2.1 States, channels and entropies

We briefly review some basic information-theoretic concepts in the noncommutative setting. Recall that a von Neumann algebra ${\mathcal M}$ is a unital $*$ -subalgebra of $B(H)$ closed under weak $^*$ -topology. A linear functional $\phi :{\mathcal M}\to \mathbb {C}$ is called a state if it is positive, meaning $\phi (x^*x)\ge 0$ for any $x\in {\mathcal M}$ , and additionally, $\phi (1)=1$ . We say $\phi $ is normal if $\phi $ is weak $^*$ -continuous. Throughout the paper, we will only consider normal states and denote $S({\mathcal M})$ as the normal state space of ${\mathcal M}$ . We write $s(\phi )$ as the support projection of a state $\phi $ , which is the minimal projection e such that $\phi (x)=\phi (exe)\hspace {.1cm}, \forall \hspace {.1cm} x\in {\mathcal M}$ . A normal state $\phi $ is faithful if $s(\phi )=1$ . For two normal states $\rho $ and $\sigma $ , the relative entropy is defined as

(2.1) $$ \begin{align}D(\rho||\sigma)=\begin{cases} \langle \xi_\rho| \log\Delta(\rho/\sigma) |\xi_\rho \rangle, & \mbox{if } s(\rho)\le s(\sigma)\\ +\infty, & \mbox{otherwise}. \end{cases},\end{align} $$

where $\xi _\rho $ is a vector implementing the state $\rho $ , and $\Delta (\rho /\sigma )$ is the relative modular operator. This form of definition (2.1) was introduced by Araki [Reference Araki2] for general von Neumann algebras.

In this section, we will focus on the case that ${\mathcal M}$ is a finite von Neumann algebra. Namely, ${\mathcal M}$ is equipped with a normal faithful tracial state $\tau $ . The tracial noncommutative $L_p$ -space $L_p({\mathcal M},\tau )$ is defined as the completion of ${\mathcal M}$ with respect to the p-norm $\parallel \! a \! \parallel _{p}=\tau (|a|^{p})^{1/p}$ . We identify $L_\infty ({\mathcal M})\cong {\mathcal M}$ , and also $ L_1({\mathcal M})\cong {\mathcal M}_*$ via the trace duality

$$ \begin{align*}d_\phi\in L_1({\mathcal M})\longleftrightarrow \phi\in {\mathcal M}_*,\hspace{.1cm} \phi(x)=\tau(d_\phi x).\end{align*} $$

We say $\rho \in L_1({\mathcal M})$ is a density operator if $\rho \ge 0$ and $\tau (\rho )=1$ , which corresponds to a normal state in the above identification. We will often identify normal states with their density operators if no ambiguity. Via this identification, relative entropy reduces to the original definition of Umegaki [Reference Umegaki76],

$$\begin{align*}D(\rho||\sigma)=\tau(\rho\log \rho-\rho\log \sigma ),\end{align*}$$

provided this trace is well defined. For example, for $\rho $ and $\sigma $ in the bounded state space

$$ \begin{align*} S_{B}({\mathcal M})=\{\rho\in S({\mathcal M})\hspace{.1cm} |\hspace{.1cm} \mu_1 1\le \rho \le \mu_2 1\text{ for some } \mu_1,\mu_2>0 \}, \end{align*} $$

the Umegaki’s formula is always well defined and finite. For this reason, we will mostly work with bounded states from $S_{B}({\mathcal M})$ and derive results for general case $S({\mathcal M})$ by approximation. When the second state $\sigma =1$ , this gives the entropy functional

$$\begin{align*}H(\rho):=D(\rho|| 1)=\tau(\rho\log\rho).\end{align*}$$

Note that the standard convention of von Neumann entropy in quantum information literature is often with an additional negative sign .

We say a linear map $T:{\mathcal M}\to {\mathcal M}$ is a quantum Markov map if T is normal, unital and completely positive. Recall that T is unital if $T( 1)=1$ . The pre-adjoint map $T_*:{\mathcal M}_*\to {\mathcal M}_*$ is called a quantum channel, which sends normal states to normal states. In the tracial setting, $T_*:L_1({\mathcal M})\to L_1({\mathcal M})$ given by

$$\begin{align*}\tau(T_*(\rho)y)=\tau(\rho T(y)), \hspace{.1cm} \forall\hspace{.1cm} y\in {\mathcal M}, \rho\in L_1({\mathcal M}),\end{align*}$$

is completely positive and trace-preserving (in short, CPTP). A fundamental inequality about quantum channel is the data processing inequality (also called monotonicity of relative entropy)

(2.2) $$ \begin{align} D(\rho||\sigma)\ge D(T_*(\rho)||T_*(\sigma)), \hspace{.1cm} \forall \rho,\sigma\in S({\mathcal M}). \end{align} $$

The data processing inequality states that two quantum states cannot become more distinguishable under a quantum channel. The data processing inequality remains valid for T being positive but not necessarily completely positive; see [Reference Müller-Hermes and Reeb56, Reference Frenkel27]. The main technical result of this work is an improved data processing inequality for quantum channels under symmetric conditions (Theorem 2.5).

2.2 Entropy contraction for unital quantum channels

We start our discussion on entropy contraction of unital quantum channels. The restriction of $\Phi $ on ${\mathcal M}$ is bounded and normal; thus, $\Phi $ can be viewed as the $L_1$ -norm extension of its restriction $\Phi :{\mathcal M}\to {\mathcal M}$ . By duality, its adjoint $\Phi ^*:{\mathcal M}\to {\mathcal M}$ is a trace-preserving quantum Markov map and hence also extends to a unital quantum channel.

For a state $\rho $ with $H(\rho )<\infty $ , we define the entropy difference of $\Phi $ ,

$$\begin{align*}D_\Phi(\rho):=H(\rho)-H(\Phi(\rho)).\end{align*}$$

Non-negativity of the entropy difference $D_\Phi (\rho )\ge 0$ follows from data processing inequality (2.2) and $\Phi (1)=1$ ,

$$\begin{align*}H(\rho)=D(\rho||1)\geq D(\Phi(\rho)||\Phi(1))= H(\Phi(\rho)).\end{align*}$$

We start with the key lemma in our argument.

Lemma 2.1 (Entropy difference lemma).

Let $\Phi :L_1({\mathcal M})\to L_1({\mathcal M})$ be a unital quantum channel and $\Phi ^*$ be its adjoint. Then for two bounded states $\rho ,\omega \in {S_B}({\mathcal M})$ ,

$$\begin{align*}D(\rho\|\Phi^*\Phi(\omega)) \le D_\Phi(\rho) + D(\rho\|\omega) \le \tau((\operatorname{\mathrm{\operatorname{id}}}-\Phi^*\Phi)(\rho)\ln \rho ) + D(\rho\|\omega) .\end{align*}$$

Proof. By duality, $\Phi ^*$ is also completely positive unital. Then,

$$ \begin{align*} D(\rho\|\Phi^*\Phi(\omega))=\hspace{.1cm}&\tau(\rho\ln \rho -\rho \ln \Phi^*\Phi(\omega)) \\ =\hspace{.1cm}&\tau(\rho\ln \rho-\Phi(\rho)\log \Phi(\rho))+\tau( \Phi(\rho)\log \Phi(\rho)-\rho \ln \Phi^*\Phi(\omega)) \\ =\hspace{.1cm}&D_\Phi(\rho)+\tau\big( \Phi(\rho)\log \Phi(\rho)-\rho \ln \Phi^*\Phi(\omega)\big) \\ \overset{(1)}{\le}\hspace{.1cm} &D_\Phi(\rho)+\tau\Big( \Phi(\rho)\log \Phi(\rho)-\rho \Phi^* \big(\ln \Phi(\omega)\big)\Big) \\ = \hspace{.1cm}&D_\Phi(\rho)+\tau\Big( \Phi(\rho)\log \Phi(\rho)-\Phi(\rho)\ln \Phi(\omega)\Big) \\ =\hspace{.1cm} &D_\Phi(\rho)+D(\Phi(\rho)\|\Phi(\omega)) \\ \overset{(2)}{\le}\hspace{.1cm} &D_\Phi(\rho)+D(\rho\|\omega), \end{align*} $$

where (2) follows from the monotonicity of relative entropy. The inequality (1) uses the operator concavity [Reference Choi18] of logarithm function $t\mapsto \ln t$ that for any positive operator $x\ge 0$ ,

$$\begin{align*}\Phi^* (\ln x)\le \ln \Phi^*(x).\end{align*}$$

This proves the first inequality. For the second part, it suffices to notice that

$$ \begin{align*} D_\Phi(\rho)=&\tau(\rho\log \rho-\Phi(\rho)\log \Phi(\rho)) \le \tau(\rho\log \rho-\Phi(\rho)\Phi(\log \rho)) =\tau(\rho\log \rho-\Phi^*\Phi(\rho)\log \rho), \end{align*} $$

where we use the operator concavity $\Phi (\ln x)\le \ln \Phi (x)$ again.

We iterate the above lemma as follows:

$$ \begin{align*}D(\rho|| (\Phi^* \Phi)^n(\rho))\le\hspace{.1cm} D_\Phi(\rho)+D(\rho|| (\Phi^* \Phi)^{n-1}(\rho)) \le\hspace{.1cm} n D_\Phi(\rho)+D(\rho|| \rho)=n D_\Phi(\rho). \end{align*} $$

Then a relevant question is what would be the limit of $(\Phi ^* \Phi )^n(\rho )$ as $n \to \infty $ . This leads to the multiplicative domain of $\Phi $ . Recall that the multiplicative domain of a unital completely positive map $\Phi $ is

$$\begin{align*}{\mathcal N}_\Phi=\{ x\in{\mathcal M} \ |\ \Phi(y) \Phi(x)=\Phi(yx),\hspace{.1cm} \Phi(x) \Phi(y)= \Phi(xy),\forall y\in\mathcal{M}\}.\end{align*}$$

When $\Phi $ is normal, ${\mathcal N}_\Phi \subset \mathcal {M}$ is a von Neumann subalgebra ([Reference Luczak52, Theorem 1]). A linear map ${E:{\mathcal M}\to {\mathcal M}}$ is called a conditional expectation if E is a unital completely positive map and idempotent $E\circ E=E$ . When ${\mathcal M}$ is a finite von Neumann algebra, for any subalgebra ${\mathcal N}\subset {\mathcal M}$ , there always exists a (unique) trace-preserving conditional expectation E onto ${\mathcal N}$ such that

(2.3) $$ \begin{align}\tau(xy)=\tau(xE(y)), \hspace{.1cm} x\in {\mathcal N}, y\in {\mathcal M}. \end{align} $$

Such E is a unital quantum channel.

Proof. It is clear that $\Phi $ is a $*$ -homomorphism on ${\mathcal N}$ . For any $x,y\in L_2({\mathcal N})\subset L_2({\mathcal M})$ ,

$$\begin{align*}\tau( y(\Phi^*\circ \Phi)(x))=\tau( \Phi(y) \Phi(x))=\tau(\Phi(xy))=\tau(xy).\end{align*}$$

Thus, $\Phi ^*\circ \Phi |_{{\mathcal N}}=\operatorname {\mathrm {\operatorname {id}}}_{\mathcal N}$ is the identity map. This verifies $(\Phi ^* \Phi )\circ E=E$ . Since $E^*=E$ , $E\circ (\Phi ^* \Phi )=E$ follows from taking the adjoint. Thus, $\Phi :{\mathcal N}\to \Phi ({\mathcal N})$ is a $*$ -isomorphism with inverse $\Phi ^*$ . Denoting ${\mathcal N}_0$ as the multiplicative domain for $\Phi ^*$ , we have $\Phi ({\mathcal N})\subset {\mathcal N}_0$ . Conversely, we also have $\Phi ^*(N_0)\subset {\mathcal N}$ by switching the role of $\Phi =(\Phi ^*)_*$ . Then $\Phi ({\mathcal N})={\mathcal N}_0$ since $\Phi $ is bijective on ${\mathcal N}$ . For ii), we note that by (2.4),

$$ \begin{align*}(\operatorname{\mathrm{\operatorname{id}}}-E)\Phi^* \Phi(\operatorname{\mathrm{\operatorname{id}}}-E)=(\operatorname{\mathrm{\operatorname{id}}}-E)(\Phi^* \Phi-E)=\Phi^* \Phi-E, \hspace{.1cm} &\hspace{.1cm} (\Phi^* \Phi-E)^n= (\Phi^* \Phi)^n-E. \end{align*} $$

Therefore,

$$ \begin{align*}&\parallel \! \Phi^* \Phi-E:L_2({\mathcal M})\to L_2({\mathcal M}) \! \parallel_{}=\parallel \! \Phi(\operatorname{\mathrm{\operatorname{id}}}-E) \! \parallel_{2}^2<1, \\ &\parallel \! (\Phi^* \Phi)^n-E:L_2({\mathcal M})\to L_2({\mathcal M}) \! \parallel_{}=\parallel \! (\Phi^* \Phi-E)^n:L_2({\mathcal M})\to L_2({\mathcal M}) \! \parallel_{}=\parallel \! \Phi(\operatorname{\mathrm{\operatorname{id}}}-E) \! \parallel_{2}^{2n}, \end{align*} $$

which goes to $0$ as $n\to \infty $ .

In order to estimate entropic quantities, we will use the approximation in terms of complete positivity. Recall that for a density operator $\sigma \in S({\mathcal M})$ with full support, the Bogoliubov-Kubo-Mori (BKM) metric for $X\in {\mathcal M}$ is defined by

$$ \begin{align*}\gamma_{\sigma}(X):=\int_{0}^\infty \tau(X^*(\sigma+s)^{-1}X(\sigma+s)^{-1})ds.\end{align*} $$

The BKM metric is a Riemannian metric on the space of states with full support that is monotone under any quantum channel $\Psi $ ,

$$\begin{align*}\gamma_{\Psi(\sigma)}(\Psi(X))\le \gamma_{\sigma}(X), \forall X\in {\mathcal M}. \end{align*}$$

It connects to the relative entropy as follows ([Reference Gao and Rouzé31, Lemma 2.2]):

(2.5) $$ \begin{align} D(\rho||\sigma)=\int_0^1 \int_{0}^s \gamma_{\rho_t}(\rho-\sigma)dtds=\int_0^1 (1-t) \gamma_{\rho_t}(\rho-\sigma)dt, \end{align} $$

where $\rho _t=t\rho +(1-t) \sigma $ for $t\in [0,1]$ . It is proved in [Reference Gao and Rouzé31, Lemma 2.1 & 2.2] that if $\rho \le c\sigma $ ,

(2.6) $$ \begin{align} &c\gamma_{\rho}(X)\le \gamma_{\sigma}(X), \hspace{.1cm} \forall X\in {\mathcal M} \nonumber\\ &k(c) \gamma_{\sigma}(\rho-\sigma) \le D(\rho||\sigma) \le \gamma_{\sigma}(\rho-\sigma), \end{align} $$

where $k(c)=\frac {c\ln c-c+1}{(c-1)^2}$ . The above discussion remains valid if $s(\rho )\le s(\sigma )$ and $X\in s(\sigma ){\mathcal M} s(\sigma )$ . For two positive maps $\Psi $ and $\Phi $ , we write $\Phi \le \Psi $ if $\Psi -\Phi $ is positive.

Lemma 2.3. Let E be a conditional expectation (not necessarily trace-preserving) and $\Psi $ be a quantum Markov map such that

$$\begin{align*}(1-\varepsilon)E\le \Psi \le (1+\varepsilon)E.\end{align*}$$

Assume that $E\circ \Psi =E$ . Then for any $\rho \in S({\mathcal M})$ ,

$$\begin{align*}D(\rho||\Psi_*(\rho))\ge \Big(\frac{1-\varepsilon}{1+\varepsilon}-\frac{\varepsilon}{(1-\varepsilon)k(2)}\Big) D(\rho||E_*(\rho)).\end{align*}$$

In particular, for $\varepsilon =\frac {1}{10}$ ,

$$\begin{align*}D(\rho||\Psi_*(\rho))\ge \frac{1}{2} D(\rho||E_*(\rho)).\end{align*}$$

Proof. By assumption, $\Psi _*=(1-\varepsilon )E_*+\varepsilon \Psi _0$ for some unital positive map $\Psi _0\le 2E_*$ . We denote $\sigma =E_*(\rho ), \tilde {\sigma }=\Phi _*(\rho )$ and $\omega =\Psi _0(\rho )$ . Then $\tilde {\sigma }=(1-\varepsilon )\sigma +\varepsilon \omega $ . Note that for any bounded state $\sigma \in S_B({\mathcal M})$ , $X\mapsto \sqrt {\gamma _\sigma (X)}$ is a Hilbert space norm. Then by the triangle inequality,

$$ \begin{align*}\sqrt{\gamma(\rho-\tilde{\sigma})}=\ &\sqrt{\gamma(\rho-(1-\varepsilon)\sigma-\varepsilon \omega)}\\ =\ &\sqrt{\gamma((\rho-\sigma)+\varepsilon (\sigma-\omega))} \\ \ge\ &\sqrt{\gamma(\rho-\sigma)}-\varepsilon\sqrt{\gamma(\sigma-\omega),} \end{align*} $$

where $\gamma $ can be $\gamma _\phi $ for any bounded state $\phi \in S_B({\mathcal M})$ . Then

$$ \begin{align*}\gamma(\rho-\tilde{\sigma}) \ge\ &\gamma(\rho-\sigma)-2\varepsilon\sqrt{\gamma(\rho-\sigma)}\sqrt{\gamma(\sigma-\omega)}+\varepsilon^2\gamma(\sigma-\omega)\\ \ge\ &\gamma(\rho-\sigma)-2\varepsilon\sqrt{\gamma(\rho-\sigma)}\sqrt{\gamma(\sigma-\omega)}\\ \ge\ &\gamma(\rho-\sigma)-\varepsilon\gamma(\rho-\sigma)-\varepsilon\gamma(\sigma-\omega)\\ =\ &(1-\varepsilon)\gamma(\rho-\sigma)-\varepsilon\gamma(\sigma-\omega). \end{align*} $$

Now take $\rho _t=t\rho +(1-t)\sigma $ and $\tilde {\rho }_t=t\rho +(1-t)\tilde {\sigma }$ ,

$$ \begin{align*} D(\rho||\tilde{\sigma})&=\int_0^1 (1-t)\gamma_{\tilde{\rho}_t}(\rho-\tilde{\sigma})dt \\ &\ge (1-\varepsilon)\int_0^1 (1-t) \gamma_{\tilde{\rho}_t}(\rho-\sigma)dt-\varepsilon\int_0^1(1-t) \gamma_{\tilde{\rho}_t}(\sigma-\omega)dt. \end{align*} $$

For the first term, because $\tilde {\rho }_t\le (1+\varepsilon )\rho _t$ ,

$$ \begin{align*} \int_0^1 (1-t) \gamma_{\tilde{\rho}_t}(\rho-\sigma)dt \ge (1+\varepsilon)^{-1}\int_0^1 (1-t) \gamma_{\rho_t}(\rho-\sigma)dtds=(1+\varepsilon)^{-1}D(\rho||\sigma). \end{align*} $$

For the second term, consider that $\tilde {\rho }_t\ge (1-\varepsilon )(1-t)\sigma $ ,

$$ \begin{align*} \int_0^1 (1-t)\gamma_{\tilde{\rho}_t}(\sigma-\omega)dt \le\ & \frac{1}{(1-\varepsilon)}\int_0^1 \gamma_{\sigma}(\omega-\sigma)dt\\ =\ &\frac{1}{(1-\varepsilon)}\gamma_{\sigma}(\omega-\sigma)\\ \overset{(1)}{\le}\ & \frac{1}{(1-\varepsilon)k(2)}D(\omega||\sigma) \\ =\ & \frac{1}{(1-\varepsilon)k(2)}D(\Psi_*(\rho)||\sigma)\overset{(2)}{\le} \frac{1}{(1-\varepsilon)k(2)}D(\rho||\sigma). \end{align*} $$

Here, the inequality (1) above uses $\omega \le 2\sigma $ and (2.6). The inequality (2) above follows from the monotonicity of relative entropy and the fact $\Psi _*(\sigma )=\sigma $ . Combining the estimated above, we obtained

$$\begin{align*}D(\rho||\tilde{\sigma})\ge \frac{1-\varepsilon}{1+\varepsilon} D(\rho||\sigma)-\varepsilon k(2)^{-1}D(\rho||\sigma)=\Big(\frac{1-\varepsilon}{1+\varepsilon}-\frac{\varepsilon}{(1-\varepsilon)k(2)}\Big)D(\rho||\sigma) ,\end{align*}$$

where $k(2)=2\ln 2-1$ . The above inequality is nontrivial for $\varepsilon $ such that

$$\begin{align*}\frac{1-\varepsilon}{1+\varepsilon}-\frac{\varepsilon}{(1-\varepsilon)k(2)}>0. \end{align*}$$

Taking $\varepsilon =0.1$ , the above expression is approximately $0.53>\frac {1}{2}$ .

Remark 2.4. This lemma is related to [Reference LaRacuente47, Corollary 2.15] and is a variant of [Reference Gao and Rouzé31, Theorem 5.3], which proves for GNS symmetric $\Phi $ ,

(2.7) $$ \begin{align}D(\rho||(\Phi_*)^2(\rho))\ge (1-\varepsilon^2 k(2)^{-1}) D(\rho||E_*(\rho)).\end{align} $$

Compared with [Reference LaRacuente47, Corollary 2.15], the above Lemma assumes a simpler condition and may achieve a stronger constant in certain regimes of interest. The above Lemma improves (2.7) from two points: 1) does not need any symmetric assumption; 2) remove the square in $\Phi _*^2$ . When $\Psi _*=\Phi _*^2$ is a square, (2.7) could yield better bound that for $\varepsilon =0.4$ ,

$$\begin{align*}(1-(0.4)^2 k(2)^{-1})>\frac{1}{2}>\frac{1}{4}> \frac{1-0.4^2}{1+0.4^2}-\frac{0.4^2}{(1-0.4^2)k(2)}.\end{align*}$$

Putting the above lemma together, we obtain the following entropy contraction of unital quantum channels.

Theorem 2.5. Let $\Phi $ be a unital quantum channel and let $E:{\mathcal M}\to {\mathcal N}$ be the trace-preserving conditional expectation onto the multiplicative domain $\mathcal {N}$ of $\Phi $ . Define the CB return time

(2.8) $$ \begin{align} k_{cb}(\Phi):=\inf \{k\in \mathbb{N}^+\hspace{.1cm} |\hspace{.1cm} 0.9 E \le_{cp} (\Phi^*\Phi)^{k}\le_{cp} 1.1 E \} \ . \end{align} $$

Then for any state $\rho \in S({\mathcal M})$ ,

(2.9) $$ \begin{align} D(\Phi(\rho)||\Phi\circ E(\rho))\le \Big (1-\frac{1}{2k_{cb}(\Phi)} \Big ) D(\rho||E(\rho))\ .\end{align} $$

Furthermore, for any finite von Neumann algebra $\mathcal {Q}$ and state $\rho \in S({\mathcal M} \overline {\otimes }\mathcal {Q}),$

(2.10) $$ \begin{align} D( \Phi\otimes \operatorname{\mathrm{\operatorname{id}}} (\rho)||(\Phi\circ E)\otimes \operatorname{\mathrm{\operatorname{id}}} (\rho))\le \Big (1-\frac{1}{2k_{cb}(\Phi)} \Big ) D(\rho|| E\otimes \operatorname{\mathrm{\operatorname{id}}} (\rho)).\end{align} $$

Proof. It suffices to consider a bounded state $\rho \in S_B({\mathcal M})$ . Note that by the conditional expectation property (2.3),

$$ \begin{align*}&D(\rho||E(\rho))=\tau(\rho\log \rho-\rho\log E(\rho))=\tau(\rho\log \rho)-\tau(E(\rho)\log E(\rho))=H(\rho)-H(E(\rho)),\\ &D(\Phi(\rho)||\Phi\circ E(\rho))\kern1.5pt{=}\kern1.5pt D(\Phi(\rho)||E_0\circ \Phi(\rho))\kern1.5pt{=}\kern1.5pt H(\Phi(\rho))\kern1.5pt{-}\kern1.5pt H(E_0 \circ \Phi(\rho)) \kern1.5pt{=}\kern1.5pt H(\Phi(\rho))\kern1.5pt{-}\kern1.5pt H(\Phi\circ E(\rho)), \end{align*} $$

where we used the property $\Phi \circ E=E_0 \circ \Phi $ from Proposition 2.2. Moreover, $H(E(\rho ))=H(\Phi \circ E(\rho ))$ as $\Phi $ is a trace-preserving $*$ -isomorphism on ${\mathcal N}$ . Thus, we have

$$\begin{align*}D_\Phi(\rho)=H(\rho)-H(\Phi(\rho))=D(\rho||E(\rho))-D(\Phi(\rho)||\Phi \circ E(\rho)).\end{align*}$$

Iterating the entropy difference Lemma 2.1, we have

$$ \begin{align*}D(\rho\|(\Phi^*\Phi)^{k}(\rho)) \hspace{.1cm} \le\hspace{.1cm} & D_\Phi(\rho) + D(\rho\|(\Phi^*\Phi)^{k-1}(\rho))\\ \le\hspace{.1cm} & k D_\Phi(\rho) + D(\rho\|\rho)\\ = \hspace{.1cm}& k (D(\rho||E(\rho))-D(\Phi(\rho)||\Phi \circ E(\rho)). \end{align*} $$

Now using Lemma 2.3, for $k=k_{cb}(\Phi )$ ,

$$\begin{align*}D(\rho||E(\rho))\le 2 D(\rho\|(\Phi^*\Phi)^{k}\rho))\le 2k\big (D(\rho||E(\rho))-D(\Phi(\rho)||\Phi \circ E(\rho))\big). \end{align*}$$

Rearranging the terms gives the assertion. The general case $\rho \in S({\mathcal M})$ can be obtained via approximation $\rho _\varepsilon =(1-\varepsilon )\rho +\varepsilon 1$ as [Reference Brannan, Gao and Junge14, Lemma A.2]. The same argument applies to $\operatorname {\mathrm {\operatorname {id}}}_{\mathcal Q}\otimes \Phi $ , because the CB return time $k_{cb}(\operatorname {\mathrm {\operatorname {id}}}_{\mathcal Q}\otimes \Phi )=k_{cb}(\Phi )$ is same as of $\Phi $ by the definition.

The above theorem is an improved data processing inequality for the relative entropy between a state $\rho $ and its conditional expectation $E(\rho )$ . Here, ${\mathcal N}$ is the ‘decoherence free’ subalgebra. Indeed, for any two states $\sigma _1,\sigma _2\in {\mathcal N},$

$$\begin{align*}D(\sigma_1||\sigma_2)\ge D(\Phi(\sigma_1)||\Phi(\sigma_2))\ge D(\Phi^*\Phi(\sigma_1)||\Phi^*\Phi(\sigma_2))=D(\sigma_1||\sigma_2)\end{align*}$$

does not decay. Outside the ‘decoherence free’ subalgebra ${\mathcal N}$ , the relative entropy from a state $\rho $ to its projection $E(\rho )$ on ${\mathcal N}$ is strictly contractive under every use of the channel $\Phi $ .

For $\Phi $ being a symmetric quantum Markov map, we have $\Phi =\Phi _*$ . Moreover, Proposition 2.2 reduces to

$$\begin{align*}\Phi\circ E=E\circ \Phi\hspace{.1cm} , \hspace{.1cm} \Phi^2\circ E= E\circ \Phi^2=E.\end{align*}$$

Then

$$ \begin{align*} D(\Phi^{2}(\rho)\|E(\rho))&=D(\Phi^{2}(\rho)\|\Phi^{2}\circ E(\rho))=D(\Phi^{2}(\rho)\|\Phi\circ E\circ \Phi (\rho))\\ &\leq (1-\frac{1}{2k_{cb}(\Phi)})D(\Phi(\rho)\|E\circ\Phi(\rho))=(1-\frac{1}{2k_{cb}(\Phi)})D(\Phi(\rho)\|\Phi\circ E(\rho))\\ &\leq (1-\frac{1}{2k_{cb}(\Phi)})^{2}D(\rho\|E(\rho)). \end{align*} $$

We can iterate the entropy contraction above and obtain the discrete time entropy decay,

$$ \begin{align*} D(\Phi^{2n}(\rho)|| E(\rho))\le (1-\frac{1}{2k_{cb}(\Phi)})^{2n} D(\rho||E(\rho)). \end{align*} $$

3.1 Functional inequalities

In this section, we discuss a continuous time relative entropy decay for symmetric quantum Markov semigroups. We first review some basics of quantum Markov semigroups. A quantum Markov semigroup $(T_t)_{t\ge 0}:{\mathcal M}\to {\mathcal M}$ is a family of maps satisfying

1. i) for each $t\ge 0$ , $T_t$ is a quantum Markov map (i.e., normal, completely positive and unital)

2. ii) $T_0=\operatorname {\mathrm {\operatorname {id}}}_{\mathcal M}$ and $T_s\circ T_t=T_{s+t}$ for any $s,t \ge 0$ .

3. iii) for $x\in {\mathcal M}$ , $t\mapsto T_t(x)$ is weak $^*$ -continuous.

The generator of the semigroup is defined as

$$\begin{align*}\hspace{.1cm} Lx=w^*\text{-}\lim_{t\to 0} \frac{1}{t}(x-T_t(x))\hspace{.1cm} \end{align*}$$

on the domain of L that the limit exists. In this section, we still consider ${\mathcal M}$ as a finite von Neumann algebra equipped with a normal faithful tracial state $\tau $ . Given $(T_t)_{t\ge 0}$ is symmetric (or more specifically, trace-symmetric), that is,

$$\begin{align*}\tau(x^*T_t(y))=\tau(T_t(x)^*y)\hspace{.1cm} , \hspace{.1cm} \forall x,y\in {\mathcal M}, t\ge 0,\end{align*}$$

the generator L is a positive, symmetric operator, densely defined on $L_2({\mathcal M})$ . Its kernel is the fixed-point subspace ${\mathcal N}:=\ker (L)=\{x\in {\mathcal M}\hspace {.1cm} | \hspace {.1cm} T_t(x)=x, \forall t\ge 0\}$ , which coincides with the common multiplicative domain of all $T_t$ – hence a von Neumann subalgebra. Moreover, each $T_t$ is an ${\mathcal N}$ -bimodule map

$$\begin{align*}T_t(axb)=aT_t(x)b, \hspace{.1cm} \forall\hspace{.1cm} a,b\in {\mathcal N} ,x\in {\mathcal M}.\end{align*}$$

In particular, we have

$$\begin{align*}T_t\circ E= E\circ T_t=E,\end{align*}$$

where $E:{\mathcal M}\to {\mathcal N}$ is the trace-preserving conditional expectation onto the fixpoint algebra ${\mathcal N}$ . We say $(T_t)$ is ergodic if ${\mathcal N}={\mathbb C} 1$ is trivial. Note that in the mathematical physics literature, it is common to use primitive instead of ergodic. In this case, the semigroup admits a unique invariant state – namely, the trace $\tau $ up to normalization. We will consider symmetric quantum Markov semigroups that are not necessarily ergodic.

Recall that a semigroup is equivalently determined by its Dirichlet form

$$\begin{align*}{\mathcal E}:L_2({\mathcal M})\to [0,\infty]\hspace{.1cm} ,\hspace{.1cm} {\mathcal E}(x,x)=\tau(x^*L x). \end{align*}$$

We write $\operatorname {\mathrm {\operatorname {dom}}} (L)$ for the domain of L and $\operatorname {\mathrm {\operatorname {dom}}} ({\mathcal E})$ for the domain of ${\mathcal E}$ . The Dirichlet subalgebra is defined as ${\mathcal A}_{\mathcal E}:=\operatorname {\mathrm {\operatorname {dom}}} ({\mathcal E})\cap {\mathcal M}$ . It was proved [Reference Davies and Lindsay21] that ${\mathcal A}_{\mathcal E}$ is a dense $*$ -subalgebra of ${\mathcal M}$ and a core of $L^{1/2}$ . We denote by

$$\begin{align*}S({\mathcal A}_{\mathcal E})=S({\mathcal M})\cap {\mathcal A}_{\mathcal E} , \hspace{.1cm} S_B({\mathcal A}_{\mathcal E})=S_B({\mathcal M})\cap {\mathcal A}_{\mathcal E}\end{align*}$$

the set of bounded density operators from ${\mathcal A}_{\mathcal E}$ . We now introduce the formal definitions of functional inequalities for quantum Markov semigroups.

The Poincaré inequality (3.1) is equivalent to the spectral gap of L as a positive operator. LSI (3.2) is equivalent to hypercontractivity [Reference Olkiewicz and Zegarlinski60]

(3.4) $$ \begin{align} \parallel \! T_t:L_2({\mathcal M})\to L_p({\mathcal M}) \! \parallel_{}\,{\le}\, 1\hspace{.1cm} \text{ if } \hspace{.1cm} p\le 1+e^{2\alpha t} .\end{align} $$

MLSI (3.3) is known to be equivalent to the exponential decay of relative entropy ([Reference Bardet5, Theorem 3.2] and [Reference Brannan, Gao and Junge14, Proposition A.3]) that

(3.5) $$ \begin{align} D(T_t(\rho)||E(\rho))\le e^{-2\alpha t}D(\rho||E(\rho)), \hspace{.1cm} \forall \rho\in S({\mathcal M}).\end{align} $$

The equivalence of (3.3) and (3.5) is obtained by differentiating the relative entropy for $T_t$ at $0$ , which leads to the entropy production on the R.H.S of MLSI

$$\begin{align*}I_L(\rho):={\mathcal E}(\rho,\ln\rho)=-\frac{d}{dt}\vert_{t=0} D(T_t(\rho)||E(\rho))=\tau(L(\rho) \ln \rho) .\end{align*}$$

It is well known that

$$\begin{align*}\alpha_{2}\leq\alpha_{1}\leq \lambda . \end{align*}$$

The main motivation to consider CMLSI over MLSI and LSI is the tensorization property

(3.6) $$ \begin{align} \alpha_{c}(L_1\otimes \operatorname{\mathrm{\operatorname{id}}}+\operatorname{\mathrm{\operatorname{id}}}\otimes L_2)=\min\{\alpha_{c}(L_1), \hspace{.1cm} \alpha_{c}(L_2)\},\end{align} $$

which in the quantum cases fails for $\alpha _1$ [Reference Brannan, Gao and Junge14, Section 4.4], and is only known to hold for $\alpha _2$ for limited examples in small dimensions. The main result of this section is Theorem 1.1, which asserts a lower bound

$$\begin{align*}\alpha_c(L)\ge \frac{1}{2t_{cb}(L)}\end{align*}$$

by the inverse of CB return time

(3.7) $$ \begin{align} t_{cb}(L)=\inf\{t>0 \hspace{.1cm} |\hspace{.1cm} (1-0.1) E\le_{cp} T_t\le_{cp} (1+0.1)E \}. \end{align} $$

Here, we set $\varepsilon =0.1$ for the notation $t_{cb}(\varepsilon )$ in Theorem 1.1 because of Lemma 2.3.

Theorem 3.2. Let $T_t=e^{-Lt}:{\mathcal M}\to {\mathcal M}$ be a symmetric quantum Markov semigroup and $E:{\mathcal M} \to {\mathcal N}$ be the trace-preserving conditional expectation onto the fixed point subalgebra ${\mathcal N}$ . Define the CB return time as

$$\begin{align*}t_{cb}(L)=\inf \Big \{ t>0\hspace{.1cm} |\hspace{.1cm} 0.9E\le_{cp} T_t\le_{cp} 1.1E \Big \}.\end{align*}$$

Then

$$\begin{align*}\frac{1}{2t_{cb}(L)}\hspace{.1cm} \le \hspace{.1cm} \alpha_{c}(L)\hspace{.1cm} \le \hspace{.1cm}\alpha_{1}(L).\end{align*}$$

Proof. Let $t_m=t_{cb}(L)/2m$ for some $m\in \mathbb {N}_+$ . Since $T_t$ is symmetric, $T_{t_m}^*T_{t_m}=T_{t_m}T_{t_m}=T_{2t_m}$ . Hence, $T_{t_m}$ has discrete return time $k_{cb}(T_{t_m})= m$ . By the Lemma 2.1, for any $\rho \in S_B({\mathcal M})$ ,

$$ \begin{align*}D(T_{t_m}(\rho)||E(\rho))\le & (1-\frac{1}{2m}) D(\rho||E(\rho)). \end{align*} $$

Write $t_{cb}=t_{cb}(L)$ . Now assume further $\rho \in \cup _{t>0} T_t({\mathcal M})\subset \operatorname {\mathrm {\operatorname {dom}}} (L)$ . We have by Theorem 2.5,

$$ \begin{align*}I(\rho)&=\lim_{t\to 0}\frac{ D(\rho||E(\rho))-D(T_{t}(\rho)||E(\rho))}{t}\\ &=\lim_{m\to \infty}\frac{ D(\rho||E(\rho))-D(T_{\frac{t_{cb}}{2m}}(\rho)||E(\rho))}{\frac{t_{cb}}{2m}}\\ &\ge \lim_{m\to \infty}\frac{\frac{1}{2m}D(\rho||E(\rho))}{\frac{t_{cb}}{2m}} =\frac{1}{t_{cb}} D(\rho||E(\rho)). \end{align*} $$

The entropy decay for general $\rho \in S({\mathcal M})$ can be obtained by approximation as in the Appendix [Reference Brannan, Gao and Junge14, Appendix]). This proves $\alpha _{1}(L)\ge \frac {1}{2t_{cb}(L)}$ . The same argument applies to $\operatorname {\mathrm {\operatorname {id}}}_{\mathcal Q}\otimes L$ yields the assertion $\alpha _{c}(L)\ge \frac {1}{2t_{cb}(L)}$ .

3.2 CB return time

We now consider a common scenario where the CB return time $t_{cb}$ is finite. The original motivation for the notion, despite defining using CP (completely positive) order (3.7), is the following characterization using CB (completely bounded) norm. Recall that a linear map $\Psi :{\mathcal M}\to {\mathcal M}$ is called a ${\mathcal N}$ -bimodule map if

$$\begin{align*}\Psi(axb)=a\Psi(x)b, \hspace{.1cm} \forall\hspace{.1cm} a,b\in {\mathcal N}, x\in {\mathcal M}.\end{align*}$$

The condition ii) above is the completely bounded norm from the space $L_\infty ^1({\mathcal N}\subset {\mathcal M})$ to ${\mathcal M}$ . $L_\infty ^1({\mathcal N}\subset {\mathcal M})$ is called a conditional $L_\infty $ space, defined as the completion of ${\mathcal M}$ with respect to the norm

$$\begin{align*}\parallel \! x \! \parallel_{L_\infty^1({\mathcal N}\subset{\mathcal M})}=\sup_{a,b\in{\mathcal N} \hspace{.1cm} ,\hspace{.1cm} \|a\|_2=\|b\|_2=1}\parallel \! axb \! \parallel_{1}, \end{align*}$$

where the supremum takes over all $a,b\in L_2({\mathcal N})$ with $\|a\|_2=\|b\|_2=1$ . The operator space structure of $L_\infty ^1({\mathcal N}\subset {\mathcal M})$ is given by the identification

$$\begin{align*}{\mathbb M}_n(L_\infty^1({\mathcal N}\subset{\mathcal M}))=L_\infty^1({\mathbb M}_n({\mathcal N})\subset {\mathbb M}_n({\mathcal M}))\hspace{.1cm}\end{align*}$$

(see [Reference Junge and Parcet42] and [Reference Gao, Junge and LaRacuente30, Appendix]). Proposition 3.4 is relatively self-evident in the ergodic case ${\mathcal N}=\mathbb {C}1$ , $L_\infty ^1({\mathcal N}\subset {\mathcal M})\cong L_1({\mathcal M})$ , which we illustrate below.

Example 3.5 (Classical case).

Let $(\Omega ,\mu )$ be a probability space. Let $P:L_\infty (\Omega )\to L_\infty (\Omega )$ be a linear map with kernel $P(f)(x)=\int _{\Omega }k(x,y)f(y)d\mu (y)$ . It is clear that P is $*$ -preserving (i.e., $P(\bar {f})=\overline {P(f)}$ if k is real); P is a positive map if and only if the kernel function $k\ge 0$ . Recall the expectation map

$$ \begin{align*}E_\mu:L_\infty(\Omega)\to \mathbb{C}\textbf{1}\hspace{.1cm} , \hspace{.1cm} E(f)=(\int_{\Omega}f\mu)\textbf{1},\end{align*} $$

where $\textbf {1}$ is the unit constant function. The kernel of $E_\mu $ is the constant function $ \textbf {1}$ on the product space $\Omega \times \Omega $ . The following equivalence is self-evident:

(3.8) $$ \begin{align} (1-\varepsilon)E\le P \le (1+\varepsilon)E \nonumber \Longleftrightarrow\hspace{.1cm} &\varepsilon E\le P-E \le \varepsilon E \nonumber \\ \Longleftrightarrow\hspace{.1cm} &\varepsilon \textbf{1}\le k-\textbf{1} \le \varepsilon \textbf{1}\nonumber \\ \Longleftrightarrow\hspace{.1cm} &\parallel \! k-\textbf{1} \! \parallel_{L_\infty(\Omega\otimes \Omega)}\le \varepsilon\nonumber \\ \Longleftrightarrow\hspace{.1cm} &\parallel \! P-E:L_1(\Omega)\to L_\infty(\Omega) \! \parallel_{}\le \varepsilon. \end{align} $$

To see the equivalence in terms of complete positivity and completely bounded norm in Proposition 3.4, it suffices to notice that every positive (resp. bounded) map to $L_\infty (\Omega )$ is automatically completely positive (resp. completely bounded with same norm [Reference Smith70]).

Example 3.6 (Quantum case).

The above argument also applies to the noncommutative ergodic case ${\mathcal N}=\mathbb {C}1\subset {\mathcal M}$ . The correspondence between the map P and its kernel k generalizes to the isomorphism between the map T and its Choi operator $C_T\in {\mathcal M}^{op}\overline {\otimes }{\mathcal M}$

$$\begin{align*}T(x)=\tau \otimes \operatorname{\mathrm{\operatorname{id}}} (C_T(x\otimes 1)),\hspace{.1cm} x\in L_1({\mathcal M})\cong ({\mathcal M}^{op})_*,\end{align*}$$

where ${\mathcal M}^{op}$ is the opposite algebra of ${\mathcal M}$ . The isomorphism $T\mapsto C_T$ is not only positivity-preserving by Choi’s Theorem ( T is CP iff $C_T\ge 0$ ), but also isometric by Effros-Ruan Theorem (see [Reference Effros and Ruan25, Reference Blecher and Paulsen10]),

$$\begin{align*}\parallel \! T:L_1({\mathcal M})\to L_\infty({\mathcal M}) \! \parallel_{cb}=\parallel \! C_T \! \parallel_{{\mathcal M}^{op}\overline{\otimes}{\mathcal M}}.\end{align*}$$

Then the equivalence in Proposition 3.4 follows as (3.8).

For the general case of a ${\mathcal N}$ -bimodule map T with a nontrivial ${\mathcal N}$ , the above isomorphism holds with more involved module Choi operator, which we refer to the discussion in Section 4.3 and also [Reference Bardet, Capel, Gao, Lucia, Pérez-García and Rouzé6, Lemma 5.1] and [Reference Gao, Junge and LaRacuente29, Lemma 3.15] for the complete proof of Proposition 3.4.

With Proposition 3.4, the CB-return time can be equivalently defined as

(3.9) $$ \begin{align} t_{cb}(L):=\inf \{\hspace{.1cm} t>0\hspace{.1cm} |\hspace{.1cm} \parallel \! T_{t}-E:L_\infty^1({\mathcal N}\subset{\mathcal M})\to L_\infty({\mathcal M}) \! \parallel_{cb}\le 0.1 \}. \end{align} $$

It is known that $t_{cb}$ is finite whenever $T_t$ satisfies the Poincaré inequality and one-point ultra-contractive estimate.

Remark 3.8. For the special case of $t_0=0$ and $T_0=\operatorname {\mathrm {\operatorname {id}}}:{\mathcal M}\to {\mathcal M}$ , we consider

$$\begin{align*}\parallel \! \operatorname{\mathrm{\operatorname{id}}}: L_\infty^1({\mathcal N}\subset {\mathcal M})\to L_\infty({\mathcal M}) \! \parallel_{cb}=\inf \{\hspace{.1cm} \mu>0\hspace{.1cm} |\hspace{.1cm} \operatorname{\mathrm{\operatorname{id}}} \le_{cp} \mu E\}:=C_{cb}(E).\end{align*}$$

$C_{cb}(E)$ was introduced in [Reference Gao, Junge and LaRacuente30] as the complete bounded version of Popa and Pimsner’s subalgebra index [Reference Pimsner and Popa67],

$$\begin{align*}C(E):=\inf\{\mu>0 \hspace{.1cm} |\hspace{.1cm} \rho\le \mu E\rho, \hspace{.1cm} \forall \rho\in {\mathcal M}_{+}\}, \hspace{.1cm} C_{cb}(E):=\sup_{n}C(E\otimes \operatorname{\mathrm{\operatorname{id}}}_{\mathbb{M}_n} ).\end{align*}$$

When ${\mathcal M}$ is finite dimensional, both the index $C(E)$ and $C_{cb}(E)$ are finite and admit the explicit formula [Reference Pimsner and Popa67, Theorem 6.1]. In this case, one can take $t_0=0$ in above Proposition 3.7 and yields

$$\begin{align*}t_{cb}\hspace{.1cm} \le \hspace{.1cm} \frac{\ln (10 C_{cb}(E))}{\lambda}.\end{align*}$$

3.3 Classical Markov semigroups

In the remainder of this section, we focus on applications toward classical Markov map. We postpone the discussion of truly noncommutative semigroups to Section 4. Let $T_t=e^{-Lt}:L_\infty (\Omega ,\mu )\to L_\infty (\Omega ,\mu )$ be an ergodic Markov semigroup symmetric to the probability measure $\mu $ . Note that in the ergodic case $L_\infty ^1(\mathbb {C}1\subset L_\infty (\Omega ) )=L_1(\Omega ,\mu )$ , and by Smith’s lemma [Reference Smith70], any bounded map $T:L_1(\Omega ,\mu )\to L_\infty (\Omega ,\mu )$ is automatic completely bounded

$$\begin{align*}\parallel \! T: L_1(\Omega,\mu)\to L_\infty(\Omega,\mu) \! \parallel_{}=\parallel \! T: L_1(\Omega,\mu)\to L_\infty(\Omega,\mu) \! \parallel_{cb}. \end{align*}$$

Then the CB return time $t_{cb}$ reduces to the standard $L_\infty $ -mixing time

$$ \begin{align*} t_{b}(\varepsilon)=&\inf \{ \hspace{.1cm} t>0\hspace{.1cm} |\hspace{.1cm} \parallel \! T_t-E:L_1(\Omega,\mu)\to L_\infty(\Omega,\mu) \! \parallel_{}\le \varepsilon \}. \end{align*} $$

Then by a combination of Theorem 1.1 and Proposition 3.7, we obtain the following theorem.

Theorem 3.9. Let $T_t=e^{-tL}:L_\infty (\Omega ,\mu )\to L_\infty (\Omega ,\mu )$ be an ergodic Markov semigroup symmetric to the probability measure $\mu $ . Suppose

1. i) $T_t$ satisfies $\lambda $ -Poincaré inequality for some $\lambda>0$ : for $f\in \operatorname {\mathrm {\operatorname {dom}}}(L^{1/2})$ ,

   (3.10) $$ \begin{align} \lambda\mu(|f-E_\mu(f)|^2)\le \int f(Lf)d\mu .\end{align} $$

2. ii) There exists $t_0>0$ such that

   (3.11) $$ \begin{align}\parallel \! T_{t_0}: L_1(\Omega,\mu)\to L_\infty(\Omega,\mu) \! \parallel_{}\le C_0.\end{align} $$

Then

(3.12) $$ \begin{align}\alpha_1\ge \alpha_{c}\ge \frac{\lambda}{2(\lambda t_0+\ln C_0+2 )}. \end{align} $$

This result can be compared to the bound of Diaconis and Saloff-Coste [Reference Diaconis and Saloff-Coste23, Theoem 3.10], which statesFootnote ¹

(3.13) $$ \begin{align} \alpha_1\ge \alpha_{2}\ge \frac{\lambda}{\lambda t_0+\ln (C_0)+1}. \end{align} $$

In particular, $\alpha _1\ge \alpha _{2}\ge \frac {2}{t_b(e^{-2})}$ for the alternative $L_\infty $ -mixing time

$$ \begin{align*} t_b(e^{-2})=\inf \{ t>0\hspace{.1cm} |\hspace{.1cm} \parallel \! T_t-E_\mu:L_1(\Omega,\mu)\to L_\infty(\Omega,\mu) \! \parallel_{}\le \frac{1}{e^2}\}. \end{align*} $$

By the comparison $e^{-3}<0.1<e^{-2}$ , we have $t_b(e^{-2})\le t_{b}(0.1)\le \frac {3}{2}t_b(e^{-2})$ . Hence, in terms of lower bound for $\alpha _1$ , (3.13) and (3.12) are equivalent up to absolute constants. The difference is that (3.13) lower bounds the LSI constant $\alpha _2$ and our estimate (3.12) bounds the CMLSI constant $\alpha _{c}$ .

For finite Markov chains with $|\Omega |<\infty $ , we have finite index

$$\begin{align*}C_{cb}(E_\mu)=C(E_\mu)=\inf\{C>0\hspace{.1cm} |\hspace{.1cm} f\le C\mu(f) \hspace{.1cm} \forall f\ge 0\}= \parallel \! \mu^{-1} \! \parallel_{\infty},\end{align*}$$

where $\mu $ is a strictly positive probability density function. It was proved in [Reference Diaconis and Saloff-Coste23] that

(3.14) $$ \begin{align} &\frac{1}{t_b(e^{-2})}\le \lambda \le \frac{2+\log\parallel \! \mu^{-1} \! \parallel_{\infty}}{2t_b(e^{-2})},\qquad \end{align} $$

(3.15) $$ \begin{align} &\frac{1}{t_b(e^{-2})}\le \alpha_2 \le \frac{4+\log\log \parallel \! \mu^{-1} \! \parallel_{\infty}}{2t_b(e^{-2})}. \end{align} $$

Combined with our Theorem 1.1, we obtain the following:

Corollary 3.10. For a finite Markov chain $T_t:l_\infty (\Omega ,\mu )\to l_\infty (\Omega ,\mu )$ symmetric to $\mu $ ,

$$\begin{align*}\min \Big \{ \frac{4\alpha_2}{3(4+\log\log \parallel \! \mu^{-1} \! \parallel_{\infty})}, \frac{\lambda}{2\log (10\parallel \! \mu^{-1} \! \parallel_{\infty})} \Big \} \le \alpha_c\le \alpha_1\le \lambda.\end{align*}$$

Proof. Note that $t_b(e^{-2})\le t_{cb}(0.1)\le \frac {3}{2}t_b(e^{-2})$ . Then by Theorem 1.1 and (3.15),

$$\begin{align*}\alpha_c\ge \frac{1}{2t_{cb}(0.1)}\ge \frac{1}{3t_{b}(e^{-2})} \ge \frac{2\alpha_2}{3(4+\log\log \parallel \! \mu^{-1} \! \parallel_{\infty})}.\end{align*}$$

The other lower bound $\alpha _c\ge \frac {\lambda }{2\log (10\parallel \! \mu ^{-1} \! \parallel _{\infty })}$ follows from Theorem 3.9 by choosing $t_0=0$ .

Example 3.11. When $\Omega $ is not finite, here is a simple example with $\alpha _c(L)>0$ , but the ultra-contractivity (3.11) is never satisfied for finite $t_0$ . Take $L=I-E_\mu $ . It generates the so-called depolarizing semigroup

$$\begin{align*}T_t=e^{-t}\operatorname{\mathrm{\operatorname{id}}}+(1-e^{-t})E_\mu, \hspace{.1cm} T_t(f)= e^{-t}f+(1-e^{-t})\mu(f)\textbf{1},\end{align*}$$

where $\textbf {1}$ is the unit constant function. Then for any $t<\infty $ ,

$$\begin{align*}\parallel \! T_{t}-E_\mu: L_1(\Omega,\mu)\to L_\infty(\Omega,\mu) \! \parallel_{}=\parallel \! e^{-t}\operatorname{\mathrm{\operatorname{id}}}: L_1(\Omega,\mu)\to L_\infty(\Omega,\mu) \! \parallel_{}=e^{-t}C(E_\mu),\end{align*}$$

which is infinite whenever $L_{\infty }(\Omega ,\mu )$ is infinite dimensional. However, it follows from direct calculation that $\alpha _{c}(I-E_\mu )\geq \frac {1}{2}$ .

3.4 Hörmander system

We now discuss the application to Markov semigroups on smooth manifolds generated by sub-Laplacians. Let $(M,g)$ be a d-dimensional compact connected Riemannian manifold without boundary and let $d\mu =\omega d\operatorname {vol}$ be a probability measure with smooth density $\omega $ w.r.t the volume form $d\operatorname {vol}$ . A family of vector fields $H=\{X_i\}_{i=1}^k\subset TM$ with $k\leqslant d$ is called a Hörmander system if at every point $x\in M$ , the tangent space at x can be spanned by the iterated Lie brackets of $X_i$ ’s

(Hörmander condition) $$ \begin{align} T_xM=\operatorname{span}\{[X_{i_1},[X_{i_2},\cdots, [X_{i_{n-1}}, X_{i_n}]]] \hspace{.1cm} | \hspace{.1cm} 1\leqslant i_1,i_2\cdots i_n\leqslant k \}. \end{align} $$

By compactness, we can assume there is a global constant $l_H$ such that for every point $x\in M$ , we need at most $l_H$ th iterated Lie bracket in above expression (also called strong Hörmander condition). Denote $\nabla =(X_1, \cdots , X_k)$ and by $X_i^*$ the adjoint of $X_i$ on $L^{2}(M,d\mu )$ . Under the Hörmander condition, the sub-Laplacian

$$\begin{align*}\Delta_H=\nabla^*\nabla= \sum_{i} X_i^*X_i=-\sum_{i}X_i^2+(\text{div}_{\mu}(X_i)+X_i(\ln \omega))X_i \end{align*}$$

is a symmetric operator on $L^{2}(M,d\mu )$ which generates an ergodic Markov semigroup $P_t=e^{-\Delta _H t}$ , often called the horizontal heat semigroup. Here, $\operatorname {div}_{\mu }(X)$ is the divergence of X w.r.t to $\mu $ . When $H=\{X_i\}_{i=1}^d$ forms an orthonormal frame to the Riemannian metric, $\Delta _H=\Delta $ recovers the (weighted) Laplace-Beltrami operator and $P_t=e^{-\Delta t}$ is the (weighted) heat semigroup on M.

The gradient form (Carré du Champ operator) of $\Delta _H$ is given by

$$\begin{align*}\Gamma(f,g):=\frac{1}{2}(f\Delta_H(g)+\Delta_H(f)g-\Delta_H(fg))= \sum_{i}\langle X_i f,X_ig\rangle. \end{align*}$$

It follows from the product rule of derivatives that $\Gamma $ is diffusive (i.e., $\Gamma (fg,h)=f\Gamma (g,h)+g\Gamma (f,h)$ ). For diffusion semigroups, it is known [Reference Bakry, Gentil and Ledoux4, Theorem 5.2.1] that the MLSI constant $\alpha _1$ and the LSI constant $\alpha _2$ coincide (i.e., $\alpha :=\alpha _1=\alpha _2$ ). The positivity

$$\begin{align*}\alpha(\Delta_H)>0 \hspace{.1cm} \end{align*}$$

for any Hörmander’s system $H=\{X_i\}_{i=1}^k$ on a compact connected Riemannian manifold without boundary was proved in [Reference Lugiewicz and Zegarliński53, Theorem 3.1]. Our Theorem 1.2 improves this to $\alpha _c(\Delta _H)>0$ .

Proof of Theorem 1.2.

Recall the following Sobolev-type inequality (see, for example, [Reference Lugiewicz and Zegarliński53, Lemma 2.1]):

(3.16) $$ \begin{align} \|f\|_{q} \hspace{.1cm} \le \hspace{.1cm} C \big( \langle \Delta_H f,f\rangle + \parallel \! f \! \parallel_{2}^2\big)^{1/2} , \end{align} $$

where $q=\frac {2dl_H}{dl_H-2}>2$ and $l_H$ is globoal Lie bracket length needed in the strong Hörmander condition. By Varopoulos’ Theorem (see [Reference Varopoulos, Saloff-Coste and Coulhon77, Chapter 2]) on the dimension of semigroups, this implies the following ultra-contractive estimate:

(3.17) $$ \begin{align} \parallel \! e^{-\Delta_Ht}:L_{1}(M,\mu)\to L_{\infty}(M,\mu) \! \parallel_{}\leqslant C^{\prime} t^{-m/2} \text{ for } 0 \le t \le 1 \text{ and some }C'>0, \end{align} $$

where $m=dl_{H}$ . Also, it was proved in [Reference Lugiewicz and Zegarliński53, Theorem 2.3] that $\Delta _H$ satisfies the Poincaré inequality: $\lambda (\Delta _H)>0$ . Combining these with our Theorem 3.9 yields the assertion.

The Sobolev-type inequality (3.16) is also used in [Reference Lugiewicz and Zegarliński53] by Lugiewicz and Zegarlínski to prove that $\alpha _2(\Delta _H)>0$ . Their proof relies on the Rothaus lemma, and so does the discrete case by Diaconis and Saloff-Coste [Reference Diaconis and Saloff-Coste23]. However, we will see in Section 3.6 that this approach is out of scope for showing the CMLSI constant $\alpha _c(\Delta _H)>0$ .

Example 3.12. The special unitary group $\operatorname {SU}\left ( 2 \right )$ is

$$\begin{align*}\operatorname{SU}\,\left( 2 \right)=\{ cI+xX+yY+zZ: c^2+x^2+y^2+z^2=1, x,y,x, c \in \mathbb{R} \}, \end{align*}$$

where $X,Y,Z$ are the skew-Hermitian Pauli unitary

$$\begin{align*}X=\left[\begin{array}{cc} 0& 1\\ -1& 0 \end{array}\right],Y=\left[\begin{array}{cc} 0& i\\ i& 0 \end{array}\right], Z=\left[\begin{array}{cc} i& 0\\ 0& -i \end{array}\right]. \end{align*}$$

The Lie algebra is $\mathfrak {su}(2)=\operatorname {span}_{\mathbb {R}}\{X,Y,Z\}$ with Lie bracket rules as

(3.18) $$ \begin{align}[X,Y]=2Z\ , [Y,Z]=2X\ , [Z,X]=2Y. \end{align} $$

The canonical sub-Riemannian structure is given by $H=\{X,Y\}$ , which is a generating set of $\mathfrak {g}$ because $[X,Y]=2Z$ . The associated sub-Laplacian is

(3.19) $$ \begin{align} \Delta_H=-(X^2+Y^2). \end{align} $$

The semigroup $P_t=e^{-\Delta _H t}$ on $\operatorname {SU}\left ( 2 \right )$ has been studied as a prototype of horizontal heat semigroups. In particular, Baudoin and Bonnefont in [Reference Baudoin and Bonnefont8] proved that

(3.20) $$ \begin{align} \Gamma(P_t f,P_t f)\leqslant C e^{-4 t}P_t(\Gamma(f, f)), \end{align} $$

for some constant $C>0$ . In [Reference Gao and Rouzé31], Gao and Gordina based on (3.20) proved the CMLSI constant that

$$\begin{align*}\alpha_c(\Delta_H)\ge (2\int_0^\infty C e^{-4 t}dt )^{-1}=\frac{2}{C}.\end{align*}$$

The gradient estimate (3.20), as a weaker variant of Bakry-Emery curvature dimension condition, has been found useful to derive CMLSI in [Reference Gao and Rouzé31]. Nevertheless, this weaker gradient estimate is only known for only a limited number of examples in the sub-Riemannian setting [Reference Driver and Melcher24, Reference Melcher54]. Our result avoids this condition and obtains CMLSI for general Hörmander systems.

Example 3.13. Let $n\ge 3$ . The special unitary group $\operatorname {SU}\left ( n \right )$ is

$$\begin{align*}\operatorname{SU}\left( n \right)=\{ u\in {\mathbb M}_n\hspace{.1cm}|\hspace{.1cm} u^*u=1, \hspace{.1cm} \det(u)=1\}. \end{align*}$$

The Lie algebra $\mathfrak {su}(n)$ is the space of all the skew-Hermitian matrices, and a natural basis $\mathfrak {su}(n)$ is given by $\{X_{j,k},Y_{j,k},Z_{k} \hspace {.1cm} | \hspace {.1cm} 1\le j< k\le n\}$ where

$$\begin{align*}X_{j,k}=e_{jk}-e_{kj}\hspace{.1cm} ,\hspace{.1cm} Y_{j,k}=i(e_{jk}+e_{kj})\hspace{.1cm} ,\hspace{.1cm} Z_k=i(e_{11}-e_{kk}), \end{align*}$$

which is $n^2-1$ dimensional. Let $V=\{1,\cdots , n\}$ be a vertex set and $E\subset V\times V$ as an edge set. The set

$$\begin{align*}H_E=\{X_{j,k},Y_{j,k} \hspace{.1cm} |\hspace{.1cm} (j,k)\in E\}\end{align*}$$

is a generating set if and only if $(V,E)$ is a connected graph. The associated sub-Laplacian

$$\begin{align*}\Delta_E=-\sum_{(j,k)\in E} X_{j,k}^2+Y_{j,k}^2\hspace{.1cm} \end{align*}$$

is a generalization of (3.19). Theorem (1.2) implies that $\alpha _c(\Delta _E)>0$ for all connected $(V,E)$ , despite the gradient estimate (3.20) is not known for this type of generator.

3.5 Transference semigroups

Let us discuss an immediate application of $\alpha _c(\Delta _H)>0$ to symmetric Quantum Markov semigroups. Let G be a compact Lie group and $H=\{X_1,\cdots ,X_k\}$ be a generating set of its Lie algebra $\mathfrak {g}$ . Then $\{X_1,\cdots ,X_k\}$ satisfies the Hörmander condition, and its sub-Laplacian $\Delta _H=-\sum _{k}X_i^2$ generates a Markov semigroup $P_t=e^{-\Delta _H t}$ symmetric to the Haar measure. Let $u:G\to {\mathbb M}_n$ be a finite dimensional unitary representation and $d_u:\mathfrak {g}\to i({\mathbb M}_n)_{s.a.}$ be the corresponding Lie algebra morphism. $P_t=e^{-\Delta _H t}$ induces a quantum Markov semigroup $T_t=e^{-L_Ht}:{\mathbb M}_m\to {\mathbb M}_m$ with generator in the Lindbladian form [Reference Lindblad50],

$$\begin{align*}L_H(\rho)=-\sum_{i=1}^k [d_u(X_i),[d_u(X_i),\rho]].\end{align*}$$

$T_t$ is called a transference semigroup of $P_t$ by the following commuting diagram:

(3.21)

where the transference map $\pi _u$ is a $*$ -endomorphism given by

$$ \begin{align*} \pi_u: {\mathbb M}_m\to L_{\infty}(G, {\mathbb M}_m),\hspace{.1cm} \pi_u(\rho)(g)=u(g)^*\rho u(g), \end{align*} $$

which embeds ${\mathbb M}_m$ into $L^{\infty }(G, {\mathbb M}_m)$ . Then the quantum semigroup $T_t$ is the restriction of the matrix-valued extension of classical semigroup $P_t\otimes \operatorname {\mathrm {\operatorname {id}}}_{{\mathbb M}_m}$ on the image of $\pi ({\mathbb M}_m)$ . Such a transference relation holds fro any unitary representation. We obtain the following dimension-free estimates both spectral gap and CMLSI constant (see [Reference Gao, Junge and LaRacuente29, Section 4]):

$$\begin{align*}\alpha_{c}(\Delta_H)\le \alpha_{c}(L_H), \lambda(\Delta_H)\le \lambda(L_H),\end{align*}$$

which are independent of the choice of the unitary representation. Then Corollary 1.3 follows immediately from Theorem 1.2.

3.6 Failure of matrix valued log-Sobolev inequality

As mentioned above, a standard analysis approach to MLSI through hypercontractivity or LSI relies on the Rothaus Lemma (see, for example, [Reference Rothaus68, Reference Bakry3])

$$\begin{align*}H(|f|^2)\le H(|f-E_\mu(f)|^2)+\parallel \! f-E_\mu(f) \! \parallel_{2}^2.\end{align*}$$

Here, we show that the Rothaus Lemma, LSI and hypercontractivity all fail for matrix-valued functions for any classical Markov semigroups. This is a strong indication that the approach by Diaconis and Saloff-Coste’s hypercontractive [Reference Diaconis and Saloff-Coste23] estimate (also used in [Reference Lugiewicz and Zegarliński53]) cannot be used in proving lower bounds for the CMLSI constants.

The following lemma calculates the derivatives of the entropy functional $H(\rho )=\tau (\rho \log \rho )$ . Recall the BKM metric of a operator $X\in {\mathcal M}$ at a base state $\rho $ is

$$\begin{align*}\gamma_\rho(X)=\int_{0}^\infty\tau(X^*(\rho+s)^{-1}X (\rho+s)^{-1}).\end{align*}$$

Lemma 3.14. Let $t\mapsto \rho _t \in S_B({\mathcal M}), t\in (a,b)$ be a smooth family of bounded density operator. Define the function $F(t)=H(\rho _t)=\tau (\rho _t\log \rho _t)$ . Then

$$ \begin{align*}F'(t)=\tau(\rho_t'(\log \rho_t+1)),\hspace{.1cm} F"(t)=\tau(\rho_t"(\log \rho_t+1))+\gamma_{\rho_t}(\rho^{\prime}_t), \end{align*} $$

where $\rho _t'$ and $\rho _t"$ are the first and second order derivative of $\rho _t$ .

Proof. The formula for $F'$ follows from [Reference Wirth79, Lemma 5.8]. For the second derivative, recall the noncommutative chain rule

$$\begin{align*}\frac{d}{dt}(\log \rho_t)=\int_{0}^\infty(\rho_t+s)^{-1}\rho_t'(\rho_t+s)^{-1}ds.\end{align*}$$

By calculating the second derivative, we obtain the second assertion

$$ \begin{align*} F"(t)=&\tau(\rho_t"(\log \rho_t+1))+\int_{0}^\infty \tau(\rho_t'(\rho_t+s)\rho_t'(\rho_t+s))ds \hspace{.1cm} = \hspace{.1cm} \tau(\rho_t"(\log \rho_t+1))+\gamma_{\rho_t}(\rho^{\prime}_t).\\[-42pt] \end{align*} $$

Proposition 3.15. Let $T_t=e^{-tL}:L_\infty (\Omega ,\mu )\to L_\infty (\Omega ,\mu )$ be an ergodic symmetric Markov semigroup. Let $\alpha _R,\alpha _2,\alpha _h$ be the optimal (largest) constant such that the following inequalities hold for any $f\in L_\infty (\Omega ,{\mathbb M}_2)\cap {\mathcal A}_{\mathcal E}$ ,

(Rothaus) $$ \begin{align} &\alpha_R\hspace{.1cm} D\Big(|f|^2 \left| \right|E_\mu(|f|^2)\Big)\le D\Big(|\hat{f}|^2 \left| \right| E_\mu(|\hat{f}|^2)\Big)+\parallel \! \hat{f} \! \parallel_{2}^2,\qquad\qquad\ \ \qquad \end{align} $$

(LSI) $$ \begin{align} &\alpha_2\hspace{.1cm} D(f^2||E_\mu(f^2))\le 2{\mathcal E}(f,f)\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \end{align} $$

(Hypercontractivity) $$ \begin{align} &\parallel \! T_tf \! \parallel_{L_2({\mathbb M}_2,L_{p(t)}(\Omega))}\le \parallel \! f \! \parallel_{L_2({\mathbb M}_2,L_2(\Omega))} \text{ for } p(t)=1+e^{2\alpha_h t} \end{align} $$

where $E_\mu (f)=(\int f d\mu )1_\Omega $ is the expectation map and $\hat {f}=f-E_\mu (f)$ is the mean zero part of f. Then $\alpha _R=\alpha _2=\alpha _h=0$ .

Proof. We write $\tau (f)=\frac {1}{2}\int {\text {tr}}(f)d\mu $ for the normalized trace on $L_\infty (\Omega ,{\mathbb M}_2)$ . We start with constant $\alpha _R$ in the Rothaus lemma. Without loss of generality, we may assume there is a measurable set $X\subset \Omega $ such that $\mu (X)=r$ for some $0<r<1$ . Let $\eta \in (0,1)$ . Then $h_0=(1-r)1_X-r 1_{X^c}$ is a real mean zero function. Consider the matrix valued function $f_\varepsilon =f+\varepsilon h$ where

$$\begin{align*}f=\left[\begin{array}{cc}1+\eta &0 \\ 0& 1-\eta \end{array}\right]\textbf{1} , \hspace{.1cm} h=\left[\begin{array}{cc}0 & h_0 \\ h_0& 0 \end{array}\right], \end{align*}$$

where f is a constant matrix valued function. Then $E_\mu f_\varepsilon =f, \hat {f}_\varepsilon =\varepsilon h$ and

$$ \begin{align*}&f_\varepsilon^2=(f+\varepsilon h)^2=f^2+\varepsilon(fh+hf)+\varepsilon^2 h^2=f^2+2\varepsilon h+\varepsilon^2 h^2. \end{align*} $$

Then $E_\mu (|f_\varepsilon |^2)=f^2+\varepsilon ^2 h^2$ and $E_\mu (|\hat {f}_\varepsilon |^2)=E_\mu ( h^2)\varepsilon ^2$ . Using Lemma 3.14, the Taylor expansion of the left-hand side of (LSI) is

$$ \begin{align*} D\Big(|f|^2 \left| \right|E_\mu(|f|^2)\Big)&=D(f^2+2\varepsilon h+\varepsilon^2 h^2\left| \right|f^2+\varepsilon^2 h^2)\\ &=H(f^2+2\varepsilon h+\varepsilon^2 h^2)-H(f^2+\varepsilon^2 h^2)\\ &=2\tau(h\log f)\varepsilon+\big(2\tau(h^2(\log f+1))+\gamma_{f}(2h)\big)\varepsilon^2+O(\varepsilon^3)\\&\quad-\tau(2E(h^2)(\log f+1))\varepsilon^2+O(\varepsilon^3)\\ &=\gamma_{f}(2h)\varepsilon^2+O(\varepsilon^3) , \end{align*} $$

where we used the fact $\tau (h\log f)=0$ and $\tau (h^2\log f-E_\mu (h^2)\log f)=0$ . For the right-hand side of the Rothaus lemma, we find

$$ \begin{align*} D(|\hat{f}_\varepsilon|^2||E_\mu(|\hat{f}_\varepsilon|^2))=D(h^2||E_\mu( h^2))\varepsilon^2, \hspace{.1cm} \parallel \! \hat{f}_\varepsilon \! \parallel_{2}^2=\parallel \! h \! \parallel_{2}^2\varepsilon^2. \end{align*} $$

While both $D(h^2||E_\mu ( h^2))$ and $\parallel \! h \! \parallel _{2}^2$ are finite, we have

$$ \begin{align*}\gamma_{f}(2h)&=4\int_{0}^\infty\tau(h(f+s)^{-1}h (f+s)^{-1})ds\\ &=4\int_{0}^\infty \int_\Omega {\text{tr}}(\left[\begin{array}{cc}\frac{1}{(1-\eta+s)(1+\eta+s)}h^2 &0 \\ 0& \frac{1}{(1-\eta+s)(1+\eta+s)}h^2 \end{array}\right]\textbf{1}_\Omega )d\mu ds\\ &=4\Big(\int_{0}^\infty \frac{1}{(1-\eta+s)(1+\eta+s)} ds\Big)\parallel \! h \! \parallel_{2}^2 \\ &= \frac{2}{\eta}\ln\frac{1+\eta}{1-\eta}\parallel \! h \! \parallel_{2}^2. \end{align*} $$

Note that we can choose $\eta \to 1$ and $\frac {1}{2\eta }\ln (\frac {1+\eta }{1-\eta })\to +\infty $ , which implies $\alpha _R=0$ . The same example applies to LSI by choosing a mean zero function $h_0$ such that ${\mathcal E}(h_0,h_0)<\infty $ . For the hypercontractivity, for $p\ge 2$ we recall the norms

$$ \begin{align*} \parallel \! f \! \parallel_{L_2({\mathbb M}_2, L_2(\Omega))}&=\parallel \! f \! \parallel_{L_2({\mathbb M}_2)\otimes L_2(\Omega)}=(\int {\text{tr}}(f^*f)d\mu)^{1/2}. \\ \parallel \! f \! \parallel_{L_2({\mathbb M}_2, L_p(\Omega))}&=\inf_{x,y\in ({\mathbb M}_2)_+,\hspace{.1cm} \parallel \! \hspace{.1cm} x\hspace{.1cm} \! \parallel_{2r}=\parallel \! \hspace{.1cm} y\hspace{.1cm} \! \parallel_{2r}=1}\parallel \! x^{-1}fy^{-1} \! \parallel_{L_2({\mathbb M}_2, L_2(\Omega))},\end{align*} $$

where the infimum takes over all positive invertible $x,y\in {\mathbb M}_2$ with unit $2r$ -norm for $\frac {1}{r}=\frac {1}{2}-\frac {1}{p}$ . Since $T_t$ is a bimodule map for $\mathbb {C}1\otimes {\mathbb M}_2\subset L_\infty (\Omega ,{\mathbb M}_2)$ , we can equivalently consider the norm

$$\begin{align*}\parallel \! T_t: L_2({\mathbb M}_2, L_2(\Omega))\to L_2({\mathbb M}_2, L_p(\Omega)) \! \parallel_{}=\parallel \! T_t: L_2({\mathbb M}_2, L_2(\Omega))\to L_2^a({\mathbb M}_2, L_p(\Omega)) \! \parallel_{},\end{align*}$$

where the asymmetric amalgamated $L_2^a({\mathbb M}_2, L_p(\Omega ))$ space is equipped with norm

$$ \begin{align*}\parallel \! f \! \parallel_{L_2^a({\mathbb M}_2, L_p(\Omega))}=\inf_{a\in ({\mathbb M}_2)_+, \parallel \! \hspace{.1cm} a\hspace{.1cm} \! \parallel_{r}=1}\parallel \! fa^{-1} \! \parallel_{L_2({\mathbb M}_2, L_2(\Omega))}. \end{align*} $$

In particular,

$$\begin{align*}\parallel \! f \! \parallel_{L_2^a({\mathbb M}_2, L_p(\Omega))}^2=\parallel \! f^*f \! \parallel_{L_1({\mathbb M}_2, L_{\frac{p}{2}}(\Omega))},\end{align*}$$

and we have

$$\begin{align*}D(|f|^2||E(|f|^2))=\lim_{q\to 1^+}\frac{\parallel \! |f|^2 \! \parallel_{L_1({\mathbb M}_2, L_q(\Omega))} -\parallel \! |f|^2 \! \parallel_{1}}{q-1 }.\end{align*}$$

Now define $p(t)=2q(t)=1+e^{2\alpha _h t}$

$$\begin{align*}G(t)=\parallel \! T_t f \! \parallel_{L_2^a({\mathbb M}_2, L_p(t)(\Omega))}^2=\parallel \! |T_tf|^2 \! \parallel_{L_1({\mathbb M}_2, L_{q(t)}(\Omega))}. \end{align*}$$

By assumption $G(t)\le 1$ , we have

$$\begin{align*}G'(0)=-2{\mathcal E}(f,f)+\alpha_hD(|f|^2||E(|f|^2))\le 0,\end{align*}$$

which implies $\alpha _h\le \alpha _2=0$ . Note, however, that $\alpha _h\ge 0$ because $T_t$ is always contractive on $L_2({\mathbb M}_2, L_2(\Omega ))$ . Hence, $\alpha _h=0$ , and the proof is complete.

4.1 State symmetric quantum channels

Let ${\mathcal M}$ be a von Neumann algebra and $\phi $ a normal faithful state. We have the GNS cyclic representation $\{\pi _\phi , H_\phi , \eta _\phi \}$ , which is a $*$ -isomorphism $\pi _\phi :{\mathcal M}\to H_\phi $ with a cyclic and separating vector $\eta _\phi $ such that

$$\begin{align*}\phi(x)=\langle \eta_\phi, \pi_\phi(x)\eta_\phi \rangle , \hspace{.1cm} x\in {\mathcal M}.\end{align*}$$

By identifying ${\mathcal M}\cong \pi _\phi ({\mathcal M})$ , the modular automorphism group $\alpha ^\phi _t$ for $t\in \mathbb {R}$ is defined as

$$\begin{align*}\alpha^\phi_t:{\mathcal M}\to {\mathcal M}\hspace{.1cm} ,\hspace{.1cm} \alpha^\phi_t(x)=\Delta^{it}x \Delta^{-it} , \hspace{.1cm} x\in {\mathcal M},\end{align*}$$

where $\Delta $ is the modular operator of $\phi $ , defined as follows:

$$\begin{align*}\Delta=S^*\bar{S}, \hspace{.1cm} S(\pi_\phi(x)\eta_\phi)=\pi_\phi(x^*)\eta_\phi .\end{align*}$$

We consider the following two symmetric conditions with respect to a state $\phi $ .

Definition 4.1. We say a quantum Markov map $\Phi :{\mathcal M} \to {\mathcal M}$ is GNS-symmetric with respect to $\phi $ (in short, GNS- $\phi $ -symmetric) if

$$\begin{align*}\phi(\Phi(x)y)=\phi(x\Phi(y)), \hspace{.1cm} \forall \hspace{.1cm} x,y\in {\mathcal M}\hspace{.1cm}; \end{align*}$$

ii) We say $\Phi $ is KMS-symmetric with respect to $\phi $ (in short, KMS- $\phi $ -symmetric) if

$$\begin{align*}\langle \Delta^{\frac{1}{4}} x\eta_\phi, \Delta^{\frac{1}{4}}\Phi(y)\eta_\phi \rangle=\langle \Delta^{\frac{1}{4}} \Phi(x)\eta_\phi, \Delta^{\frac{1}{4}}y\eta_\phi \rangle, \hspace{.1cm} \forall \hspace{.1cm} x,y\in {\mathcal M}. \end{align*}$$

Correspondingly, we call the pre-adjoint $\Phi _*:{\mathcal M}_*\to {\mathcal M}_*$ a GNS- or KMS- $\phi $ -symmetric quantum channel.

Both definitions are generalizations of the detailed balance condition for classical Markov chains and imply that $\phi =\phi \circ \Phi =\Phi _*(\phi )$ is an invariant state of $\Phi $ . It is proven in [Reference Giorgetti, Parzygnat, Ranallo and Russo32, Reference Wirth80] that the GNS- $\phi $ -symmetric quantum Markov map is equivalent to KMS- $\phi $ -symmetric plus that $\Phi $ commutes with the modular group

$$\begin{align*}\alpha_t^{\phi}\circ\Phi\hspace{.1cm} = \hspace{.1cm} \Phi\circ\alpha_t^{\phi}, \hspace{.1cm} t\in {\mathbb R}.\end{align*}$$

The commutation to modular group is also called Accardi-Cecchini condition in [Reference Giorgetti, Parzygnat, Ranallo and Russo32] for a study of quantum Bayes rule [Reference Parzygnat62, Reference Parzygnat and Russo65, Reference Parzygnat and Buscemi63, Reference Parzygnat and Fullwood64].

For simplicity, we will consider a semifinite von Neumann algebra ${\mathcal M}$ equipped with a normal faithful semi-finite trace $\tau $ , but our discussion applies to general von Neumann algebras with proper interpretation of notations. In the tracial setting, we can write $\phi (x)=\tau (d_\phi x)$ using the density operator $d_\phi $ of $\phi $ . Then the modular automorphism group is given by

$$\begin{align*}\alpha_t^{\phi}(x) \hspace{.1cm} = \hspace{.1cm} d_{\phi}^{-it}xd_{\phi}^{it}, \hspace{.1cm} x\in {\mathcal M} , t\in \mathbb{R}.\end{align*}$$

Let $\Phi _*:L_1({\mathcal M})\to L_1({\mathcal M})$ be the pre-adjoint quantum channel via trace duality. The KMS- $\phi $ -symmetry is equivalent to

(4.1) $$ \begin{align} \Phi_*(d_\phi^{1/2}xd_\phi^{1/2}) \hspace{.1cm} = \hspace{.1cm} d_\phi^{1/2}\Phi(x)d_\phi^{1/2}, \hspace{.1cm} \forall x\in {\mathcal M}. \end{align} $$

For $1\le p\le \infty $ , the weighted $L_p$ -space $L_p({\mathcal M},\phi )$ is the completion of ${\mathcal M}$ under the norm

$$\begin{align*}\parallel \! x \! \parallel_{p,\phi}=\parallel \! d_\phi^{1/2p} x d_\phi^{1/2p} \! \parallel_{p},\end{align*}$$

where $\parallel \! y \! \parallel _{p}=\tau (|y|^p)^{1/p}$ is the tracial p-norm. For $p=2$ , $L_p({\mathcal M},\phi )$ is a Hilbert space with KMS-inner product $\parallel \! x \! \parallel _{2,\phi }^2=\langle \Delta ^{\frac {1}{4}}x\eta _\phi ,\Delta ^{\frac {1}{4}}x\eta _\phi \rangle $ . By equation (4.1), $\Phi $ is also a contraction on $L_1({\mathcal M},\phi )$ , and hence a contraction on $L_p({\mathcal M},\phi )$ for all $1\le p\le \infty $ by complex interpolation.

The lemma below is an analog of Proposition 2.2.

Proof. It suffices to explain i). The rest follows similar as Proposition 2.2 (see also [Reference Gao and Rouzé31, Lemma 2.5] for the finite dimensional case). Indeed, since $\Phi $ commutes with $\alpha _t^\phi $ , for $x\in {\mathcal N}$ ,

$$ \begin{align*} \Phi\big(\alpha_t^\phi(x)y\big)&=\Phi\big(\alpha_t^\phi(x \alpha_{t}^{\phi}\circ\alpha_{-t}^\phi(y) )\big)=\alpha_t^\phi\circ\Phi\big(x\alpha_{t}^{\phi}\circ \alpha_{-t}^\phi(y) \big)\\ &=\alpha_t^\phi\Big(\Phi(x)\Phi(\alpha_{t}^{\phi}\circ\alpha_{-t}^\phi(y)) \Big)=\alpha_t^\phi\circ\Phi(x) \alpha_t^\phi\circ \Phi\circ \alpha_{-t}^\phi(y)=\Phi(\alpha_t^\phi(x))\Phi(y). \end{align*} $$

Multiplicativity on the other side is similar, implying $ \alpha _t^\phi (x)\in {\mathcal N}$ . By Takesaki’s theorem [Reference Takesaki72], there exists $\phi $ -preserving conditional expectation satisfying the defining property

$$\begin{align*}\phi(xy)=\phi(xE(y)) \hspace{.1cm} \forall \hspace{.1cm} x\in {\mathcal N}, y\in {\mathcal M},\end{align*}$$

from which the GNS- $\phi $ -symmetry follows.

4.2 Haagerup’s reduction

A von Neumann algebra ${\mathcal M}$ is called type III if it does not admit a nontrivial semifinite trace. We briefly review the basics of Haagerup’s construction and refer to [Reference Haagerup, Junge and Xu36] for more details. The key idea is to consider the additive subgroup $G=\bigcup _{n\in {\mathbb N}} 2^{-n}\mathbb {Z} \subset \mathbb {R}$ of the automorphism group. Let ${\mathcal M}\subset B(H)$ be a von Neumann algebra and $\phi $ be a normal faithful state. One can define the crossed product by the action $\alpha ^\phi :G\curvearrowright {\mathcal M}$

$$\begin{align*}\hat{{\mathcal M}}\hspace{.1cm} = \hspace{.1cm} {\mathcal M}\rtimes_{\alpha^{\phi}}G . \end{align*}$$

$\hat {{\mathcal M}}$ can be considered as the von Neumann subalgebra $ \hat {{\mathcal M}}=\{\pi ({\mathcal M}),\lambda (G)\}" \subset {\mathcal M}\overline \otimes B(\ell _2(G))$ generated by the embeddings

(4.3) $$ \begin{align} \pi:{\mathcal M}\to {\mathcal M}\rtimes_{\alpha^{\phi}}G, \hspace{.1cm} &\pi(a)\hspace{.1cm} = \hspace{.1cm} \sum_{g} \alpha_{g^{-1}}(a) \otimes |{g}\rangle\langle{g}|\nonumber\\ \lambda:G\to {\mathcal M}\rtimes_{\alpha^{\phi}}G,\hspace{.1cm} &\lambda(g)(|{x}\rangle\otimes |{h}\rangle)=|{x}\rangle\otimes |{gh}\rangle\hspace{.1cm} , \hspace{.1cm} \forall \hspace{.1cm} |{x}\rangle\in H, |{h}\rangle\in \ell_2(G) . \end{align} $$

Basically, $\pi $ is the transference homomorphism ${\mathcal M}\to \ell _\infty (G,{\mathcal M})$ , and $\lambda $ is the left regular representation on $\ell _2(G)$ . The set of finite sums $\{\sum _g a_g \lambda (g)\hspace {.1cm} | \hspace {.1cm} a_g \in {\mathcal M} \}\subset \hat {{\mathcal M}}$ forms a dense $w^*$ -subalgebra of $\hat {{\mathcal M}}$ . In the following, we identify ${\mathcal M}$ with $\pi ({\mathcal M})\subset \hat {M}$ (resp. a with $\pi (a)$ ) and view ${\mathcal M}\subset \hat {{\mathcal M}}$ as a subalgebra. The state $\phi $ admits a natural extension as a normal faithful state on $\hat {{\mathcal M}}$

$$\begin{align*}\hat{\phi}(\sum_g a_g \lambda(g)) \hspace{.1cm} = \hspace{.1cm} \phi(a_0) .\end{align*}$$

Moreover,

$$\begin{align*}E_{\mathcal M}:\hat{{\mathcal M}}\to {\mathcal M}\hspace{.1cm}, \hspace{.1cm} E_{{\mathcal M}}(\sum_g a_g\lambda(g))=a_0\end{align*}$$

is the canonical normal conditional expectation such that $\phi \circ E_{\mathcal M}\hspace {.1cm} = \hspace {.1cm} \hat {\phi }$ .

The main object in Haagerup’s construction is an increasing family of subalgebras

$$\begin{align*}{\mathcal M}_n \hspace{.1cm} = \hspace{.1cm} \hat{{\mathcal M}}_{\psi_n}:=\{x\in \hat{{\mathcal M}} \hspace{.1cm} |\hspace{.1cm} \alpha_t^{\psi_n}(x)=x \hspace{.1cm}, \hspace{.1cm}\forall\hspace{.1cm} t\in {\mathbb R}\},\end{align*}$$

given by the centralizer algebra $\hat {{\mathcal M}}_{\psi _n}$ for a suitable family of states $\psi _n$ so that $\bigcup _n {\mathcal M}_n$ is $w^*$ -dense in $\hat {{\mathcal M}}$ . The state $\psi _n$ is defined via a Radon-Nikodym density w.r.t to $\hat {\phi }$

$$\begin{align*}\psi_n(x) \hspace{.1cm} = \hspace{.1cm} \hat{\phi}(e^{-a_n}x) \hspace{.1cm} ,\hspace{.1cm} a_n \hspace{.1cm} = \hspace{.1cm} -i2^n\text{Log} (\lambda(2^{-n})) .\end{align*}$$

Here, $\text {Log}$ is the principal branch of the logarithmic function with $0\le \text {Log}(z)<2\pi $ . Each subalgebra ${\mathcal M}_n$ contains $\lambda (G)$ , and there exists normal conditional expectation $E_{{\mathcal M}_n}:\hat {{\mathcal M}}\to {\mathcal M}_n$ . Indeed, by the definition of $\psi _n$ , the modular group $\alpha _t^{\psi _n}$ is $2^{-n}$ periodic. The explicit form (see [Reference Haagerup, Junge and Xu36, Lemma 2.3]) is given by

$$\begin{align*}E_{{\mathcal M}_n}=2^{n}\int_0^{2^{-n}}\alpha_t^{\psi_n} dt.\end{align*}$$

The normalized state $\tau _n=\frac {\psi _n}{\psi _n(1)}$ is a normalized trace on ${\mathcal M}_n$ . The key properties of ${\mathcal M}_n$ are summarized in [Reference Haagerup, Junge and Xu36, Theorem 2.1 & Lemma 2.7], which we state below.

We now look at the Haagerup reduction on the states. For a state $\rho \in S({\mathcal M})$ , $\hat {\rho }=\rho \circ E_{\mathcal M}$ is the canonical extension on $\hat {{\mathcal M}}$ . We denote $\rho _n:=\hat {\rho }|_{{\mathcal M}_n}\in {\mathcal M}_{n,*}$ as the restriction state of $\hat {\rho }$ on the subalgebra ${\mathcal M}_n \subset \hat {{\mathcal M}}$ . Note that the predual ${\mathcal M}_{n,*}$ can be viewed as a subspace of $\hat {{\mathcal M}}_*$ via the embedding

$$\begin{align*}\iota_{n,*}: \hat{{\mathcal M}}_{n, *}\to \hat{{\mathcal M}}_*\hspace{.1cm}, \iota_{n,*}(\omega)=\omega\circ E_{{\mathcal M}_n}. \end{align*}$$

Via this identification, $\rho _n=\hat {\rho }|_{{\mathcal M}_n}\circ E_{{\mathcal M}_n}=\hat {\rho }\circ E_{{\mathcal M}_n}=E_{{\mathcal M}_n,*}(\hat {\rho })\in \hat {{\mathcal M}}_*$ . Moreover, by the weak $^*$ -density of the family ${\mathcal M}_n$ , $\rho _n\to \hat {\rho }$ converges in the weak topology. An immediate consequence is the following approximation of relative entropy.

Lemma 4.4. Let $\rho $ and $\sigma $ be two normal states of ${\mathcal M}$ . Then

$$\begin{align*}D(\rho||\sigma) \hspace{.1cm} = \hspace{.1cm} D(\hat{\rho}||\hat{\sigma})=\lim_{n\to \infty} D(\rho_n||\sigma_n) .\end{align*}$$

Proof. Let $\iota :{\mathcal M}\subset \hat {{\mathcal M}}$ be the inclusion map. Because $\hat {\rho }=\rho \circ E_{\mathcal M}$ is an extension of $\rho $ , $\iota _*(\hat {\rho })=\hat {\rho }|_{\mathcal M}=\rho $ , and similarly for $\sigma $ . Both $\iota : {\mathcal M} \to \hat {{\mathcal M}}$ and $E_{\mathcal M}:\hat {{\mathcal M}}\to {\mathcal M}$ are quantum Markov maps. Then by the data processing inequality,

$$\begin{align*}D(\rho||\sigma)=D(\iota_*(\hat{\rho})||\iota_*(\hat{\sigma}))\le D(\hat{\rho}||\hat{\sigma})=D(E_{{\mathcal M},*}(\rho)||E_{{\mathcal M},*}(\sigma))\le D(\rho||\sigma). \end{align*}$$

Thus, $D(\rho ||\sigma )=D(\hat {\rho }||\hat {\sigma })$ . As for the limit, we have

$$ \begin{align*} D(\hat{\rho}\|\hat{\sigma})&\leq \liminf_{n}D(\rho_{n}\|\sigma_{n})\\ &=\liminf_{n} D(E_{{\mathcal M}_n,*}(\rho_n)||E_{{\mathcal M}_n,*}(\sigma_{n,*}))\\ &\le D(\hat{\rho}||\hat{\sigma}) , \end{align*} $$

where the equality follows from the lower semi-continuity of relative entropy (see, for example, [Reference Hiai37, Theorem 2.7]). The second inequality is another use the data processing inequality.

We shall also apply the Haagerup’s reduction on GNS-symmetric maps. Let $\Phi :{\mathcal M}\to {\mathcal M}$ be a GNS- $\phi $ -symmetric quantum Markov map. Its canonical extension map

$$\begin{align*}\hat{\Phi}:\hat{{\mathcal M}}\to \hat{{\mathcal M}}\hspace{.1cm}, \hspace{.1cm} \hat{\Phi}(\sum_g a_g\lambda(g)) \hspace{.1cm} = \hspace{.1cm} \sum_g \Phi(a_g)\lambda(g) \hspace{.1cm} \end{align*}$$

is also a GNS- $\hat {\phi }$ -symmetric quantum Markov map. Indeed, $\hat {\Phi }=\Phi \otimes \operatorname {\mathrm {\operatorname {id}}}_{B(\ell _2(G))}|_{\hat {{\mathcal M}}}$ is the restriction of $\Phi \otimes \operatorname {\mathrm {\operatorname {id}}}_{B(\ell _2(G))}$ on $\hat {{\mathcal M}}\subset {\mathcal M}\overline {\otimes } B(\ell _2(G)$ . It is clear that $\hat {\Phi }$ has the multiplicative domain

(4.4) $$ \begin{align}\hat{{\mathcal N}}:={\mathcal N}\rtimes_{\alpha^{\phi}} G, \end{align} $$

where ${\mathcal N}$ is the multiplicative domain of $\Phi $ . In particular, this crossed product is well defined because $\alpha ^\phi _t({\mathcal N})={\mathcal N}$ . Moreover, the $\hat {\phi }$ -preserving conditional expectation $\hat {E}:\hat {{\mathcal M}}\to \hat {{\mathcal N}}$ is nothing but the canonical extension of $E:{\mathcal M}\to {\mathcal N}$ .

Recall that we write $E_{\mathcal M}$ and $E_n$ as the normal conditional expectations from $\hat {{\mathcal M}}$ onto ${\mathcal M}$ and ${\mathcal M}_n$ , respectively. The next lemma shows that the extension $\hat {\Phi }$ is well compatible with the approximation family ${\mathcal M}_n$ .

Proof. The relation $\hat {\Phi }\circ E_{\mathcal M}=E_{\mathcal M}\circ \hat {\Phi }$ is clear from the definition of $\hat {\Phi }$ , and $\hat {\Phi }\circ \hat {E}=\hat {E}\circ \hat {\Phi }$ follows from Lemma 4.2. Recall that $\psi _n(x)=\hat {\phi }(e^{-a_n}x)$ with density operator $e^{-a_n}\in \lambda (G)"$ and $\lambda (G)$ is in the centralizer of $\hat {\phi }$ [Reference Haagerup, Junge and Xu36, Lemma 2.3]. Then

$$\begin{align*}\alpha_t^{\psi_n} \hspace{.1cm} = \hspace{.1cm} u(t)^*\alpha_t^{\hat{\phi}}u(t)=\text{ad}_{u(t)}\alpha_t^{\hat{\phi}} \end{align*}$$

for the unitary $u(t)=e^{-ita_n}$ . Note that $\hat {\Phi }$ commutes with $\alpha _t^{\hat {\phi }}$ by GNS- $\hat {\phi }$ -symmetry, and also commutes with $\text {ad}_{u(t)}$ because $u(t)\in \lambda (G)"$ is in $\hat {\Phi }$ ’s multiplicative domain. Thus, $\hat {\Phi }$ commutes with $\alpha _t^{\psi _n}$ and hence the conditional expectation $E_{{\mathcal M}_n}=2^{-n}\int _0^{2^{-n}}\alpha _t^{\psi _n}$ . This proves i).

For ii), we note that for $x,y\in {\mathcal M}_n$ ,

$$\begin{align*}\psi_n(x\Phi_n(y))=\hat{\phi}(e^{-a_n}x\hat{\Phi}(y))=\hat{\phi}(\hat{\Phi}(e^{-a_n}x)y)=\hat{\phi}(e^{-a_n}\hat{\Phi}(x)y)=\psi_n(\Phi_n(x)y),\end{align*}$$

where we use the fact that $\hat {\Phi }$ is GNS- $\hat {\phi }$ -symmetric and $e^{-a_n}\in \lambda (G)"$ is in the fixed point subspace of $\hat {\Phi }$ . Finally, iii) follows from applying i) and ii) to $\hat {E}$ .

To summarize the lemma above, we have the following commuting diagrams:

Figure 1 Haagerup reduction of quantum Markov map and conditional expectation.

Basically, $\Phi _n$ is a family of trace symmetric channels approximating $\hat {\Phi }$ , which is in turn a natural extension of $\Phi $ . The same picture holds for the conditional expectations $E_n,\hat {E}$ and E.

4.3 Entropy contraction

We shall now discuss the entropy contraction of GNS- $\phi $ -symmetric channels. The first step is to extend the entropy difference Lemma 2.1. Define the state space that is bounded with respect to $\phi $ ,

$$\begin{align*}S_B({\mathcal M},\phi)=\{\rho \in S({\mathcal M})\hspace{.1cm} |\hspace{.1cm} c^{-1}\phi\le \rho\le c\phi\hspace{.1cm}, \hspace{.1cm} \text{for some } c>0 \}.\end{align*}$$

For all $\rho \in S_B({\mathcal M},\phi )$ , $D(\rho ||\phi ) <\infty $ is finite. Such $S_B({\mathcal M},\phi )$ is a dense subset of $S({\mathcal M})$ because for any $\rho $ and $0<\varepsilon <1$ , $\rho _\varepsilon =(1-\varepsilon )\rho +\varepsilon \phi \in S_B({\mathcal M})$ . For $\rho \in S_B({\mathcal M})$ , we define the entropy difference for a GNS- $\phi $ -symmetric quantum channel $\Phi _*$ as

$$\begin{align*}D_{\Phi_*}(\rho):=D(\rho||\phi)-D(\Phi_*(\rho)||\phi).\end{align*}$$

By data processing inequality and $\Phi _*(\phi )=\phi $ , $D_{\Phi _*}(\rho )\ge 0$ . In the trace symmetric case, $D_{\Phi _*}(\rho )=D(\rho ||1)-D(\Phi _*(\rho )||1)=H(\rho )-H(\Phi _*(\rho ))$ as in Section 2. Let E be the conditional expectation onto the multiplicative domain of $\Phi $ . By the chain rule [Reference Petz66, Theorem 2] that for any E invariant state $\psi \circ E=\psi $ ,

$$\begin{align*}D(\rho||\psi)=D(\rho||E_*(\rho))+D(E_*(\rho)||\psi), \end{align*}$$

we have the alternative expressions $D_{\Phi _*}(\rho )=D(\rho ||E_*(\rho ))-D(\Phi _*(\rho )||\Phi _*E_*(\rho )), $ where we used the property $\Phi _*E_*=E_*\Phi _*$ in Proposition 4.2.

Lemma 4.6. Let $\Phi _*$ be a GNS- $\phi $ -symmetric quantum channel. For any state $\rho ,\omega \in S_{b}({\mathcal M},\phi )$ ,

$$\begin{align*}D(\rho||\Phi_*^2(\omega))\hspace{.1cm} \le \hspace{.1cm} D_{\Phi_*}(\rho)+D(\rho||\omega) .\end{align*}$$

Proof. Recall that we use $\rho _n=\hat {\rho }|_{{\mathcal M}_n}=E_{{\mathcal M}_n,*}(\hat {\rho })$ and $\omega _n=\hat {\omega }|_{{\mathcal M}_n}=E_{{\mathcal M}_n,*}(\hat {\omega })$ as the restriction states on finite von Neumann algebra ${\mathcal M}_n\subset \hat {{\mathcal M}}$ obtained from the Haagerup reduction. By Lemma 4.5, we know that $\Phi _n=\hat {\Phi }|_{{\mathcal M}_n}$ is a quantum Markov map symmetric with respect to the tracial state $\tau _n$ . Thus, by Lemma 2.1 in the tracial case,

$$\begin{align*}D(\rho_n||\Phi_{n}^2(\omega_n)) \hspace{.1cm} \le \hspace{.1cm} D_{\Phi_n}(\rho_n) +D(\rho_n||\omega_n) ,\end{align*}$$

where we identify $\Phi _{n}=\Phi _{n,*}$ by trace symmetry. Here, since $\Phi _n=\Phi |_{{\mathcal M}_n}$ is GNS-symmetric to $\phi _n=\phi |_{{\mathcal M}_n}$ ,

$$\begin{align*}D_{\Phi_n}(\rho_n)=D(\rho_n|| \tau_n)-D(\Phi_n(\rho_n)|| \tau_n)=D(\rho_n||\phi_n)-D(\Phi_n(\rho_n)|| \phi_n) .\end{align*}$$

By the definitions of $\Phi _n$ and $\rho _n$ , and the hat “ $\:\hat {}\:$ ” notation for states on $\hat {{\mathcal M}}$ ,

$$ \begin{align*} &\Phi_{n}(\rho_n)=\hat{\rho}|_{{\mathcal M}_n}\circ \Phi_{n} =\hat{\rho}\circ \hat{\Phi}|_{{\mathcal M}_n}= \widehat{\Phi_*(\rho)}|_{{\mathcal M}_n}= \Phi_*(\rho)_n\hspace{.1cm} ,\\ &\Phi_{n}^2(\rho_n)=\Phi_{n}(\Phi_*(\rho)_n)=\Phi_*^2(\rho)_n . \end{align*} $$

Then by Lemma 4.4, we can approximate every entropic term

$$ \begin{align*}\lim_{n}D(\rho_n||(\Phi_{n})^2(\omega_n))=&\lim_{n}D(\rho_n||\Phi_*^2(\omega)_n)=D(\rho||\Phi_*^2(\omega))\hspace{.1cm} ,\\ \lim_{n} D_{\Phi_n}(\rho_n)=&\lim_{n}D(\rho_n|| \phi_n)-D(\Phi_n(\rho_n)|| \phi_n)\\=&\lim_{n}D(\rho_n||\phi_n)-D(\Phi_*(\rho)_n|| \phi_n)=D_{\Phi_*}(\rho)\hspace{.1cm} ,\\ \lim_{n}D(\rho_n||\omega_n)=&D(\rho||\omega).\\[-42pt] \end{align*} $$

The next lemma shows the CB-return time is also compatible with Haargerup reduction.

Lemma 4.7. Let $\Psi :{\mathcal M}\to {\mathcal M}$ be a GNS- $\phi $ -symmetric quantum Markov map and E be the conditional expectation on its multiplicative domain. Suppose

(4.5) $$ \begin{align} (1-\varepsilon) E \le_{cp} \Psi \le_{cp} (1+\varepsilon)E . \end{align} $$

Then for all $n\in {\mathbb N}$ ,

$$\begin{align*}(1-\varepsilon)E_{n} \le_{cp} \Psi_n \le_{cp} (1+\varepsilon)E_{n} . \end{align*}$$

Moreover, if $0.9 E \le _{cp} \Psi \le _{cp} 1.1E$ and $\Psi \circ E=E$ , then for any $\rho \in S_{B}({\mathcal M},\phi )$ ,

$$\begin{align*}\frac{1}{2}D(\rho||E_*(\rho))\le D(\rho||\Psi_*(\rho)).\end{align*}$$

We then extend the entropy contraction to the GNS-symmetric case.

Theorem 4.8. Let $\Phi :{\mathcal M}\to {\mathcal M}$ be a GNS- $\phi $ -symmetric quantum Markov map and E be the $\phi $ -preserving conditional expectation onto its multiplicative domain ${\mathcal N}$ . Define

$$\begin{align*}k_{cb}(\Phi) \hspace{.1cm} = \hspace{.1cm} \inf\{k\in {\mathbb N}^+ \hspace{.1cm} |\hspace{.1cm} 0.9 E\le_{cp}\Phi^{2k} \le_{cp} 1.1 E\hspace{.1cm} \}. \end{align*}$$

Then, for any $\sigma $ -finite von Neumann algebra ${\mathcal Q}$ , state $\rho \in S({\mathcal M}\overline {\otimes } {\mathcal Q})$ ,

$$\begin{align*}D(\Phi_*\otimes \operatorname{\mathrm{\operatorname{id}}}_{{\mathcal Q}}(\rho)||(\Phi_*\circ E_*)\otimes \operatorname{\mathrm{\operatorname{id}}}_{{\mathcal Q}}(\rho)) \hspace{.1cm} \le \hspace{.1cm} \Big (1-\frac{1}{2k_{cb}(\Phi)}\Big )D(\rho||E_*\otimes \operatorname{\mathrm{\operatorname{id}}}_{{\mathcal Q}}(\rho) ).\end{align*}$$

Recall that in finite dimensions, the MLSI is defined as the supremum of $\alpha $ such that

$$\begin{align*}2\alpha D(\rho||E_*(\rho))\le I_L(\rho):=\tau(L_*(\rho)(\ln \rho-\ln\phi) ).\end{align*}$$

The right-hand side $I_L(\rho )$ is the entropy production, and the equivalence to entropy decay relies on the de Bruijn identity

(4.6) $$ \begin{align} I_L(\rho)=-\frac{d}{dt}D(T_*(\rho)||E_*(\rho))|_{t=0}.\end{align} $$

In infinite dimensions, the de Bruijn identity (4.6) is less justified even in $B(H)$ with $\dim (H)=+\infty $ (see discussions in [Reference Huber, K¨onig and Vershynina40, Reference König and Smith44]). To avoid this issue, we define the MLSI on Type III von Neumann algebra as follows.

Definition 4.9. For a GNS- $\phi $ -symmetric quantum Markov semigroup $T_t=e^{-tL}:{\mathcal M}\to {\mathcal M}$ , we define the modified log-Sobolev (MLSI) constant $\alpha _1(L)$ as the largest constant $\alpha $ such that

(4.7) $$ \begin{align}D(T_{t,*}(\rho)||E_*(\rho)) \hspace{.1cm} \le \hspace{.1cm} e^{-2\alpha t}D(\rho||E_*(\rho))\hspace{.1cm}, \hspace{.1cm} \forall \rho\in S({\mathcal M}),\end{align} $$

where E is the $\phi $ -preserving conditional expectation onto the fixed point subalgebra ${\mathcal N}$ . The complete MLSI constant is then defined as $\alpha _c(L):=\sup _{\mathcal {{\mathcal Q}}}\alpha (L\otimes \operatorname {\mathrm {\operatorname {id}}}_{{\mathcal Q}})$ , where the supremum is over all $\sigma $ -finite von Neumann algebra ${\mathcal Q}$ .

This definition of MLSI also does not depend on any choice of reference state $\phi $ (see Lemma 4.16). With this definition, we obtain the first half of Theorem 1.1, which is restated below.

Theorem 4.10. Let $T_t=e^{-tL}:{\mathcal M}\to {\mathcal M}$ be a GNS- $\phi $ -symmetric quantum Markov semigroup. Denote $t_{cb} \hspace {.1cm} = \hspace {.1cm} \inf \{t>0 \hspace {.1cm}|\hspace {.1cm} 0.9 E\le _{cp}T_t \le _{cp} 1.1 E\} $ . Then

$$\begin{align*}\alpha_1\ge \alpha_c\ge \frac{1}{2t_{cb}}.\end{align*}$$

Namely, for any $\sigma $ -finite von Neumann algebra ${\mathcal Q}$ and state $\rho \in S({\mathcal M}\overline {\otimes } {\mathcal Q})$ , we have the exponential decay of relative entropy

$$\begin{align*}D(T_{t,*}\otimes \operatorname{\mathrm{\operatorname{id}}}_{{\mathcal Q}}(\rho)||E_*\otimes \operatorname{\mathrm{\operatorname{id}}}_{{\mathcal Q}}(\rho)) \hspace{.1cm} \le \hspace{.1cm} e^{-\frac{t}{t_{cb}}}D(\rho||E_*\otimes \operatorname{\mathrm{\operatorname{id}}}_{{\mathcal Q}}(\rho)) \hspace{.1cm},\hspace{.1cm} t\ge 0.\end{align*}$$

As we have seen in Proposition 3.7 for the tracial case, a combination of heat kernel estimates and spectral gap allows us to bound CB return time. The same analysis remains valid in the GNS- $\phi $ -symmetric case. For $1\le p\le \infty $ , we define the $\phi $ -weighted conditional $L_\infty ^p({\mathcal N}\subset {\mathcal M}, \phi )$ space as the completion of ${\mathcal M}$ under the norm

$$\begin{align*}\parallel \! x \! \parallel_{L_\infty^p({\mathcal N}\subset {\mathcal M}, \phi)}=\sup\{\parallel \! axb \! \parallel_{p,\phi}\hspace{.1cm} |\hspace{.1cm} a,b\in {\mathcal N},\hspace{.1cm} \parallel \! aa^* \! \parallel_{p,\phi}=\parallel \! b^*b \! \parallel_{p,\phi}=1\hspace{.1cm}\}.\end{align*}$$

For a GNS-symmetric ${\mathcal N}$ -bimodule map $\Psi :{\mathcal M}\to {\mathcal M}$ , the equivalence in Proposition 3.4 also holds,

(4.8) $$ \begin{align}(1-\varepsilon) E\le_{cp} \Psi \le_{cp} (1+\varepsilon) E\hspace{.1cm} \Longleftrightarrow \hspace{.1cm} \parallel \! \Psi-E:L_\infty^1({\mathcal N}\subset{\mathcal M}, \phi)\to L_\infty({\mathcal M}) \! \parallel_{cb}\le \varepsilon. \end{align} $$

Based on that, we have an analog of Proposition 3.7.

4.4 Applications to finite quantum Markov chains

Let $T_t=e^{-Lt}:{\mathbb M}_d\to {\mathbb M}_d$ be a quantum Markov semigroup on matrix algebra ${\mathbb M}_d$ . Its generator L admits the following Lindbladian form ([Reference Gorini, Kossakowski and Sudarshan33, Reference Lindblad50]):

$$\begin{align*}L(x)= i[h,x]+\sum_{j} \gamma_j(V_j^*[x,V_j]+[ V_j^*,x]V_j), \end{align*}$$

where $h,V_j\in {\mathbb M}_d$ and $h=h^*$ is Hermitian. When $T_t$ is GNS-symmetric, one has the following simplified form [Reference Alicki1, Reference Kossakowski, Frigerio, Gorini and Verri45] that

$$\begin{align*}L(x)=\sum_{j}e^{-w_j/2}\Big(V_j^*[x,V_j]+[ V_j^*,x]V_j\Big), \end{align*}$$

where $\{V_j\}=\{V_j\}^*$ is an orthogonal set with respect to trace inner product and the eigenvector of modular group $\alpha _t^{\phi }(V_j)=e^{-iw_j t}V_j\hspace {.1cm}.$ In finite dimensions, the completely Pimsner-Popa index $C_{cb}(E)$ is always finite. Combining Theorem 4.10 and Proposition 4.12, we obtain the second half of Theorem 1.1 restated as below.

Corollary 4.13. For finite dimensional GNS-symmetric quantum Markov semigroups,

(4.9) $$ \begin{align}\alpha_1\ge \alpha_{c}\ge \frac{\lambda}{2\ln (10 C_{cb}(E))}.\end{align} $$

Corollary 4.13 improves the bound $\alpha _c \ge \frac {\lambda }{2C_{cb}(E)}$ in the previous work of Gao and Rouzé [Reference Gao and Rouzé31].

Remark 4.14. In the ergodic case ${\mathcal N}=\mathbb {C}1$ , the conditional expectation $E_\phi (x)=\phi (x)1$ has index

$$\begin{align*}C(E_\phi)=\parallel \! \phi^{-1} \! \parallel_{\infty}\hspace{.1cm}, C_{cb}(E_\phi)\le \parallel \! \phi^{-1} \! \parallel_{\infty}^2.\end{align*}$$

The above bound (4.9) gives

$$\begin{align*}\alpha_1\ge \alpha_{c}\ge \frac{\lambda}{2\ln10+4 \ln\parallel \! \phi^{-1} \! \parallel_{\infty}}. \end{align*}$$

This can be compared to the bound

(4.10) $$ \begin{align} \alpha_1\ge \alpha_2\ge \frac{2(1-\frac{2}{\parallel \! \phi^{-1} \! \parallel_{\infty}})\lambda}{\ln(\parallel \! \phi^{-1} \! \parallel_{\infty}-1)} \hspace{.1cm}\end{align} $$

proved by Diaconis and Saloff-Coste [Reference Diaconis and Saloff-Coste23] for symmetric classical Markov semigroups. In the quantum case, it is only obtained for unital semigroups [Reference Kastoryano and Temme43] and $d=2$ [Reference Beigi, Datta and Rouzé9]. For both classical and quantum depolarizing semigroups $L(x)=x-\phi (x)1$ , this bound is known to be optimal for $\alpha _2$ , which lower bounds $\alpha _1$ . Our results gives a general $\mathcal {O}(\frac {\lambda }{\parallel \! \phi ^{-1} \! \parallel _{\infty }})$ lower bound for $\alpha _1$ for non-ergodic cases and also the complete constant $\alpha _c$ .

Remark 4.15. The Corollary 3.10 shows that the CMLSI constant $\alpha _{c}$ for a classical Markov semigroup is lower bounded by LSI constant $\alpha _2$ up to a $O(\log \log \parallel \! \mu ^{-1} \! \parallel _{\infty })$ term. This argument does not work for Quantum Markov semigroup $T_t:\mathbb {M}_d\to {\mathbb M}_d$ on matrix algebras, although (3.15) remains valid for ergodic quantum Markov semigroups. The difference is that for matrix algebra, the bounded return time

$$\begin{align*}t_b(e^{-2}):=\frac{1}{2}\inf\{t>0\hspace{.1cm} |\hspace{.1cm} \parallel \! T_t-E:L_1({\mathbb M}_d,\phi)\to L_\infty({\mathbb M}_d) \! \parallel_{}<1/e^2\}\end{align*}$$

and the CB return time of completely bounded norm

$$ \begin{align*} t_{cb}=\inf\{t>0 \hspace{.1cm} | \hspace{.1cm} \parallel \! T_t-E_\mu:L_1({\mathbb M}_d,\phi)\to L_\infty({\mathbb M}_d) \! \parallel_{cb}<1/10\hspace{.1cm}\} \end{align*} $$

are quite different. In the classical setting, we used the fact

$$\begin{align*}\parallel \! T:L_1(\Omega)\to L_\infty(\Omega) \! \parallel_{}=\parallel \! T:L_1(\Omega)\to L_\infty(\Omega) \! \parallel_{cb}.\end{align*}$$

So the $t_b(e^{-2})$ and $t_{cb}(0.1)$ are comparable by absolute constants. In the noncommutative setting, we only have

$$\begin{align*}\parallel \! T_t-E_\mu:L_1({\mathbb M}_d,\phi)\to L_\infty({\mathbb M}_d) \! \parallel_{cb}\le d\parallel \! T_t-E_\mu:L_1({\mathbb M}_d)\to L_\infty({\mathbb M}_d) \! \parallel_{}.\end{align*}$$

In the trace symmetric case, $\parallel \! \mu ^{-1} \! \parallel _{\infty }=d$ and $t_{cb}(0.1)\le \frac {3}{2}t_b(e^{-2}) +\ln d$ ,

$$\begin{align*}\alpha_{c}\ge \frac{1}{2t_{cb}(0.1)}\ge \frac{1}{3t_b(e^{-2})+ 2\ln d}\sim O(\frac{\alpha_2}{\ln d}),\end{align*}$$

which is worse than the lower bound in the previous remark as $\alpha _2\le \lambda $ .

4.5 Independence of invariant state

The next lemma shows that the GNS-symmetry is also independent of the choice of invariant state $\phi $ .

Proof. Without loss of generality, we assume $\psi \le C\phi $ for some $C>0$ . We first view them as the states on the subalgebra ${\mathcal N}$ by restriction. By [Reference Takesaki73, Theorem 3.17], there exists $h\in {\mathcal N}$ such that

$$\begin{align*}\psi( x )=\phi(h^*xh ) \hspace{.1cm}, \hspace{.1cm} \forall x\in {\mathcal N}. \end{align*}$$

This identity actually also holds for $y\in {\mathcal M}$ . Indeed, because of $\phi \circ E=\phi $ and $\psi \circ E=\psi $ ,

$$\begin{align*}\psi( y )=\psi( E(y) )=\phi(h^*E(y)h )=\phi(E(h^*yh) )=\phi(h^*yh ) \hspace{.1cm}, \hspace{.1cm} \forall y\in {\mathcal M}. \end{align*}$$

Moreover, one can replace h by $T(h)$ , because

$$\begin{align*}\psi( x )=\psi( T(x) )=\phi(h^*T(x)h )=\phi\circ T(T(h^*)xT(h) )=\phi(T(h^*)xT(h) ),\end{align*}$$

where we use the fact that $T^2(h)=h$ . Thus, the GNS-symmetry with respect to $\psi $ follows that for $x,y\in {\mathcal M}$ ,

$$ \begin{align*} \psi( xT(y) )&=\phi(h^*xT(y)h )=\phi(h^*xT(yT(h))) =\phi(T(h^*x)yT(h))=\phi(T(h^*)T(x)yT(h))\\ &=\psi( T(x)y ) ,\end{align*} $$

where we used the multiplicative property of $T(axb)=T(a)T(x)T(b)$ for $a,b\in {\mathcal N}$ . The general case can be obtained via $\psi _{\varepsilon }=(1-\varepsilon )\psi +\varepsilon \phi $ .

We remark that if one has convergence $\lim _{n}\Phi ^{2n}=E$ in $L_2$ -norm, the above E-invariant condition $\phi \circ E=\phi $ can be replaced by $\phi =\Phi ^{2}\circ \phi $ .

Note that the left-hand side of (4.8) only relies on complete positivity. Indeed, the $L_\infty ^1({\mathcal N}\subset {\mathcal M}, \phi )$ norm at the right hand-side is also independent of the choice of the invariant state $\phi =\phi \circ E$ .

Lemma 4.17. Let $\phi $ be a normal faithful state and $E:{\mathcal M}\to {\mathcal N}$ be a $\phi $ -preserving conditional expectation. Suppose $\psi =\psi \circ E$ is another normal faithful state preserved by E. Then,

$$\begin{align*}\parallel \! x \! \parallel_{L_\infty^p({\mathcal N}\subset{\mathcal M}, \phi)}=\parallel \! x \! \parallel_{L_\infty^p({\mathcal N}\subset{\mathcal M}, \psi)}\hspace{.1cm}, \hspace{.1cm} \forall x\in {\mathcal M}.\end{align*}$$

The identity extends to all $x\in L_\infty ^p({\mathcal N}\subset {\mathcal M}, \phi ) $ .

Proof. Note that if both $\phi $ and $\psi $ are E invariant, then $d_{\psi }^{-\frac {1}{2p}}d_\phi ^{\frac {1}{2p}}$ is affiliated to ${\mathcal N}$ . Indeed, as argued in Lemma 4.16, if $\psi \le C\phi $ , then $d_\psi = hd_\phi h^*$ for some $h\in {\mathcal N}$ , and the general case follows from approximation $\psi \le \frac {1}{\varepsilon }((1-\varepsilon )\phi +\varepsilon \psi )$ . Then we have

$$\begin{align*}\parallel \! aa^* \! \parallel_{\phi, p}=\parallel \! d_{\phi}^{\frac{1}{2p}}aa^*d_{\phi}^{\frac{1}{2p}} \! \parallel_{p}=\parallel \! d_{\psi}^{-\frac{1}{2p}}d_{\phi}^{\frac{1}{2p}}aa^*d_{\phi}^{\frac{1}{2p}}d_{\psi}^{-\frac{1}{2p}} \! \parallel_{\psi,2p} .\end{align*}$$

Denote $a_1=d_{\psi }^{-\frac {1}{2p}}d_\phi ^{\frac {1}{2p}}a$ and $b_1=bd_\phi ^{\frac {1}{2p}}d_{\psi }^{-\frac {1}{2p}}$ . For $x\in {\mathcal M}$ ,

$$ \begin{align*} \parallel \! x \! \parallel_{L_\infty^p({\mathcal N}\subset{\mathcal M}, \phi)}=&\sup_{\parallel \! \hspace{.1cm} aa^*\hspace{.1cm} \! \parallel_{\phi,2p}=\parallel \! \hspace{.1cm} b^*b\hspace{.1cm} \! \parallel_{\phi,2p}=1}\parallel \! axb \! \parallel_{\phi,p} =\sup_{\parallel \! \hspace{.1cm} aa^*\hspace{.1cm} \! \parallel_{\phi,p}=\parallel \! \hspace{.1cm} b^*b\hspace{.1cm} \! \parallel_{\phi,p}=1}\parallel \! d_{\psi}^{-\frac{1}{2p}}d_\phi^{\frac{1}{2p}}axbd_{\phi}^{\frac{1}{2p}}d_{\psi}^{-\frac{1}{2p}} \! \parallel_{\psi,p} \\=& \sup_{\parallel \! \hspace{.1cm} a_1a_1^*\hspace{.1cm} \! \parallel_{2p,\psi}=\parallel \! \hspace{.1cm} b_1b_1^*\hspace{.1cm} \! \parallel_{2p,\psi}=1}\parallel \! a_1xb_1 \! \parallel_{\psi, p}=\parallel \! x \! \parallel_{L_\infty^p({\mathcal N}\subset{\mathcal M},\psi)}, \end{align*} $$

where the supremum are for $a,b\in {\mathcal N}$ .

Remark 4.18. For finite ${\mathcal M}$ , one particular invariant state of E used in [Reference Bardet and Rouzé7, Reference Bardet, Capel, Gao, Lucia, Pérez-García and Rouzé6] is $\phi _{{\text {tr}}}=E_*(1)$ . This state is convenient because $\phi _{{\text {tr}}}|_{\mathcal N}$ is a trace. Then by Lemma 4.17, we have

$$\begin{align*}\parallel \! x \! \parallel_{L_\infty^p({\mathcal N}\subset {\mathcal M}, \phi)}=\parallel \! x \! \parallel_{L_\infty^p({\mathcal N}\subset {\mathcal M}, \phi_{\text{tr}})}=\sup\{\parallel \! axb \! \parallel_{p,\phi_{\text{tr}}}\hspace{.1cm} |\hspace{.1cm} a,b\in {\mathcal N},\hspace{.1cm} \parallel \! a \! \parallel_{p,\phi_{\text{tr}}}=\parallel \! b \! \parallel_{p,\phi_{\text{tr}}}=1\hspace{.1cm}\},\end{align*}$$

where we used the fact $L_p({\mathcal N},\phi _{\text {tr}})$ is a tracial $L_p$ -space. We will use this point to simplify the discussion in Section 5.4.

5.1 Entropy contraction coefficients

In this section, we discuss the implications of our results on contraction coefficients studied in [Reference Diaconis and Saloff-Coste23, Reference Del Moral, Ledoux and Miclo22, Reference Müller-Hermes, Stilck França and Wolf57, Reference Gao and Rouzé31]. These are analogs of functional inequalities for a single quantum channel.

The condition $\lambda (\Phi )<1$ can be viewed as a Poincaré inequality for a quantum channel $\Phi $ , which implies the exponential convergence in $L_2$ ,

$$\begin{align*}\parallel \! \Phi^n(X)-E(X) \! \parallel_{L_2({\mathcal M},\phi)}\le \lambda(\Phi)^n\parallel \! X-E(X) \! \parallel_{L_2({\mathcal M},\phi)}\to 0.\end{align*}$$

Similarly, the entropy contraction coefficient gives the convergence in relative entropy

$$\begin{align*}D(\Phi^n(\rho)||\Phi^n\circ E(\rho))\le \alpha(\Phi)^n D(\rho||E(\rho)).\end{align*}$$

The complete constant $\alpha _c(\Phi )$ controls not only the entropy contraction of $\Phi $ but also $\operatorname {\mathrm {\operatorname {id}}}_{\mathcal Q}\otimes \Phi $ with any environment system $\mathcal {Q}$ . This leads to the tensorization property of $\alpha _c$ that for two GNS-symmetric quantum channels [Reference Gao and Rouzé31],

(5.2) $$ \begin{align}\alpha_c(\Phi_1\otimes \Phi_2)=\max\{\alpha_c(\Phi_1),\alpha_c(\Phi_2)\}.\end{align} $$

For classical Markov maps, the tensorization property (5.2) is known to also hold for the non-complete constant $\alpha $ . Nevertheless, for the quantum Markov map (channel), this is not the case, and $\alpha (\Phi )$ in general can be strictly less than $\alpha _c(\Phi )$ (see [Reference Brannan, Gao and Junge14, Section 4.4]).

In finite dimensions, the existence of strictly contractive constant $\alpha _c(\Phi )<1$ was obtained in [Reference Gao and Rouzé31, Theorem 4.1]. Our results give an explicit estimate for $\alpha _c(\Phi )$ .

Corollary 5.2. Let $\Phi $ be a GNS-symmetric quantum Markov map,

$$\begin{align*}\lambda(\Phi)\le \alpha(\Phi)\le \alpha_c(\Phi)\le (1-\frac{1}{2k_{cb}(\Phi)})\le \left(1-\frac{-\ln \lambda(\Phi)}{\ln (10 C_{cb}(E))}\right).\end{align*}$$

Remark 5.3. In the ergodic trace symmetric case ${\mathcal N}=\mathbb {C}1$ and ${\mathcal M}={\mathbb M}_d$ , we have the trace map $E(x)={\text {tr}}(x)\frac {1}{d}$ and the CB-index $C_{cb}(E)= d^2$ . The above estimate implies

(5.3) $$ \begin{align} \lambda(\Phi)\le \alpha(\Phi)\le \alpha_c(\Phi)\le (1-\frac{-\ln \lambda(\Phi)}{\ln (10 d^2)}). \end{align} $$

This can be compared to [Reference Müller-Hermes, Stilck França and Wolf57, Theorem 4.2] and [Reference Kastoryano and Temme43, Corollary 27],

(5.4) $$ \begin{align} \alpha(\Phi)\le 1-\frac{1}{2}\alpha_{2}(\operatorname{\mathrm{\operatorname{id}}}-\Phi^*\Phi)\le 1-\frac{(1-\lambda(\Phi)^2)^2(1-\frac{2}{d})}{\ln(d-1)},\end{align} $$

where $\alpha _{2}(\operatorname {\mathrm {\operatorname {id}}}-\Phi ^2)$ is the LSI constant of $\operatorname {\mathrm {\operatorname {id}}}-\Phi ^2$ as a generator of quantum Markov semigroup. The two upper bounds in (5.3) and (5.4) are comparable, as both are asymptotically $\Theta (\frac {-\ln \lambda (\Phi )}{\ln d})$ . The strength of our results is that (5.3) also bounds the complete constant $\alpha _c(\Phi )$ which has the tensorization property.

Remark 5.4. Our Lemma 2.1 implies

$$\begin{align*}1-\alpha_1(\operatorname{\mathrm{\operatorname{id}}}-\Phi^2)\le \alpha(\Phi),\end{align*}$$

where $\alpha _1$ is MLSI constant of the semigroup generator $(\operatorname {\mathrm {\operatorname {id}}}-\Phi ^*\Phi )$ . For a classical Markov map, it was proved by Del Moral, Ledoux and Miclo [Reference Del Moral, Ledoux and Miclo22] that there exists a universal constant $0<c<1$ such that

(5.5) $$ \begin{align} 1-\alpha_1(\operatorname{\mathrm{\operatorname{id}}}-\Phi^*\Phi)\le \alpha(\Phi)\le 1-c\alpha_1(\operatorname{\mathrm{\operatorname{id}}}-\Phi^*\Phi).\end{align} $$

To the best of our knowledge, the above upper bound in (5.5) is open in the quantum case.

5.2 Graph random walks

Let $G=(V,E)$ be a finite undirected graph with $|V|=d$ and the edge set $E\subset V\times V$ . The discrete time random walk on G is a finite Markov chain given by the stochastic matrix

$$\begin{align*}K_G(u,v)=\begin{cases} \frac{1}{d(u)}, & \mbox{if } (u,v)\in E \\ 0, & \mbox{otherwise}. \end{cases}\end{align*}$$

Here, $d(u)$ is the degree of vertex $u\in V$ . Then $K_G:l_\infty (V)\to l_\infty (V)$ is a Markov map. The $K_G$ admits a unique station distribution $\pi (u)=\frac {d(u)}{2m}$ , where $|E|=m$ . It is clear that $K_G$ is symmetric to the measure $\pi $ , also called reversible. Hence, $K_G$ is an ergodic unital channel on $L_\infty (V,\pi )$ as $\pi (K_G(f))=\pi (f)$ . The expectation map is $E_\pi (f)=\pi (f)1$ whose index is

$$\begin{align*}C_{cb}(E_\pi)=\parallel \! \pi^{-1} \! \parallel_{\infty}.\end{align*}$$

$K_G$ is connected and $E_\pi $ are symmetric operators on $L_2(V,\pi )$ and

$$\begin{align*}\lambda(K_G)=\parallel \! K_G-E_\pi:L_2(V,\pi)\to L_2(V,\pi) \! \parallel_{}<1\end{align*}$$

if $K_G$ not bipartite (in the bipartite case $K_G$ has eigenvalue $-1$ ). Then our results imply

(5.6) $$ \begin{align}\alpha(K_G)\le \alpha_c(K_G)\le (1-\frac{1}{2k_{cb}(K_G)})\le (1-\frac{-\ln \lambda(K_G)}{\ln (10 \parallel \! \pi^{-1} \! \parallel_{\infty})}). \end{align} $$

Example 5.5 (Cyclic graphs).

Let us consider the cyclic graph $C_d=(V,E)$ with $d\ge 4$ where $V=\{1,\cdots ,d\}$ and $E=\{(j,j+1)| j=1,\cdots , d \}$ . Here, the addition is understood in the sense of ‘mod d’. Then

$$\begin{align*}K_{C_d}(i,j)=\begin{cases} \frac{1}{2}, & \mbox{if } |i-j|=1 \\ 0, & \mbox{otherwise}. \end{cases}\end{align*}$$

As $C_d$ is 2-regular, $K_{C_d}$ is symmetric to the uniform distribution $\pi (i)=1/d$ . It is known that $K_{C_d}$ has spectrum

$$\begin{align*}\lambda_j=\cos(\frac{2\pi j}{d})\hspace{.1cm} ,\hspace{.1cm} j=0,\cdots, d-1.\end{align*}$$

The associated eigenvector is $e_j=\frac {1}{\sqrt {d}}(1,\omega ^j,\omega ^{2j},\cdots , \omega ^{(d-1)j})$ where $\omega =\exp (\frac {2\pi i}{d})$ . When $d=2m+1$ is odd, $\pi $ is the unique stationary measure, and $E_\pi $ is the projection onto the vector $e_0$ . We have

$$ \begin{align*} K_G^k-E_\pi= (K_G-E_\pi)^k=\sum_{j=1}^{2m}\lambda_j^k |{e_j}\rangle\langle{e_j}|. \end{align*} $$

By triangle inequality, we have

$$ \begin{align*} \parallel \! K_G^k-E_\pi:L_1(V,\pi)\to L_\infty(V,\pi) \! \parallel_{}&\le \sum_{j=1}^{2m}|\lambda_j|^k= 2\sum_{j=1}^{m}\cos(\frac{\pi j}{d})^k\\ &\le 2\frac{d}{\pi}\int_{0}^{\pi/2}\cos^k(x)dx =2\frac{d}{\pi}W_k\le 2Cd\sqrt{\frac{1}{2k\pi }}, \end{align*} $$

where $C>0$ is some absolute constant by fact that the Wallis integrals $W_k=\int _{0}^{\pi /2}\cos ^k(x)dx\sim \sqrt {\frac {\pi }{2k}}$ . Thus,

$$\begin{align*}k_{cb}(K_{C_d})\le \frac{(10Cd)^2}{\pi}\sim \mathcal{O}(d^2), \end{align*}$$

and (5.6) implies

$$\begin{align*}\alpha(K_{C_d})\ge \alpha_c(K_{C_d})\ge 1-\mathcal{O}(d^{-2}).\end{align*}$$

By Miclo’s result (5.5), this is asymptotically tight because the MLSI constant $\alpha _1(I-K_G^2)\sim \mathcal {O}(d^{-2})$ (see Example 5.6 below for detials). The similar asymptotic estimate also holds for even circle $d=2m$ .

For the continuous time random walk, we consider $w:E\to (0,\infty )$ to be a positive weighted function on the edge set E. The (weighted) graph Laplacian is given by the matrix

$$ \begin{align*} L_G(u,v)=\begin{cases} \sum_{e=(u,u')\in E}w_e, & \mbox{if } u=v \\ -w_e, & \mbox{if } (u,v)\in E \\ 0, & \mbox{otherwise}. \end{cases} \end{align*} $$

$L_G$ generates the continuous time random walk $T_t=e^{-L_G t}$ as a Markov semigroup and is symmetric with to the uniform distribution $\pi $ on V. $T_t$ is ergodic if and only if $G=(V,E)$ is connected. The expectation map $E_\pi (f)=\pi (f)\textbf {1}$ has index $C_{cb}(E_\pi )=d$ . Then Corollary 4.13,

(5.7) $$ \begin{align}\frac{ \lambda(L_G)}{2(\ln d+\ln 10)}\le \alpha_c(L_G)\le \alpha(L_G)\le \lambda(L_G). \end{align} $$

This lower bound of $\alpha _c(L_G)$ has better dependence on the dimension d than [Reference Li, Junge and LaRacuente49, Lemma 5.2].

Example 5.6 (Cyclic graphs).

Let us again consider the cyclic graph $C_d$ with d vertices. For the uniformly weighted case $w_e\equiv 1$ , $L_{C_d}$ is a circulant matrix

$$\begin{align*}L_{C_d}(i,j)=\begin{cases} 2, & \mbox{if } i=j \\ -1, & \mbox{if } |i-j|=1 \\ 0, & \mbox{otherwise}. \end{cases}\end{align*}$$

Thus, $L_{C_d}=2(I-K_{C_d})$ where $K_{C_d}$ is the random walk kernel in Example 5.5, and $ L_{C_d}$ has spectrum $\lambda _j=2(1-\cos \frac {2\pi j}{d})$ . As discussed in [Reference Diaconis and Saloff-Coste23, Example 3.6],

$$\begin{align*}\parallel \! T_t-E:L_1(V,\pi)\to L_\infty(V,\pi) \! \parallel_{}\le 2\exp(-\frac{4t}{d^2})(\sqrt{1+d^2/4t}).\hspace{.1cm}\end{align*}$$

Choosing $t_0=d^2$ , we have

$$\begin{align*}\parallel \! T_t-E:L_1(V,\pi)\to L_\infty(V,\pi) \! \parallel_{}\le 2e^{-4}\sqrt{5/4}<\frac{1}{10}.\end{align*}$$

Thus, by Theorem 1.1,

$$\begin{align*}\frac{1}{2d^2}\le \alpha_c(L_{C_d})\le \alpha_1(L_{C_d})\le 2(1-\cos\frac{2\pi}{d})=\frac{8\pi^2}{d^2}+\mathcal{O}(\frac{1}{d^4}). \end{align*}$$

This shows that for this example, our inverse of $t_{cb}$ bound for $\alpha _c$ is tight up to absolute constant. Note that the LSI constant $\alpha _2(L_{C_d})$ is also of $\Theta (\frac {1}{d^2})$ .