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Lower semicontinuity of relative entropy disturbance and its consequences M. E. Shirokov Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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    § 1. IntroductionQuantum relative entropy is one of the basic characteristics of quantum states, which is used essentially in the study of the information and statistical properties of quantum systems and channels [1]–[6]. From the mathematical point of view the quantum relative entropy $D(\rho\,\|\,\sigma)$ is a jointly convex function of the pair $(\rho,\sigma)$ of quantum states (or, more generally, positive trace-class operators) taking values in $[0,+\infty]$. One of the fundamental properties of quantum relative entropy is its monotonicity under the action of quantum operations (completely positive trace-nonincreasing linear maps), which means that
$$
\begin{equation*}
D(\Phi(\rho)\,\|\, \Phi(\sigma))\leqslant D(\rho\,\|\,\sigma)
\end{equation*}
\notag
$$
(with the possible value $+\infty$ in one side or both) for an arbitrary quantum operation $\Phi$ from a quantum system $A$ to a quantum system $B$ and any states $\rho$ and $\sigma$ of $A$ [7]. Another important and widely used property of quantum relative entropy is its (joint) lower semicontinuity, which means that the set of all pairs $(\rho,\sigma)$ such that $D(\rho\,\|\,\sigma)\leqslant c$ is a closed subset of $\mathfrak{T}_+(\mathcal{H})\times\mathfrak{T}_+(\mathcal{H})$ for any $c\geqslant0$, where $\mathfrak{T}_+(\mathcal{H})$ denotes the cone of positive trace-class operators on the Hilbert space $\mathcal{H}$ [4], [5], [8]. In this article, given an arbitrary quantum operation $\Phi\colon A\to B$, we analyse the nonnegative function
$$
\begin{equation*}
\Delta_{\Phi}(\rho,\sigma)\doteq D(\rho\,\|\,\sigma)-D(\Phi(\rho)\,\|\,\Phi(\sigma)),
\end{equation*}
\notag
$$
which is well defined on the set of all pairs $(\rho,\sigma)$ in $\mathfrak{T}_+(\mathcal{H}_A)\times\mathfrak{T}_+(\mathcal{H}_A)$ such that $D(\Phi(\rho)\,\|\,\Phi(\sigma))$ is finite. If $\Phi$ is a quantum channel (trace-preserving quantum operation), then the function $\Delta_{\Phi}(\rho,\sigma)$ characterizes the degree of reversivility of $\Phi$. By Petz’s theorem (see [9]) the equality $\Delta_{\Phi}(\rho,\sigma)=0$ holds for some quantum states $\rho$ and $\sigma$ if and only if the channel $\Phi$ is (completely) reversible with respect to these states (this means that there is a channel $\Psi=\Psi(\Phi,\rho,\sigma)$ such that $\rho=\Psi\circ\Phi(\rho)$ and $\sigma=\Psi\circ\Phi(\sigma)$). If $\Delta_{\Phi}(\rho,\sigma)\leqslant\varepsilon$, then the channel $\Phi$ is $\varepsilon$-reversible with respect to the states $\rho$ and $\sigma$ in the following sense: there is a quantum channel $\Psi=\Psi(\Phi,\rho,\sigma)$ such that
$$
\begin{equation*}
F(\rho,\Psi\circ\Phi(\rho))\geqslant \exp\biggl(-\frac{\varepsilon}2\biggr) \quad\text{and}\quad \sigma=\Psi\circ\Phi(\sigma),
\end{equation*}
\notag
$$
where $F(\,\cdot\,{,}\,\cdot\,)$ is the fidelity of quantum states [10]–[12]. We will prove that for any quantum operation $\Phi$ the function $\Delta_{\Phi}(\rho,\sigma)$ is lower semicontinuous on its domain of definition. Moreover, we will prove that the function is lower semicontinuous on the set
$$
\begin{equation}
\bigl\{(\rho,\sigma,\Phi)\in\mathfrak{T}_+(\mathcal{H}_A) \times\mathfrak{T}_+(\mathcal{H}_A)\times\mathfrak{F}_{\leqslant1}(A,B) \bigm| D(\Phi(\rho)\,\|\,\Phi(\sigma))<+\infty\bigr\},
\end{equation}
\tag{1.1}
$$
where $\mathfrak{F}_{\leqslant1}(A,B)$ is the set of all quantum operations from $A$ to $B$ equipped with the strong convergence topology [13], [14]. The last property means that the set of all triplets $(\rho,\sigma,\Phi)$ such that ${\Delta_{\Phi}(\rho,\sigma)\!\leqslant\!\varepsilon}$ is a closed subset of the set (1.1) for any $\varepsilon\geqslant0$. It also implies that the equality
$$
\begin{equation}
D(\rho\,\|\,\sigma)=D(\Phi(\rho)\,\|\,\Phi(\sigma))+\Delta_{\Phi}(\rho,\sigma)
\end{equation}
\tag{1.2}
$$
is a decomposition of $D(\rho\,\|\,\sigma)$ into a sum of two nonnegative lower semicontinuous functions on the set (1.1), since the lower semicontinuity of the function $(\rho,\sigma,\Phi)\mapsto D(\Phi(\rho)\,\|\,\Phi(\sigma))$ follows from the lower semicontinuity of quantum relative entropy and the definition of strong convergence. It follows immediately from the decomposition (1.2) that the local continuity of $D(\rho\,\|\,\sigma)$ (as a function of the pair $(\rho,\sigma)$) implies the local continuity of $D(\Phi(\rho)\,\|\,\Phi(\sigma))$ and $\Delta_{\Phi}(\rho,\sigma)$ (considered either as functions of the pair $(\rho,\sigma)$ for a fixed operation $\Phi$ or as functions of the triplet $(\rho,\sigma,\Phi)$). This property was originally proved in [15] in a direct and rather technical way using the convergence criterion for quantum relative entropy proposed there. Moreover, the decomposition (1.2) allows us to show that
$$
\begin{equation}
\limsup_{n\to+\infty}D(\Phi(\rho_n)\,\|\,\Phi(\sigma_n))-D(\Phi(\rho_0) \,\|\,\Phi(\sigma_0))\leqslant\limsup_{n\to+\infty}D(\rho_n\,\|\,\sigma_n)-D(\rho_0\,\|\,\sigma_0)
\end{equation}
\tag{1.3}
$$
for an arbitrary quantum operation $\Phi\colon A\to B$ and any sequences $\{\rho_n\}$ and $\{\sigma_n\}$ in $\mathfrak{T}_+(\mathcal{H}_A)$ converging to operators $\rho_0$ and $\sigma_0$ such that $D(\rho_0\,\|\,\sigma_0)<+\infty$. Since the quantity in the right-hand side of (1.3) characterizes the local discontinuity of quantum relative entropy for given convergent sequences $\{\rho_n\}$ and $\{\sigma_n\}$ (it is always nonnegative, and it is equal to zero if and only if $D(\rho_n\,\|\,\sigma_n)$ converges to $D(\rho_0\,\|\,\sigma_0)$), inequality (1.3) can be interpreted as a ‘contraction’ of possible discontinuities of the quantum relative entropy by quantum operations. An essential part of this article is devoted to various applications of the above general properties to the analysis of the local continuity of quantum relative entropy and some related functions (quantum mutual information, conditional relative entropy and so on). § 2. Preliminaries2.1. The basic notationLet $\mathcal{H}$ be a separable Hilbert space, $\mathfrak{B}(\mathcal{H})$ the algebra of all bounded operators on $\mathcal{H}$ with operator norm $\|\,{\cdot}\,\|$ and $\mathfrak{T}(\mathcal{H})$ the Banach space of all trace-class operators on $\mathcal{H}$ the trace norm $\|\,{\cdot}\,\|_1$. Let $\mathfrak{S}(\mathcal{H})$ be the set of quantum states (positive operators in $\mathfrak{T}(\mathcal{H})$ with unit trace) [14], [16], [17]. We denote the identity operator on a Hilbert space $\mathcal{H}$ by $I_{\mathcal{H}}$ and the identity transformation of the Banach space $\mathfrak{T}(\mathcal{H})$ by $\mathrm{Id}_{\mathcal{\mathcal{H}}}$. Trace-class operators will be denoted by Greek letters $\rho,\sigma,\omega,\dots$ . All other linear operators will be denoted by Latin letters $A,B,H,\dots$ . For vectors and operators of rank 1 on a Hilbert space we use the Dirac notation $|\phi\rangle$ and $|\chi\rangle\langle\psi|$ (where the action of an operator $|\chi\rangle\langle\psi|$ on a vector $|\phi\rangle$ gives the vector $\langle\psi|\phi\rangle|\chi\rangle$) [14], [16]. The support $\operatorname{supp}\rho$ of an operator $\rho$ in $\mathfrak{T}_+(\mathcal{H})$ is the closed subspace spanned by the eigenvectors of $\rho$ corresponding to its positive eigenvalues. The dimension of the subspace $\operatorname{supp}\rho$ is called the rank of $\rho$ and denoted by $\operatorname{rank}\rho$. If quantum systems $A$ and $B$ are described by Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, then the bipartite system $AB$ is described by the tensor product of these spaces, that is, $\mathcal{H}_{AB}\doteq\mathcal{H}_A\otimes\mathcal{H}_B$. The marginal states $\operatorname{Tr}_{B}\rho\doteq\operatorname{Tr}_{\mathcal{H}_B}\rho$ and $\operatorname{Tr}_{A}\rho\doteq\operatorname{Tr}_{\mathcal{H}_A}\rho$ of the state $\rho\in\mathfrak{S}(\mathcal{H}_{AB})$ are denoted by $\rho_{A}$ and $\rho_{B}$, respectively.1 The von Neumann entropy of a quantum state $\rho \in \mathfrak{S}(\mathcal{H})$ is defined by $S(\rho)=\operatorname{Tr}\eta(\rho)$, where $\eta(x)=-x\ln x$ if $x>0$ and $\eta(0)=0$. It is a concave lower semicontinuous function on the set $\mathfrak{S}(\mathcal{H})$ taking values in $[0,+\infty]$ [5], [8], [14]. The von Neumann entropy satisfies the inequality
$$
\begin{equation}
S(p\rho+(1-p)\sigma)\leqslant pS(\rho)+(1-p)S(\sigma)+h_2(p),
\end{equation}
\tag{2.1}
$$
which is valid for any states $\rho$ and $\sigma$ in $\mathfrak{S}(\mathcal{H})$ and $p\in(0,1)$, where $h_2(p)=\eta(p)+\eta(1-p)$ is the binary entropy [4], [16]. We will use the homogeneous extension of von Neumann entropy to the positive cone $\mathfrak{T}_+(\mathcal{H})$ defined by
$$
\begin{equation}
S(\rho)\doteq(\operatorname{Tr}\rho)S(\rho/\operatorname{Tr}\rho) =\operatorname{Tr}\eta(\rho)-\eta(\operatorname{Tr}\rho)
\end{equation}
\tag{2.2}
$$
at any nonzero operator $\rho$ in $\mathfrak{T}_+(\mathcal{H})$ and set equal to $0$ at the zero operator [8]. By using the concavity of entropy and inequality (2.1) it is easy to show that
$$
\begin{equation}
S(\rho)+S(\sigma)\leqslant S(\rho+\sigma)\leqslant S(\rho)+S(\sigma)+H(\{\operatorname{Tr}\rho,\operatorname{Tr}\sigma\})
\end{equation}
\tag{2.3}
$$
for any $\rho$ and $\sigma$ in $\mathfrak{T}_+(\mathcal{H})$, where $H(\{\operatorname{Tr}\rho,\operatorname{Tr}\sigma\}) =\eta(\operatorname{Tr}\rho)+\eta(\operatorname{Tr}\sigma) -\eta(\operatorname{Tr}(\rho+\sigma))$ is the homogeneous extension of binary entropy to the positive cone in $\mathbb{R}^2$. Note that both sides of (2.1) and all parts of (2.3) can be equal to $+\infty$. For two quantum states $\rho$ and $\sigma$ in $\mathfrak{S}(\mathcal{H})$ the quantum relative entropy is defined by
$$
\begin{equation}
D(\rho\,\|\,\sigma)=\sum_i\langle i|\rho\ln\rho-\rho\ln\sigma|i\rangle,
\end{equation}
\tag{2.4}
$$
where $\{|i\rangle\}$ is the orthonormal basis of eigenvectors of the state $\rho$ and it is assumed that $D(\rho\,\|\,\sigma)=+\infty$ if $\operatorname{supp}\sigma$ does not contain $\operatorname{supp}\rho$ [1], [14], [16]. Let $H$ be a positive semidefinite operator on a Hilbert space $\mathcal{H}$ (we always assume that positive operators are self-adjoint). We denote the domain of $H$ by $\mathcal{D}(H)$. For any positive operator $\rho\in\mathfrak{T}(\mathcal{H})$ we define the quantity $\operatorname{Tr} H\rho$ by
$$
\begin{equation}
\operatorname{Tr} H\rho= \begin{cases} \displaystyle \sup_n \operatorname{Tr} P_n H\rho &\text{if } \operatorname{supp}\rho\subseteq \operatorname{cl}(\mathcal{D}(H)), \\ +\infty & \text{otherwise,} \end{cases}
\end{equation}
\tag{2.5}
$$
where $P_n$ is the spectral projector of $H$ corresponding to the interval $[0,n]$ and $\operatorname{cl}(\mathcal{D}(H))$ is the closure of $\mathcal{D}(H)$. We use the following notion introduced in [15]. Definition. A double sequence $\{P^n_m\}_{n\geqslant0,\,m\geqslant m_0}$ ($m_0\in\mathbb{N}$) of finite-rank projectors2 is completely consistent with a sequence $\{\sigma_n\}\subset\mathfrak{T}_+(\mathcal{H})$ converging to an operator $\sigma_0$ if where $Q_n$ is the projector onto the support of $\sigma_n$, and also for all $m\geqslant m_0$ and $n\geqslant0$, where the limit is in the operator norm topology. It is essential that for any sequence $\{\sigma_n\}\subset\mathfrak{T}_+(\mathcal{H})$ converging to an operator $\sigma_0$ there exists a double sequence $\{P^n_m\}_{n\geqslant0,\,m\geqslant m_0}$ of finite-rank projectors completely consistent with the sequence $\{\sigma_n\}$; see [15], Lemma 4. A finite or countable set $\{\rho_i\}$ of quantum states with probability distribution $\{p_i\}$ is called a (discrete) ensemble and denoted by $\{p_i,\rho_i\}$. The state $\overline{\rho}=\sum_{i} p_i\rho_i$ is called the average state of $\{p_i,\rho_i\}$. The Holevo quantity of an ensemble $\{p_i,\rho_i\}$ is defined by
$$
\begin{equation}
\chi(\{p_i,\rho_i\})= \sum_{i} p_i D(\rho_i\,\|\,\overline{\rho})=S(\overline{\rho})-\sum_{i} p_iS(\rho_i),
\end{equation}
\tag{2.8}
$$
where the second formula is valid provided that $S(\overline{\rho})$ is finite. This quantity gives an upper bound for the amount of classical information that can be obtained from quantum measurements over the ensemble [18], [14], [16]. Given a lower semicontinuous function $f$ on a metric space $X$ and a sequence $\{x_n\}\subset X$ converging to a point $x_0\in X$ such that $f(x_0)<+\infty$, we use the quantity
$$
\begin{equation}
\operatorname{dj}\{f(x_n)\}\doteq\limsup_{n\to+\infty}f(x_n)-f(x_0)
\end{equation}
\tag{2.9}
$$
characterizing the discontinuity jump of the function $f$ corresponding to the sequence $\{x_n\}$. The lower semicontinuity of $f$ implies that $\operatorname{dj}(\{f(x_n)\})\geqslant0$ and that
$$
\begin{equation*}
\operatorname{dj}\{f(x_n)\}=0\quad\Longleftrightarrow\quad \exists\,\lim_{n\to+\infty}f(x_n)=f(x_0).
\end{equation*}
\notag
$$
We will use the following simple result. Lemma 1. If $f$ and $g$ are lower semicontinuous functions on a metric space $X$ taking values in $(-\infty,+\infty]$, then for any sequence $\{x_n\}\subset X$ converging to $x_0\in X$ such that $(f+g)(x_0)<+\infty$. In particular, if then A quantum operation $\Phi$ from a system $A$ to a system $B$ is a completely positive trace-nonincreasing linear map from $\mathfrak{T}(\mathcal{H}_A)$ to $\mathfrak{T}(\mathcal{H}_B)$. A trace-preserving quantum operation is called a quantum channel [14], [16]. For any quantum operation $\Phi\colon A\to B$ Stinespring’s theorem [19] ensures the existence of a Hilbert space $\mathcal{H}_E$ and a contraction $V_{\Phi}\colon \mathcal{H}_A\to\mathcal{H}_B\otimes\mathcal{H}_E$ such that
$$
\begin{equation}
\Phi(\rho)=\operatorname{Tr}_{E}V_{\Phi}\rho V_{\Phi}^{*}, \qquad \rho\in\mathfrak{T}(\mathcal{H}_A).
\end{equation}
\tag{2.10}
$$
If $\Phi$ is a channel, then $V_{\Phi}$ is an isometry [14], [16]. The mutual information $I(\Phi,\rho)$ of a quantum channel $\Phi\colon A\to B$ at a state $\rho$ in $\mathfrak{S}(\mathcal{H}_A)$ can be defined by
$$
\begin{equation*}
I(\Phi,\rho)=I(B:R)_{\Phi\otimes\operatorname{Id}_R(\overline{\rho})}\doteq D(\Phi\otimes\operatorname{Id}_R(\overline{\rho})\,\|\,\Phi(\rho)\otimes\overline{\rho}_R),
\end{equation*}
\notag
$$
where $\overline{\rho}$ is a pure state in $\mathfrak{S}(\mathcal{H}_A\otimes\mathcal{H}_R)$ such that $\operatorname{Tr}_R\overline{\rho}=\rho$ [14], [16]. We write $\mathfrak{F}_{\leqslant 1}(A,B)$ for the set of all quantum operations from $A$ to $B$ equipped with the topology of strong convergence defined by the family of seminorms $\Phi\mapsto\|\Phi(\rho)\|_1$, $\rho\in\mathfrak{S}(\mathcal{H}_A)$ [13], [14]. The strong convergence of a sequence $\{\Phi_n\}$ of operations in $\mathfrak{F}_{\leqslant 1}(A,B)$ to an operation $\Phi_0\in\mathfrak{F}_{\leqslant 1}(A,B)$ means that
$$
\begin{equation*}
\lim_{n\to\infty}\Phi_n(\rho)=\Phi_0(\rho) \quad \forall\, \rho\in\mathfrak{S}(\mathcal{H}_A).
\end{equation*}
\notag
$$
We will use the following important result. Lemma 2 ([20]). If a sequence $\{\rho_n\}$ of states converges to a state $\rho_0$ with respect to the weak operator topology, then the sequence $\{\rho_n\}$ converges to the state $\rho_0$ with respect to the trace norm. 2.2. Lindblad’s extension of quantum relative entropyLindblad’s extension of the quantum relative entropy of positive operators $\rho$ and $\sigma$ in $\mathfrak{T}(\mathcal{H})$ is defined by
$$
\begin{equation}
D(\rho\,\|\,\sigma)=\sum_i\langle\varphi_i|\rho\ln\rho-\rho\ln\sigma+\sigma-\rho|\varphi_i\rangle,
\end{equation}
\tag{2.11}
$$
where $\{\varphi_i\}$ is the orthonormal basis of eigenvectors of the operator $\rho$ and it is assumed that $D(0\,\|\,\sigma)=\operatorname{Tr}\sigma$ and $D(\rho\,\|\,\sigma)=+\infty$ if $\operatorname{supp}\rho$ is not contained in $\operatorname{supp}\sigma$ (in particular, if $\rho\neq0$ and $\sigma=0$) [8]. It is easy to show that all terms in the right-hand side of (2.11) are nonnegative. So $D(\rho\,\|\,\sigma)$ is well defined (as a nonnegative number or $+\infty$) for any positive operators $\rho$ and $\sigma$. If the extended von Neumann entropy $S(\rho)$ of $\rho$ (defined in (2.2)) is finite, then
$$
\begin{equation}
D(\rho\,\|\,\sigma)=\operatorname{Tr}\rho(-\ln\sigma) -S(\rho)-\eta(\operatorname{Tr}\rho)+\operatorname{Tr}\sigma-\operatorname{Tr}\rho,
\end{equation}
\tag{2.12}
$$
where $\operatorname{Tr}\rho(-\ln\sigma)$ is defined according to (2.5) and $\eta(x)=-x\ln x$. The function $(\rho,\sigma)\mapsto D(\rho\,\|\,\sigma)$ is nonnegative lower semicontinuous and jointly convex on $\mathfrak{T}_+(\mathcal{H}) \times\mathfrak{T}_+(\mathcal{H})$. We use the following properties of this function:
Inequalities (2.15) and (2.16) are easily proved by using representation (2.12) if the extended von Neumann entropy of the operators $\rho$, $\sigma$ and $\omega$ is finite. Indeed, inequality (2.15) follows from the operator monotonicity of the logarithm, and inequality (2.16) follows from (2.3). In the general case these inequalities can be proved by approximation, using Lemma 4 in [8]. Inequality (2.17) is a direct consequence of the joint convexity of relative entropy and identity (2.13). Equality (2.18) follows from the definition [8]. We will use Donald’s identity
$$
\begin{equation}
pD(\rho\,\|\,\omega)+\overline{p}D(\sigma\,\|\,\omega)=pD(\rho\,\|\,p\rho +\overline{p}\sigma)+\overline{p}D(\sigma\,\|\,p\rho+\overline{p}\sigma) +D(p\rho+\overline{p}\sigma\,\|\,\omega),
\end{equation}
\tag{2.19}
$$
where $\overline{p}=1-p$, which is valid for arbitrary operators4 $\rho$, $\sigma$ and $\omega$ in $\mathfrak{T}_+(\mathcal{H})$ and any $p\in[0,1]$ [21]. Note that both sides in (2.19) can be equal to $+\infty$.
§ 3. The main results3.1. The relative entropy disturbance and its lower semicontinuityA basic property of quantum relative entropy is its monotonicity under quantum operations (completely positive trace-nonincreasing linear maps), which means that
$$
\begin{equation}
D(\Phi(\rho)\,\|\, \Phi(\sigma))\leqslant D(\rho\,\|\,\sigma)
\end{equation}
\tag{3.1}
$$
for an arbitrary quantum operation $\Phi\colon \mathfrak{T}(\mathcal{H}_A)\to\mathfrak{T}(\mathcal{H}_B)$ and any operators $\rho$ and $\sigma$ in $\mathfrak{T}_+(\mathcal{H}_A)$ [7].5 The monotonicity property (3.1) means the nonnegativity of the function
$$
\begin{equation*}
\Delta_{\Phi}(\rho,\sigma)\doteq D(\rho\,\|\,\sigma)-D(\Phi(\rho)\,\|\,\Phi(\sigma)),
\end{equation*}
\notag
$$
which is well defined on the set of all pairs $(\rho,\sigma)$ in $\mathfrak{T}_+(\mathcal{H}_A)\times\mathfrak{T}_+(\mathcal{H}_A)$ such that $D(\Phi(\rho)\,\|\,\Phi(\sigma))$ is finite. This function can be called6 the relative entropy disturbance by $\Phi$. It turns out that the function $\Delta_{\Phi}(\rho,\sigma)$ appears (explicitly or implicitly) in various problems in quantum information theory. To show this it suffices to consider the following ‘special realizations’ of this function:
In addition to the above examples, we must say that in many basic inequalities in quantum information theory the gap (difference between the right- and left-hand sides) can be expressed in terms of the function $\Delta_{\Phi}(\rho,\sigma)$. This holds, for example, for the first and second chain rules for the mutual information of a quantum channel $I(\Phi,\rho)$ (defined in § 2.1). Indeed, it is easy to see that for any channels $\Phi\colon A\to B$ and $\Psi\colon B\to C$ and an arbitrary state $\rho$ in $\mathfrak{S}(\mathcal{H}_A)$ the following expressions hold:
$$
\begin{equation*}
I(\Phi,\rho)-I(\Psi\circ\Phi,\rho)=\Delta_{\Psi\otimes\operatorname{Id}_R} (\omega_{BR},\omega_{B}\otimes\omega_{R})
\end{equation*}
\notag
$$
and
$$
\begin{equation}
I(\Psi,\Phi(\rho))-I(\Psi\circ\Phi,\rho)=\Delta_{\operatorname{Tr}_E(\,\cdot\,)} (\Psi\otimes\operatorname{Id}_{RE}(\omega),\Psi(\omega_{B})\otimes\omega_{RE}),
\end{equation}
\tag{3.3}
$$
where $\omega=(V_{\Phi}\otimes I_R)\,\overline{\rho}\,(V_{\Phi}^*\otimes I_R)$ is a state in $\mathfrak{S}(\mathcal{H}_{BER})$ defined by means of the Stinespring representation (2.10) of the channel $\Phi$ and a given purification $\overline{\rho}\in\mathfrak{S}(\mathcal{H}_{AR})$ of the state $\rho$. For the first three of the above ‘special realizations’ of $\Delta_{\Phi}(\rho,\sigma)$ it is known that the corresponding characteristic is a lower semicontinuous function of $\rho$. This was proved, respectively, in [27], [13] and [28]. It turns out that the lower semicontinuity of all characteristics expressed in terms of the function $\Delta_{\Phi}(\rho,\sigma)$ is a consequence of one general result presented in part (A) of the following theorem. Theorem. (A) For an arbitrary quantum operation $\Phi\colon \mathfrak{T}(\mathcal{H}_A)\to\mathfrak{T}(\mathcal{H}_B)$ the function $\Delta_{\Phi}(\rho,\sigma)=D(\rho\,\|\,\sigma)-D(\Phi(\rho)\,\|\,\Phi(\sigma))$ is lower semicontinuous on the set (B) The function $(\rho,\sigma,\Phi)\mapsto\Delta_{\Phi}(\rho,\sigma)$ is lower semicontinuous on the set
$$
\begin{equation*}
\bigl\{(\rho,\sigma,\Phi)\in \mathfrak{T}_+(\mathcal{H}_A)\times\mathfrak{T}_+(\mathcal{H}_A) \times\mathfrak{F}_{\leqslant1}(A,B)\mid D(\Phi(\rho)\,\|\,\Phi(\sigma))<+\infty\bigr\},
\end{equation*}
\notag
$$
where $\mathfrak{F}_{\leqslant1}(A,B)$ is the set of all quantum operations from $A$ to $B$ equipped with the strong convergence topology (see § 2.1). Proof. (A) Assume first that $\Phi$ is a quantum channel with the Stinespring representation $\Phi(\varrho)=\operatorname{Tr}_E V\varrho V^*$, $\varrho\in\mathfrak{T}(\mathcal{H}_A)$, where $V$ is an isometry from $\mathcal{H}_{A}$ to $\mathcal{H}_{BE}$.
Let $\{\rho_n\}$ and $\{\sigma_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H}_A)$ converging, respectively, to operators $\rho_0$ and $\sigma_0$ such that $D(\Phi(\rho_n)\,\|\,\Phi(\sigma_n))<+\infty$ for all $n\geqslant0$. We have to show that
$$
\begin{equation}
\liminf_{n\to+\infty}\Delta_{\Phi}(\rho_n,\sigma_n)\geqslant\Delta_{\Phi}(\rho_0,\sigma_0).
\end{equation}
\tag{3.4}
$$
If $\sigma_0=0$, then $\rho_0=0$ (otherwise $D(\Phi(\rho_0)\,\|\,\Phi(\sigma_0))=+\infty$) and (3.4) follows directly from the monotonicity property (3.1). So we assume that $\sigma_0\neq0$, and therefore $\Phi(\sigma_0)\neq0$ (as $\Phi$ is a channel). Since $V$ is an isometry, we have
$$
\begin{equation*}
a_n\doteq D(V\rho_nV^*\,\|\, V\sigma_nV^*)=D(\rho_n\,\|\,\sigma_n) \quad \forall\, n\geqslant0.
\end{equation*}
\notag
$$
Let $\{P^n_m\}_{n\geqslant0,\,m\geqslant m_0}$ be a double sequence of finite-rank projectors in $\mathfrak{B}(\mathcal{H}_B)$ completely consistent with the sequence $\{\Phi(\sigma_n)\}$ (see the definition in § 2.1), which exists by Lemma 4 in [15]. Consider the double sequences
$$
\begin{equation*}
\begin{gathered} \, a_n^m=D((P^n_m\otimes I_E)V\rho_nV^*(P^n_m\otimes I_E)\,\|\, (P^n_m\otimes I_E)V\sigma_nV^*(P^n_m\otimes I_E))\leqslant+\infty, \\ b_n^m=D((\overline{P}^{\,n}_m\otimes I_E)V\rho_nV^*(\overline{P}^{\,n}_m\otimes I_E)\,\|\, (\overline{P}^{\,n}_m\otimes I_E)V\sigma_nV^*(\overline{P}^{\,n}_m\otimes I_E))\leqslant+\infty \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, \widehat{a}_n^m=D(P^n_m\Phi(\rho_n)P^n_m\,\|\, P^n_m\Phi(\sigma_n))<+\infty, \\ \widehat{b}_n^m=D(\overline{P}^{\,n}_m\Phi(\rho_n)\overline{P}^{\,n}_m\,\|\, \overline{P}^{\,n}_m\Phi(\sigma_n))<+\infty \end{gathered}
\end{equation*}
\notag
$$
for $n\geqslant0$ and $m\geqslant m_0$, where $\overline{P}^n_m=I_B-P^n_m$. The quantities $\widehat{a}_n^m$ and $\widehat{b}_n^m$ are finite by Lemma 4 in [8] and because $D(\Phi(\rho_n)\,\|\,\Phi(\sigma_n))$ is assumed to be finite for each $n\geqslant0$. By the same result of Lemma 4 in [8] we have
$$
\begin{equation}
a_n\geqslant a_n^m+b_n^m \quad \forall\, n\geqslant0, \quad m\geqslant m_0.
\end{equation}
\tag{3.5}
$$
Lemma 3 below implies that
$$
\begin{equation}
\widehat{a}_n\doteq D(\Phi(\rho_n)\,\|\,\Phi(\sigma_n))=\widehat{a}_n^m+\widehat{b}_n^m +D\biggl(\Phi(\rho_n)\biggm\|\frac{1}{2}(\Phi(\rho_n)+U^n_m\Phi(\rho_n)[U^n_m]^*)\biggr)
\end{equation}
\tag{3.6}
$$
for all $n\geqslant0$ and $m\geqslant m_0$, where $U^n_m=2P^n_m-I_B$ is a unitary operator.
Since $\widehat{b}_n^m\leqslant b_n^m$ by the monotonicity property (3.1) of quantum relative entropy, it follows from (3.5) and (3.6) that
$$
\begin{equation}
\begin{aligned} \, c_n^m &\doteq a_n^m-\widehat{a}_n^m-D\biggl(\Phi(\rho_n)\biggm\|\frac{1}{2}(\Phi(\rho_n) +U^n_m\Phi(\rho_n)[U^n_m]^*)\biggr) \nonumber \\ &\leqslant a_n-\widehat{a}_n=\Delta_{\Phi}(\rho_n,\sigma_n) \end{aligned}
\end{equation}
\tag{3.7}
$$
for all $n\geqslant0$ and $m\geqslant m_0$.
By Lemma 4 below the properties of the double sequence $\{P^n_m\}_{n\geqslant0,\,m\geqslant m_0}$ imply that
$$
\begin{equation}
\lim_{n\to+\infty}\widehat{a}_n^m=\widehat{a}_0^m<+\infty \quad \forall\, m\geqslant m_0.
\end{equation}
\tag{3.8}
$$
Since $P^n_m$ tends to $P^0_m$ in the operator norm topology as $n\to+\infty$ and $\Phi(\rho_n)\leqslant \Phi(\rho_n)+U^n_m\Phi(\rho_n)[U^n_m]^*$ for all $n\geqslant0$ and $m\geqslant m_0$, using Proposition 2 in [29] and the obvious equality $D(\Phi(\rho_n)\,\|\,\Phi(\rho_n))=0$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to+\infty} D\biggl(\Phi(\rho_n)\biggm\|\frac{1}{2}(\Phi(\rho_n)+U^n_m\Phi(\rho_n)[U^n_m]^*)\biggr) \\ &\qquad =D\biggl(\Phi(\rho_0)\biggm\|\frac{1}{2}(\Phi(\rho_0)+U^0_m\Phi(\rho_0)[U^0_m]^*)\biggr)<+\infty. \end{aligned}
\end{equation}
\tag{3.9}
$$
Since $(P^n_m\otimes I_E)V\omega_nV^*(P^n_m\otimes I_E)$ tends to $(P^0_m\otimes I_E)V\omega_0V^*(P^0_m\otimes I_E)$ as $n\to+\infty$ for all $m\geqslant m_0$, $\omega=\rho,\sigma$, the lower semicontinuity of quantum relative entropy implies that
$$
\begin{equation*}
\liminf_{n\to+\infty}a_n^m\geqslant a_0^m \quad \forall\, m\geqslant m_0.
\end{equation*}
\notag
$$
This and the limit relations (3.8) and (3.9) show that
$$
\begin{equation*}
\liminf_{n\to+\infty}c_n^m\geqslant c_0^m \quad \forall\, m\geqslant m_0.
\end{equation*}
\notag
$$
Thus we will prove (3.4) once we will have shown that $\Delta_{\Phi}(\rho_n,\sigma_n)=\sup_{m\geqslant m_0}c_n^m$ for each $n\geqslant0$. To do this, by (3.7) it is sufficient to show that
$$
\begin{equation*}
\lim_{m\to+\infty}c_n^m=\Delta_{\Phi}(\rho_n,\sigma_n)\leqslant+\infty\quad \forall\, n\geqslant0.
\end{equation*}
\notag
$$
This can be done by noting that
$$
\begin{equation}
\lim_{m\to+\infty}D\biggl(\Phi(\rho_n)\biggm\| \frac{1}{2}(\Phi(\rho_n)+U^n_m\Phi(\rho_n)[U^n_m]^*)\biggr)=0\quad \forall\, n\geqslant0,
\end{equation}
\tag{3.10}
$$
as Lemma 4 in [8] (and its proof) imply that
$$
\begin{equation*}
\lim_{m\to+\infty}\widehat{a}_n^m=\widehat{a}_n<+\infty \quad\text{and}\quad \lim_{m\to+\infty}a_n^m=a_n\leqslant+\infty \quad \forall\, n\geqslant0.
\end{equation*}
\notag
$$
Since the condition $D(\Phi(\rho_n)\|\Phi(\sigma_n))\!<\!+\infty$ implies that $\operatorname{supp}\Phi(\rho_n)\!\subseteq\!\operatorname{supp}\Phi(\sigma_n)$ for all $n\geqslant0$, the properties of the double sequence $\{P^n_m\}_{n\geqslant0,\,m\geqslant m_0}$ show that $U^n_m\Phi(\rho_n)[U^n_m]^*$ tends to $\Phi(\rho_n)$ as $m\to+\infty$ for each $n\geqslant0$. Hence the limit relation (3.10) is proved easily by using Proposition 2 in [29] (in view of the obvious inequality $\Phi(\rho_n)\leqslant \Phi(\rho_n)+U^n_m\Phi(\rho_n)[U^n_m]^*$ and the trivially holding equality $D(\Phi(\rho_n)\,\|\,\Phi(\rho_n))=0$). The validity of claim (A) in the case when $\Phi$ is an arbitrary quantum operation follows from claim (B) proved below (by using the part of (A) proved above). (B) Let $\{\rho_n\}$ and $\{\sigma_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H}_A)$ converging, respectively, to operators $\rho_0$ and $\sigma_0$. Let $\{\Phi_n\}$ be a sequence of quantum operations in $\mathfrak{F}_{\leqslant1}(A,B)$ converging strongly to a quantum operation $\Phi_0$ such that $D(\Phi_n(\rho_n)\,\|\,\Phi_n(\sigma_n))<+\infty$ for all $n\geqslant0$. We have to show that
$$
\begin{equation}
\liminf_{n\to+\infty}\Delta_{\Phi_n}(\rho_n,\sigma_n)\geqslant\Delta_{\Phi_0}(\rho_0,\sigma_0).
\end{equation}
\tag{3.11}
$$
Assume first that $\{\Phi_n\}$ is a sequence of quantum channels converging strongly to a quantum channel $\Phi_0$. By Theorem 7 in [13] there exist a system $E$ and a sequence $\{V_n\}$ of isometries from $\mathcal{H}_{A}$ to $\mathcal{H}_{BE}$ converging strongly to an isometry $V_0$ such that $\mathrm{\Phi}_n(\varrho)=\operatorname{Tr}_E V_n\varrho V^*_n$ for all $n\geqslant0$. It is clear that the operators $\varrho_n=V_n\rho_nV_n^*$ and $\varsigma_n=V_n\sigma_nV_n^*$ in $\mathfrak{T}_+(\mathcal{H}_{BE})$ tend, respectively, to the operators $\varrho_0=V_0\rho_0V_0^*$ and $\varsigma_0=V_0\sigma_0V_0^*$ as $n\to+\infty$. Since all operators $V_n$ are isometries, we have
$$
\begin{equation*}
D(\varrho_n\,\|\, \varsigma_n)=D(\rho_n\,\|\, \sigma_n) \quad \forall\,n\geqslant0.
\end{equation*}
\notag
$$
Hence $\Delta_{\Phi_n}(\rho_n,\sigma_n)=\Delta_{\Theta}(\varrho_n,\varsigma_n)$ for all $n\geqslant0$, where $\Theta=\operatorname{Tr}_E(\,\cdot\,)$ is a channel from $BE$ to $B$. Thus, the limit relation (3.11) follows from the part of claim (A) proved already.
Assume now that $\{\Phi_n\}$ is a sequence of quantum operations converging strongly to a quantum operation $\Phi_0$. Lemma 2 in [15] and its proof show the existence of a system $C$ and a sequence $\{\widetilde{\Phi}_n\}$ of quantum channels from $\mathfrak{T}(\mathcal{H}_A)$ to $\mathfrak{T}(\mathcal{H}_B\oplus\mathcal{H}_C)$ converging strongly to a quantum channel $\widetilde{\Phi}_0$ such that
$$
\begin{equation*}
\Phi_n(\varrho)=P_B\widetilde{\Phi}_n(\varrho)=\widetilde{\Phi}_n(\varrho)P_B \quad \forall\, \varrho\in\mathfrak{T}(\mathcal{H}_A), \quad \forall\, n\geqslant0,
\end{equation*}
\notag
$$
where $P_B$ is the projection onto the subspace $\mathcal{H}_B$ of $\mathcal{H}_{B}\oplus\mathcal{H}_{C}$. By identity (2.18) we have
$$
\begin{equation}
D(\widetilde{\Phi}_n(\rho_n)\,\|\, \widetilde{\Phi}_n(\sigma_n))=D(\Phi_n(\rho_n)\,\|\, \Phi_n(\sigma_n))+D(\Psi_n(\rho_n)\,\|\, \Psi_n(\sigma_n)) \quad \forall\, n\geqslant0,
\end{equation}
\tag{3.12}
$$
where $\Psi_n(\varrho)\!=\!P_C\widetilde{\Phi}_n(\varrho)$ and $P_C$ is the projection onto the subspace $\mathcal{H}_C$ of ${\mathcal{H}_{B}\!\oplus\!\mathcal{H}_{C}}$.
Since we do not assume that $D(\rho_n\,\|\, \sigma_n)$ is finite for all $n\geqslant0$, we cannot guarantee that $D(\widetilde{\Phi}_n(\rho_n)\,\|\, \widetilde{\Phi}_n(\sigma_n))$ is finite for all $n\geqslant0$. So we cannot apply the above part of the proof directly to the sequence $\{\widetilde{\Phi}_n\}$. Nevertheless, we can limit our attention to the case when
$$
\begin{equation*}
\liminf_{n\to+\infty}D(\widetilde{\Phi}_n(\rho_n)\,\|\, \widetilde{\Phi}_n(\sigma_n))\leqslant\liminf_{n\to+\infty}D(\rho_n\,\|\, \sigma_n)<+\infty,
\end{equation*}
\notag
$$
since otherwise the limit relation (3.11) holds trivially. Thus, by passing to a subsequence we may assume that $D(\widetilde{\Phi}_n(\rho_n)\,\|\, \widetilde{\Phi}_n(\sigma_n))<+\infty$ for all $n>0$.
If $D(\widetilde{\Phi}_0(\rho_0)\,\|\, \widetilde{\Phi}_0(\sigma_0))<+\infty$, then applying the above part of the proof to the sequence $\{\widetilde{\Phi}_n\}$ of quantum channels we obtain
$$
\begin{equation*}
\liminf_{n\to+\infty}\Delta_{\widetilde{\Phi}_n}(\rho_n,\sigma_n)\geqslant\Delta_{\widetilde{\Phi}_0}(\rho_0,\sigma_0).
\end{equation*}
\notag
$$
This and (3.12) imply (3.11), since
$$
\begin{equation}
\liminf_{n\to+\infty}D(\Psi_n(\rho_n)\,\|\, \Psi_n(\sigma_n))\geqslant D(\Psi_0(\rho_0)\,\|\, \Psi_0(\sigma_0))
\end{equation}
\tag{3.13}
$$
by the lower semicontinuity of quantum relative entropy.
If $D(\widetilde{\Phi}_0(\rho_0)\,\|\, \widetilde{\Phi}_0(\sigma_0))=+\infty$ then the expression (3.12) and the assumed finiteness of $D(\Phi_0(\rho_0)\,\|\, \Phi_0(\sigma_0))$ show that $D(\Psi_0(\rho_0)\,\|\, \Psi_0(\sigma_0))=+\infty$. So it follows from (3.13) that
$$
\begin{equation*}
\lim_{n\to+\infty}D(\Psi_n(\rho_n)\,\|\, \Psi_n(\sigma_n))=+\infty.
\end{equation*}
\notag
$$
This implies that
$$
\begin{equation*}
\lim_{n\to+\infty}\Delta_{\Phi_n}(\rho_n,\sigma_n)=+\infty,
\end{equation*}
\notag
$$
since $\Delta_{\Phi_n}(\rho_n,\sigma_n)\geqslant D(\Psi_n(\rho_n)\,\|\, \Psi(\sigma_n))$ for all $n\geqslant0$ by (3.12) and the obvious inequality $D(\widetilde{\Phi}_n(\rho_n)\,\|\, \widetilde{\Phi}_n(\sigma_n))\leqslant D(\rho_n\,\|\,\sigma_n)$, $n\geqslant0$. Hence the limit relation (3.11) holds trivially in this case.
The theorem is proved. Lemma 3. Let $\rho$ and $\sigma$ be operators in $\mathfrak{T}_+(\mathcal{H})$ and $P$ a projector in $\mathfrak{B}(\mathcal{H})$ such that $P\sigma=\sigma P$. Then Proof. We assume that the operators $\rho$ and $\sigma$ are nonzero, since otherwise (3.14) holds trivially.
Since $U\sigma U^*=\sigma$ and $U^*=U$, using Donald’s identity (2.19) we obtain
$$
\begin{equation*}
\begin{aligned} \, D(\rho\,\|\, \sigma) &=\frac{1}{2}D(\rho\,\|\, \sigma)+\frac{1}{2}D(U\rho U^*\,\|\, \sigma) =D\biggl(\frac{1}{2}(\rho+U\rho U^*)\biggm\| \sigma\biggr) \\ &\qquad+ \frac{1}{2}D\biggl(\rho\biggm\|\frac{1}{2}(\rho+U\rho U^*)\biggr) +\frac{1}{2}D\biggl(U\rho U^*\biggm\|\frac{1}{2}(\rho+U\rho U^*)\biggr) \\ &=D(P\rho P+\overline{P}\rho\overline{P}\,\|\, \sigma) +D\biggl(\rho\biggm\|\frac{1}{2}(\rho+U\rho U^*)\biggr) \\ &=D(P\rho P\,\|\, P\sigma)+D(\overline{P}\rho\overline{P}\,\|\, \overline{P}\sigma)+D\biggl(\rho\biggm\|\frac{1}{2}(\rho+U\rho U^*)\biggr) \\ &=D(P\rho P\,\|\, P\sigma) +D(\overline{P}\rho \overline{P}\,\|\,\overline{P}\sigma)+ D(\rho\,\|\, P\rho P+\overline{P}\rho\overline{P}), \end{aligned}
\end{equation*}
\notag
$$
where the 4the equality follows from identity (2.18).
The lemma is proved. Lemma 4. Let $\{\rho_n\}$ and $\{\sigma_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H})$ converging, respectively, to operators $\rho_0$ and $\sigma_0\neq0$. If $\{P^n_m\}_{n\geqslant0,\,m\geqslant m_0}$ is a double sequence of finite-rank projectors completely consistent with the sequence $\{\sigma_n\}$ (see the definition in § 2.1), then Proof. Since $P^n_m\rho_n P^n_m$ tends to $P^0_m\rho_0 P^0_m$ as $n\!\to\!+\infty$ and ${\sup_{n\geqslant0}\operatorname{rank} P^n_m\rho_n P^n_m\mkern-1mu\!<\!\mkern-1.5mu +\infty}$ by the conditions in (2.6) and (2.7), we have
$$
\begin{equation*}
\lim_{n\to+\infty}S(P^n_m\rho_nP^n_m)=S(P^0_m\rho_0P^0_m)<+\infty \quad \forall\, m\geqslant m_0.
\end{equation*}
\notag
$$
Therefore, using representation (2.12) we see that to prove (3.15) it suffices to show that
$$
\begin{equation}
\lim_{n\to+\infty}\|P^n_m \ln (P^n_m\sigma_n)-P^0_m \ln (P^0_m\sigma_0)\|=0
\end{equation}
\tag{3.16}
$$
for any fixed $m\geqslant m_0$. By the second condition in (2.7) we have $\operatorname{rank} P^n_m\sigma_n=\operatorname{rank} P^n_m$ for all $n\geqslant0$. Hence the sequence $\{P^n_m\sigma_n+\overline{P}^{\,n}_m\}_n$ consists of bounded nondegenerate operators and converges to the nondegenerate operator $P^0_m\sigma_0+\overline{P}^{\,0}_m$ in the operator norm by the third condition in (2.7). It follows that $P^n_m\sigma_n+\overline{P}^{\,n}_m\geqslant\epsilon I_{\mathcal{H}}$ for all $n\geqslant0$ and some $\epsilon>0$. So, by Proposition VIII.20 in [30] the sequence $\{\ln (P^n_m\sigma_n+\overline{P}^{\,n}_m)\}_n$ converges to the operator $\ln (P^0_m\sigma_0+\overline{P}^{\,0}_m)$ in the operator norm. This implies (3.16) since
The lemma is proved. 3.2. The nonincrease of local discontinuity jumps of quantum relative entropy under the action of quantum operationsThe lower semicontinuity of the relative entropy disturbance by quantum operations implies that the local discontinuity jumps of quantum relative entropy do not increase under action of quantum operations. Proposition 1. Let $\{\rho_n\}$ and $\{\sigma_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H}_A)$ converging, respectively, to operators $\rho_0$ and $\sigma_0$ such that $D(\rho_0\,\|\,\sigma_0)<+\infty$. (A) If $\Phi$ is an arbitrary quantum operation from $A$ to $B$, then
$$
\begin{equation}
\begin{aligned} \, &\limsup_{n\to+\infty}D(\Phi(\rho_n)\,\|\,\Phi(\sigma_n))-D(\Phi(\rho_0)\,\|\, \Phi(\sigma_0)) \nonumber \\ &\qquad\leqslant\limsup_{n\to+\infty}D(\rho_n\,\|\,\sigma_n)-D(\rho_0\,\|\,\sigma_0). \end{aligned}
\end{equation}
\tag{3.17}
$$
(B) If $\{\Phi_n\}$ is a sequence of quantum operations from $A$ to $B$ converging strongly to a quantum operation $\Phi_0$, then Remark 1. The left-hand sides of (3.17) and (3.18) are well defined due to the finiteness of $D(\rho_0\,\|\,\sigma_0)$ and the monotonicity of quantum relative entropy. By the lower semicontinuity of quantum relative entropy the quantities in the right- and left-hand sides of (3.17) characterize the discontinuity of the functions $(\rho,\sigma)\mapsto D(\rho\,\|\,\sigma)$ and $(\rho,\sigma)\mapsto D(\Phi(\rho)\,\|\,\Phi(\sigma))$ for the given convergent sequences $\{\rho_n\}$ and $\{\sigma_n\}$. Thus, claim (A) of Proposition 1 can be interpreted as the ‘contraction’ of possible discontinuities of quantum relative entropy by quantum operations, and claim (B) can be interpreted as the ‘contraction’ of possible discontinuities of quantum relative entropy under the action of a strongly convergent sequence of quantum operations. Proof of Proposition 1. Since the quantum relative entropy is a lower semicontinuous function of its arguments, claims (A) and (B) of the proposition follow by Lemma 1 from claims (A) and (B), respectively, of the theorem in § 3.1. We must only observe that the assumption in claim (B) ensures the convergence of the sequences $\{\Phi_n(\rho_n)\}$ and $\{\Phi_n(\sigma_n)\}$ to the operators $\Phi_0(\rho_0)$ and $\Phi_0(\sigma_0)$.
The proposition is proved. Proposition 1 implies the following observations, which can be treated as the preservation of the local continuity of quantum relative entropy by a single quantum operation and by a strongly convergent sequence of quantum operations. It was originally proved in [15] by a quite different method (based on the convergence criterion for quantum relative entropy proposed there). Corollary 1. Let $\{\rho_n\}$ and $\{\sigma_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H}_A)$ converging, respectively, to operators $\rho_0$ and $\sigma_0$ such that (A) If $\Phi$ is an arbitrary quantum operation from $A$ to $B$, then
$$
\begin{equation}
\lim_{n\to+\infty}D(\Phi(\rho_n)\,\|\,\Phi(\sigma_n))=D(\Phi(\rho_0)\,\|\,\Phi(\sigma_0))<+\infty.
\end{equation}
\tag{3.20}
$$
(B) If $\{\Phi_n\}$ is a sequence of quantum operations from $A$ to $B$ strongly converging to a quantum operation $\Phi_0$, then Remark 2. Roughly speaking, the quantum relative entropy of two states $\rho$ and $\sigma$ can be expressed as Thus, if $\{\rho_n\}$ and $\{\sigma_n\}$ are some convergent sequences of states such that the limit relations (3.21) hold, but (3.23) is not valid, then Corollary 1, (A), implies that the limit relation (3.22) is not valid either and the discontinuity jump of the function $(\varrho,\varsigma)\mapsto F(\Phi(\varrho),\Phi(\varsigma))$ compensates for the discontinuity jump of the function $(\varrho,\varsigma)\mapsto S(\Phi(\varrho))$ in such a way that
$$
\begin{equation*}
\lim_{n\to+\infty}(F-S)(\Phi(\rho_n),\Phi(\sigma_n)) =(F-S)(\Phi(\rho_0),\Phi(\sigma_0))<+\infty.
\end{equation*}
\notag
$$
Note that the monotonicity property of quantum relative entropy means a similar compensation for the possible decrease of von Neumann entropy $S$ under the action of a quantum channel. This compensation is necessary for the nonincrease of the quantum relative entropy, the function ${F-S}$. Remark 3. At first glance, Corollary 1 is the only contribution of Proposition 1 to the task of proving the local continuity (convergence) of quantum relative entropy and related functions. In fact, the more general claim of Proposition 1 opens way to the proof of the local continuity of a given function $f$ by uniform approximation of this function by some appropriate functions $f_n$ for which a relation similar to (3.17), with right-hand side tending to zero as $n\to+\infty$, can be established. This approach is used in the proof of Proposition 9 in § 4.6. There is a class of quantum operations $\Phi$ for which relation (3.23) holds, provided that the second limit relation in (3.21) is valid. The quantum operations in this class are characterized by the following equivalent properties (see [31], Theorem 1):
Property (a) means that such operations preserve the finiteness of von Neumann entropy, so they are called PFE-operations in [31], where their detailed classification can be found. Corollary 2. Let $\{\rho_n\}$ and $\{\sigma_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H}_A)$ converging, respectively, to operators $\rho_0$ and $\sigma_0$ such that Then for any PFE-operation $\Phi$. Proof. Using the expression (2.12) and Lemma 1 we can show that condition (3.24) implies the (3.19) and the second limit relation in (3.21). Hence (3.20) holds by Corollary 1, (A), and (3.23) is valid, since $\Phi$ is a PFE-operation. It is clear that (3.25) follows from (3.20) and (3.23) in view of (2.12).
The corollary is proved. 3.3. The lower semicontinuity of the joint convexity modulus of quantum relative entropy3.3.1. The case of convex mixturesThe joint convexity of quantum relative entropy means that
$$
\begin{equation*}
D\biggl(\sum_ip_i\rho_i\biggm\|\sum_ip_i\sigma_i\biggr)\leqslant \sum_i p_iD(\rho_i\,\|\,\sigma_i)
\end{equation*}
\notag
$$
for any ensembles $\{p_i,\rho_i\}$ and $\{p_i,\sigma_i\}$ of quantum states in $\mathfrak{S}(\mathcal{H})$ with the same probability distributions [8], [14], [16]. We denote the set of all discrete ensembles of states in $\mathfrak{S}(\mathcal{H})$ by $\mathcal{P}_{d}(\mathcal{H})$. We say that a sequence of ensembles $\mu_n=\{p^n_i,\rho^n_i\}$ in $\mathcal{P}_{d}(\mathcal{H})$ $D_0$-converges to an ensemble $\mu_0=\{p^0_i,\rho^0_i\}$ if
$$
\begin{equation*}
\lim_{n\to+\infty}p^n_i\rho^n_i=p^0_i\rho^0_i \quad \forall\, i.
\end{equation*}
\notag
$$
Part (A) of the theorem in § 3.1 implies the following result. Proposition 2. The nonnegative function is lower semicontinuous on the set with respect to $D_0$-convergence in $\mathcal{P}_{d}(\mathcal{H})\times\mathcal{P}_{d}(\mathcal{H})$. Proof. Given a pair of ensembles $\mu=\{p_i,\rho_i\}$ and $\nu=\{p_i,\sigma_i\}$ belonging to the set in (3.26), we introduce the QC states
$$
\begin{equation*}
\widehat{\mu}=\sum_i p_i\rho_i\otimes|i\rangle\langle i| \quad\text{and}\quad \widehat{\nu}=\sum_i p_i\sigma_i\otimes|i\rangle\langle i|
\end{equation*}
\notag
$$
in $\mathfrak{S}(\mathcal{H}\otimes\mathcal{H}_E)$, where $\{|i\rangle\}$ is a basis in a separable Hilbert space $\mathcal{H}_E$. Then identities (2.13) and (2.18) imply that
$$
\begin{equation*}
\sum_i p_iD(\rho_i\,\|\,\sigma_i)=D(\widehat{\mu}\,\|\,\widehat{\nu}),
\end{equation*}
\notag
$$
while
$$
\begin{equation*}
D\biggl(\sum_ip_i\rho_i\biggm\|\sum_ip_i\sigma_i\biggr)=D(\Phi(\widehat{\mu})\,\|\,\Phi(\widehat{\nu})),
\end{equation*}
\notag
$$
where $\Phi=\operatorname{Tr}_E(\,\cdot\,)$. Thus, to deduce the required result from part (A) of the theorem in § 3.1, it is sufficient to note that if a sequence $\{\{p^n_i,\varrho^n_i\}_i\}_n$ of ensembles in $\mathcal{P}_{d}(\mathcal{H})$ $D_0$-converges to an ensemble $\{p^0_i,\varrho^0_i\}_i$ , then the corresponding sequence $\{\sum_i p^n_i\varrho^n_i\otimes|i\rangle\langle i|\}_n$ of QC states converges to the QC state $\sum_i p^0_i\varrho^0_i\otimes|i\rangle\langle i|$ in the trace norm by Lemma 2.
The proposition is proved. Using the lower semicontinuity of quantum relative entropy and the last observation in the proof of Proposition 2 it is easy to show that the function $(\{p_i,\rho_i\},\{p_i,\sigma_i\})\mapsto D\bigl(\sum_ip_i\rho_i\bigm\|\sum_ip_i\sigma_i\bigr)$ is lower semicontinuous with respect to $D_0$-convergence. Thus, Proposition 2 implies (by Lemma 1) the following convergence condition for quantum relative entropy. Corollary 3. Let $\{\{p^n_i,\rho^n_i\}\}_n$ and $\{\{p^n_i,\sigma^n_i\}\}_n$ be sequences of ensembles in $\mathcal{P}_{d}(\mathcal{H})$ $D_0$-converging to ensembles $\{p^0_i,\rho^0_i\}$ and $\{p^0_i,\sigma^0_i\}$, respectively. If then In § 4.5 we obtain a generalization of Corollary 3 to the case of arbitrary $D_0$-convergent sequences of discrete ensembles. 3.3.2. The case of finite and countable sumsThe joint convexity and lower semicontinuity of quantum relative entropy, along with identity (2.13), imply that
$$
\begin{equation}
D\biggl(\sum_i\rho_i\biggm\|\sum_i\sigma_i\biggr)\leqslant \sum_i D(\rho_i\,\|\,\sigma_i)
\end{equation}
\tag{3.27}
$$
for any finite or countable sets $\{\rho_i\}$ and $\{\sigma_i\}$ of operators in $\mathfrak{T}_+(\mathcal{H})$ such that $\sum_i\operatorname{Tr}\rho_i$ and $\sum_i\operatorname{Tr}\sigma_i$ are finite. We denote by $\mathfrak{T}^{\infty}_+(\mathcal{H})$ the set $\{\{\varrho_i\}_{i=1}^{+\infty}\subset\mathfrak{T}_+(\mathcal{H}) |\sum_i\operatorname{Tr}\varrho_i<+\infty\}$. We say that a sequence $\{\{\varrho^n_i\}_i\}_n\subset\mathfrak{T}^{\infty}_+(\mathcal{H})$ converges to $\{\varrho^0_i\}\in\mathfrak{T}^{\infty}_+(\mathcal{H})$ if
$$
\begin{equation}
\lim_{n\to+\infty}\varrho^n_i=\varrho^0_i \quad \forall\, i \quad\text{and}\quad \lim_{n\to+\infty}\sum_i\operatorname{Tr}\varrho^n_i=\sum_i\operatorname{Tr}\varrho^0_i.
\end{equation}
\tag{3.28}
$$
By Lemma 2 this is equivalent to the trace norm convergence of the sequence $\bigl\{\sum_i \varrho^n_i\otimes|i\rangle\langle i|\bigr\}\subset\mathfrak{T}(\mathcal{H}\otimes\mathcal{H}_E)$ to the operator $\sum_i \varrho^0_i\otimes|i\rangle\langle i|\in\mathfrak{T}(\mathcal{H}\otimes\mathcal{H}_E)$, where $\{|i\rangle\}$ is a basis in a separable Hilbert space $\mathcal{H}_E$. Thus, the same arguments as in the proof of Proposition 2 allow us to derive the following result from part (A) of the theorem in § 3.1. Proposition 3. The function $(\{\rho_i\},\{\sigma_i\})\mapsto\sum_i D(\rho_i\|\sigma_i)-D(\sum_i\rho_i\,\|\,\sum_i\sigma_i)$ is lower semicontinuous on the set This result and Lemma 1 imply directly the following observation concerning the convergence of quantum relative entropy for finite and countable sums. Corollary 4. Let $\{\{\rho^n_i\}_{n\geqslant 0}\}_{i\in I}$ and $\{\{\sigma^n_i\}_{n\geqslant0}\}_{i\in I}$ be finite or countable sets of convergent sequences of operators in $\mathfrak{T}_+(\mathcal{H})$ such that $\sum_{i\in I}D(\rho^0_i\|\sigma^0_i)<+\infty$, Remark 4. The left-hand side of (3.29) is well defined, since $D(\sum_{i\in I}\rho^0_i\|\sum_i\sigma^0_i)<+\infty$ by the condition $\sum_{i\in I}D(\rho^0_i\|\sigma^0_i)<+\infty$ and inequality (3.27). If the set $I$ is finite, then condition (3.30) means that
$$
\begin{equation}
\lim_{n\to+\infty}D(\rho^n_i\,\|\,\sigma^n_i)=D(\rho^0_i\,\|\,\sigma^0_i)<+\infty \quad \forall\, i\in I.
\end{equation}
\tag{3.31}
$$
Thus, in this case the last claim of Proposition 4 agrees with Corollary 3 in [29], which claims the preservation of the convergence of quantum relative entropy under finite summation. If the set $I$ is countable, then (3.31) is necessary but not sufficient for (3.30). So, in this case we can treat (3.30) as a sufficient condition for the preservation of the convergence of quantum relative convergence under countable summation. It can be compared with the condition given by Corollary 4 in [15]. Using inequality (3.27) and Dini’s lemma, we can show that the former condition is slightly weaker than the latter one. The advantage of Corollary 4 in this article is the general relation (3.29), which provides additional tools for the analysis of the local continuity of quantum relative entropy (see the proof of Proposition 9 in § 4.6, which is based on the main claim of Corollary 7 in § 4.5, proved by using (3.29)). § 4. Local continuity analysis of quantum relative entropy and related functions4.1. Generalization of Lindblad’s lemmaThe best known and widely used result concerning the convergence (local continuity) of quantum relative entropy was presented in Lemma 4 in [8]. It states that
$$
\begin{equation*}
\lim_{n\to+\infty}D(P_n\rho P_n\,\|\,P_n\sigma P_n)=D(\rho\,\|\,\sigma)\leqslant+\infty
\end{equation*}
\notag
$$
for any operators $\rho$ and $\sigma$ in $\mathfrak{T}_+(\mathcal{H})$, where $\{P_n\}$ is an arbitrary nondecreasing sequence of projectors converging to the identity operator in the strong operator topology. The results of § 3 imply the following strengthened version of this claim. Proposition 4. If $\rho$ and $\sigma$ are operators in $\mathfrak{T}_+(\mathcal{H})$ such that $D(\rho\,\|\,\sigma)<+\infty$, then Proof. The uniform boundedness principle (see [30], Theorem III.9) implies that $\sup_n\|A_n\|<+\infty$. So by identity (2.13) we may assume that $\|A_n\|\leqslant1$ for all $n\geqslant0$. Then the sequence of quantum operations $\Phi_n=A_n(\,\cdot\,)A_n$ converges strongly to the quantum operation $\Phi_0=A_0(\,\cdot\,)A_0$. Hence the limit relation (4.1) follows from Corollary 1, (B).
The validity of (4.1) in the case when $D(\rho\,\|\,\sigma)=+\infty$, provided that $A_0=I_{\mathcal{H}}$ and $\|A_n\|\leqslant1$ for all $n$, follows from the lower semicontinuity of quantum relative entropy, since $D(A_n\rho A_n\|A_n\sigma A_n)\leqslant D(\rho\,\|\,\sigma)$ by the monotonicity of quantum relative entropy under the quantum operation $\Phi_n=A_n(\,\cdot\,)A_n$. The proposition is proved. Remark 5. The condition $D(\rho\,\|\,\sigma)<+\infty$ in the main claim of Proposition 4 is essential. Indeed, let $\rho$ and $\sigma$ be operators diagonizable in a given basis $\{|i\rangle\}$ in $\mathcal{H}$ and such that $D(\rho\,\|\,\sigma)=+\infty$, and let $A_n=\sum_{i>n}|i\rangle\langle i|$ for all $n\in \mathbb{N}$. Then it is easy to see that the sequence $\{A_n\}$ converges strongly to $A_0=0$ and while 4.2. The function $\Phi\mapsto D(\Phi(\rho)\,\|\,\Phi(\sigma))$From part (B) of Corollary 1, in the case when $\rho_n=\rho_0$ and $\sigma_n=\sigma_0$ for all $n$ we obtain the following result on the properties of the function $\Phi\mapsto D(\Phi(\rho)\,\|\,\Phi(\sigma))$. Proposition 5. If $\rho$ and $\sigma$ are operators in $\mathfrak{T}_+(\mathcal{H}_A)$ such that $D(\rho\,\|\,\sigma)<+\infty$, then the function $\Phi\mapsto D(\Phi(\rho)\,\|\,\Phi(\sigma))$ is continuous on the set of all quantum operations from $A$ to an arbitrary quantum system $B$ equipped with the strong convergence topology. The usefulness of this assertion can be illustrated by the following example. Example 1. Let $\{\Phi_t\}_{t\in \mathbb{R}_+}$ be an arbitrary strongly continuous family of quantum channels (for instance, a quantum dynamical semigroup). Then Proposition 5 implies that the function is continuous on $\mathbb{R}_+$ for any input states $\rho$ and $\sigma$ such that $D(\rho\,\|\,\sigma)$ is finite. If $\{\Phi_t\}$ is a semigroup, then the general properties of the quantum relative entropy show that the above function is nonincreasing and lower semicontinuous on $\mathbb{R}_+$. This ensures only that it is right continuous on $\mathbb{R}_+$. Thus, the claim about the continuity of this function is not trivial even in this case. 4.3. The function $(\rho,\sigma,\eta,\theta)\mapsto D(\rho+\eta\,\|\,\sigma)-D(\rho\,\|\,\sigma+\theta)$Inequalities (2.15) and (2.16) show that
$$
\begin{equation*}
D(\rho+\eta\,\|\,\sigma)\geqslant D(\rho\,\|\,\sigma+\theta)-\operatorname{Tr}\theta-\operatorname{Tr}\sigma
\end{equation*}
\notag
$$
for any operators $\rho,\sigma,\eta$ and $\theta$ in $\mathfrak{T}_+(\mathcal{H})$. Part (A) of the theorem in § 3.1 implies the following result. Proposition 6. The function $(\rho,\sigma,\eta,\theta)\mapsto D(\rho+\eta\,\|\,\sigma)-D(\rho\,\|\,\sigma+\theta)$ is lower semicontinuous on the set Proof. Let $\mathcal{H}_R$ be a two-dimensional Hilbert space. Given operators $\rho$, $\sigma$ and $\theta$ in $\mathfrak{T}_+(\mathcal{H})$, consider the operators
$$
\begin{equation*}
\widehat{\rho}=\rho\otimes |0\rangle\langle0|\quad\text{and} \quad \widehat{\sigma}=\sigma\otimes |0\rangle\langle0|+\theta\otimes |1\rangle\langle1|
\end{equation*}
\notag
$$
in $\mathfrak{T}_+(\mathcal{H}\otimes\mathcal{H}_R)$, where $\{|0\rangle,|1\rangle\}$ is a basis in $\mathcal{H}_R$. Then identity (2.18) implies that
$$
\begin{equation*}
D(\widehat{\rho}\,\|\,\widehat{\sigma})=D(\rho\,\|\,\sigma)+D(0\,\|\,\theta) =D(\rho\,\|\,\sigma)+\operatorname{Tr}\theta,
\end{equation*}
\notag
$$
while Hence
Using Donald’s identity (2.19) and identities (2.13) and (2.14) we obtain
$$
\begin{equation}
D(\rho+\eta\,\|\,\sigma)+f(\rho,\eta)=D(\rho\,\|\,\sigma)+D(\eta\,\|\,\sigma) +\operatorname{Tr}(\rho+\eta)\ln2-\operatorname{Tr}\sigma,
\end{equation}
\tag{4.3}
$$
where
$$
\begin{equation*}
f(\rho,\eta)=D\biggl(\rho\biggm\|\frac{1}{2}(\rho+\eta)\biggr) +D\biggl(\eta\biggm\|\frac{1}{2}(\rho+\eta)\biggr).
\end{equation*}
\notag
$$
Applying Proposition 2 in [29] it is easy to show that $f(\rho,\eta)$ is a continuous function on $[\mathfrak{T}_+(\mathcal{H})]^{\times2}$. Identity (2.14) and inequality (2.15) show that $f(\rho,\eta)\leqslant\operatorname{Tr}(\rho+ \eta)\ln2$.
It follows from (4.2) and (4.3) that
$$
\begin{equation*}
\begin{aligned} \, &D(\rho+\eta\,\|\,\sigma)-D(\rho\,\|\,\sigma+\theta) \\ &\qquad=D(\eta\,\|\,\sigma) +\Delta_{\operatorname{Tr}_R(\,\cdot\,)}(\widehat{\rho},\widehat{\sigma}) +[\operatorname{Tr}(\rho+\eta)\ln2-f(\rho,\eta)]-\operatorname{Tr}(\sigma+\theta) \end{aligned}
\end{equation*}
\notag
$$
for all $\rho,\sigma,\eta$ and $\theta$ in $\mathfrak{T}_+(\mathcal{H})$ such that $D(\rho\,\|\,\sigma+\theta)<+\infty$. Thus, the claim of the proposition follows from the lower semicontinuity of quantum relative entropy, part (A) of the theorem in § 3.1 and the continuity of the function $f(\rho,\eta)$ mentioned above.
The proposition is proved. Proposition 6, the lower semicontinuity of quantum relative entropy and Lemma 1 imply the following result. Corollary 5. Let $\{\rho^1_n\}$, $\{\rho^2_n\}$, $\{\sigma^1_n\}$ and $\{\sigma^2_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H})$ converging, respectively, to operators $\rho^1_0$, $\rho^2_0$, $\sigma^1_0$ and $\sigma^2_0$ such that $\rho^2_n\leqslant \rho^1_n$ and $\sigma^1_n\leqslant \sigma^2_n$ for all $n\geqslant0$. If $D(\rho^1_0\,\|\,\sigma^1_0)<+\infty$, then $D(\rho^2_0\,\|\,\sigma^2_0)<+\infty$ and9 The last claim of Corollary 5 coincides with the claim of Proposition 2 in [29], proved there in a rather indirect and complex way.10 4.4. The function $(\rho,\sigma,\eta,\theta)\mapsto \operatorname{Tr} (\rho+\eta)(-\ln\sigma)-\operatorname{Tr} \rho(-\ln(\sigma+\theta))$The operator monotonicity of the logarithm and the obvious inequality $\sigma\leqslant I_{\mathcal{H}}\operatorname{Tr}\sigma $ show that
$$
\begin{equation*}
\operatorname{Tr} (\rho+\eta)(-\ln\sigma)\geqslant \operatorname{Tr} \rho(-\ln(\sigma+\theta))-\operatorname{Tr}\eta\ln\operatorname{Tr}\sigma
\end{equation*}
\notag
$$
for any operators $\rho$, $\sigma$, $\eta$ and $\theta$ in $\mathfrak{T}_+(\mathcal{H})$.11 Proposition 6 implies the following result. Proposition 7. The function $(\rho,\sigma,\eta,\theta)\mapsto \operatorname{Tr} (\rho+\eta)(-\ln\sigma)-\operatorname{Tr} \rho(-\ln(\sigma+\theta))$ is lower semicontinuous on the set Proof. The function is lower semicontinuous on the set $[\mathfrak{T}_+(\mathcal{H})]^{\times 2}$ by the lower semicontinuity of quantum relative entropy and extended von Neumann entropy. Thus, to prove the proposition it suffices to show that the function $(\rho,\sigma,\theta)\mapsto \operatorname{Tr} \rho(-\ln\sigma)-\operatorname{Tr} \rho(-\ln(\sigma+\theta))$ is lower semicontinuous on the set of all triplets $(\rho,\sigma,\theta)$ such that ${\operatorname{Tr} \rho(-\ln(\sigma+\theta))<+\infty}$.
Since the finiteness of $\operatorname{Tr} \rho(-\ln(\sigma+\theta))$ implies the finiteness of the entropy $S(\rho)$, using representation (2.12) we see that
$$
\begin{equation*}
\operatorname{Tr}\rho(-\ln\sigma)-\operatorname{Tr} \rho(-\ln(\sigma+\theta))=D(\rho\,\|\,\sigma)-D(\rho\,\|\,\sigma+\theta)+\operatorname{Tr}\theta.
\end{equation*}
\notag
$$
By Proposition 6 the function $(\rho,\sigma,\theta)\mapsto D(\rho\,\|\,\sigma)-D(\rho\,\|\,\sigma+\theta)$ is lower semicontinuous on the set of triplets $(\rho,\sigma,\theta)$ such that $D(\rho\,\|\,\sigma+\theta)<+\infty$, which contains the set of triplets $(\rho,\sigma,\theta)$ such that $\operatorname{Tr} \rho(-\ln(\sigma+\theta))<+\infty$.
The proposition is proved. Using Proposition 7, the lower semicontinuity of the function $(\rho,\sigma)\!\mapsto\! \operatorname{Tr} \rho(-\ln\sigma)$ on $[\mathfrak{T}_+(\mathcal{H})]^{\times 2}$ (mentioned in the proof of Proposition 7) and Lemma 1 we prove the following result. Corollary 6. Let $\{\rho^1_n\}$, $\{\rho^2_n\}$, $\{\sigma^1_n\}$ and $\{\sigma^2_n\}$ be sequences of operators in $\mathfrak{T}_+(\mathcal{H})$ converging, respectively, to operators $\rho^1_0$, $\rho^2_0$, $\sigma^1_0$ and $\sigma^2_0$ such that $\rho^1_n\geqslant \rho^2_n$ and $\sigma^1_n\leqslant \sigma^2_n$ for all $n\geqslant0$. If $\operatorname{Tr} \rho^1_0(-\ln \sigma_0^1)<+\infty$, then $\operatorname{Tr} \rho^2_0(-\ln \sigma_0^2)<+\infty$ and The direct proof of Corollary 6 obtained in [29], the appendix, requires a lot of technical efforts. 4.5. The function $(\{p_i,\rho_i\},\{q_i,\sigma_i\})\mapsto D\bigl(\sum_ip_i\rho_i\,{\bigm\|}\sum_iq_i\sigma_i\bigr)$Using the joint convexity of quantum relative entropy (in the form of inequality (3.27)) and identities (2.13) and (2.14) it is easy to show that
$$
\begin{equation*}
D\biggl(\sum_ip_i\rho_i\biggm\|\sum_iq_i\sigma_i\biggr)\leqslant\sum_i p_iD(\rho_i\,\|\,\sigma_i)+D_{\mathrm{KL}}(\{p_i\}\,\|\,\{q_i\})
\end{equation*}
\notag
$$
for any ensembles $\{p_i,\rho_i\}$ and $\{q_i,\sigma_i\}$ of quantum states in $\mathfrak{S}(\mathcal{H})$, where
$$
\begin{equation*}
D_{\mathrm{KL}}(\{p_i\}\,\|\,\{q_i\})\doteq \sum_i p_i \ln \biggl(\frac{p_i}{q_i}\biggr)
\end{equation*}
\notag
$$
is the Kullback–Leibler divergence between the probability distributions $\{p_i\}$ and $\{q_i\}$ [32] (we assume that $D_{\mathrm{KL}}(\{p_i\}\|\{q_i\})=+\infty$ if there is $i$ such that $p_i\neq0$ and $q_i=0$). Indeed, using identities (2.13) and (2.14) it is easy to show that
$$
\begin{equation}
\sum_i p_iD(\rho_i\,\|\,\sigma_i)+D_{\mathrm{KL}}(\{p_i\}\,\|\,\{q_i\})=\sum_i D(p_i\rho_i\,\|\, q_i\sigma_i).
\end{equation}
\tag{4.4}
$$
Part (A) of the theorem in § 3.1 implies the following result. Proposition 8. The nonnegative function For the proof it suffices to use (4.4) and apply Proposition 3. Proposition 8 allows us to generalize the claims of Corollary 3. Corollary 7. Let $\{\{p^n_i,\rho^n_i\}\}_n$ and $\{\{q^n_i,\sigma^n_i\}\}_n$ be sequences of ensembles in $\mathcal{P}_{d}(\mathcal{H})$ $D_0$-converging to ensembles $\{p^0_i,\rho^0_i\}$ and $\{q^0_i,\sigma^0_i\}$ such that $\sum_i p^0_iD(\rho^0_i\|\sigma^0_i)<+\infty$ and $D_{\mathrm{KL}}(\{p^0_i\}\|\{q^0_i\})<+\infty$. Then13 Proof. By Lemma 1 the main claim follows from Proposition 8 and the lower semicontinuity of the function $(\{p_i,\rho_i\},\{q_i,\sigma_i\})\mapsto D\bigl(\sum_i p_i\rho_i\,{\bigm\|}\sum_i q_i\sigma_i\bigr)$ with respect to $D_0$-convergence in $\mathcal{P}_{d}(\mathcal{H})\times\mathcal{P}_{d}(\mathcal{H})$ (which is easy to prove using Lemma 2).
If $\rho^n_i\rho^n_j=\rho^n_i\sigma^n_j=\sigma^n_i\sigma^n_j=0$ for all $i\neq j$ and each $n$, then identity (2.18) implies that
$$
\begin{equation*}
D\biggl(\sum_ip^n_i\rho^n_i\biggm\|\sum_iq^n_i\sigma^n_i\biggr) =\sum_i p^n_iD(\rho^n_i\,\|\,\sigma^n_i)+D_{\mathrm{KL}}(\{p^n_i\}\,\|\,\{q^n_i\}) \quad \forall\, n.
\end{equation*}
\notag
$$
So the last claim follows by Lemma 1, from the lower semicontinuity of the function $(\{p_i,\rho_i\},\{q_i,\sigma_i\})\mapsto \sum_i p_iD(\rho_i\,\|\,\sigma_i)$ and the Kullback–Leibler divergence.
The corollary is proved. Example 2. Let $\{\rho_i\}$ and $\{\sigma_i\}$ be countable collections of quantum states such that $\sup_iD(\rho_i\|\sigma_i)<+\infty$. If $\{\{p^n_i\}_i\}_n$ and $\{\{q^n_i\}_i\}_n$ are sequences of probability distributions converging, respectively, to probability distributions $\{p^0_i\}$ and $\{q^0_i\}$ such that condition (4.6) holds, then Corollary 7 implies that 4.6. The function $\displaystyle(\mu,\nu)\mapsto D\biggl(\int \rho(x)\,\mu(dx)\biggm\|\int \rho(x)\,\nu(dx)\biggr)$Assume that $\rho(x)$ is an $\mathfrak{S}(\mathcal{H})$-valued measurable function on a separable metric space $X$ and that $\{\mu_n\}$ and $\{\nu_n\}$ are sequences of probability measures on $X$ converging (in some sense) to probability measures $\mu_0$ and $\nu_0$ such that
$$
\begin{equation}
\lim_{n\to+\infty}D_{\mathrm{KL}}(\mu_n\,\|\,\nu_n)=D_{\mathrm{KL}}(\mu_0\,\|\,\nu_0)<+\infty,
\end{equation}
\tag{4.8}
$$
where
$$
\begin{equation*}
D_{\mathrm{KL}}(\mu\,\|\,\nu)=\int_X \ln\biggl(\frac{d\mu}{d\nu}\biggr)\,\mu(dx)
\end{equation*}
\notag
$$
is the Kullback–Leibler divergence between the probability measures $\mu$ and $\nu$, and $\frac{d\mu}{d\nu}$ denotes the Radon–Nikodym derivative of $\mu$ with respect to $\nu$ (if $\mu$ is not absolutely continuous with respect to $\nu$, then $D_{\mathrm{KL}}(\mu\|\nu)=+\infty$) [32]–[34]. Motivated by Example 2, we conjecture that condition (4.8) implies that
$$
\begin{equation}
\begin{aligned} \, \notag &\lim_{n\to+\infty} D\biggl(\int_X \rho(x)\,\mu_n(dx)\biggm\|\int_X \rho(x)\,\nu_n(dx)\biggr) \\ &\qquad = D\biggl(\int_X \rho(x)\,\mu_0(dx)\biggm\|\int_X \rho(x)\,\nu_0(dx)\biggr)<+\infty. \end{aligned}
\end{equation}
\tag{4.9}
$$
Using the main claim of Corollary 7 and simple approximation technique we can prove the following result. Proposition 9. Let $\rho(x)$ be an $\mathfrak{S}(\mathcal{H})$-valued continuous function on a separable metric space $X$. Let $\{\mu_n\}$ and $\{\nu_n\}$ be sequences of probability measures converging setwise14 to probability measures $\mu_0$ and $\nu_0$, respectively, such that $\mu_n$ is absolutely continuous with respect to $\nu_n$ for all $n\geqslant0$ and the family $\left\{\frac{d\mu_n}{d\nu_n}\right\}_{n\geqslant 0}$ of Radon–Nikodym derivatives is uniformly bounded and uniformly equicontinuous on $X$, that is, Proof. Assume that (4.8) holds and $D_{\mathrm{KL}}(\mu_n\|\nu_n)<+\infty$ for all $n$.
For fixed $\varepsilon>0$ consider a countable decomposition $\{X_i\}$ of $X$ into disjoint Borel subsets with diameter not exceeding $\varepsilon$. For each $n$ consider the ensembles $\{p^{n}_i,\rho^{n}_i\}$ and $\{q^{n}_i,\sigma^{n}_i\}$ of states in $\mathfrak{S}(\mathcal{H})$ such that
$$
\begin{equation*}
p^{n}_i=\mu_n(X_i), \qquad \rho^{n}_i=\frac{1}{p^{n}_i}\int_{X_i}\rho(x)\,\mu_n(dx)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
q^{n}_i=\nu_n(X_i), \qquad \sigma^{n}_i=\frac{1}{q^{n}_i}\int_{X_i}\rho(x)\,\nu_n(dx).
\end{equation*}
\notag
$$
If $\mu_n(X_i)=0$ (respectively, $\nu_n(X_i)=0$), then we assume that $p^n_i=0$ and $\rho^n_i=\tau$ (respectively, $q^n_i=0$ and $\sigma^n_i=\tau$), where $\tau$ is any state in $\mathfrak{S}(\mathcal{H})$. Note that $q^n_i=0$ implies by assumption that $p^n_i=0$.
Noting that setwise convergence is not weaker than weak convergence and using Lemma 2 it is easy to show that the setwise convergence of the sequences $\{\mu_n\}$ and $\{\nu_n\}$ to the probability measures $\mu_0$ and $\nu_0$ implies the $D_0$-convergence15 of the sequences $\{\{p^{n}_i,\rho^{n}_i\}_i\}_n$ and $\{\{q^{n}_i,\sigma^{n}_i\}_i\}_n$ to the ensembles $\{p^{0}_i,\rho^{0}_i\}$ and $\{q^{0}_i,\sigma^{0}_i\}$, respectively. To simplify the notation we denote the function $\frac{d\mu_n}{d\nu_n}$ on $X$ by $f_n$. Condition (4.10) implies that
$$
\begin{equation}
\sup_{n\geqslant 0}\sup_{x\in X}f_n(x)\leqslant b \quad\text{and}\quad \sup_{n\geqslant 0}\sup_{d(x_1,x_2)\leqslant\varepsilon}\bigl|f_n(x_1)-f_n(x_2)\bigr|\leqslant \alpha(\varepsilon),
\end{equation}
\tag{4.11}
$$
where $b\in\mathbb{R}_+$ and $\alpha(\varepsilon)$ is a function tending to zero as $\varepsilon\to0^+$.
Let $\omega_{\eta,b}(\varepsilon)\doteq \sup\{|\eta(x_1)-\eta(x_2)\mid x_1,x_2\in[0,b],\,|x_1-x_2|\leqslant\varepsilon\}$ be the modulus of continuity of the function $\eta(x)=-x\ln x$ on $[0,b]$ (we assume that $\eta(0)=0$). We will show that
$$
\begin{equation}
\sum_i p^n_iD(\rho^n_i\,\|\,\sigma^n_i)\leqslant \omega_{\eta,b}(\alpha(\varepsilon)) \quad \forall\, n\geqslant 0
\end{equation}
\tag{4.12}
$$
and
$$
\begin{equation}
0\leqslant D_{\mathrm{KL}}(\mu_n\,\|\,\nu_n)-D_{\mathrm{KL}}(\{p^n_i\}\,\|\,\{q^n_i\})\leqslant \omega_{\eta,b}(\alpha(\varepsilon)) \quad \forall\, n\geqslant 0.
\end{equation}
\tag{4.13}
$$
For any given $n\geqslant0$ and each $i$ let $a^n_i=\inf_{x\in X_i}f_n(x)$ and $b^n_i=\sup_{x\in X_i}f_n(x)$. Since the diameter of each set $X_i$ does not exceed $\varepsilon$, it follows from (4.11) that Since $\displaystyle p_i=\mu_n(X_i)=\int_{X_i}f_n(x)\,\nu_n(dx)$ and $q_i=\nu_n(X_i)$, we have
$$
\begin{equation}
a^n_i\leqslant \frac{p^n_i}{q^n_i}\leqslant b^n_i \quad \forall\, i\colon q^n_i\neq0.
\end{equation}
\tag{4.15}
$$
It follows from (4.14) and (4.15) that
$$
\begin{equation}
\sup_{x\in X_i}\biggl|f_n(x)- \frac{p^n_i}{q^n_i}\biggr|\leqslant \alpha(\varepsilon) \quad \forall\, i\colon q^n_i\neq0.
\end{equation}
\tag{4.16}
$$
To prove (4.12) note that $\nu_n/q^n_i$ is a probability measure on $X_i$ for all $i$ and $n$ such that $q_i^n\neq0$. So, using Lemma 5 below and identities (2.13) and (2.14) we obtain
$$
\begin{equation*}
\begin{aligned} \, p^n_iD(\rho^n_i\,\|\,\sigma^n_i) &=p^n_iD\biggl(\int_{X_i}\frac{q^n_i}{p^n_i}\rho(x)f_n(x)\,\frac{\nu_n(dx)}{q^n_i} \biggm\|\int_{X_i}\rho(x)\,\frac{\nu_n(dx)}{q^n_i}\biggr) \\ &\leqslant\int_{X_i} D(q^n_i\rho(x)f_n(x)\,\|\, p^n_i\rho(x))\,\frac{\nu_n(dx)}{q^n_i} \\ &=\int_{X_i} f_n(x)\ln \biggl(\frac{q^n_if_n(x)}{p_i^n}\biggr)\, \nu_n(dx) \\ &=\int_{X_i}\biggl(\eta\biggl(\frac{p^n_i}{q_i^n}\biggr)-\eta(f_n(x))\biggr)\,\nu_n(dx) \leqslant q^n_i\omega_{\eta,b}(\alpha(\varepsilon)) \end{aligned}
\end{equation*}
\notag
$$
for all $i$ and $n$ such that $p_i^n\neq0$, where the last inequality follows from (4.16). This inequality implies (4.12).
The left-hand inequality in (4.13) follows from the monotonicity of Kullback–Leibler divergence under the positive linear map $\mu\mapsto \{\mu(X_i)\}_i$ from the space of all signed Borel measures on $X$ to the space $\ell_1$ defined in terms of the decomposition $\{X_i\}_i$ of $X$. To prove the right-hand inequality in (4.13) note that (4.16) implies that
$$
\begin{equation*}
\begin{aligned} \, D_{\mathrm{KL}}(\mu_n\,\|\,\nu_n) &=-\int_{X}\eta(f_n(x))\, \nu_n(dx) \\ &\leqslant\sum_i\int_{X_i}\biggl(-\eta\biggl(\frac{p_i^n}{q_i^n}\biggr) +\omega_{\eta,b}(\alpha(\varepsilon))\biggr)\, \nu_n(dx) \\ &=D_{\mathrm{KL}}(\{p^n_i\}\,\|\,\{q^n_i\})+\omega_{\eta,b}(\alpha(\varepsilon)). \end{aligned}
\end{equation*}
\notag
$$
Inequalities (4.12) and (4.13) show that
$$
\begin{equation*}
\sup_{n\geqslant0} \biggl|\sum_i p^n_iD(\rho^n_i\,\|\,\sigma^n_i)+D_{\mathrm{KL}}(\{p^n_i\}\,\|\,\{q^n_i\})-D_{\mathrm{KL}} (\mu_n\,\|\,\nu_n)\biggr|\leqslant \omega_{\eta,b}(\alpha(\varepsilon)).
\end{equation*}
\notag
$$
So it follows from condition (4.8) that
$$
\begin{equation*}
\operatorname{dj}\biggl\{\sum_i p^n_iD(\rho^n_i\,\|\,\sigma^n_i)+D_{\mathrm{KL}}(\{p^n_i\}\,\|\,\{q^n_i\})\biggr\}_{n}\leqslant 2\omega_{\eta,b}(\alpha(\varepsilon)).
\end{equation*}
\notag
$$
Hence, by Corollary 7 we have
$$
\begin{equation*}
\operatorname{dj}\biggl\{D\biggl(\sum_ip^n_i\rho^n_i\biggm\|\sum_iq^n_i\sigma^n_i\biggr)\biggr\}_{n} \leqslant 2\omega_{\eta,b}(\alpha(\varepsilon)).
\end{equation*}
\notag
$$
Since $\omega_{\eta,b}(\alpha(\varepsilon))$ can be made arbitrarily small by choosing a sufficiently small $\varepsilon$, we obtain (4.9) by noting that
$$
\begin{equation*}
\sum_ip_i^n\rho_i^n=\int_X \rho(x)\,\mu_n(dx) \quad\text{and}\quad \sum_iq_i^n\sigma_i^n=\int_X \rho(x)\,\nu_n(dx).
\end{equation*}
\notag
$$
The proposition is proved. Remark 6. The setwise convergence of the sequences $\{\mu_n\}$ and $\{\nu_n\}$ to the measures $\mu_0$ and $\nu_0$ in Proposition 9 can be relaxed to weak convergence, provided that for each $\varepsilon>0$ there exists a countable decomposition $\{X_i\}$ of the space $X$ into disjoint Borel subsets with diameter $\leqslant \varepsilon$ such that $\mu_0(\partial X_i)=\nu_0(\partial X_i)=0$ for all $i$, where $\partial X_i$ is the boundary of $X_i$. This follows from the above proof and the portmanteau theorem [36], [35]. The condition (4.10) is quite restrictive, but it seems that it can be relaxed by using more subtle estimates in the proof of inequalities (4.12) and (4.13). Example 3. Let $\mathcal{H}$ be the Hilbert space describing $n$-mode quantum oscillator and $\mathfrak{S}_\mathrm{cl}(\mathcal{H})$ be the set of classical states — the convex closure of the family $\{|\overline{z}\rangle\langle \overline{z}|\}_{\overline{z}\in\mathbb{C}^n}$ of coherent states [14], [37]. Every state $\rho$ in $\mathfrak{S}_\mathrm{cl}(\mathcal{H})$ can be represented as Proposition 9 for $X=\mathbb{C}^n$ and $\rho(\overline{z})=|\overline{z}\rangle\langle \overline{z}|$ gives us a sufficient condition for convergence of the quantum relative entropy between classical states of the $n$-mode quantum oscillator in terms of their representing measures. Remark 6 allows us to replace the setwise convergence of the sequences $\{\mu_n\}$ and $\{\nu_n\}$ to the measures $\mu_0$ and $\nu_0$ in Proposition 9 by weak convergence in many cases, in particular, if the measures $\mu_0$ and $\nu_0$ are absolutely continuous with respect to the Lebesgue measure on $\mathbb{C}^n$. The following lemma contains a ‘continuous’ version of the joint convexity of quantum relative entropy. Lemma 5. Let $\rho(x)$ and $\sigma(x)$ be $\mathfrak{S}(\mathcal{H})$-valued continuous functions on a separable metric space $X$. Then for any Borel probability measure $\mu$ on $X$. Remark 7. Since quantum relative entropy is lower semicontinuous, the function $x\mapsto D(\rho(x)\|\sigma(x))$ is lower semicontinuous on $X$, and therefore it is measurable with respect to the Borel $\sigma$-algebra on $X$. So the right-hand side of (4.18) is well defined. Proof of Lemma 5. The joint convexity of quantum relative entropy implies that (4.18) holds for any purely atomic (discrete) probability measure $\mu$, that is, a measure of the form $\sum_ip_i\delta(x_i)$, where $\{p_i\}$ is a probability distribution, $\{x_i\}$ is a finite or countable subset of $X$ and $\delta(x)$ denotes the Dirac measure concentrated at $x$.
Given a probability measure $\mu$ on $X$, it is easy16 to construct a sequence $\{\mu_n\}$ of purely atomic probability measures on $X$ weakly converging to the measure $\mu$ on $X$ and such that
$$
\begin{equation}
\int_X D(\rho(x)\,\|\,\sigma(x))\,\mu_n(dx)\leqslant \int_X D(\rho(x)\,\|\,\sigma(x))\,\mu(dx) \quad \forall\, n.
\end{equation}
\tag{4.19}
$$
Using the definition of weak convergence (cf. [36] and [35]) and Lemma 2 it is easy to prove that
$$
\begin{equation*}
\lim_{n\to+\infty}\int_X \rho(x)\,\mu_n(dx)=\int_X \rho(x)\,\mu(dx)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\lim_{n\to+\infty}\int_X \sigma(x)\,\mu_n(dx)=\int_X \sigma(x)\,\mu(dx)
\end{equation*}
\notag
$$
(the limits with respect to the trace norm). So the lower semicontinuity of quantum relative entropy shows that
$$
\begin{equation*}
\begin{aligned} \, &\liminf_{n\to+\infty}D\biggl(\int_X \rho(x)\,\mu_n(dx)\biggm\|\int_X \sigma(x)\,\mu_n(dx)\biggr) \\ &\qquad\geqslant D\biggl(\int_X \rho(x)\,\mu(dx)\biggm\|\int_X \sigma(x)\,\mu(dx)\biggr). \end{aligned}
\end{equation*}
\notag
$$
Since (4.18) holds for $\mu=\mu_n$ for all $n$, the last inequality and (4.19) implies (4.18).
Lemma 5 is proved. 4.7. Quantum mutual informationThe lower semicontinuity of the function $(\rho,\sigma)\mapsto D(\rho\,\|\,\sigma)-D(\Phi(\rho)\,\|\,\Phi(\sigma))$ proved in this article in the general form was established before in the special case when $\rho$ is a state of a composite quantum system $A_1\dotsb A_n$, $\sigma=\rho_{A_1}\otimes\dotsb \otimes\rho_{A_n}$ and $\Phi$ is a local channel [40]. In this case $D(\rho\,\|\,\sigma)=I(A_1:\dotsb:A_n)_{\rho}$ is the quantum mutual information of the state $\rho$, which describes the total correlation of this state [41], [11]. The lower semicontinuity of the decrease of quantum mutual information under the action of a local channel is related (in view of the Stinespring representation (2.10)) to the lower semicontinuity of the quantum conditional mutual information (3.2), which was proved in [27], § 6. In § 5 of [40] this property was used to obtain new conditions for the convergence of (conditional and unconditional) quantum mutual information and for squashed entanglement, whose effectiveness was shown in concrete examples. All these results can also be treated as applications of the theorem in § 3.1. Here we present one observation, which was not mentioned in [40]. Proposition 10. Let $A_1\dotsb A_n$ be an $n$-partite quantum system, $n>2$, and $B_1=A_{i^1_1}\dotsb A_{i^1_{k(1)}}$, …, $B_m=A_{i^m_1}\dotsb A_{i^m_{k(m)}}$ its subsystems determined by disjoint subsets $\{i^1_1,\dots,i^1_{k(1)}\}$, …, $\{i^m_1,..,i^m_{k(m)}\}$ of $[1,n]\cap\mathbb{N}$, $m<n$.17 Then the nonnegative function $\rho\mapsto I(A_1:\dotsb:A_n)_{\rho}-I(B_1:\dotsb:B_m)_{\rho}$ is lower semicontinuous on the set
$$
\begin{equation}
\bigl\{\rho\in\mathfrak{S}(\mathcal{H}_{A_1\dots A_n}) \mid I(B_1:\dots :B_m)_{\rho}<+\infty\bigr\}.
\end{equation}
\tag{4.20}
$$
If the function $\rho\mapsto I(A_1:\dotsb:A_n)_{\rho}$ is continuous on some subset $\mathfrak{S}_0$ of $\mathfrak{S}(\mathcal{H}_{A_1\dotsb A_n})$ then the function $\rho\mapsto I(B_1:\dotsb:B_m)_{\rho}$ is continuous on $\mathfrak{S}_0$. Proof. We may assume without loss of generality that $\bigcup_{j=1}^m\{i^j_1,\dots,i^j_{k(j)}\}\!=\![1,n']\cap \mathbb{N}$ for some $n'\leqslant n$.
Part (A) of the theorem in § 3.1 implies that $\rho\!\mapsto\! I(A_1\mkern-1.5mu:\!\dotsb\!:\mkern-1.5mu A_n)_{\rho}-I({A_1\mkern-1.5mu:\!\dotsb\!:\mkern-1.5mu A_{n'}})_{\rho}$ is a lower semicontinuous function on the set
$$
\begin{equation*}
\mathfrak{A}=\bigl\{\rho\in\mathfrak{S}(\mathcal{H}_{A_1\dots A_n})\mid I(A_1:\dots :A_{n'})_{\rho}<+\infty\bigr\}.
\end{equation*}
\notag
$$
Since
$$
\begin{equation}
I(A_1:\dots :A_{n'})_{\rho}=I(B_1:\dots :B_m)_{\rho}+\sum_{j=1}^m I(A_{i^j_1}:\dots :A_{i^j_{k(j)}})_{\rho},
\end{equation}
\tag{4.21}
$$
the lower semicontinuity of quantum mutual information implies that $\rho\mapsto I(A_1: \dotsb:A_{n'})_{\rho}-I(B_1:\dotsb:B_m)_{\rho}$ is a lower semicontinuous function on the set in (4.20), which we denote by $\mathfrak{B}$.
Thus, the function $f(\rho)=I(A_1:\dotsb:A_n)_{\rho}-I(B_1:\dotsb:B_m)_{\rho}$ is lower semicontinuous on the set $\mathfrak{A}\subseteq\mathfrak{B}$. It is clear that $f(\rho)=+\infty$ for any $\rho$ in $\mathfrak{B}\setminus\mathfrak{A}$. So to prove the lower semicontinuity of $f$ on $\mathfrak{B}$ it suffices to show that $\lim_{n\to+\infty}f(\rho_n)=+\infty$ for each sequence $\{\rho_n\}\subset\mathfrak{A}$ converging to a state $\rho_0\in\mathfrak{B}\setminus\mathfrak{A}$. This can be done by noting that identity (4.21) implies that
$$
\begin{equation*}
f(\rho_n)\geqslant \sum_{j=1}^m I(A_{i^j_1}:\dots :A_{i^j_{k(j)}})_{\rho_n} \quad \forall\, n,
\end{equation*}
\notag
$$
since the right-hand side of this inequality tends to
$$
\begin{equation*}
\sum_{j=1}^m I(A_{i^j_1}:\dots :A_{i^j_{k(j)}})_{\rho_0}=+\infty
\end{equation*}
\notag
$$
by the lower semicontinuity of quantum mutual information.
The second claim of the proposition follows from the first by Lemma 1 and the lower semicontinuity of quantum mutual information. The proposition is proved. By Proposition 10 the local continuity (convergence) of total correlation in the ‘large’ composite system $A_1\dotsb A_n$ implies the local continuity (convergence) of the total correlation in any composite system $B_1\dotsb B_m$ whose components $B_1,\dots,B_m$ are obtained by combining some of the subsystems $A_1,\dots,A_n$. For example, the local continuity of $I(A:B:C)$ implies the local continuity of $I(A:B)$, $I(B:C)$, $I(A:C)$, $I(A:BC)$, $I(AB:C)$ and $I(AC:B)$. Remark 8. The claim of Proposition 10 is also valid for multipartite quantum conditional mutual information, that is, it holds for the functions $I(A_1:\dotsb:A_n)_{\rho}$ and $I(B_1:\dotsb:B_m)_{\rho}$ replaced by $I(A_1:\dotsb:A_n\,|\,C)_{\rho}$ and $I(B_1:\dotsb:B_m\,|\,C)_{\rho}$. This can be shown by similar arguments, using Theorem 3 in [40] and the lower semicontinuity of multipartite quantum conditional mutual information [27]. 4.8. Quantum conditional relative entropyThe quantum conditional relative entropy of states $\rho$ and $\sigma$ of a composite system $AB$ was defined by Capel, Lucia and Perez-Garcia by
$$
\begin{equation*}
D_A(\rho\,\|\,\sigma)=D(\rho\,\|\,\sigma)-D(\rho_B\,\|\,\sigma_B).
\end{equation*}
\notag
$$
In [25], where this notion was introduced, the reader can find a number of its interesting properties. The monotonicity of quantum relative entropy shows that $D_A(\rho\,\|\,\sigma)\geqslant0$ for any $\rho$ and $\sigma$. The theorem in § 3.1 and Corollary 1 imply directly the following result. Proposition 11. The nonnegative function $(\rho,\sigma)\mapsto D_A(\rho\,\|\,\sigma)$ is lower semicontinuous on the set If the function $(\rho,\sigma)\mapsto D(\rho\,\|\,\sigma)$ is continuous on a subset $\mathfrak{S}_0$ of $\mathfrak{S}(\mathcal{H}_{AB})\times\mathfrak{S}(\mathcal{H}_{AB})$, then the function $(\rho,\sigma)\mapsto D_A(\rho\,\|\,\sigma)$ is continuous on $\mathfrak{S}_0$. § 5. Concluding remarks and open questionsIn this article we have proved the lower semicontinuity of the function
$$
\begin{equation*}
(\rho,\sigma)\mapsto D(\rho\,\|\,\sigma)-D(\Phi(\rho)\,\|\,\Phi(\sigma))
\end{equation*}
\notag
$$
for any given quantum operation $\Phi$ and have considered its consequences and applications. It turns that this property allows us to obtain many new local continuity (convergence) conditions for quantum relative entropy and related functions and to reprove a number of results of this type obtained before by other methods. So we can say that the property of quantum relative entropy we have established plays a central role in the analysis of the local continuity of entropic characteristics of quantum systems and channels which are either defined in terms of quantum relative entropy (like quantum mutual information) or can be expressed in terms of quantum relative entropy (like von Neumann entropy). Of course, the question arises as to whether the same property can be proved for other divergence-type functions (Belavkin–Staszewski relative entropy, Renyi relative entropy and so on). Below we describe the basic properties of (Lindblad’s extension of Umegaki) quantum relative entropy which were used essentially in the proof of the theorem in § 3.1:
The first three of the above properties of quantum relative entropy are typical for a divergence-type function, but the last three of them are quite specific. For example, it seems that the fourth property does not hold for Belavkin–Staszewski relative entropy in view of its discontinuity in the finite-dimensional settings, which was mentioned in [42], Proposition 6.7. Of course, we do not assert that all above properties are necessary for the proof of an analogue of the theorem in § 3.1 for a divergence-type function $D$. AcknowledgementsI am grateful to A. S. Holevo and G. G. Amosov for useful discussions and comments. I am grateful to M. M. Wilde for the valuable communication. Special thanks are due to the referee for their useful comments and recommendations.
Citation in format AMSBIB
\Bibitem{Shi24} |
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