Convergence Conditions for the Quantum Relative Entropy and Other Applications of the Deneralized Quantum Dini Lemma

Abstract

We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows to expand the scope of possible applications of the method. We prove two general dominated convergence theorems and the theorem about preserving local continuity under convex mixtures. By using these theorems we obtain several convergence conditions for the quantum relative entropy and for the mutual information of a quantum channel considered as a function of a pair (channel, input state).

APPENDIX A

We assume below that the value of $\text{Tr}H\rho$ (finite or infinite) for a positive (semi-definite) operator $H$ on a Hilbert space $\mathcal{H}$ and any positive operator $\rho$ in $\mathfrak{T}(\mathcal{H})$ is defined according to the rule (8).

Proposition 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 nonzero operators $\rho^{1}_{0}$, $\rho^{2}_{0}$, $\sigma^{1}_{0}$ and $\sigma^{2}_{0}$ such that $\rho^{1}_{n}\geq\rho^{2}_{n}$ and $\sigma^{1}_{n}\leq\sigma^{2}_{n}$ for all $n\geq 0$. If

$$A_{1}\doteq\limsup_{n\to+\infty}\text{Tr}\rho^{1}_{n}(-\ln\sigma_{n}^{1})<+\infty\quad and\quad A_{1}-\text{Tr}\rho^{1}_{0}(-\ln\sigma_{0}^{1})=\Delta,$$

(103)

then

$$A_{2}\doteq\limsup_{n\to+\infty}\text{Tr}\rho^{2}_{n}(-\ln\sigma_{n}^{2})<+\infty\quad and\quad A_{2}-\text{Tr}\rho^{2}_{0}(-\ln\sigma_{0}^{2})\leq\Delta.$$

Remark 5. The lower semicontinuity of the von Neumann entropy and the relative entropy implies, due to representation (71), that the function $(\rho,\sigma)\mapsto\text{Tr}\rho(-\ln\sigma)$ is lower semicontinuous on $\mathfrak{T}_{+}(\mathcal{H})\times\mathfrak{T}_{+}(\mathcal{H})$. It follows that the quantities $A_{i}-\text{Tr}\rho^{i}_{0}(-\ln\sigma_{0}^{i})$, $i=1,2$, in Proposition 5 are nonnegative.

Proof. By the condition (103) we may assume that $\text{Tr}\rho^{1}_{n}(-\ln\sigma_{n}^{1})<+\infty$ for all $n\geq 0$ and hence

$$\text{supp}\rho_{n}^{i}\subseteq\text{supp}\sigma_{n}^{i}\quad\forall n\geq 0,\quad i=1,2.$$

(104)

Let $a_{n}^{i}\doteq\text{Tr}\rho^{i}_{n}(-\ln\sigma_{n}^{i})$, $n\geq 0$, $i=1,2$. For any natural $k$ introduce the quantities

$$a_{k,n}^{i}=-\text{Tr}\rho^{i}_{n}\ln(\sigma_{n}^{i}+k^{-1}I_{\mathcal{H}}),\quad n\geq 0,\quad i=1,2.$$

By Theorems VIII.18 and VIII.20 in [22] for each $i$ and $k$ the sequence of bounded operators $\ln(\sigma_{n}^{i}+k^{-1}I_{\mathcal{H}})$ tends to the bounded operator $\ln(\sigma_{0}^{i}+k^{-1}I_{\mathcal{H}})$ as $n\to+\infty$ in the operator norm. It follows that

$$\lim_{n\to+\infty}a_{k,n}^{i}=a_{k,0}^{i}<+\infty,\quad\forall k\in\mathbb{N},\quad i=1,2.$$

(105)

By using the operator monotonicity of the logarithm it is easy to show that

$$\sup_{k}a_{k,n}^{i}=a_{n}^{i},\quad\forall n\geq 0,\quad i=1,2.$$

(106)

Thus, for any $\varepsilon>0$ there is $k_{\varepsilon}$ such that $a_{0}^{1}-a_{k_{\varepsilon},0}^{1}<\varepsilon$. Condition (103) and relation (105) with $i=1$ imply that there is $n_{\varepsilon}$ such that $a_{n}^{1}-a_{0}^{1}\leq\Delta+\varepsilon$ and $a_{0,k_{\varepsilon}}^{1}-a_{n,k_{\varepsilon}}^{1}\leq\varepsilon$ for all $n\geq n_{\varepsilon}$. It follows that

$$a_{n}^{1}-a_{k_{\varepsilon},n}^{1}=(a_{n}^{1}-a_{0}^{1})+(a_{0}^{1}-a_{k_{\varepsilon},0}^{1})+(a_{k_{\varepsilon},0}^{1}-a_{k_{\varepsilon},n}^{1})<\Delta+3\varepsilon\quad\forall n\geq n_{\varepsilon}.$$

(107)

By using expression (9) and taking (104) into account it is easy to show that

$$a_{n}^{i}-a_{k,n}^{i}=\text{Tr}\rho^{i}_{n}\ln(I_{\mathcal{H}}+k^{-1}[\sigma_{n}^{i}]^{-1}),\quad\forall k,\forall n\geq 0,\quad i=1,2,$$

where $[\sigma_{n}^{i}]^{-1}$ is the Moore–Penrose inverse of $\sigma_{n}^{i}$. So, since $\rho_{n}^{1}\geq\rho_{n}^{2}$ and $\sigma_{n}^{1}\leq\sigma_{n}^{2}$, Lemmas 4 and 5 below show that

$$0\leq a_{n}^{2}-a_{k,n}^{2}\leq a_{n}^{1}-a_{k,n}^{1},\quad\forall k,\quad\forall n\geq 0.$$

Thus, it follows from (105) with $i=2$, (106) and (107) that

$$a_{n}^{2}-a_{0}^{2}\leq a_{n}^{2}-a_{0,k}^{2}=(a_{n}^{2}-a_{n,k_{\varepsilon}}^{2})+(a_{n,k_{\varepsilon}}^{2}-a_{0,k_{\varepsilon}}^{2})\leq\Delta+4\varepsilon$$

for all sufficiently large $n$. This implies the assertion of the proposition. $\Box$

Lemma 4. Let $\sigma_{1}$ and $\sigma_{2}$ be nonzero positive operators in $\mathfrak{T}(\mathcal{H})$ such that $\sigma_{1}\leq\sigma_{2}$ and $c>0$ be arbitrary. Then $H_{1}=\ln(I_{\mathcal{H}}+c\sigma^{-1}_{1})$ and $H_{2}=\ln(I_{\mathcal{H}}+c\sigma^{-1}_{2})$ are positive semidefinite operators on $\mathcal{H}$ such that

$$\mathcal{D}(\sqrt{-\ln\sigma_{1}})\subseteq\mathcal{D}(\sqrt{H_{1}})\cap\mathcal{D}(\sqrt{H_{2}})\quad and\quad\langle\varphi|H_{1}|\varphi\rangle\geq\langle\varphi|H_{2}|\varphi\rangle$$

for any $\varphi\in\mathcal{D}(\sqrt{-\ln\sigma_{1}})$, where $\langle\varphi|H_{i}|\varphi\rangle\doteq||\sqrt{H_{i}}\varphi||^{2}$ and $\sigma^{-1}_{i}$ denotes the Moore–Penrose inverse of $\sigma_{i}$, $i=1,2$.⁷

If the operators $\sigma_{1}$ and $\sigma_{2}$ are non-degenerate then Lemma 4 can be deduced easily from Corollary 10.12 in [23] and the operator monotonicity of the logarithm.

Proof. Denote $I_{\mathcal{H}}$ by $I$ for brevity. For a given vector $\varphi\in\mathcal{D}(\sqrt{-\ln\sigma_{1}}\hskip 1.0pt)$ and any natural $k$ and $n$ introduce the quantities

$$a_{k,n}^{i}=\langle\varphi|P_{n}^{i}\ln(I+c(\sigma_{i}+k^{-1}I)^{-1})|\varphi\rangle,\quad a_{k,*}^{i}=\langle\varphi|\ln(I+c(\sigma_{i}+k^{-1}I)^{-1})|\varphi\rangle$$

and

$$a_{*,n}^{i}=\langle\varphi|P_{n}^{i}\ln(I+c\sigma_{i}^{-1})|\varphi\rangle,\quad i=1,2,$$

where $P_{n}^{i}$ is the spectral projector of the operator $\sigma_{i}$ corresponding to its $n$ maximal nonzero eigenvalues (taking multiplicity into account), $i=1,2$ (if $\text{rank}\sigma_{i}<n$ we assume that $P_{n}^{i}$ is the projector on the support $\text{supp}\sigma_{i}$ of $\sigma_{i}$). Since $\varphi\in\text{supp}\sigma_{1}\subseteq\text{supp}\sigma_{2}$ by the condition $\sigma_{1}\leq\sigma_{2}$, we have

$$\lim_{n\rightarrow+\infty}P_{n}^{1}|\varphi\rangle=\lim_{n\rightarrow+\infty}P_{n}^{2}|\varphi\rangle=|\varphi\rangle.$$

(108)

It is easy to see that $\varphi\in\mathcal{D}(\sqrt{H_{1}}\hskip 1.0pt)$, which means that $\sup_{n}a_{*,n}^{1}$ is finite. Thus, to prove the assertion of the lemma it suffices to show that

$$\sup_{n}a_{*,n}^{2}\leq\sup_{n}a_{*,n}^{1}.$$

The positivity of the operator $\ln(I+c(\sigma_{i}+k^{-1}I)^{-1})$ and the operator monotonicity of the logarithm imply that

$$a_{k,n}^{i}\leq a_{k,n+1}^{i},\quad a_{k,n}^{i}\leq a_{k+1,n}^{i},\quad i=1,2,$$

(109)

for all $n$ and $k$. Since $\sigma_{1}\leq\sigma_{2}$, we have $(\sigma_{1}+k^{-1}I)^{-1}\geq(\sigma_{2}+k^{-1}I)^{-1}$ for any $k$ [23, Corollary 10.13], So, the operator monotonicity of the logarithm also implies

$$a_{k,*}^{1}\geq a_{k,*}^{2}\quad\forall k.$$

(110)

Since $P_{n}^{i}\ln(I+c\sigma_{i}^{-1})$ and $\ln(I+c(\sigma_{i}+k^{-1}I)^{-1})$ are bounded positive operators, it follows from (108) and (109) that

$$\sup_{n}a_{k,n}^{i}=\lim_{n\rightarrow+\infty}a_{k,n}^{i}=a_{k,*}^{i},\quad\sup_{k}a_{k,n}^{i}=\lim_{k\rightarrow+\infty}a_{k,n}^{i}=a_{*,n}^{i}$$

and hence

$$\sup_{n}a_{*,n}^{i}=\sup_{n}\sup_{k}a_{k,n}^{i}=\sup_{k}\sup_{n}a_{k,n}^{i}=\sup_{k}a_{k,*}^{i},\quad i=1,2.$$

Thus, inequality (110) shows that

$$\sup_{n}a_{*,n}^{2}=\sup_{k}a_{k,*}^{2}\leq\sup_{k}a_{k,*}^{1}=\sup_{n}a_{*,n}^{1}.$$

$\Box$

Lemma 5. Let $H$ be a positive (semi-definite) operator on a Hilbert space $\mathcal{H}$ and $\rho=\sum_{i}|\varphi_{i}\rangle\langle\varphi_{i}|$ the spectral decomposition of a positive operator $\rho$ in $\mathfrak{T}(\mathcal{H})$ . If $\text{Tr}H\rho$ is finite then all the vectors $\varphi_{i}$ belong to the domain of $\sqrt{H}$ and

$$\text{Tr}H\rho=\sum_{i}\langle\varphi_{i}|H|\varphi_{i}\rangle=\sum_{i}||\sqrt{H}\varphi_{i}||^{2}.$$

Proof. Let $P_{n}$ be the spectral projector of $H$ corresponding to the interval $[0,n]$. Since $P_{n}H$ is a bounded operator for each $n$, we have

$$\text{Tr}HP_{n}\rho=\sum_{i}||\sqrt{H}P_{n}\varphi_{i}||^{2}.$$

Since the sequence $\{||\sqrt{H}P_{n}\varphi_{i}||^{2}\}_{n}$ is nondecreasing for each given $i$, it follows from the assumption $\text{Tr}H\rho=\sup_{n}\text{Tr}HP_{n}\rho<+\infty$ that $a_{i}=\sup_{n}||\sqrt{H}P_{n}\varphi_{i}||^{2}<+\infty$ for all $i$ and $\text{Tr}H\rho=\sum_{i}a_{i}$. The finiteness of $a_{i}$ means, by the spectral theorem, that $\varphi_{i}\in\mathcal{D}(\sqrt{H})$ and $a_{i}=||\sqrt{H}\varphi_{i}||^{2}$. $\Box$