mathcal {E}}_{{\mathcal {N}},\omega }(x)=J_0PJxJPJ_0=J_0PJxJJ_0,\qquad x\in {\mathcal {M}}, \end{aligned}$$

(D.4)

which is also determined by ${\mathcal {E}}_{{\mathcal {N}},\omega }(x)\Omega =J_0PJx\Omega$, $x\in {\mathcal {M}}$. As is well known [67], there exists a (genuine) conditional expectation (i.e., a norm-one projection) $E:{\mathcal {M}}\rightarrow {\mathcal {N}}$ such that $\omega \circ E=\omega$ on ${\mathcal {M}}$, if and only if ${\mathcal {N}}$ is globally invariant under the modular automorphism group $\sigma _t^\omega$ (see Appendix B) of ${\mathcal {M}}$ with respect to $\omega$, i.e., $\sigma _t^\omega ({\mathcal {N}})={\mathcal {N}}$, $t\in {\mathbb {R}}$. If this is the case, $J_0=J|_{{\mathcal {H}}_0}$ and $JP=PJ$ hold so that ${\mathcal {E}}_{{\mathcal {N}},\omega }=E$. An important property of E is the bimodule property $E(axb)=aE(x)b$ for $a,b\in {\mathcal {N}}$ and $x\in {\mathcal {M}}$, which ${\mathcal {E}}_{{\mathcal {N}},\omega }$ does not satisfy in general. A merit of ${\mathcal {E}}_{{\mathcal {N}},\omega }$ is that it always exists, while the existence of E is very restrictive as stated above.

Injective von Neumann Algebras and Nuclear $C^*$-algebras

A von Neumann algebra ${\mathcal {M}}$ on a Hilbert space ${\mathcal {H}}$ is injective if and only if there exists a (not necessarily normal) conditional expectation (i.e., a projection of norm one [73]) from ${\mathcal {B}}({\mathcal {H}})$ onto ${\mathcal {M}}$; see, e.g., [70, Corollary XV.1.3]. A fundamental result of Connes [14] (see also [70, Theorem XVI.1.9]) says that a von Neumann algebra ${\mathcal {M}}$ of separable predual is injective if and only if ${\mathcal {M}}$ is AFD (approximately finite dimensional), i.e., there exists an increasing sequence $\{{\mathcal {M}}_j\}_{j=1}^\infty$ of finite-dimensional *-subalgebras of ${\mathcal {M}}$ such that ${\mathcal {M}}=\bigl (\bigcup _{j=1}^\infty {\mathcal {M}}_j\bigr )''$. In [18], the result was furthermore extended in such a way that a (general) von Neumann algebra ${\mathcal {M}}$ is injective if and only if there is an increasing net $\{{\mathcal {M}}_i\}_{i\in {\mathcal {I}}}$ of finite-dimensional *-subalgebras of ${\mathcal {M}}$ with ${\mathcal {M}}=\bigl (\bigcup _{i\in {\mathcal {I}}}{\mathcal {M}}_i\bigr )''$. (Here, ${\mathcal {A}}''$ denotes the double commutant, i.e., the commutant of ${\mathcal {A}}'$, for any ${\mathcal {A}}\subseteq {\mathcal {B}}({\mathcal {H}})$.)

Next, a $C^*$-algebra ${\mathcal {A}}$ is said to be nuclear if, for every $C^*$-algebra ${\mathcal {B}}$, there is a unique $C^*$-cross-norm on ${\mathcal {A}}\odot {\mathcal {B}}$, i.e., ${\mathcal {A}}\otimes _{\min }{\mathcal {B}}={\mathcal {A}}\otimes _{\max }{\mathcal {B}}$; see, e.g., [70, Chap. XV]. Concerning nuclear $C^*$-algebras, among many others, the most fundamental result is that ${\mathcal {A}}$ is nuclear if and only if ${\mathcal {A}}^{**}$ is injective. Here, ${\mathcal {A}}^{*}$ denotes the Banach space dual of ${\mathcal {A}}$, and ${\mathcal {A}}^{**}$ the second Banach space dual of ${\mathcal {A}}$. Note that ${\mathcal {A}}^{**}$ is isometrically isomorphic to the universally enveloping von Neumann algebra of ${\mathcal {A}}$, and so it is customary to use ${\mathcal {A}}^{**}$ to denote the latter as well; see [69, Sect. III.2]. Therefore, if ${\mathcal {A}}$ is nuclear, then $\pi ({\mathcal {A}})''$ is injective for every representation $\pi$ of ${\mathcal {A}}$. Typical examples of nuclear $C^*$-algebras are AF $C^*$-algebras, in particular, the compact operator ideal ${\mathcal {C}}({\mathcal {H}})$ (or rather ${\mathcal {C}}({\mathcal {H}})+{\mathbb {C}}\textbf{1}$ in our present setting; see Example 4.6). Here, recall that a $C^*$-algebra ${\mathcal {A}}$ is AF if there exists an increasing sequence $\{{\mathcal {A}}_k\}_{k=1}^\infty$ of finite-dimensional *-subalgebras of ${\mathcal {A}}$ such that $\bigcup _{k=1}^\infty {\mathcal {A}}_k$ is norm-dense in ${\mathcal {A}}$. More intricate examples are provided by groups. For a discrete group G, the $C^*$-algebra generated by the left regular representation on $\ell ^2(G)$ is the (reduced) group $C^*$-algebra $C_r^*(G)$, while the generated von Neumann algebra is the group von Neumann algebra $W^*(G)$. Then, G is amenable $\iff$ $C_r^*(G)$ is nuclear $\iff$ $W^*(G)$ is injective.

Strong Converse Exponent in the Finite-Dimensional Case

In this appendix, we assume that a von Neumann algebra ${\mathcal {M}}$ is finite-dimen-sional, so ${\mathcal {M}}\subseteq {\mathcal {B}}({\mathcal {H}})$ with a finite-dimensional Hilbert space ${\mathcal {H}}$. Note that ${\mathcal {M}}$ is isomorphic to $\bigoplus _{i=1}^m{\mathcal {B}}({\mathcal {H}}_i)$, a finite direct sum of finite-dimensional ${\mathcal {B}}({\mathcal {H}}_i)$, so it is clear that all the arguments in [48] are valid with ${\mathcal {M}}$ in place of ${\mathcal {B}}({\mathcal {H}})$. Let $\textrm{Tr}$ be the usual trace on ${\mathcal {M}}$ (such that $\textrm{Tr}(e)=1$ of all minimal projections $e\in {\mathcal {M}}$). Below, to designate states of ${\mathcal {M}}$, we use density operators $\rho ,\sigma$ with respect to $\textrm{Tr}$ rather than positive functionals. Recall that both of the relative entropy $D(\rho \Vert \sigma )$ and the max-relative entropy $D_{\max }(\rho \Vert \sigma )$ showing up in (2.4)–(2.6) play an important role to describe $\psi (s):=\psi ^*(\rho \Vert \sigma |s+1)$ and $H_r^*(\rho \Vert \sigma )$ in [48, Sect. 4].

The aim of this appendix is to give Proposition F.2, which is used in Sect. 3.2. The main assertion is that for finite-dimensional density operators we have $sc_r^0(\rho \Vert \sigma )\le H_r^*(\rho \Vert \sigma )$, $r\ge 0$, which in turn can be obtained easily from the weaker inequalities $sc_r(\rho \Vert \sigma )\le H_r^*(\rho \Vert \sigma )$, $r\ge 0$. The latter was proved in [48]; however, the proof contains a gap, as it is implicitly assumed there that $D(\rho \Vert \sigma )<D_{\max }(\rho \Vert \sigma )$. Our main contribution in Proposition F.2 is filling this gap; for this, we give a characterization of the case $D(\rho \Vert \sigma )=D_{\max }(\rho \Vert \sigma )$, which may be of independent interest.

Lemma F.1

For density operators $\rho ,\sigma$ in ${\mathcal {M}}$ with $s(\rho )\le s(\sigma )$, the following conditions are equivalent:

• (a) $s\mapsto \psi (s):=\psi ^*(\rho \Vert \sigma |s+1)$ is affine on $(0,+\infty )$;

• (b) $D(\rho \Vert \sigma )=D_{\max }(\rho \Vert \sigma )$;

• (c) $\rho$ and $\sigma$ commute, and $\rho \sigma ^{-1}=\gamma s(\rho )$ for some constant $\gamma >0$;

• (d) $s(\rho )\sigma =\sigma s(\rho )$ and $\rho =\gamma \sigma s(\rho )$ for some constant $\gamma >0$.

Moreover, if the above hold, then we have $\gamma \ge 1$, $D(\rho \Vert \sigma )=\log \gamma$ and

$$\begin{aligned} H_r^*(\rho \Vert \sigma )=(r-D(\rho \Vert \sigma ))_+,\qquad r\ge 0. \end{aligned}$$

(F.1)

Proof

(a)$\iff$(b). Since $\psi (s)$ is a differentiable convex function on $[0,+\infty )$, this is clear from [48, Lemma 4.2].

(b)$\implies$(c). Consider $D_2(\rho \Vert \sigma ):=\log \textrm{Tr}\rho ^2\sigma ^{-1}$, the standard (or Petz-type) Rényi 2-divergence of $\rho ,\sigma$. Note that

$$\begin{aligned} D(\rho \Vert \sigma )\le D_2^*(\rho \Vert \sigma )\le D_2(\rho \Vert \sigma )\le D_{\max }(\rho \Vert \sigma ), \end{aligned}$$

where the first inequality is seen from the properties noted in Sect. 2, the second is due to the Araki–Lieb–Thirring inequality, and the last was shown in [11, Lemma 7]. Hence, (b) implies that $D_2^*(\rho \Vert \sigma )=D_2(\rho \Vert \sigma )$, i.e.,

$$\textrm{Tr}(\sigma ^{-1/4}\rho \sigma ^{-1/4})^2 =\textrm{Tr}\,\sigma ^{-1/2}\rho ^2\sigma ^{-1/2}.$$

Using [26, Theorem 2.1], we find that $\rho ,\sigma$ commute. Hence, (a) says that $\psi (s)=\log \textrm{Tr}\,\rho ^{s+1}\sigma ^{-s}$ is affine on $(0,+\infty )$. From [32, Lemma 3.2] for the commutative case, it follows that $\rho \sigma ^{-1}=\gamma s(\rho )$, implying (c).

(c)$\implies$(d) is obvious.

(d)$\implies$(b). From condition (c), it easily follows that $\gamma \ge 1$ and $D_{\max }(\rho \Vert \sigma )=\log \gamma$. Moreover, since

$$\begin{aligned} D(\rho \Vert \sigma )&=\textrm{Tr}(\rho \log \rho -\rho \log (s(\rho )\sigma )) \\&=\textrm{Tr}(\rho \log \rho -\rho \log (\gamma ^{-1}\rho ))=\log \gamma , \end{aligned}$$

(b) follows.

Finally, if (b) and hence (a) hold, then $\psi (s)=D(\rho \Vert \sigma )s$ for all $s>0$, from which (F.1) follows immediately. $\square$

The next proposition is used in the proof of Theorem 3.7, while the former is a specialized case of the latter.

Proposition F.2

For every density operators $\rho ,\sigma$ in ${\mathcal {M}}$ with $s(\rho )\le s(\sigma )$ and any $r\ge 0$, we have

$$\begin{aligned} {\underline{sc}}_r(\rho \Vert \sigma )=sc_r^0(\rho \Vert \sigma )=H_r^*(\rho \Vert \sigma ). \end{aligned}$$

Proof

Since $sc_r^0(\rho \Vert \sigma )\ge {\underline{sc}}_r(\rho \Vert \sigma )\ge H_r^*(\rho \Vert \sigma )$, where the second inequality is by [48, Lemma 4.7], it suffices to prove that $sc_r^0(\rho \Vert \sigma )\le H_r^*(\rho \Vert \sigma )$, $r\ge 0$. Moreover, the last inequality follows if we can prove that $sc_r(\rho \Vert \sigma )\le H_r^*(\rho \Vert \sigma )$, $r\ge 0$, since then

$$\begin{aligned} sc_r^0(\rho \Vert \sigma )\le \inf _{r'>r}sc_r(\rho \Vert \sigma ) \le \inf _{r'>r} H_{r'}^*(\rho \Vert \sigma ) = H_r^*(\rho \Vert \sigma ), \end{aligned}$$

where the first inequality is obvious by definition, the second inequality is to be proved below, and the equality follows from the fact that $r\mapsto H_r^*(\rho \Vert \sigma )$ is a monotone increasing finite-valued convex function on ${\mathbb {R}}$, whence it is also continuous.

Let us therefore prove $sc_r(\rho \Vert \sigma )\le H_r^*(\rho \Vert \sigma )$, $r\ge 0$. The proof of [48, Theorem 4.10] gives this when $D(\rho \Vert \sigma )<D_{\max }(\rho \Vert \sigma )$. Assume thus that $D(\rho \Vert \sigma )=D_{\max }(\rho \Vert \sigma )=:D$. For any $r\ge 0$, the test sequence $T_{n,r}:=e^{-n(r-D)_+}s(\rho )^{\otimes n}$, $n\in {\mathbb {N}}$, yields

$$\begin{aligned} -\frac{1}{n}\log \textrm{Tr}\,\rho _nT_{n,r}=(r-D)_+=H_r^{*}(\rho \Vert \sigma ), \end{aligned}$$

where the last equality is by (F.1), and

$$\begin{aligned} -\frac{1}{n}\log \textrm{Tr}\,\sigma _nT_{n,r} =-\frac{1}{n}\log \bigl ( e^{-n(r-D)_+}\textrm{Tr}(\sigma s(\rho ))^{\otimes n}\bigr ) =D+(r-D)_+\ge r, \end{aligned}$$

where we have used that $\sigma s(\rho )=e^{-D}\rho$ by Lemma F.1. This proves $sc_r(\rho \Vert \sigma )\le H_r^{*}(\rho \Vert \sigma )$. $\square$

Boundary Values of Convex Functions on (0, 1)

Let $\{\phi _i\}_{i\in {\mathcal {I}}}$ be a set of convex functions on (0, 1) with values in $(-\infty ,+\infty ]$. Define

$$\begin{aligned} \phi (u):=\sup _{i\in {\mathcal {I}}}\phi _i(u),\qquad u\in (0,1), \end{aligned}$$

which is obviously convex on (0, 1) with values in $(-\infty ,+\infty ]$. We extend $\phi _i$ and $\phi$ to [0, 1] by continuity as

$$\begin{aligned} \phi _i(u)&:=\lim _{u\searrow 0}\phi _i(u),\qquad \phi _i(1):=\lim _{u\nearrow 1}\phi _i(u), \\ \phi (0)&:=\lim _{u\searrow 0}\phi (u),\qquad \ \ \phi (1):=\lim _{u\nearrow 1}\phi (u). \end{aligned}$$

We then give the next lemma to use it in the proof of Theorem 3.7.

Lemma G.1

In the situation stated above, if $\phi (u)<+\infty$ for some $u\in (0,1)$, then

$$\begin{aligned} \phi (0)=\sup _{i\in {\mathcal {I}}}\phi _i(0),\qquad \phi (1)=\sup _{i\in {\mathcal {I}}}\phi _i(1). \end{aligned}$$

Proof

By assumption, we have a $u_0\in (0,1)$ with $\phi (u_0)<+\infty$. Obviously, $\phi _i(0)\le \phi (0)$ and $\phi _i(1)\le \phi (1)$ for all $i\in {\mathcal {I}}$. Hence, it suffices to show that $\phi (0)\le \sup _i\phi _i(0)$ and $\phi (1)\le \sup _i\phi _i(1)$. Let us prove the first inequality (the proof of the latter is similar). Set $\xi :=\sup _i\phi _i(0)$. If $\xi =+\infty$, the assertion holds trivially. So assume $\xi <+\infty$. By convexity, for every $i\in {\mathcal {I}}$ we have

$$\begin{aligned} \phi _i(u)\le {u_0-u\over u_0}\,\xi +{u\over u_0}\,\phi (u_0),\qquad u\in (0,u_0), \end{aligned}$$

so that

$$\begin{aligned} \phi (u)\le {u_0-u\over u_0}\,\xi +{u\over u_0}\,\phi (u_0),\qquad u\in (0,u_0). \end{aligned}$$

This implies that $\phi (0)\le \xi =\sup _i\phi _i(0)$. $\square$

Proof of Theorem 4.3

Let $\rho ,\sigma \in {\mathcal {A}}_+^*$ and $\pi$ be any $(\rho ,\sigma )$-normal representation of ${\mathcal {A}}$ with ${\tilde{\rho }}=\rho _\pi$ and ${\tilde{\sigma }}=\sigma _\pi$, the normal extensions to ${\mathcal {M}}:=\pi ({\mathcal {A}})''$. Also, let ${\overline{\rho }},{\overline{\sigma }}$ be the normal extensions of $\rho ,\sigma$ to the enveloping von Neumann algebra ${\mathcal {A}}^{**}$ and ${\overline{\pi }}:{\mathcal {A}}^{**}\rightarrow {\mathcal {M}}$ be the normal extension of $\pi$ to ${\mathcal {A}}^{**}$ (see [69, p. 121]). Let $s({\overline{\pi }})$ be the support projection of ${\overline{\pi }}$. Concerning the support projections $s({\tilde{\rho }})$ and $s({\overline{\rho }})$, we have $s({\tilde{\rho }})={\overline{\pi }}(s({\overline{\rho }}))$ with $s({\overline{\rho }})\le s({\overline{\pi }})$. Therefore, $s({\tilde{\rho }})\le s({\tilde{\sigma }})$ is equivalent to $s({\overline{\rho }})\le s({\overline{\sigma }})$. This means that the condition $s({\tilde{\rho }})\le s({\tilde{\sigma }})$ is independent of the choice of a representation $\pi$ as above. (The condition is called the absolute continuity of $\rho$ with respect to $\sigma$ [25].)

Now, let ${\hat{\pi }}$ be another $(\rho ,\sigma )$-normal representation of ${\mathcal {A}}$ with ${\hat{\rho }}:=\rho _{{\hat{\pi }}}$ and ${\hat{\sigma }}:=\sigma _{{\hat{\pi }}}$, the normal extensions to ${\hat{{\mathcal {M}}}}:={\hat{\pi }}({\mathcal {A}})''$. The next lemma is a main ingredient of the proof of Theorem 4.3.

Lemma H.1

In the situation stated above, assume that $s({\tilde{\rho }})\le s({\tilde{\sigma }})$ (hence $s({\hat{\rho }})\le s({\hat{\sigma }})$ as well). Let $z_0,{\hat{z}}_0$ denote the central supports of $s({\tilde{\sigma }}),s({\hat{\sigma }})$, respectively. Then, there exists an isomorphism $\Lambda :{\mathcal {M}}z_0\rightarrow {\hat{{\mathcal {M}}}}{\hat{z}}_0$ for which we have

$$\begin{aligned} {\tilde{\rho }}(x)={\hat{\rho }}\circ \Lambda (x),\quad {\tilde{\sigma }}(x)={\hat{\sigma }}\circ \Lambda (x), \qquad x\in {\mathcal {M}}z_0, \end{aligned}$$

(H.1)

and for every $p\in [1,+\infty )$,

$$\begin{aligned} \textrm{tr}(h_{{\tilde{\sigma }}}^{1/2p}xh_{{\tilde{\sigma }}}^{1/2p})^p =\textrm{tr}(h_{{\hat{\sigma }}}^{1/2p}\Lambda (x)h_{{\hat{\sigma }}}^{1/2p})^p,\qquad x\in {\mathcal {M}}z_0, \end{aligned}$$

(H.2)

where $h_{{\tilde{\sigma }}}\in L^1({\mathcal {M}})_+$ and $h_{{\hat{\sigma }}}\in L^1({\hat{{\mathcal {M}}}})_+$ are Haagerup’s $L^1$-elements corresponding to ${\tilde{\sigma }}\in {\mathcal {M}}_*^+$ and ${\hat{\sigma }}\in {\hat{{\mathcal {M}}}}_*^+$, respectively.

Proof

We will work in the standard forms

$$\begin{aligned} ({\mathcal {M}},L^2({\mathcal {M}}),J=\,^*,L^2({\mathcal {M}})_+),\qquad ({\hat{{\mathcal {M}}}},L^2({\hat{{\mathcal {M}}}}),{\hat{J}}=\,^*,L^2({\hat{{\mathcal {M}}}})_+). \end{aligned}$$

For brevity, we write

$$\begin{aligned} {\left\{ \begin{array}{ll} h_0:=h_{{\tilde{\rho }}}\in L^1({\mathcal {M}})_+, \\ k_0:=h_{{\tilde{\sigma }}}\in L^1({\mathcal {M}})_+, \\ e_0:=s({\tilde{\sigma }})=s(k_0)\in {\mathcal {M}}, \\ e_0':=Je_0J\in {\mathcal {M}}', \end{array}\right. }\qquad {\left\{ \begin{array}{ll} {\hat{h}}_0:=h_{{\hat{\rho }}}\in L^1({\hat{{\mathcal {M}}}})_+, \\ {\hat{k}}_0:=h_{{\hat{\sigma }}}\in L^1({\hat{{\mathcal {M}}}})_+, \\ {\hat{e}}_0:=s({\hat{\sigma }})=s({\hat{k}}_0)\in {\hat{{\mathcal {M}}}}, \\ {\hat{e}}_0':={\hat{J}}{\hat{e}}_0{\hat{J}}\in {\hat{{\mathcal {M}}}}'. \end{array}\right. } \end{aligned}$$

Below the proof is divided into several steps.

Step 1. Note that

$$\begin{aligned} \overline{\pi ({\mathcal {A}})k_0^{1/2}}=\overline{{\mathcal {M}}k_0^{1/2}}=L^2({\mathcal {M}})e_0=e_0'L^2({\mathcal {M}}) \end{aligned}$$

and for every $a\in {\mathcal {A}}$,

$$\begin{aligned} \langle k_0^{1/2},\pi (a)e_0'k_0^{1/2}\rangle =\langle k_0^{1/2},\pi (a)k_0^{1/2}\rangle ={\tilde{\sigma }}\circ \pi (a)=\sigma (a). \end{aligned}$$

Hence, $\{\pi (\cdot )e_0',e_0'L^2({\mathcal {M}}),k_0^{1/2}\}$ is the cyclic representation of ${\mathcal {A}}$ with respect to $\sigma$, and similarly $\{{\hat{\pi }}(\cdot ){\hat{e}}_0',{\hat{e}}_0'L^2({\hat{{\mathcal {M}}}}),{\hat{k}}_0^{1/2}\}$ is the same. By the uniqueness (up to unitary conjugation) of the cyclic representation, there exists a unitary $V:L^2({\mathcal {M}})e_0\rightarrow L^2({\hat{{\mathcal {M}}}}){\hat{e}}_0$ such that

$$\begin{aligned} Vk_0^{1/2}={\hat{k}}_0^{1/2},\qquad V(\pi (a)e_0')V^*={\hat{\pi }}(a){\hat{e}}_0',\quad a\in {\mathcal {A}}. \end{aligned}$$

(H.3)

We hence have an isomorphism $V\cdot V^*:{\mathcal {M}}e_0'\rightarrow {\hat{{\mathcal {M}}}}{\hat{e}}_0'$.

Step 2. Since $z_0$ is the central support of $e_0'$, note that $xz_0\in {\mathcal {M}}z_0\mapsto xe_0'\in {\mathcal {M}}e_0'$ ($x\in {\mathcal {M}}$) is an isomorphism, and similarly so is ${\hat{x}}{\hat{z}}_0\in {\hat{{\mathcal {M}}}}{\hat{z}}_0\mapsto {\hat{x}}{\hat{e}}_0'\in {\hat{{\mathcal {M}}}}{\hat{e}}_0'$ (${\hat{x}}\in {\hat{{\mathcal {M}}}}$). Hence, one can define an isomorphism $\Lambda :{\mathcal {M}}z_0\rightarrow {\hat{{\mathcal {M}}}}{\hat{z}}_0$ as follows:

$$\begin{aligned} \Lambda :\,{\mathcal {M}}z_0\,\cong \,{\mathcal {M}}e_0'\,\cong \,{\hat{{\mathcal {M}}}}{\hat{e}}_0' \,\cong \,{\hat{{\mathcal {M}}}}{\hat{z}}_0,\qquad xz_0\,\mapsto xe_0'\,\mapsto \,V(xe_0')V^*={\hat{x}}{\hat{e}}_0'\,\mapsto \,{\hat{x}}{\hat{z}}_0. \end{aligned}$$

(H.4)

Note [21, Lemma 2.6] that the standard forms of ${\mathcal {M}}z_0$ and ${\hat{{\mathcal {M}}}}{\hat{z}}_0$ are, respectively, given by

$$\begin{aligned}&({\mathcal {M}}z_0,\,z_0L^2({\mathcal {M}})z_0=L^2({\mathcal {M}})z_0,\,J=\,^*,\,z_0L^2({\mathcal {M}})_+z_0=L^2({\mathcal {M}})_+z_0), \\&({\hat{{\mathcal {M}}}}{\hat{z}}_0,\,{\hat{z}}_0L^2({\hat{{\mathcal {M}}}}){\hat{z}}_0=L^2({\hat{{\mathcal {M}}}}){\hat{z}}_0,\,{\hat{J}}=\,^*,\, {\hat{z}}_0L^2({\hat{{\mathcal {M}}}})_+{\hat{z}}_0=L^2({\hat{{\mathcal {M}}}})_+{\hat{z}}_0). \end{aligned}$$

By the uniqueness (up to unitary conjugation) of the standard form (under isomorphism) [21, Theorem 2.3], there exists a unitary $U:L^2({\mathcal {M}})z_0\rightarrow L^2({\hat{{\mathcal {M}}}}){\hat{z}}_0$ such that

$$\begin{aligned}&\Lambda (x)=UxU^*,\qquad x\in {\mathcal {M}}z_0, \end{aligned}$$

(H.5)

$$\begin{aligned}&(U\xi )^*=U(\xi ^*),\qquad \xi \in z_0L^2({\mathcal {M}})z_0. \end{aligned}$$

(H.6)

$$\begin{aligned}&U(L^2({\mathcal {M}})_+z_0)=L^2({\hat{{\mathcal {M}}}})_+{\hat{z}}_0. \end{aligned}$$

(H.7)

Step 3. Since $s(h_0)\le e_0\le z_0$ by assumption, one has $h_0^{1/2},k_0^{1/2}\in L^2({\mathcal {M}})z_0$, and similarly ${\hat{h}}_0^{1/2},{\hat{k}}_0^{1/2}\in L^2({\hat{{\mathcal {M}}}}){\hat{z}}_0$. By (H.7) one has $Uh_0^{1/2},Uk_0^{1/2}\in L^2({\hat{{\mathcal {M}}}})_+{\hat{z}}_0$. Here, we confirm that

$$\begin{aligned} Uh_0^{1/2}={\hat{h}}_0^{1/2},\qquad Uk_0^{1/2}={\hat{k}}_0^{1/2}. \end{aligned}$$

(H.8)

To show this, for every $a\in {\mathcal {A}}$ we find that

$$\begin{aligned} \langle Uh_0^{1/2},({\hat{\pi }}(a){\hat{z}}_0)Uh_0^{1/2}\rangle&=\langle h_0^{1/2},\Lambda ^{-1}({\hat{\pi }}(a){\hat{z}}_0)h_0^{1/2}\rangle \quad (\text{ by } (H.5)) \\&=\langle h_0^{1/2},(\pi (a)z_0)h_0^{1/2}\rangle \quad (\text{ by } (H.3) \hbox { and } (H.4)) \\&={\tilde{\rho }}\circ \pi (a)=\rho (a)={\hat{\rho }}\circ {\hat{\pi }}(a) \\&=\langle {\hat{h}}_0^{1/2},({\hat{\pi }}(a){\hat{z}}_0){\hat{h}}_0^{1/2}\rangle , \end{aligned}$$

which implies that $Uh_0^{1/2}={\hat{h}}_0^{1/2}$. The proof of $Uk_0^{1/2}={\hat{k}}_0^{1/2}$ is similar. By (H.5) and (H.8), we have also

$$\begin{aligned} \Lambda (x){\hat{h}}_0^{1/2}=U(xh_0^{1/2}),\quad \Lambda (x){\hat{k}}_0^{1/2}=U(xk_0^{1/2}), \qquad x\in {\mathcal {M}}z_0. \end{aligned}$$

(H.9)

These imply (H.1). Furthermore, by (H.8) and (H.9) we have $\Lambda (e_0){\hat{k}}_0^{1/2}=Uk_0^{1/2}={\hat{k}}_0^{1/2}$, from which $\Lambda (e_0)\ge {\hat{e}}_0$ follows. Applying the same argument to $\Lambda ^{-1}({\hat{x}})=U^*{\hat{x}}U$ (${\hat{x}}\in {\hat{{\mathcal {M}}}}{\hat{z}}_0$) with $k_0^{1/2},{\hat{k}}_0^{1/2}$ exchanged gives $\Lambda ^{-1}({\hat{e}}_0)\ge e_0$ as well. Therefore,

$$\begin{aligned} \Lambda (e_0)={\hat{e}}_0. \end{aligned}$$

(H.10)

Step 4. We define

$$\begin{aligned} (\Lambda ^{-1})_*:L^1({\mathcal {M}}z_0)=L^1({\mathcal {M}})z_0\,\rightarrow \,L^1({\hat{{\mathcal {M}}}}{\hat{z}}_0)=L^1({\hat{{\mathcal {M}}}}){\hat{z}}_0 \end{aligned}$$

by transforming $\psi \in ({\mathcal {M}}z_0)_*\mapsto \psi \circ \Lambda ^{-1}\in ({\hat{{\mathcal {M}}}}{\hat{z}}_0)_*$ via $L^1({\mathcal {M}}z_0)\cong ({\mathcal {M}}z_0)_*$ and $L^1({\hat{{\mathcal {M}}}}{\hat{z}}_0)\cong ({\hat{{\mathcal {M}}}}{\hat{z}}_0)_*$, that is, $(\Lambda ^{-1})_*:h_\psi \in L^1({\mathcal {M}}z_0)\mapsto h_{\psi \circ \Lambda ^{-1}}\in L^1({\hat{{\mathcal {M}}}}{\hat{z}}_0)$ for $\psi \in ({\mathcal {M}}z_0)_*$. Of course, $(\Lambda ^{-1})_*$ is an isometry with respect to $\left\| \cdot \right\| _1$. Now, Kosaki’s (symmetric) interpolation $L^p$-spaces enter into our discussions. Here, we confirm that

$$\begin{aligned} (\Lambda ^{-1})_*(k_0^{1/2}xk_0^{1/2})={\hat{k}}_0^{1/2}\Lambda (x){\hat{k}}_0^{1/2}, \qquad x\in {\mathcal {M}}z_0. \end{aligned}$$

(H.11)

Indeed, for every $x,y\in {\mathcal {M}}z_0$ we find that

which yields (H.11).

Step 5. Thanks to (H.11) we see that the isometry $(\Lambda ^{-1})_*:L^1({\mathcal {M}}z_0)\rightarrow L^1({\hat{{\mathcal {M}}}}{\hat{z}}_0)$ with respect to $\left\| \cdot \right\| _1$ is restricted to an isometry from $k_0^{1/2}({\mathcal {M}}z_0)k_0^{1/2}$ (embedded into $L^1({\mathcal {M}}z_0)$) onto ${\hat{k}}_0^{1/2}({\hat{{\mathcal {M}}}}{\hat{z}}_0){\hat{k}}_0^{1/2}$ (embedded into $L^1({\hat{{\mathcal {M}}}}{\hat{z}}_0)$) with respect to $\left\| \cdot \right\| _\infty$, i.e.,

$$\begin{aligned} \Vert k_0^{1/2}xk_0^{1/2}\Vert _\infty =\Vert x\Vert =\Vert \Lambda (x)\Vert =\Vert {\hat{k}}_0^{1/2}\Lambda (x){\hat{k}}_0^{1/2}\Vert _\infty ,\qquad x\in {\mathcal {M}}z_0. \end{aligned}$$

By Kosaki’s construction in [42] (or the Riesz–Thorin theorem), it follows that $(\Lambda ^{-1})_*$ gives rise to an isometry

$$\begin{aligned} (\Lambda ^{-1})_*:&\,L^p({\mathcal {M}}z_0,{\tilde{\sigma }}) =C_{1/p}((k_0^{1/2}({\mathcal {M}}z_0)k_0^{1/2},L^1({\mathcal {M}}z_0)) \\&\qquad \rightarrow \,L^p({\hat{{\mathcal {M}}}}{\hat{z}}_0,{\hat{\sigma }}) =C_{1/p}({\hat{k}}_0^{1/2}({\hat{{\mathcal {M}}}}{\hat{z}}_0){\hat{k}}_0^{1/2},L^1({\hat{{\mathcal {M}}}}{\hat{z}}_0)) \end{aligned}$$

with respect to the interpolation norms $\left\| \cdot \right\| _{p,{\tilde{\sigma }}}$ and $\left\| \cdot \right\| _{p,{\hat{\sigma }}}$ for any $p\in [1,+\infty )$. For every $x\in {\mathcal {M}}z_0$, applying this to $k_0^{1/2}xk_0^{1/2}$ with (H.11) gives

$$\begin{aligned} \Vert k_0^{1/2}xk_0^{1/2}\Vert _{p,{\tilde{\sigma }}}=\Vert {\hat{k}}_0^{1/2}\Lambda (x){\hat{k}}_0^{1/2}\Vert _{p,{\hat{\sigma }}}. \end{aligned}$$

By [42, Theorem 9.1], for every $p\in [1,+\infty )$ the above equality is rephrased as Haagerup’s $L^p$-norm equality

$$\begin{aligned} \Vert k_0^{1/2p}xk_0^{1/2p}\Vert _p=\Vert {\hat{k}}_0^{1/2p}\Lambda (x){\hat{k}}_0^{1/2p}\Vert _p, \end{aligned}$$

which is (H.2), as asserted. $\square$

We are now in a position to prove Theorem 4.3.

Proof of (i)

We use the variational expressions in Proposition 2.3 based on Lemma H.1. Assume first that $\alpha >1$. If $s({\tilde{\rho }})\not \le s({\tilde{\sigma }})$, then $s({\hat{\rho }})\not \le s({\hat{\sigma }})$ (as mentioned at the beginning of this appendix) so that both of $D_\alpha ^*({\tilde{\rho }}\Vert {\tilde{\sigma }})$ and $D_\alpha ^*({\hat{\rho }}\Vert {\hat{\sigma }})$ are $+\infty$. Hence, we assume that $s({\tilde{\rho }})\le s({\tilde{\sigma }})$ (and $s({\hat{\rho }})\le s({\hat{\sigma }})$). Using (2.7), we have

$$\begin{aligned}&Q_\alpha ^*({\tilde{\rho }}\Vert {\tilde{\sigma }})\\&\quad =\sup _{x\in {\mathcal {M}}_+}\Bigl [\alpha {\tilde{\rho }}(x) -(\alpha -1)\textrm{tr}\bigl (h_{{\tilde{\sigma }}}^{\alpha -1\over 2\alpha }xh_{\tilde{\sigma }}^{\alpha -1\over 2\alpha }\bigr )^{\alpha \over \alpha -1}\Bigr ] \\&\quad =\sup _{x\in ({\mathcal {M}}z_0)_+}\Bigl [\alpha {\tilde{\rho }}(x)-(\alpha -1) \textrm{tr}\bigl (h_{{\tilde{\sigma }}}^{\alpha -1\over 2\alpha }xh_{{\tilde{\sigma }}}^{\alpha -1\over 2\alpha } \bigr )^{\alpha \over \alpha -1}\Bigr ]\quad (\text{ since } s({\tilde{\rho }})\le s({\tilde{\sigma }})\le z_0)\\&\quad =\sup _{x\in ({\mathcal {M}}z_0)_+}\Bigl [\alpha {\hat{\rho }}(\Lambda (x))-(\alpha -1) \textrm{tr}\bigl (h_{{\hat{\sigma }}}^{\alpha -1\over 2\alpha }\Lambda (x)h_{{\hat{\sigma }}}^{\alpha -1\over 2\alpha } \bigr )^{\alpha \over \alpha -1}\Bigr ]\quad (\text{ by } \text{ Lemma } {H.1})\\&\quad =\sup _{{\hat{x}}\in ({\hat{{\mathcal {M}}}}{\hat{z}}_0)_+}\Bigl [\alpha {\hat{\rho }}({\hat{x}})-(\alpha -1) \textrm{tr}\bigl (h_{{\hat{\sigma }}}^{\alpha -1\over 2\alpha }\hat{x}h_{{\hat{\sigma }}}^{\alpha -1\over 2\alpha } \bigr )^{\alpha \over \alpha -1}\Bigr ] \\&\quad =Q_\alpha ^*({\hat{\rho }}\Vert {\hat{\sigma }}). \end{aligned}$$

Next, assume that $1/2\le \alpha <1$. When $s({\tilde{\rho }})\le s({\tilde{\sigma }})$, we use (2.8) as above to have

$$\begin{aligned}&Q_\alpha ^*({\tilde{\rho }}\Vert {\tilde{\sigma }}) \\&\quad =\inf _{x\in ({\mathcal {M}}z_0)_{++}}\Bigl [\alpha {\tilde{\rho }}(x)+(1-\alpha ) \textrm{tr}\bigl (h_{{\tilde{\sigma }}}^{1-\alpha \over 2\alpha }x^{-1}h_{{\tilde{\sigma }}}^{1-\alpha \over 2\alpha } \bigr )^{\alpha \over 1-\alpha }\Bigr ] \\&\quad =\inf _{x\in ({\mathcal {M}}z_0)_{++}}\Bigl [\alpha {\hat{\rho }}(\Lambda (x))+(1-\alpha ) \textrm{tr}\bigl (h_{{\hat{\sigma }}}^{1-\alpha \over 2\alpha }\Lambda (x^{-1})h_{{\hat{\sigma }}}^{1-\alpha \over 2\alpha } \bigr )^{\alpha \over 1-\alpha }\Bigr ]\quad (\text{ by } \text{ Lemma } {H.1})\\&\quad =\inf _{{\hat{x}}\in ({\hat{{\mathcal {M}}}}{\hat{z}}_0)_{++}}\Bigl [\alpha {\hat{\rho }}({\hat{x}})+(1-\alpha ) \textrm{tr}\bigl (h_{{\hat{\sigma }}}^{1-\alpha \over 2\alpha }{\hat{x}}^{-1}h_{{\hat{\sigma }}}^{1-\alpha \over 2\alpha } \bigr )^{\alpha \over 1-\alpha }\Bigr ]\quad (\text{ since } \Lambda (x^{-1})=\Lambda (x)^{-1}) \\&\quad =Q_\alpha ^*({\hat{\rho }}\Vert {\hat{\sigma }}). \end{aligned}$$

For general $\rho ,\sigma \in {\mathcal {A}}_+^*$, let $\sigma _\varepsilon :=\sigma +\varepsilon \rho$ for any $\varepsilon >0$. Then, $\sigma _\varepsilon$ has the normal extensions ${\tilde{\sigma }}_\varepsilon ={\tilde{\sigma }}+\varepsilon {\tilde{\rho }}$ to ${\mathcal {M}}$ and ${\hat{\sigma }}_\varepsilon ={\hat{\sigma }}+\varepsilon {\hat{\rho }}$ to ${\hat{{\mathcal {M}}}}$. The above case yields $Q_\alpha ^*({\tilde{\rho }}\Vert {\tilde{\sigma }}_\varepsilon )=Q_\alpha ^*({\hat{\rho }}\Vert {\hat{\sigma }}_\varepsilon )$ for all $\varepsilon >0$. From the continuity of $Q_\alpha ^*$ on ${\mathcal {M}}_*^+\times {\mathcal {M}}_*^+$ in the norm topology when $1/2\le \alpha <1$ (see [29, Theorem 3.16 (3)]), letting $\varepsilon \searrow 0$ gives $Q_\alpha ^*({\tilde{\rho }}\Vert {\tilde{\sigma }})=Q_\alpha ^*({\hat{\rho }}\Vert {\hat{\sigma }})$, implying (4.4).

Proof of (ii)

Assume first that $s({\tilde{\rho }})\le s({\tilde{\sigma }})$ (hence, $s({\hat{\rho }})\le s({\hat{\sigma }})$). Below let us use the same symbols as in the proof of Lemma H.1. Recall [3] that the relative modular operator $\Delta _{{\tilde{\rho }},{\tilde{\sigma }}}$ is defined as $\Delta _{{\tilde{\rho }},{\tilde{\sigma }}}:=S_{{\tilde{\rho }},{\tilde{\sigma }}}^*\overline{S_{{\tilde{\rho }},{\tilde{\sigma }}}}$, where $S_{{\tilde{\rho }},{\tilde{\sigma }}}$ is a closable conjugate linear operator defined by

$$\begin{aligned} S_{{\tilde{\rho }},{\tilde{\sigma }}}(xk_0^{1/2}+\zeta ):=e_0x^*h_0^{1/2},\qquad x\in {\mathcal {M}},\ \zeta \in (L^2({\mathcal {M}})e_0)^\perp . \end{aligned}$$

Similarly, $\Delta _{{\hat{\rho }},{\hat{\sigma }}}:=S_{{\hat{\rho }},{\hat{\sigma }}}^*\overline{S_{{\hat{\rho }},{\hat{\sigma }}}}$ is given, where

$$\begin{aligned} S_{{\hat{\rho }},{\hat{\sigma }}}({\hat{x}}{\hat{k}}_0^{1/2}+{\hat{\zeta }}):={\hat{e}}_0{\hat{x}}^*{\hat{h}}_0^{1/2},\qquad {\hat{x}}\in {\hat{{\mathcal {M}}}},\ {\hat{\zeta }}\in (L^2({\hat{{\mathcal {M}}}}){\hat{e}}_0)^\perp . \end{aligned}$$

Since $s(h_0)\le e_0\le z_0$, we can consider $S_{{\tilde{\rho }},{\tilde{\sigma }}}$ and $\Delta _{{\tilde{\rho }},{\tilde{\sigma }}}$ as operators on $L^2({\mathcal {M}})z_0$ (they are zero operators on $(L^2({\mathcal {M}})z_0)^\perp$). Similarly, $S_{{\hat{\rho }},{\hat{\sigma }}}$ and $\Delta _{{\hat{\rho }},{\hat{\sigma }}}$ are considered on $L^2({\hat{{\mathcal {M}}}}){\hat{z}}_0$. Let us use an isomorphism $\Lambda :{\mathcal {M}}z_0\rightarrow {\hat{{\mathcal {M}}}}{\hat{z}}_0$ and a unitary $U:L^2({\mathcal {M}})z_0\rightarrow L^2({\hat{{\mathcal {M}}}}){\hat{z}}_0$. Since $\overline{{\mathcal {M}}k_0^{1/2}}=L^2({\mathcal {M}})e_0$ and $\overline{{\hat{{\mathcal {M}}}}{\hat{k}}_0^{1/2}}=L^2({\hat{{\mathcal {M}}}}){\hat{e}}_0$, it follows from (H.9) that $U(L^2({\mathcal {M}})e_0)=L^2({\hat{{\mathcal {M}}}}){\hat{e}}_0$ and hence, $U((L^2({\mathcal {M}})e_0)^\perp )=(L^2({\hat{{\mathcal {M}}}}){\hat{e}}_0)^\perp$. For every ${\hat{x}}=\Lambda (x)\in {\hat{{\mathcal {M}}}}{\hat{z}}_0$ (with $x\in {\mathcal {M}}z_0$) and ${\hat{\zeta }}\in (L^2({\hat{{\mathcal {M}}}}){\hat{e}}_0)^\perp$, we find that

$$\begin{aligned} S_{{\hat{\rho }},{\hat{\sigma }}}({\hat{x}}{\hat{k}}_0^{1/2}+{\hat{\zeta }})&=Ue_0U^*Ux^*U^*Uh_0^{1/2}\quad (\text{ by } H.10, H.5 \hbox { and } H.8) \\&=Ue_0x^*h_0^{1/2}=US_{{\tilde{\rho }},{\tilde{\sigma }}}(xk_0^{1/2}+U^*{\hat{\zeta }}) \\&=US_{{\tilde{\rho }},{\tilde{\sigma }}}(U^*{\hat{x}}{\hat{k}}_0^{1/2}+U^*{\hat{\zeta }})\quad (\text{ by } (H.9)) \\&=US_{{\tilde{\rho }},{\tilde{\sigma }}}U^*({\hat{x}}{\hat{k}}_0^{1/2}+{\hat{\zeta }}). \end{aligned}$$

This implies that $S_{{\hat{\rho }},{\hat{\sigma }}}=US_{{\tilde{\rho }},{\tilde{\sigma }}}U^*$ and hence, $\Delta _{{\hat{\rho }},{\hat{\sigma }}}=U\Delta _{{\tilde{\rho }},{\tilde{\sigma }}}U^*$. Therefore, for every $\alpha \in [0,+\infty )$, ${\hat{k}}_0^{1/2}$ is in ${\mathcal {D}}(\Delta _{{\hat{\rho }},{\hat{\sigma }}}^{\alpha /2})$ if and only if $k_0^{1/2}=U^*{\hat{k}}_0^{1/2}$ is in ${\mathcal {D}}(\Delta _{{\tilde{\rho }},{\tilde{\sigma }}}^{\alpha /2})$, and in this case,

$$\begin{aligned} Q_\alpha ({\tilde{\rho }}\Vert {\tilde{\sigma }})=\Vert \Delta _{{\tilde{\rho }},{\tilde{\sigma }}}^{\alpha /2}k_0^{1/2}\Vert ^2 =\Vert \Delta _{{\hat{\rho }},{\hat{\sigma }}}^{\alpha /2}{\hat{k}}_0^{1/2}\Vert ^2=Q_\alpha ({\hat{\rho }}\Vert {\hat{\sigma }}). \end{aligned}$$

Otherwise, $Q_\alpha ({\tilde{\rho }}\Vert {\tilde{\sigma }})=Q_\alpha ({\hat{\rho }}\Vert {\hat{\sigma }})=+\infty$.

Next, assume that $s({\tilde{\rho }})\not \le s({\tilde{\sigma }})$, equivalently $s({\hat{\rho }})\not \le s({\hat{\sigma }})$. Then, $Q_\alpha ({\tilde{\rho }}\Vert {\tilde{\sigma }})=Q_\alpha ({\hat{\rho }}\Vert {\hat{\sigma }})=+\infty$ for $\alpha >1$. When $0\le \alpha <1$, let $\sigma _\varepsilon :=\sigma +\varepsilon \rho$ for every $\varepsilon >0$. The above case yields

$$\begin{aligned} Q_\alpha ({\tilde{\rho }}\Vert {\tilde{\sigma }})=\lim _{\varepsilon \searrow 0}Q_\alpha ({\tilde{\rho }}\Vert {\tilde{\sigma }}_\varepsilon ) =\lim _{\varepsilon \searrow 0}Q_\alpha ({\hat{\rho }}\Vert {\hat{\sigma }}_\varepsilon )=Q_\alpha ({\hat{\rho }}\Vert {\hat{\sigma }}), \end{aligned}$$

implying (4.5), where the continuity of $Q_\alpha$ in the second variable holds for $0\le \alpha <1$. $\square$

Remark H.2

The notion of standard f-divergences $S_f(\rho \Vert \sigma )$ with a parametrization of operator convex functions f on $(0,+\infty )$ has been studied in [27] in the von Neumann algebra setting. From the above proof of (ii), we observe that $S_f(\rho \Vert \sigma )$ can be extended to $\rho ,\sigma \in {\mathcal {A}}_+^*$ as

$$\begin{aligned} S_f(\rho \Vert \sigma ):=S_f(\rho _\pi \Vert \sigma _\pi ) \end{aligned}$$

independently of the choice of a $(\rho ,\sigma )$-normal representation $\pi$ of ${\mathcal {A}}$. Then, we can easily extend properties of $S_f(\rho \Vert \sigma )$ given in [27] to the $C^*$-algebra setting (like Proposition 4.5 for the sandwiched and the standard Rényi divergences).