Simple characterization of positive linear maps preserving continuity of the von Neumann entropy

TL;DR

This paper characterizes positive linear maps that preserve the local continuity of von Neumann entropy, showing they must also preserve entropy finiteness, which relates to boundedness on pure states.

Contribution

It provides a precise criterion for when positive linear maps preserve the continuity of von Neumann entropy, linking it to the preservation of entropy finiteness.

Findings

Maps preserving entropy continuity also preserve finiteness of entropy.

Finiteness preservation is equivalent to boundedness of output entropy on pure states.

Characterization applies to positive linear maps in quantum information theory.

Abstract

We show that a positive linear map preserves local continuity (convergence) of the entropy if and only if it preserves finiteness of the entropy, i.e. transforms operators with finite entropy to operators with finite entropy. The last property is equivalent to the boundedness of the output entropy of a map on the set of pure states.

Equations

$$H(\rho)\doteq[\mathrm{Tr}\rho]H\biggl{(}\frac{\rho}{\mathrm{Tr}\rho}\biggr{)}=\mathrm{Tr}\eta(\rho)-\eta(\mathrm{Tr}\rho).$$

$$H\left(\sum _k{p}_k{\rho }_k\right)\le \sum _k{p}_{k}H({\rho }_k)+S(\{p_k{\}}_k),$$

$$S(\{p_k\})\doteq \sum _k\eta (p_k)-\eta \left(\sum _k{p}_k\right)$$

$$H\left(\sum _k{\rho }_k\right)\le \sum _{k}H({\rho }_k)+S\left({\left\{\mathrm{Tr}{\rho }_k\right\}}_k\right),$$

$$H(\rho \Vert \sigma )=\sum _i⟨i|\rho \log \rho -\rho \log \sigma +\sigma -\rho |i⟩,$$

$$\underset{n\to \infty }{\lim }H({\rho }_n)=H({\rho }_0)<+\infty \Rightarrow \underset{n\to \infty }{\lim }{H}_{\Phi }({\rho }_n)=H_{\Phi }({\rho }_0)<+\infty$$

$$H(\rho )<+\infty \Rightarrow H_{\Phi }(\rho )<+\infty$$

$$H_{\Phi }(\rho )\le \sum _i{p}_i{H}_{\Phi }(|{\phi }_i⟩⟨{\phi }_i|)+H(\rho )\le C\Vert \rho {\Vert }_1+H(\rho ),$$

$${\mathfrak{S}}_k({\mathcal{H}}_A)=\left\{\rho \in \mathfrak{S}({\mathcal{H}}_A)\mid \mathrm{rank}\rho \le k\right\}$$

$$\frac{1}{1+\epsilon }\rho +\frac{\epsilon }{1+\epsilon }{\tau }_{-}={\omega }^{\ast }=\frac{1}{1+\epsilon }\sigma +\frac{\epsilon }{1+\epsilon }{\tau }_{+},$$

$$H_{\Phi }({\tau }_{\pm })\le C+\log (2k-1).$$

$$p{H}_{\Phi }(\varrho )+(1-p)H_{\Phi }(\varsigma )\le H_{\Phi }\left(p\varrho +(1-p)\varsigma \right)\le p{H}_{\Phi }(\varrho )+(1-p)H_{\Phi }(\varsigma )+h_2(p)$$

$$(1-p)\left[H_{\Phi }(\rho )-H_{\Phi }(\sigma )\right]\le p\left[H_{\Phi }({\tau }_{+})-H_{\Phi }({\tau }_{-})\right]+h_2(p)$$

$$(1-p)\left[H_{\Phi }(\sigma )-H_{\Phi }(\rho )\right]\le p\left[H_{\Phi }({\tau }_{-})-H_{\Phi }({\tau }_{+})\right]+h_2(p),$$

$$|H_{\Phi }(\rho )-H_{\Phi }(\sigma )|\le \epsilon (C+\log (2k-1))+(1+\epsilon )h_2\left(\frac{\epsilon }{1+\epsilon }\right).$$

$${\hat{f}}_k(\rho )=\underset{\{{\pi }_i,{\rho }_i\}\in {\mathcal{P}}_k(\rho )}{\sup }\sum _i{\pi }_{i}f({\rho }_i),\rho \in \mathfrak{S}(\mathcal{H}),$$

$${\Delta }_k^H(\rho )=\underset{\{{\pi }_i,{\rho }_i\}\in {\mathcal{P}}_k(\rho )}{\inf }\sum _i{\pi }_{i}H({\rho }_i\Vert \rho ).\text{vskip}3.0ptplus1.0ptminus1.0pt$$

$${\Delta }_k^{H_{\Phi }}(\rho )\doteq H_{\Phi }(\rho )-[\widehat{H_{\Phi }}{]}_k(\rho )\le \underset{\{{\pi }_i,{\rho }_i\}\in {\mathcal{P}}_k(\rho )}{\inf }\sum _i{\pi }_{i}H(\Phi ({\rho }_i)\Vert \Phi (\rho )).\text{vskip}3.0ptplus1.0ptminus1.0pt$$

$${\Delta }_k^{H_{\Phi }}(\rho )\le {\Delta }_k^H(\rho )\text{ for all }\rho \in \mathfrak{S}({\mathcal{H}}_A).$$

$$|H_{\Phi }(\rho )-H_{\Phi }(\sigma )|\le \epsilon (C+\log (\mathrm{rank}\rho +\mathrm{rank}\sigma -1))+(1+\epsilon )h_2\left(\frac{\epsilon }{1+\epsilon }\right)$$

$$H(\Phi (\rho )\Vert \Phi (\sigma ))\le H(\rho \Vert \sigma )\text{ for any states }\rho \text{ and }\sigma .$$

$$H({\Phi }^{\prime }(\rho )\Vert {\Phi }^{\prime }(\sigma ))=H(\Phi (\rho )\Vert \Phi (\sigma ))+H(\Psi (\rho )\Vert \Psi (\sigma )).$$

$$H(\Phi (\rho )\Vert \Phi (\sigma ))\le H({\Phi }^{\prime }(\rho )\Vert {\Phi }^{\prime }(\sigma ))\le H(\rho \Vert \sigma )$$

Full text

1 Preliminaries

Linear completely positive maps between Banach spaces of trace-class operators play a basic role in description of evolution of quantum systems and of measurements of their parameters. Such maps are called quantum channels (respectively, quantum operations) if they preserve (respectively, don’t increase) the trace of any input positive operator [3].

In analysis of information properties of infinite-dimensional quantum channels and operations the question about continuity of their output entropy as a function of input states naturally appears [9]. In this note we give a characterization of positive maps (in particular, quantum channels and operations) for which continuity of the entropy on a set of input operators (states) implies continuity of the output entropy on this set.

Let $\mathcal{H}$ be a separable Hilbert space, $\mathfrak{T}(\mathcal{H})$ the Banach space of trace class operators on $\mathcal{H}$, $\mathfrak{T}_{+}(\mathcal{H})$ the cone of positive operators in $\mathfrak{T}(\mathcal{H})$ and $\mathfrak{S}(\mathcal{H})$ the set of quantum states – operators in  $\mathfrak{T}_{+}(\mathcal{H})$ with unit trace [3].

The von Neumann entropy of a quantum state $\rho\in\mathfrak{S}(\mathcal{H})$ is defined by the formula $H(\rho)=\operatorname{Tr}\eta(\rho)$, where $\eta(x)=-x\log x$ for $x>0$ and $\eta(0)=0$. It is a concave lower semicontinuous function on the set $\mathfrak{S}(\mathcal{H})$ taking values in $[0,+\infty]$ (see [3, 4, 11]). The homogeneous extension of the von Neumann entropy to the cone $\mathfrak{T}_{+}(\mathcal{H})$ is given by the formula:

[TABLE]

By using theorem 11.10 in [6] and a simple approximation it is easy to obtain the following inequality

[TABLE]

valid for any finite or countable collection $\{\rho_{k}\}$ of positive operators in the unit ball of $\mathfrak{T}(\mathcal{H})$ and any probability distribution $\{p_{k}\}$, where

[TABLE]

is the homogeneous extension of the Shannon entropy to the positive cone of $\ell_{1}$. Inequality (1) implies the following one

[TABLE]

valid for any finite or countable collection $\{\rho_{k}\}$ of positive operators in $\mathfrak{T}(\mathcal{H})$ with finite $\sum_{k}\!\mathrm{Tr}\rho_{k}$. If $\mathrm{supp}\rho_{k}\perp\mathrm{supp}\rho_{j}$ for all $k\neq j$ then $"="$ holds in (3).222The support $\,\mathrm{supp}\rho\,$ of a positive operator $\rho$ is the orthogonal complement to its kernel.

The quantum relative entropy for positive trace class operators $\rho$ and $\sigma$ can be defined as

[TABLE]

where $\{|i\rangle\}$ is the orthonormal basis of eigenvectors of the state $\rho$ and it is assumed that $H(\rho\hskip 1.0pt\|\hskip 1.0pt\sigma)=+\infty$ if $\,\mathrm{supp}\rho\hskip 1.0pt$ is not contained in $\hskip 1.0pt\mathrm{supp}\hskip 1.0pt\sigma$ [4].

2 PCE-property and its characterization

For an arbitrary positive linear map $\Phi\colon\mathfrak{T}(\mathcal{H}_{A})\to\mathfrak{T}(\mathcal{H}_{B})$ the output entropy $H_{\Phi}(\rho)\doteq H(\Phi(\rho))$ is a concave lower semicontinuous function on the cone $\mathfrak{T}_{+}(\mathcal{H}_{A})$ taking values in $[0,+\infty]$.

The class of positive maps characterised by continuity of the output entropy $H_{\Phi}(\rho)$ on any subset of the cone $\mathfrak{T}_{+}(\mathcal{H}_{A})$ on which the input entropy $H(\rho)$ is continuous was considered in [9]. This property can be called preserving continuity of the entropy (PCE) under action of a map $\Phi$. A simple characterization of the PCE-property is given by the following theorem (which is a strengthened version of Theorem 2 in [9]).

Theorem 1.  Let $\,\Phi$ be a positive linear map from $\mathfrak{T}(\mathcal{H}_{A})$ into $\mathfrak{T}(\mathcal{H}_{B})$. The following properties are equivalent:

(i) $\,\Phi$ preserves continuity of the entropy, i.e.

[TABLE]

for any sequence $\{\rho_{n}\}\subset\mathfrak{T}_{+}(\mathcal{H}_{A})$ converging to an operator $\rho_{0}$;

(ii) $\,\Phi$ preserves finiteness of the entropy, i.e.

[TABLE]

for any state $\rho\in\mathfrak{S}(\mathcal{H}_{A})$;

(iii) the function $H_{\Phi}(\rho)$ is bounded on the set $\,\mathrm{ext}\mathfrak{S}(\mathcal{H}_{A})$ of pure states.

Proof. We may assume that $\Phi$ does not increase a trace of any positive operator.

It is clear that (i) implies (ii).

By using inequality (1) and the spectral decomposition $\rho=\sum_{i}p_{i}|\varphi_{i}\rangle\langle\varphi_{i}|$ we obtain

[TABLE]

where $\,C\doteq\sup_{\rho\in\mathrm{ext}\mathfrak{S}(\mathcal{H}_{A})}H_{\Phi}(\rho)$. This shows that (iii) implies (ii).

To prove the implication $\,\mathrm{(ii)\Rightarrow(iii)}$ assume that for each natural $n$ there exists a pure state $\rho_{n}$ such that $H_{\Phi}(\rho_{n})\geq 2^{n}$. It follows from (1) that the state $\rho_{0}=\sum_{n\in\mathbb{N}}2^{-n}\rho_{n}$ has finite entropy, while the concavity of the function $H_{\Phi}$ implies $H_{\Phi}(\rho_{0})\geq\sum_{n\in\mathbb{N}}2^{-n}H_{\Phi}(\rho_{n})=+\infty$.

To prove the nontrivial implication $\,\mathrm{(iii)\Rightarrow(i)}$ it suffices to show that (iii) implies continuity of the function $H_{\Phi}$ on the set $\mathrm{ext}\mathfrak{S}(\mathcal{H}_{A})$ and to apply Theorem 2 in [9]. But we will give an independent proof of this implication based on the recently established monotonicity of the relative entropy under positive linear maps [5].

Show first that (iii) implies continuity of the function $H_{\Phi}$ on the set

[TABLE]

for any $\,k$ by using Winter’s modification of the Alicki-Fannes method [1, 13].

Let $\rho$ and $\sigma$ be different states in $\mathfrak{S}_{k}(\mathcal{H}_{A})$ and $\,\varepsilon=\frac{1}{2}\|\rho-\sigma\|_{1}$. Following [13] introduce the state $\,\omega^{*}=(1+\varepsilon)^{-1}(\rho+[\hskip 1.0pt\sigma-\rho\hskip 1.0pt]_{+})$. Then

[TABLE]

where $\,\tau_{+}=\varepsilon^{-1}[\hskip 1.0pt\rho-\sigma\hskip 1.0pt]_{+}\,$ and $\,\tau_{-}=\varepsilon^{-1}[\hskip 1.0pt\rho-\sigma\hskip 1.0pt]_{-}\,$ are states in $\mathfrak{S}(\mathcal{H}_{A})$. Since $\rho$ and $\sigma$ are different states in $\mathfrak{S}_{k}(\mathcal{H}_{A})$, the rank of the states $\tau_{+}$ and $\tau_{-}$ does not exceed $2k-1$. It follows from (5) that

[TABLE]

The concavity of the entropy and inequality (1) imply

[TABLE]

for any $p\in(0,1)$ and any states $\varrho$ and $\varsigma$, where $h_{2}(p)=\eta(p)+\eta(1-p)$.

By applying this double inequality to the above convex decompositions of $\,\omega_{*}$ we obtain

[TABLE]

and

[TABLE]

where $p=\frac{\varepsilon}{1+\varepsilon}$.

These inequalities and upper bound (7) show that

[TABLE]

This implies (uniform) continuity of the function $H_{\Phi}(\rho)$ on the set $\mathfrak{S}_{k}(\mathcal{H}_{A})$.

We will prove (i) by using the approximation technique proposed in [8]. By Proposition 3 in [8] any concave lower semicontinuous nonnegative function $f$ on $\mathfrak{S}(\mathcal{H})$ is a pointwise limit of the nondecreasing sequence $\{\hat{f}_{k}\}$ of concave lower semicontinuous nonnegative functions on $\mathfrak{S}(\mathcal{H})$ defined as follows

[TABLE]

where $\mathcal{P}_{k}(\rho)$ is the set of all countable ensembles of states in $\mathfrak{S}_{k}(\mathcal{H})$ with the average state $\rho$. If the function $f$ is continuous and bounded on the set $\mathfrak{S}_{k}(\mathcal{H})$ for any natural $k$ then all the functions $\hat{f}_{k}$ are continuous on $\mathfrak{S}(\mathcal{H})$.333This is a corollary of the strong stability of the set $\mathfrak{S}(\mathcal{H})$ [8]. So, in this case Dini’s lemma implies the following criterion for local continuity of the function $f$: the function $f$ is continuous on a compact subset of $\mathfrak{S}(\mathcal{H})$ if and only if the sequence $\{\hat{f}_{k}\}$ uniformly converges to the function $f$ on this subset.

This criterion gives a powerful method for analysis of continuity of the von Neumann entropy (described in [8]), since in the case $f=H$ the difference $\,\Delta^{H}_{k}(\rho)=H(\rho)-\hat{H}_{k}(\rho)\,$ between the von Neumann entropy and its $k$-approximator $\hat{H}_{k}$ is expressed via the quantum relative entropy as follows

[TABLE]

Since we have shown that (iii) implies continuity and boundedness of the function $H_{\Phi}$ on the set $\mathfrak{S}_{k}(\mathcal{H}_{A})$ for each $k$, we can use the above criterion of local continuity in the case $f=H_{\Phi}$. The concavity of the function $\eta(x)=-x\log x$ implies that

[TABLE]

By the monotonicity of the relative entropy (defined by formula (4)) under trace-non-increasing positive linear maps (see the Appendix) it follows from (9) and (10) that

[TABLE]

Hence, uniform convergence of the sequence $\,\{\hat{H}_{k}\}\,$ to the function $H$ on any subset of $\,\mathfrak{S}(\mathcal{H}_{A})$ implies uniform convergence of the sequence $\,\{[\widehat{H_{\Phi}}]_{k}\}\,$ to the function $H_{\Phi}$ on this subset. By combining this observation with the above criterion of local continuity in the cases $f=H$ and $f=H_{\Phi}$ we obtain the implication in (i) for any sequence $\{\rho_{n}\}$ of states converging to a state  $\rho_{0}$. Since the functions $H$ and $H_{\Phi}$ are homogeneous (of degree 1) to prove the general form of (i) it suffices to show that $\,\lim_{n}H_{\Phi}(\rho_{n})=0\,$ for any sequence $\{\rho_{n}\}$ in $\mathfrak{T}_{+}(\mathcal{H}_{A})$ converging to the zero operator such that $\,\lim_{n}H(\rho_{n})=0$. This can be done by using (5). $\square$

The arguments from the proof of Theorem 2 imply the following

Corollary 1. If $H_{\Phi}(\rho)\leq C<+\infty$ for any pure state $\rho\in\mathfrak{S}(\mathcal{H}_{A})$ then the function $H_{\Phi}$ is uniformly continuous on the set $\,\mathfrak{S}_{k}(\mathcal{H}_{A})$ defined in (6) for any natural $\,k$ and

[TABLE]

for any finite rank states $\,\rho$ and $\,\sigma$ in $\,\mathfrak{S}(\mathcal{H}_{A})$, where $\;\varepsilon=\frac{1}{2}\|\hskip 1.0pt\rho-\sigma\|_{1}$.

The simplest PCE-maps are maps with finite-dimensional output and unitary transformations, i.e. maps of the form $\Phi(\rho)=U\rho U^{*}$, where $U$ is an isometry from $\mathcal{H}_{A}$ into $\mathcal{H}_{B}$. More interesting examples are quantum channels and operations with finite Choi rank. The PCE-channel with infinite input and output spaces and infinite Choi rank is discribed after Proposition 2 in [7].

Further results in this direction, in particular, the classification ofPCE-channels and quantitative continuity analysis of their output entropy, are obtained in [10].

Appendix: On monotonicity of the relative entropy under trace-non-increasing positive maps

Recently Muller-Hermes and Reeb established (essentially basing on Beigi’s results [2]) the following fundamental property.

Theorem 2.  [5] If $\,\Phi$ is a positive trace-preserving linear map then

[TABLE]

Muller-Hermes and Reeb mentioned in [5] that this result is not generalized to trace-non-increasing positive linear maps until we use for all positive trace-class operators the same definition of the relative entropy as for quantum states. But using Lindblad’s definition (4) of the relative entropy it is easy to obtain such generalization.

Corollary 2. If the relative entropy is defined by formula (4) then (11) is valid for any trace-non-increasing positive linear map $\Phi$.

Proof. Consider the trace-preserving positive map $\Phi^{\prime}(\rho)=\Phi(\rho)\oplus\Psi(\rho)$ from $\mathfrak{T}(\mathcal{H}_{A})$ to $\mathfrak{T}(\mathcal{H}_{B}\oplus\mathcal{H}_{C})$, where $\Psi(\rho)=[\mathrm{Tr}(\rho-\Phi(\rho))]\tau\,$ is a positive linear map from $\mathfrak{T}(\mathcal{H}_{A})$ to $\mathfrak{T}(\mathcal{H}_{C})$ determined by a given state $\tau$ in $\mathfrak{S}(\mathcal{H}_{C})$. Then

[TABLE]

The nonnegativity of the relative entropy defined by formula (4) and Theorem Appendix: On monotonicity of the relative entropy under trace-non-increasing positive maps imply

[TABLE]

for any states $\rho$ and $\sigma$. $\square$

I am grateful to A.S.Holevo and G.G.Amosov for useful discussion.