Upper bounds for the Holevo quantity and their use

TL;DR

This paper introduces new, easily computable upper bounds for the Holevo quantity in quantum information, which improve existing estimates and are applicable to both finite and infinite-dimensional quantum channels.

Contribution

It proposes a family of tight upper bounds for the Holevo quantity based on probabilistic and metric ensemble characteristics, applicable to various quantum systems.

Findings

Upper bounds depend on a reference state and ensemble metrics.

Bounds are tight for large energy in multi-mode oscillators.

Results improve estimates for quantum channel capacities.

Abstract

We present a family of easily computable upper bounds for the Holevo quantity of ensemble of quantum states depending on a reference state as a free parameter. These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve existing estimates in the literature. We also present upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on metric divergence of the ensemble. The specification of this upper bound for the multi-mode quantum oscillator is tight for large energy. The above results are used to obtain tight upper bounds for the Holevo capacity of finite-dimensional and infinite-dimensional energy-constrained quantum channels depending on metric characteristics of the channel output.

Equations

$$\chi \left(\{p_i,{\rho }_i\}\right)\doteq \sum _i{p}_{i}H({\rho }_i\Vert \bar{\rho })=H(\bar{\rho })-\sum _i{p}_{i}H({\rho }_i),\bar{\rho }=\sum _i{p}_i{\rho }_i,$$

$$\chi (\{p_i,{\rho }_i\})\le {\upsilon }_{\mathrm{m}}S(\{p_i\}),$$

$$\chi (\{p_i,{\rho }_i\})\le {\upsilon }_{\mathrm{m}}\log n,$$

$${\epsilon }_{\mathrm{m}}=\frac{1}{2}\underset{\sigma }{\inf }\underset{i}{\sup }\Vert {\rho }_i-\sigma {\Vert }_1\text{and}{\epsilon }_{\mathrm{av}}=\frac{1}{2}\underset{\sigma }{\inf }\sum _i{p}_i\Vert {\rho }_i-\sigma {\Vert }_1$$

$$S(\{p_i\})={\sum }_i\eta (p_i)-\eta \left({\sum }_i{p}_i\right)\text{and}H(\rho )=\mathrm{Tr}\eta (\rho )-\eta (\mathrm{Tr}\rho ).$$

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

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

$$\sum _i{p}_{i}H({\rho }_i\Vert \sigma )=\sum _i{p}_{i}H({\rho }_i\Vert \bar{\rho })+H(\bar{\rho }\Vert \sigma )$$

$$t_i=\frac{p_i\Vert {\rho }_i-\sigma {\Vert }_1}{{\sum }_i{p}_i\Vert {\rho }_i-\sigma {\Vert }_1}\text{ and }{\tau }_i^{\pm }=2\frac{[{\rho }_i-\sigma {]}_{\pm }}{\Vert {\rho }_i-\sigma {\Vert }_1},i=\overline{1,n}$$

$$\left|\chi (\{p_i,{\rho }_i\})-\epsilon \left(\chi (\{t_i,{\tau }_i^{+}\})-\chi (\{t_i,{\tau }_i^{-}\})\right)\right|\le g(\epsilon ),$$

$$\chi (\{p_i,{\rho }_i\})\le \epsilon \left(\chi (\{t_i,{\tau }_i^{+}\})-\chi (\{t_i,{\tau }_i^{-}\})\right)+g(\epsilon )\le \epsilon \chi (\{t_i,{\tau }_i^{+}\})+g(\epsilon ),$$

$$\chi (\{p_i,{\rho }_i\})\le \epsilon S\left(\{t_i\}\right)+g(\epsilon )=S\left(\{\frac{1}{2}{p}_i\Vert {\rho }_i-\sigma {\Vert }_1\}\right)+g(\epsilon )$$

$$\chi (\{p_i,{\rho }_i\})\le \epsilon H\left(\sum _i{t}_i{\tau }_i^{+}\right)+g(\epsilon )=H\left(\sum _i{p}_i[{\rho }_i-\sigma {]}_{+}\right)+g(\epsilon ).$$

$$\chi (\{p_i,{\rho }_i\})-\epsilon \left(\chi (\{t_i,{\tau }_i^{+}\})-\chi (\{t_i,{\tau }_i^{-}\})\right)=h_2(\epsilon ).$$

$${\rho }_{AB}=\sum _{i=1}^n{p}_i{\rho }_i\otimes |i⟩⟨i|\text{and}{\sigma }_{AB}=\sum _{i=1}^n{p}_i\sigma \otimes |i⟩⟨i|,$$

$$I(A:B{)}_{\rho }=\chi (\{p_i,{\rho }_i\}),I(A:B{)}_{\sigma }=0\text{and}I(A:B{)}_{{\tau }_{\pm }}=\chi (\{t_i,{\tau }_i^{\pm }\}),$$

$$\frac{1}{2}{\sup }_i\Vert {\rho }_i-\sigma {\Vert }_{1}S(\{p_j\})+g(\epsilon ),\epsilon \log n+g(\epsilon ),\epsilon \log d+g(\epsilon ),$$

$$T_{\chi }(\{p_i,{\rho }_i\}|\sigma )\doteq \epsilon \left(\chi (\{t_i,{\tau }_i^{+}\})-\chi (\{t_i,{\tau }_i^{-}\})\right)$$

$$\epsilon =\frac{1}{2}\sum _i{p}_i\Vert {\rho }_i-\sigma {\Vert }_1$$

$$[\rho -{\rho }_c{]}_{+}=\sum _{{\lambda }_k>1/d}({\lambda }_k-1/d\text{d})|{\phi }_k⟩⟨{\phi }_k|,[\rho -{\rho }_c{]}_{-}=\sum _{{\lambda }_k<1/d}(1/d-{\lambda }_k)|{\phi }_k⟩⟨{\phi }_k|$$

$$T_{\chi }(\{p_i,{\rho }_i\}|{\rho }_c)=(1-1/d)\left(\chi (\{p_i,{\rho }_i\})-\chi (\{p_i,{\tilde{\rho }}_i\})\right)$$

$$\chi (\{p_i,{\rho }_i\})-T_{\chi }(\{p_i,{\rho }_i\}|{\rho }_c)=(1/d)\chi (\{p_i,{\rho }_i\})+(1-1/d)\chi (\{p_i,{\tilde{\rho }}_i\})$$

$$0\le \chi (\{p_i,{\rho }_i\})-T_{\chi }(\{p_i,{\rho }_i\}|{\rho }_c)\le \frac{\log d}{d}+(1-1/d)\log \frac{d}{d-1}=h_2(1/d),$$

$$\chi (\{p_i,{\rho }_i\})\le (1-1/d)S(\{p_i\})+g(1-1/d)$$

$$\chi (\{p_i,{\rho }_i\})\le (1-1/d)H(\bar{\rho })+g(1-1/d),$$

$$\chi (\{p_i,{\rho }_i\})-T_{\chi }(\{p_i,{\rho }_i\}|{\rho }_c)=\frac{k}{d}\log \frac{d}{k}+\frac{d-k}{d}\log \frac{d}{d-k}=h_2(\epsilon ).$$

$${\tau }_i^{\pm }=\frac{2[{\rho }_i-{\hat{\rho }}_i{]}_{\pm }}{\Vert {\rho }_i-{\hat{\rho }}_i{\Vert }_1}\text{and}{t}_i=\frac{1}{2\epsilon }{p}_i\Vert {\rho }_i-\bar{\rho }{\Vert }_1=\frac{1}{2\epsilon }{p}_i(1-p_i)\Vert {\rho }_i-{\hat{\rho }}_i{\Vert }_1$$

$$\epsilon =D(\{p_i,{\rho }_i\}|\bar{\rho })\doteq \frac{1}{2}\sum _{i=1}^n{p}_i\Vert {\rho }_i-\bar{\rho }{\Vert }_1=\frac{1}{2}\sum _{i=1}^n{p}_i(1-p_i)\Vert {\rho }_i-{\hat{\rho }}_i{\Vert }_1.$$

$$\Vert {\rho }_i-{\hat{\rho }}_i{\Vert }_1\le {\upsilon }_{\mathrm{m}}\text{and hence}\epsilon \le {\upsilon }_{\mathrm{m}}\left(1-\sum _{i=1}^n{p}_i^2\right),$$

$$\begin{array}[]{rl}\displaystyle T_{\chi}(\{p_{i},\rho_{i}\}|\hskip 1.0pt\bar{\rho}\hskip 1.0pt)&\displaystyle\!\!=\;\varepsilon\sum_{i=1}^{n}t_{i}(H(\tau^{-}_{i})-H(\tau^{+}_{i}))\\ &\displaystyle\!\!=\;\sum_{i=1}^{n}p_{i}(1-p_{i})\left(H([\rho_{i}-\hat{\rho}_{i}]_{-})-H([\rho_{i}-\hat{\rho}_{i}]_{+})\right).\end{array}$$

Full text

We also present upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on metric divergence of the ensemble. The specification of this upper bound for the multi-mode quantum oscillator is tight for large energy.

The above results are used to obtain tight upper bounds for the Holevo capacity of finite-dimensional and infinite-dimensional energy-constrained quantum channels depending on metric characteristics of the channel output.

1 Introduction and preliminaries

The Holevo quantity of ensemble of quantum states (also called Holevo information) is the upper bound for the classical information obtained from quantum measurements over the ensemble [8]. It plays a basic role in analysis of information properties of quantum systems and channels [9, 12, 18].

The Holevo quantity of a discrete (finite or countable) ensemble $\{p_{i},\rho_{i}\}$ of quantum states is defined as

[TABLE]

where $H(\cdot\|\cdot)$ is the quantum relative entropy, $H(\cdot)$ is the von Neumann entropy (introduced below) and the second formula is valid if $\,H(\rho_{i})<+\infty\,$ for all $i$. So, the exact value of the Holevo quantity can be found by calculation of the entropy (relative entropy) for a collection of quantum states, which requires some efforts, especially, in the infinite-dimensional case. Therefore it is useful to have easily computable estimates for the Holevo quantity.

A problem of finding easily computable estimates (in particular, upper estimates) for the Holevo quantity was considered by several authors [3, 4, 6, 14, 20]. The main idea of works in this direction is to use geometrical and probabilistic features of the ensemble to obtain effective estimates. For example, it is shown in [6] that in finite dimensions the Holevo quantity is upper bounded by the entropy of the matrices with entries depending on mutual fidelities of states of the ensemble and their probabilities. Recently Audenaert obtained in [3] the following upper bound:

[TABLE]

where $\upsilon_{\mathrm{m}}=\frac{1}{2}\sup_{i,j}\|\rho_{i}-\rho_{j}\|_{1}$ is the maximal trace norm distance between the states of the ensemble and $S(\{p_{i}\})$ is the Shannon entropy of the probability distribution $\{p_{i}\}$. It implies that

[TABLE]

where $n$ is the number of states in the ensemble $\{p_{i},\rho_{i}\}$.222In the case $\,n=2\,$ inequality (2) is originally proved in [4].

Audenaert’s upper bound (1) refines the well-known rough estimate$\chi(\{p_{i},\rho_{i}\})\leq S(\{p_{i}\})$ by taking metric relations between states of the ensemble into account.

In this paper we present a family of upper bounds for the Holevo quantity depending on a reference state as a free parameter. These upper bounds are proved by applying the Alicki-Fannes-Winter technique (generally used for proving uniform continuity bounds) [1, 19]. In particular, we obtain several modifications of Audenaert’s upper bound (1) and of its corollary (2). We show that the maximal distance $\upsilon_{\mathrm{m}}$ between states of the ensemble in (1) and in (2) can be replaced, respectively, by the quantities

[TABLE]

called maximal metric divergence and average metric divergence of the ensemble $\{p_{i},\rho_{i}\}$, which can be significantly less than $\upsilon_{\mathrm{m}}$. The cost of such replacement is the appearance of (nonavidable) additional term independent of the size of the ensemble and of the dimension of underlying Hilbert space (Corollaries 2.1 and 2.1).

In the last part of the paper the above results are used to obtain upper bound for the Holevo capacity of a finite-dimensional quantum channel depending on the Chebyshev raduis of its output set. This upper bound gives relatively sharp estimates of the Holevo capacity for several types of channels (in particular, for depolarising and erasure channels).

We also present upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on metric divergence of the ensemble and consider its specification for the multi-mode quantum oscillator. This upper bound is used to obtain upper bound for the Holevo capacity of infinite-dimensional quantum channels with energy constraints.

Let $\mathcal{H}$ be a finite-dimensional or separable infinite-dimensional Hilbert space, $\mathfrak{B}(\mathcal{H})$ the algebra of all bounded operators with the operator norm $\|\cdot\|$ and $\mathfrak{T}(\mathcal{H})$ the Banach space of all trace-class operators in $\mathcal{H}$ with 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) [9, 12, 18].

We denote by $I_{\mathcal{H}}$ the unit operator in a Hilbert space $\mathcal{H}$ and by $\mathrm{Id}_{\mathcal{\mathcal{H}}}$ the identity transformation of the Banach space $\mathfrak{T}(\mathcal{H})$.

A finite or countable collection $\{\rho_{i}\}$ of states with a probability distribution $\{p_{i}\}$ is conventionally called (discrete) ensemble and denoted $\{p_{i},\rho_{i}\}$. The state $\bar{\rho}\doteq\sum_{i}p_{i}\rho_{i}$ is called the average state of this ensemble.

The Shannon entropy $\,S(\{p_{i}\})=\sum_{i}\eta(p_{i})\,$ of a probability distribution $\{p_{i}\}$ and the von Neumann entropy $\,H(\rho)=\mathrm{Tr}\eta(\rho)\,$ of a state $\rho\in\mathfrak{S}(\mathcal{H})$, where $\eta(x)=-x\log x$, have concave homogeneous333A function $f(x)$ is called homogeneous (of degree 1) if $f(cx)=cf(x)$ for $c\geq 0$. extensions to the positive cones in $\ell_{1}$ and in $\mathfrak{T}(\mathcal{H})$ defined, respectively, by the formulas (cf.[11])

[TABLE]

The extended von Neumann entropy satisfies the following inequality

[TABLE]

valid for any finite or countable collection $\{\rho_{i}\}$ of positive operators in $\mathfrak{T}(\mathcal{H})$ with finite $\sum_{i}\mathrm{Tr}\rho_{i}$ [12, 13]. Denote by $h_{2}(p)$ the binary entropy $S(\{p,1-p\})$.

The quantum relative entropy for two states $\rho$ and $\sigma$ in $\mathfrak{S}(\mathcal{H})$ is defined as follows

[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 $\hskip 1.0pt\mathrm{supp}\hskip 1.0pt\sigma$ [11, 13].

We will use Donald’s identity

[TABLE]

valid for arbitrary ensemble $\{p_{i},\rho_{i}\}$ of states with the average state $\bar{\rho}$ and arbitrary state $\sigma$ [5, 13].

Throughout the paper we will use the following

Definition 1.  An upper bound $g(x)$ for a nonnegative function $f(x)$ on a set $X$ is called tight if $\;\sup_{x\in X}\frac{f(x)}{g(x)}=1$.

2 Estimates for the Holevo quantity

2.1 Discrete ensembles

For arbitrary given ensemble $\{p_{i},\rho_{i}\}$ of $\,n\leq\infty\,$ states in $\mathfrak{S}(\mathcal{H})$ and any state $\sigma\in\mathfrak{S}(\mathcal{H})$ consider two ensembles $\{t_{i},\tau^{+}_{i}\}$ and $\{t_{i},\tau^{-}_{i}\}$ of $\,n\,$ states in $\mathfrak{S}(\mathcal{H})$, where

[TABLE]

($[\rho_{i}-\sigma]_{+}$ and $[\rho_{i}-\sigma]_{-}$ are, respectively, the positive and negative parts of the operator $\rho_{i}-\sigma$). If $\sigma=\rho_{i_{0}}$ for some $i_{0}$ then we assume that both ensembles have no states in the $i_{0}$-th position.

Proposition 1.  The Holevo quantities of the above ensembles $\{p_{i},\rho_{i}\}$, $\{t_{i},\tau^{+}_{i}\}$ and $\{t_{i},\tau^{-}_{i}\}$ are related by the inequality

[TABLE]

which implies that

[TABLE]

where $\varepsilon=\frac{1}{2}\sum_{i}p_{i}\|\rho_{i}-\sigma\|_{1}$ and $\,g(\varepsilon)\!\doteq\!(1+\varepsilon)h_{2}\!\left(\frac{\varepsilon}{1+\varepsilon}\right)$. It follows that 444$S$ and $H$ are the homogeneous extensions of the Shannon entropy and of the von Neumann entropy to the positive cones in $\ell_{1}$ and in $\mathfrak{T}(\mathcal{H})$ defined by the formulae in (3).

[TABLE]

and

[TABLE]

Upper bounds (6)-(9) are tight in the sense of Def.1. For any $\,\varepsilon>0$ there is an ensemble $\{p_{i},\rho_{i}\}$ and a state $\sigma$ such that $\,\varepsilon=\frac{1}{2}\sum_{i}p_{i}\|\rho_{i}-\sigma\|_{1}$ and

[TABLE]

Remark 1. The last assertion of Proposition 2.1 shows that the right hand side of (6) can not be less than $h_{2}(\varepsilon)$, which is equivalent to $g(\varepsilon)$ for small $\varepsilon$.

Proof. Inequality (6) directly follows from Proposition 1 in [16] (with trivial C). It suffices to take the $qc$-states

[TABLE]

where $\mathcal{H}_{A}=\mathcal{H}$ and $\{|i\rangle\}$ is an orthonormal basis in $n$-dimensional Hilbert space $\mathcal{H}_{B}$, and to note that $\rho_{B}=\sigma_{B}$,

[TABLE]

where $\tau_{\pm}=\varepsilon^{-1}[\rho-\sigma]_{\pm}$. Inequalities (8) and (9) directly follow from (7).

The tightness of upper bounds (6)-(9) and the last assertion of the proposition can be shown by using Examples 2.1 and 2.1 below. $\square$

Note first that Proposition 2.1 implies the following easily computable upper bounds for the Holevo quantity.

Corollary 1. The Holevo quantity $\chi(\{p_{i},\rho_{i}\})$ of an arbitrary ensemble $\{p_{i},\rho_{i}\}$ of $\,n\leq\infty\,$ states in $\mathfrak{S}(\mathcal{H})$ is upper bounded by any of the quantities

[TABLE]

where $\sigma$ is any state in $\,\mathfrak{S}(\mathcal{H})$, $\,\varepsilon=\frac{1}{2}\sum_{i}p_{i}\|\rho_{i}-\sigma\|_{1}$ and $\,d=\dim\mathcal{H}\leq\infty$.

The first and the second upper bound in (10) may be stronger than Audenaert’s upper bound (1) and its corollary (2) correspondingly (despite nonavoidable term $g(\varepsilon)$ in the formers), since the values of $\sup_{i}\|\rho_{i}-\sigma\|_{1}$ and $\sum_{i}p_{i}\|\rho_{i}-\sigma\|_{1}$ may be significantly less than $\,\sup_{i,j}\|\rho_{i}-\rho_{j}\|_{1}$ for ensembles with arbitrary large Holevo quantity (see Examples 2.1 and 2.1 below).

Proposition 2.1 shows that the quantity

[TABLE]

can be considered as an approximation of $\,\chi(\{p_{i},\rho_{i}\})$.

We will call the quantity

[TABLE]

metric divergence of an ensemble $\{p_{i},\rho_{i}\}$ with respect to a state $\sigma$ and will denote it by $D(\{p_{i},\rho_{i}\}|\hskip 1.0pt\sigma)$.

The reference state $\sigma$ is a free parameter which can be used to optimise upper bounds (6)-(10). Below we will specify these upper bounds and analyse the quantity $T_{\chi}(\{p_{i},\rho_{i}\}|\hskip 1.0pt\sigma)$ in the following cases:

• •

$\sigma=\rho_{c}\doteq I_{\mathcal{H}}/d\,$ is the chaotic state in $d$-dimensional Hilbert space $\mathcal{H}$;

• •

$\sigma=\bar{\rho}\doteq\sum_{i}p_{i}\rho_{i}\,$ is the average state of the ensemble $\{p_{i},\rho_{i}\}$;

• •

$\sigma=\rho_{i_{0}}\,$ is one of the states of the ensemble $\{p_{i},\rho_{i}\}$;

• •

$\sigma\,$ is the state minimazing the value of $\,\frac{1}{2}\sum_{i}p_{i}\|\rho_{i}-\sigma\|_{1}$;

• •

$\sigma\,$ is the state minimazing the value of $\,\frac{1}{2}\sup_{i}\|\rho_{i}-\sigma\|_{1}$.

Note: The minimazing states $\sigma$ in the last two cases may not coincide with each other and with the average state $\bar{\rho}$ even for ensemble $\{p_{i},\rho_{i}\}$ of isomorphic states with uniform probability distribution $\{p_{i}\}$ (see Example 4 below).

The case $\sigma=\rho_{c}$. In this case the values of $\|\rho_{i}-\sigma\|_{1}$ and the ensembles $\{t_{i},\tau^{+}_{i}\}$ and $\{t_{i},\tau^{-}_{i}\}$ are easily determined. Indeed, if $\rho=\sum_{k}\lambda_{k}|\varphi_{k}\rangle\langle\varphi_{k}|$ is a spectral decomposition of a state $\rho$ in $d$-dimensional Hilbert space $\mathcal{H}$ then

[TABLE]

and $\|\rho-\rho_{c}\|_{1}=\sum_{k}|\lambda_{k}-1/d|$. It follows, in particular, that in this case the probability distribution $\{t_{i}\}$ is completely determined by eigenvalues of the states $\rho_{i}$ and by the probability distribution $\{p_{i}\}$.

The above formulae show that $\{t_{i},\tau^{+}_{i}\}=\{p_{i},\rho_{i}\}$ for any ensemble $\{p_{i},\rho_{i}\}$ consisting of states proportional to projectors of the same rank.

Example 1. Let $\{p_{i},\rho_{i}\}$ be an arbitrary ensemble of pure states. Then $\frac{1}{2}\|\rho_{i}-\rho_{c}\|_{1}=1-1/d$, $\,t_{i}=p_{i}$, $\,\tau^{+}_{i}=\rho_{i}\,$ and $\,\tau^{-}_{i}=\tilde{\rho}_{i}\doteq(d-1)^{-1}(I_{\mathcal{H}}-\rho_{i})$. So,

[TABLE]

and hence

[TABLE]

Since $\chi(\{p_{i},\tilde{\rho}_{i}\})\leq\log d-\log(d-1)$, we have

[TABLE]

where an equality holds in the second inequality if and only if $\,\bar{\rho}=\rho_{c}$.

The upper bounds (8) and (9) imply, respectively,

[TABLE]

and

[TABLE]

where $\bar{\rho}\doteq\sum_{i}p_{i}\rho_{i}$. We see that the second upper bound is closer to the exact value $H(\bar{\rho})$ of $\chi(\{p_{i},\rho_{i}\})$.

Example 2. Let $\{p_{i},\rho_{i}\}$ be an ensembles of states proportional to $k$-rank projectors in $d$-dimensional Hilbert space $\mathcal{H}$ such that $\sum_{i}p_{i}\rho_{i}=\rho_{c}$. If $\sigma=\rho_{c}$ then it is easy to see that $\,\varepsilon=\frac{d-k}{d}$, $\,\{t_{i},\tau^{+}_{i}\}=\{p_{i},\rho_{i}\}$ and that the ensemble $\,\{t_{i},\tau^{-}_{i}\}$ consists of states proportional to $(d-k)$-rank projectors and has the average state $\rho_{c}$. It follows that

[TABLE]

This is the first example proving the last assertion of Proposition 2.1.

The case $\sigma=\bar{\rho}$. For each $i$ let $\hat{\rho}_{i}=(1-p_{i})^{-1}\sum_{j\neq i}p_{j}\rho_{j}$ be the complementary state to the state $\rho_{i}$ [3]. Then $\rho_{i}-\bar{\rho}=(1-p_{i})(\rho_{i}-\hat{\rho}_{i})$. So, in this case

[TABLE]

for $i=\overline{1,n}$, where

[TABLE]

By convexity of the trace norm we have

[TABLE]

where $\upsilon_{\mathrm{m}}=\frac{1}{2}\sup_{i,j}\|\rho_{i}-\rho_{j}\|_{1}$.

In the case $\,\sigma=\bar{\rho}\,$ the ensembles $\,\{t_{i},\tau^{+}_{i}\}$ and $\,\{t_{i},\tau^{-}_{i}\}$ have the same average state. So, if this average state has finite entropy then

[TABLE]

If the ensemble $\{p_{i},\rho_{i}\}$ consists of mutually orthogonal states then

[TABLE]

and hence

[TABLE]

We see again that the quantity $T_{\chi}$ may be less than the Holevo quantity. Since in this case $\,\varepsilon=1-\sum_{i}p_{i}^{2}$, by the concavity of $h_{2}$ we have

[TABLE]

in accordance with (7).

By using (12)-(14) the upper bounds in Proposition 2.1 and Corollary 2.1 in the case $\sigma=\bar{\rho}\hskip 1.0pt$ can be specified as follows

Corollary 2. Let $\,\{p_{i},\rho_{i}\}$ be an ensembles of $\,n\leq+\infty$ states in $\,\mathfrak{S}(\mathcal{H})$ and $\,d=\dim\mathcal{H}\leq+\infty$. Then 555$S$ is the homogenious enstensions of the Shannon entropy to the positive cone in $\ell_{1}$ defined by the first formula in (3).

[TABLE]

and

[TABLE]

where $\,\varepsilon=D(\{p_{i},\rho_{i}\}|\hskip 1.0pt\bar{\rho}\hskip 1.0pt)$ determined in (13) and $\,\upsilon_{\mathrm{m}}=\frac{1}{2}\sup_{i,j}\|\rho_{i}-\rho_{j}\|_{1}$. The term $g(\varepsilon)$ in all the inequalities in (15) can be replaced by $\,g(\upsilon_{\mathrm{m}}(1-\sum_{i}p_{i}^{2}))$.

The last upper bound in (15) is stronger than (2) for ensembles with significantly non-uniform probability distribution (for which $1-\sum_{i}p_{i}^{2}\ll 1$).

Example 3. Let $\{p_{i},\rho_{i}\}$ be an ensembles of $\,n+1\,$ mutually orthogonal states, where $p_{1}=1-\delta$ and $p_{i}=\delta/n$ for $i=\overline{2,n+1}$. Then $\upsilon_{\mathrm{m}}=1$ and $1-\sum_{i}p_{i}^{2}=2\delta-(1+1/n)\delta^{2}$. So, the last upper bound in (15) gives

[TABLE]

while $\chi(\{p_{i},\rho_{i}\})=S(\{p_{i}\})=\delta\log n+h_{2}(\delta)$. We see that the term $1-\sum_{i}p_{i}^{2}$ allows to take degeneracy of the probability distribution $\{p_{i}\}$ into account.

The case $\sigma=\rho_{i_{0}}$. We will assume that $i_{0}=1$. In this case

[TABLE]

where

[TABLE]

If the state $\rho_{1}$ is orthogonal to all other states of the ensemble then $\varepsilon=1-p_{1}$ and

[TABLE]

So, in this case $\,\chi(\{t_{i},\tau^{+}_{i}\})=\chi(\{\tilde{p}^{1}_{i},\rho_{i}\}_{i>1})$ and $\,\chi(\{t_{i},\tau^{-}_{i}\})=0$. Hence

[TABLE]

while Donald’s identity (5) implies that

[TABLE]

This is the second example proving the last assertion of Proposition 2.1.

By using (17)-(18) and the equality $\,S(\{p_{i}\}_{i\geq 0})=S(\{p_{i}\}_{i>0})+h_{2}(p_{1})\,$ the upper bounds in Proposition 2.1 and Corollary 2.1 in the case $\sigma=\rho_{1}$ can be specified as follows

Corollary 3. Let $\,\{p_{i},\rho_{i}\}$ be an ensembles of $\,n\leq+\infty$ states in $\,\mathfrak{S}(\mathcal{H})$ and $\,d=\dim\mathcal{H}\leq+\infty$. Then

[TABLE]

and

[TABLE]

where $\,\varepsilon_{1}=\frac{1}{2}\sum_{i>1}p_{i}\|\rho_{i}-\rho_{1}\|_{1}\,$ and $\,\upsilon_{1}=\frac{1}{2}\sup_{i>1}\|\rho_{i}-\rho_{1}\|_{1}$.

Upper bounds in (19) are modifications of Audenaert’s upper bound (1). The term in square brackets in the second of them is equal to

[TABLE]

for $p_{1}$ close to $1$. This term is the cost for replacing the maximal distance $\upsilon_{\mathrm{m}}$ between all states of ensemble in (1) by the maximal distance $\upsilon_{1}$ from the first state of ensemble to all others. It is easy to find an ensemble $\{p_{i},\rho_{i}\}$ with arbitrary $S(\{p_{i}\})$ such that $\upsilon_{1}$ is significantly less than $\upsilon_{\mathrm{m}}$ (such ensemble can be obtained by adding the state $|1\rangle\langle 1|$ to the ensemble in Example 2.1 below).

The average metric divergence. For a given ensemble $\{p_{i},\rho_{i}\}$ consider the quantity

[TABLE]

which can be called average metric divergence of the ensemble $\{p_{i},\rho_{i}\}$. In finite dimensions the infimum in (21) is always achieved at some state $\sigma$ which will be called AMD-optimal state for the ensemble $\{p_{i},\rho_{i}\}$. For the ensemble of two states $\rho_{1}$ and $\rho_{2}$ with probabilities $p_{1}$ and $p_{2}=1-p_{1}$ AMD-optimal states are easily determined: if $p_{1}>p_{2}$ (correspondingly, $p_{1}<p_{2}$) then $\rho_{2}$ (correspondingly, $\rho_{1}$) is a unique AMD-optimal state, if $p_{1}=p_{2}$ then any convex mixture of the states $\rho_{1}$ and $\rho_{2}$ is an AMD-optimal state for this ensemble. In this case $\varepsilon_{\mathrm{av}}=\frac{1}{2}\min\{p_{1},p_{2}\}\|\rho_{1}-\rho_{2}\|_{1}$. In general, continuity and convexity of the function $\sigma\mapsto\sum_{i}p_{i}\|\rho_{i}-\sigma\|_{1}$ implies that the set of all AMD-optimal states for a given ensemble is closed and convex. The below example shows (contrary to intuition) that the average state $\bar{\rho}$ of an ensemble of isomorphic states with uniform probability distribution may be not AMD-optimal.

Example 4. Let $\{p_{i},|\varphi_{i}\rangle\langle\varphi_{i}|\}_{i=1}^{4}$ be the ensemble of four pure states in $\mathrm{3\textup{-}D}$ Hilbert space $\mathcal{H}$, where $p_{i}\equiv 1/4,$ $|\varphi_{1}\rangle=|1\rangle$, $|\varphi_{2}\rangle=-\frac{1}{2}|1\rangle+\frac{\sqrt{3}}{2}|2\rangle$, $|\varphi_{3}\rangle=-\frac{1}{2}|1\rangle-\frac{\sqrt{3}}{2}|2\rangle$ and $|\varphi_{4}\rangle=|3\rangle$ (here $\{|1\rangle,|2\rangle,|3\rangle\}$ is an orthonormal basis in $\mathcal{H}$). Then $\bar{\rho}=\frac{3}{8}(|1\rangle\langle 1|+|2\rangle\langle 2|)+\frac{1}{4}|3\rangle\langle 3|$. It is easy to see that

[TABLE]

where $\,\sigma=\frac{1}{2}(|1\rangle\langle 1|+|2\rangle\langle 2|)$ is a unique AMD-optimal state for this ensemble.

By taking AMD-optimal state666If $\dim\mathcal{H}=+\infty$ and there are no AMD-optimal states, it suffices to take for given $\epsilon>0$ a state $\sigma_{\epsilon}$ such that $\,\frac{1}{2}\sum_{i}p_{i}\|\rho_{i}-\sigma_{\epsilon}\|_{1}$ is $\epsilon$-close to $\varepsilon_{\mathrm{av}}$. in the role of the reference state $\sigma$ in Corollary 2.1 we obtain the following

Corollary 4. Let $\,\{p_{i},\rho_{i}\}$ be an ensembles of $\,n\leq+\infty$ states in $\,\mathfrak{S}(\mathcal{H})$ and $\,d=\dim\mathcal{H}\leq+\infty$. Then

[TABLE]

where $\varepsilon_{\mathrm{av}}$ is the average metric divergence of $\,\{p_{i},\rho_{i}\}$ defined in (21).

Since $\varepsilon_{\mathrm{av}}$ may be significantly less than the maximal distance $\upsilon_{\mathrm{m}}$ between states of an ensemble $\{p_{i},\rho_{i}\}$, the first upper bound in Corollary 2.1 may be stronger than upper bound (2) despite (nonavoidable) additional term $g(\varepsilon_{\mathrm{av}})$.

The maximal metric divergence. For a given ensemble $\{p_{i},\rho_{i}\}$ consider the quantity

[TABLE]

which can be called maximal metric divergence of the ensemble $\{p_{i},\rho_{i}\}$. In finite dimensions the infimum in (22) is always achieved at some state $\sigma$ which will be called MMD-optimal state for the ensemble $\{p_{i},\rho_{i}\}$. For ensemble of two states $\rho_{1}$ and $\rho_{2}$ with any probabilities $p_{1}$ and $p_{2}=1-p_{1}$ the state $\frac{1}{2}(\rho_{1}+\rho_{2})$ is a unique MMD-optimal state. In this case $\varepsilon_{\mathrm{m}}=\frac{1}{4}\|\rho_{1}-\rho_{2}\|_{1}$.

The ensemble of four pure states in Example 2.1 has a unique MMD-optimal state $\rho_{c}\doteq I_{\mathcal{H}}/3$ not coinciding with the average state and with the AMD-optimal state of this ensemble. For this ensemble $\varepsilon_{\mathrm{m}}=2/3>\varepsilon_{\mathrm{av}}=5/8$.

By taking MMD-optimal state in the role of the reference state $\sigma$ in Corollary 2.1 we obtain the following

Corollary 5. Let $\,\{p_{i},\rho_{i}\}$ be an ensembles of $\,n\leq+\infty$ states in $\,\mathfrak{S}(\mathcal{H})$, where $\,\dim\mathcal{H}\leq+\infty$. Then

[TABLE]

where $\varepsilon_{\mathrm{m}}$ is the maximal metric divergence of $\{p_{i},\rho_{i}\}$ defined in (22).

We will show that in some cases this upper bound is stronger than the Audenaert’s upper bound (1) despite (nonavoidable) extra term $g(\varepsilon_{\mathrm{m}})$ (bounded by $g(1)=2\log 2$). Note first that

[TABLE]

by convexity of the trace norm.

For any ensemble of two states we have $\upsilon_{\mathrm{m}}/\varepsilon_{\mathrm{m}}=2$, but for multi-state ensembles the difference between $\varepsilon_{\mathrm{m}}$ and $\upsilon_{\mathrm{m}}$ are not so large.777I would be grateful for any comments concerning possible values of $\upsilon_{\mathrm{m}}/\varepsilon_{\mathrm{m}}$ in general case. The following example shows existence of ensemble with arbitrary large Holevo quantity for which $\upsilon_{\mathrm{m}}/\varepsilon_{\mathrm{m}}$ is close to $\sqrt{2}$.

Example 5. Let $\{p_{i},|\varphi_{i}\rangle\langle\varphi_{i}|\}_{i=1}^{n}$ be the ensemble of $n$ pure states in $(n+1)\textup{-}$dimensional Hilbert space $\mathcal{H}$, where $\{p_{i}\}$ is an arbitrary probability distribution and $|\varphi_{i}\rangle=\sqrt{1-a^{2}}|1\rangle+a|i+1\rangle$, $a\in[0,1]$ (here $\{|1\rangle,...,|n+1\rangle\}$ is an orthonormal basis in $\mathcal{H}$). Then

[TABLE]

and

[TABLE]

It follows that $\upsilon_{\mathrm{m}}=a\sqrt{2-a^{2}}$, while $\varepsilon_{\mathrm{m}}\leq a$.888One can show that $\,n^{-1}\sum_{i=1}^{n}|\varphi_{i}\rangle\langle\varphi_{i}|\,$ is a unique MMD-optimal state for this ensemble and that $\,\varepsilon_{\mathrm{m}}=a-o(a)<a$. So, in this case Audenaert’s upper bound (1) and the upper bound in Corollary 2.1 give, respectively,

[TABLE]

and

[TABLE]

It is clear that the latter upper bound is stronger than the former for small $a$ and large $S(\{p_{i}\})$.

Direct calculation of eigenvalues of the state $\bar{\rho}\doteq\sum_{i=1}^{n}p_{i}|\varphi_{i}\rangle\langle\varphi_{i}|\,$ in the case $\,p_{i}\equiv 1/n\,$ shows that

[TABLE]

2.2 Generalized ensembles with finite average energy

In analysis of infinite-dimensional quantum systems and channels it is necessary to consider generalized ensembles of quantum states defined as Borel probability measures on the set of quantum states [9, 10]. A discrete ensemble $\{p_{i},\rho_{i}\}$ corresponds to the measure $\sum_{i}p_{i}\delta(\rho_{i})$, where $\delta(\rho)$ is the Dirac measure concentrating at a state $\rho$. The average state of a generalized ensemble $\mu$ is the barycenter of the measure $\mu$ defined by the Bochner integral

[TABLE]

The Holevo quantity of a generalized ensemble $\mu$ is defined as

[TABLE]

where the second formula is valid under the condition $H(\bar{\rho}(\mu))<+\infty$ [9, 10].

In this subsection we consider upper bounds for the Holevo quantity of generalised ensembles $\,\mu\,$ with finite average energy

[TABLE]

provided that the Hamiltonian $H$ of the system satisfies the condition

[TABLE]

Condition (23) implies that all spectral projectors of $H$ corresponding to finite intervals are finite-dimensional and that the von Neumann entropy $H(\rho)$ is bounded on the sets of states $\rho$ with bounded energy $E(\rho)\doteq\mathrm{Tr}H\rho$ [15, Pr.1]. It follows that

[TABLE]

is a finite function on $[E_{0},+\infty)$, where $E_{0}\doteq\inf_{\|\varphi\|=1}\langle\varphi|H|\varphi\rangle$.

Let $\widehat{F}_{H}$ be a smooth function on $[0,+\infty)$ such that $\widehat{F}_{H}(E)\geq F_{H}(E)$ for all $E\geq E_{0}$ possessing the properties

[TABLE]

At least one such function $\widehat{F}_{H}$ always exists: the function $E\mapsto F_{H}(E+E_{0})$ satisfies all the above conditions by Proposition 1 in [15].

The metric divergence of a generalized ensemble $\mu$ with respect to a state $\sigma$ is naturally defined as

[TABLE]

If $\mu=\{p_{i},\rho_{i}\}$ then (26) coincides with (11).

Proposition 2.  Let $\,\mu\,$ be a generalized ensembles of states in $\,\mathfrak{S}(\mathcal{H})$ with finite average energy $\bar{E}(\mu)\doteq E(\bar{\rho}(\mu))$ and $\,\sigma$ a state in $\,\mathfrak{S}(\mathcal{H})$ with finite energy $E(\sigma)$. Let $\,\varepsilon=D(\mu|\hskip 1.0pt\sigma)$ be the metric divergence of $\,\mu$ with respect to $\sigma$ defined in (26). Then

[TABLE]

*where $\,a=1/(2\varepsilon)$, $\widehat{F}_{H}$ is any upper bound for the function $F_{H}$ (defined in (24)) satisfying conditions (25) and $\,\kappa=\frac{1}{2}(1+E(\sigma)/\bar{E}(\mu))$.*999$h_{2}(p)$ is the binary entropy, $g(p)=(1+p)h_{2}\!\left(\frac{p}{1+p}\right)=(p+1)\log(p+1)-p\log p$.

Remark 2. The right hand side of (27) is an increasing function of $\varepsilon$. It tends to zero as $\,\varepsilon\rightarrow 0\,$ if and only if $\,\widehat{F}_{H}(E)=o\hskip 1.0pt(E)$ as $E\rightarrow+\infty$. By Proposition 1 in [15] the function $\,\widehat{F}_{H}(E)=F_{H}(E+E_{0})$ satisfies the last condition if and only if

[TABLE]

It is interesting that (28) is a necessary and sufficient condition of continuity of the Holevo quantity on the set of all generalized ensembles $\mu$ with bounded average energy $\bar{E}(\mu)$ with respect to the weak convergence topology. This follows from Proposition 8 in [16], since (28) is a necessary and sufficient condition of continuity of the von Neumann entropy on the set of states $\rho$ with bounded energy $E(\rho)=\mathrm{Tr}H\rho$ [17, 15].

Proof. Assume first that $\mu$ is a discrete ensemble $\{p_{i},\rho_{i}\}$ with the average state $\bar{\rho}$.

Following the proofs of Lemmas 16,17 in [19] take any $\delta\in(0,\frac{1}{2}]$ and denote by $P_{\delta}$ the spectral projector of the operator $H$ corresponding to the interval $[0,\delta^{-1}\bar{E}(\mu)]$. By condition (23) $\mathrm{Tr}P_{\delta}<+\infty$. Since $\mathrm{Tr}H\bar{\rho}=\bar{E}(\mu)$ and $\mathrm{Tr}H\sigma=E(\sigma)$, it is easy to show that

[TABLE]

Consider the ensemble $\{\hat{p}_{i},\hat{\rho}_{i}\}$, where $\hat{\rho}_{i}=r_{i}^{-1}P_{\delta}\rho_{i}P_{\delta}$, $\hat{p}_{i}=r_{i}p_{i}/r$, $r_{i}=\mathrm{Tr}P_{\delta}\rho_{i}$, $r=\mathrm{Tr}P_{\delta}\bar{\rho}$. Corollary 2.1 implies

[TABLE]

where $\hat{\varepsilon}$ is the average metric divergence of the ensemble $\{\hat{p}_{i},\hat{\rho}_{i}\}$.

Let $\hat{\sigma}=s^{-1}P_{\delta}\sigma P_{\delta}$, where $s=\mathrm{Tr}P_{\delta}\sigma$. Then

[TABLE]

where the last inequality follows from (29).

By using (29) and the arguments from the proof of Lemma 16 in [19] (based on properties (25) of the function $\widehat{F}_{H}$) we obtain

[TABLE]

This inequality and Lemma 2 in [16] imply that

[TABLE]

Since the energy of the state $[\mathrm{Tr}P_{\delta}]^{-1}P_{\delta}$ does not exceed $\bar{E}(\mu)/\delta$, its entropy $\log\mathrm{Tr}P_{\delta}$ is upper bounded by $\widehat{F}_{H}(\bar{E}(\mu)/\delta)$. So, it follows from (30), (31) and (32) that

[TABLE]

Now assume that $\delta=\varepsilon t$, where $t\in(0,\frac{1}{2\varepsilon}]$. Then $\varepsilon^{\prime}=\varepsilon(1+\kappa t)/(1-\varepsilon t)$ and hence (33) implies (27) for $\mu=\{p_{i},\rho_{i}\}$.

For arbitrary generalized ensemble $\mu$ there exists a sequence $\{\mu_{n}\}$ of discrete ensembles weakly101010The weak convergence of a sequence $\{\mu_{n}\}$ to an ensemble $\mu_{0}$ means that $\,\lim_{n\rightarrow\infty}\int f(\rho)\mu_{n}(d\rho)=\int f(\rho)\mu_{0}(d\rho)\,$ for any continuous bounded function $f$ on $\,\mathfrak{S}(\mathcal{H})$. converging to $\mu$ such that

[TABLE]

Such sequence can be obtained by using the construction from the proof of Lemma 1 in [10] and taking into account the lower semicontinuity of the function $\mu\mapsto\chi(\mu)$ [10, Pr.1]. Since $D(\mu_{n}|\hskip 1.0pt\sigma)$ tends to $D(\mu|\hskip 1.0pt\sigma)$ (due to the weak convergence of $\mu_{n}$ to $\mu$), the validity of inequality (27) for the ensemble $\mu$ follows from its validity for all the ensembles $\mu_{n}$ proved before. $\square$

Consider specification of the upper bound in Proposition 2.2 for the $\,\ell$-mode quantum oscillator. In this case

[TABLE]

where $\,a_{i}\,$ and $\,a^{+}_{i}\,$ are the annihilation and creation operators and $\,\omega_{i}\,$ is the frequency of the $i$-th oscillator [9, Ch.12]. Since condition (28) holds, for any $E>E_{0}$ the von Neumann entropy $H(\rho)$ is continuous on the sets of states determined by the inequality $\mathrm{Tr}H\rho\leq E$ and attains maximum on this set at the Gibbs state $\gamma(E)=[\mathrm{Tr}e^{-\lambda(E)H}]^{-1}e^{-\lambda(E)H}$, where $\lambda(E)$ is the solution of the equation $\mathrm{Tr}He^{-\lambda H}=E\mathrm{Tr}e^{-\lambda H}$ [17].

The exact value of $F_{H}(E)\doteq\sup_{\mathrm{Tr}H\rho\leq E}H(\rho)$ can be found by solving a transcendental equation. But one can show that $F_{H}(E)$ is upper bounded by the function

[TABLE]

on $[0,+\infty)$ satisfying conditions (25) such that $\,\widehat{F}_{\ell,\omega}(E)-F_{H}(E)$ tends to zero as $\,E\rightarrow+\infty$ [16, Sect.3.2].

Corollary 6. Let $\,\mu\,$ be a generalized ensembles of states of the $\,\ell$-mode quantum oscillator with finite average energy $\bar{E}(\mu)\doteq E(\bar{\rho}(\mu))$ and $\,\sigma$ a state with finite energy $E(\sigma)$. Let $\,\varepsilon=D(\mu|\hskip 1.0pt\sigma)$ be the metric divergence of $\,\mu$ with respect to $\,\sigma$ defined in (26). Then

[TABLE]

where $\,a=1/(2\varepsilon)$, $\widehat{F}_{\ell,\omega}(E)$ is defined in (35) and $\,\kappa=\frac{1}{2}(1+E(\sigma)/\bar{E}(\mu))$.

This upper bound is tight (for large $E$ and appropriate choice of $\,\sigma$).

Proof. Since $\widehat{F}_{\ell,\omega}(E/x)\leq\widehat{F}_{\ell,\omega}(E)-\ell\log x$ for any positive $E$ and $x\leq 1$, the main assertion of the corollary directly follows from Proposition 2.2.

Let $E>E_{0}$ and $\{p_{i},\rho_{i}\}$ be any pure state ensemble with the average state $\gamma(E)$. Consider the ensemble $\{p_{i},\rho^{\varepsilon}_{i}\}$, where $\rho^{\varepsilon}_{i}=\varepsilon\rho_{i}+(1-\varepsilon)\gamma(E)$. Then

[TABLE]

while concavity of the entropy implies

[TABLE]

This shows tightness of the upper bound, since $\,\widehat{F}_{\ell,\omega}(E)-F_{H}(E)=o(1)\,$ as $\,E\rightarrow+\infty$ and the quantity

[TABLE]

can be made not greater than $\,\varepsilon(\widehat{F}_{\ell,\omega}(E)+o\hskip 1.0pt(\widehat{F}_{\ell,\omega}(E)))\,$ as $E\rightarrow+\infty$ by appropriate choice of $\,t$. This follows from Lemma 2.2 below proved by elementary methods. $\square$

Lemma 1. Let $\,f(t)=\frac{1+t}{1-t}+t$, $\,b>0\,$ and $\,c\,$ be arbitrary. Then

[TABLE]

3 Upper bounds for the Holevo capacity

3.1 Finite-dimensional channels

A quantum channel $\,\Phi$ from a system $A$ to a system $B$ is a completely positive trace preserving linear map $\mathfrak{T}(\mathcal{H}_{A})\rightarrow\mathfrak{T}(\mathcal{H}_{B})$, where $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are Hilbert spaces associated with these systems [9, 12, 18].

The Holevo capacity of a quantum channel $\Phi:A\rightarrow B$ is defined as follows

[TABLE]

where the supremum is over all ensembles of input states. This quantity determines the ultimate rate of transmission of classical information trough the channel $\Phi$ with non-entangled input encoding, it is closely related to the classical capacity of a quantum channel [9, 12, 18].

For a given subset $\mathfrak{S}_{0}$ of $\mathfrak{S}(\mathcal{H})$ consider the quantity

[TABLE]

called Chebyshev radius of $\mathfrak{S}_{0}$ with respect to the metric $\Delta(\rho,\sigma)=\frac{1}{2}\|\rho-\sigma\|_{1}$ [2, 7]. For example, $Cr(\{\rho,\sigma\})=\frac{1}{4}\|\rho-\sigma\|_{1}$ and $Cr(\mathfrak{S}(\mathcal{H}))=1-1/d$, where $d=\dim\mathcal{H}$. The Chebyshev radius of a set $\mathfrak{S}_{0}$ does not exceed its diameter $D(\mathfrak{S}_{0})\doteq\textstyle\frac{1}{2}\displaystyle\sup_{\rho,\sigma\in\mathfrak{S}_{0}}\|\rho-\sigma\|_{1}$, but $Cr(\mathfrak{S}_{0})$ may be significantly less than $D(\mathfrak{S}_{0})$ even for multi-dimensional sets $\mathfrak{S}_{0}$: the diameter of the set of vectors in Example 2.1 is equal to $a\sqrt{2-a^{2}}$ while its Chebyshev radius is less than $a$.

Corollary 2.1 implies the following

Proposition 3.  Let $\,\Phi:A\rightarrow B\,$ be a quantum channel. Then

[TABLE]

where $\,r_{\Phi}=Cr(\Phi(\mathfrak{S}(\mathcal{H}_{A})))\,$ and $\,d_{B}=\dim\mathcal{H}_{B}$.111111$g(p)=(1+p)h_{2}\!\left(\frac{p}{1+p}\right)=(p+1)\log(p+1)-p\log p$. Upper bound (38) is tight.

Proof. Inequality (38) follows from the second inequality in Corollary 2.1, since the average metric divergence $\varepsilon_{\mathrm{av}}$ of the image of any input ensemble $\,\{p_{i},\rho_{i}\}$ under the channel $\Phi$ does not exceed $r_{\Phi}$.

The tightness of upper bound (38) follows from Examples 3.1 and 3.1 below. $\square$

Remark 3. By Corollary 2.1 the quantity $\,r_{\Phi}=Cr(\Phi(\mathfrak{S}(\mathcal{H}_{A})))\,$ in (38) can be replaced by the quantity $\frac{1}{2}\sup_{\{p_{i},\rho_{i}\}}\inf_{\sigma\in\mathfrak{S}(\mathcal{H}_{B})}\sum_{i}p_{i}\|\Phi(\rho_{i})-\sigma\|_{1}$ which formally may be less than $r_{\Phi}$. But we have not found examples for which this quantity is really less than $r_{\Phi}$.

The following example shows that the extra term $g(r_{\Phi})$ in (38) can not be removed.

Example 6. Let $\Phi:A\rightarrow B$ be a quantum channel such that the set $\Phi(\mathfrak{S}(\mathcal{H}_{A}))$ contains a collection of pure states corresponding to some orthonormal basis in $\mathcal{H}_{B}$ (for example, $\Phi$ is the identity channel or the channel $\rho\mapsto\sum_{k}\langle\varphi_{k}|\rho|\varphi_{k}\rangle|\psi_{k}\rangle\langle\psi_{k}|$, where $\{|\varphi_{k}\rangle\}$ and $\{|\psi_{k}\rangle\}$ are orthonormal base in $\mathcal{H}_{A}$ and $\mathcal{H}_{B}\cong\mathcal{H}_{A}$ correspondingly). Then $C_{\chi}(\Phi)=\log d_{B}$ and $r_{\Phi}=1-1/d_{B}$. So, in this case inequality (38) has the form

[TABLE]

which would not be valid without the term $g(1-1/d_{B})$. $\square$

Despite the fact that upper bound (38) depends only on the Chebyshev radius of the output set of a channel $\Phi$, it gives relatively sharp estimates for the Holevo capacity of some nontrivial channels.

Example 7. Let $\Phi_{p}$ be a depolarizing channel from $d$-dimensional quantum system to itself, i.e. $\Phi_{p}(\rho)=(1-p)\rho+p\rho_{c}$, where $\rho_{c}$ is the chaotic state and $p\in[0,1]$. Then

[TABLE]

where $\,c=1-1/d$ [9, 18], while the upper bound (38) implies

[TABLE]

since $\|\Phi_{p}(\rho)-\rho_{c}\|_{1}=(1-p)\|\rho-\rho_{c}\|_{1}\leq(1-p)c\,$ for any input state $\rho$.

Another example for which upper bound (38) gives asymptotically sharp estimates for the Holevo capacity is the erasure channel

[TABLE]

from $d$-dimensional quantum system to its $(d+1)$-dimensional extension, since in this case $\,C_{\chi}(\Psi_{p})=(1-p)\log d\,$ and $\,r_{\Psi_{p}}=(1-p)(1-1/d)$.

The following example shows that accuracy of the upper bound (38) varies significantly within one class of channels.

Example 8. Let $\Phi:A\rightarrow B$ be a quantum channel such that the set $\Phi(\mathfrak{S}(\mathcal{H}_{A}))$ coincides with the convex hull of a set $\mathfrak{S}_{0}$ of isomorphic states in $\mathfrak{S}(\mathcal{H}_{B})$ and contains the chaotic state $\rho_{c}\doteq I_{\mathcal{H}_{B}}/d_{B}$, where $d_{B}=\dim\mathcal{H}_{B}$ (for example, $\Phi$ is the channel $\rho\mapsto\sum_{k}\langle\varphi_{k}|\rho|\varphi_{k}\rangle\sigma_{k}$, where $\{|\varphi_{k}\rangle\}$ is an orthonormal basis in $\mathcal{H}_{A}$ and $\{\sigma_{k}\}$ is a collection of isomorphic states in $\mathfrak{S}(\mathcal{H}_{B})$ such that $\,\rho_{c}=\sum_{k}p_{k}\sigma_{k}\,$ for some probability distribution $\{p_{k}\}$). Then $C_{\chi}(\Phi)=\log d_{B}-H_{\mathrm{min}}(\Phi)$, where $H_{\mathrm{min}}(\Phi)=H(\sigma),\;\sigma\in\mathfrak{S}_{0}$.

We will show that accuracy of the upper bound (38) strongly depends on the form of spectrum of the states in $\mathfrak{S}_{0}$.

Assume first that all the states in $\mathfrak{S}_{0}$ have the spectrum

[TABLE]

where $d=d_{B}$ and $r<d-1$. In this case $H_{\mathrm{min}}(\Phi)=(r/d)\log d+\eta(1-r/d)$ and hence

[TABLE]

while upper bound (38) implies

[TABLE]

since $\,\|\sigma-\rho_{c}\|_{1}=2(d-r-1)/d\,$ for all $\sigma\in\mathfrak{S}_{0}$. We see again that upper bound (38) gives asymptotically sharp estimate for the Holevo capacity for large $d$ and any $r$.

Now assume that all the states in $\mathfrak{S}_{0}$ are proportional to $r$-rank projectors. Then

[TABLE]

while the upper bound (38) implies

[TABLE]

So, in this case the upper bound (38) gives too rough estimate for the Holevo capacity.

3.2 Infinite-dimensional channels with energy constraints

The Holevo capacity of an infinite-dimensional quantum channel $\Phi:A\rightarrow B$ with energy constraint can be defined as follows

[TABLE]

where $H_{A}$ is the Hamiltonian of the system $A$, the supremum is over all generalized input ensembles $\mu$ with the average energy $\bar{E}(\mu)\doteq\mathrm{Tr}H_{A}\bar{\rho}(\mu)$ not exceeding $E$ and $\Phi(\mu)$ is the image of $\mu$ under the channel $\Phi$ (defined as the measure $\mu\circ\Phi^{-1}$ on $\mathfrak{S}(\mathcal{H}_{B})$). In fact, the supremum in (39) can be taken only over discrete ensembles [10]. This quantity determines the ultimate rate of transmission of classical information trough the channel $\Phi$ under the constraint on mean energy of a code if only non-entangled input encoding is used [9, Ch.12].

For given channel $\Phi:A\rightarrow B$ and state $\sigma$ in $\,\mathfrak{S}(\mathcal{H}_{B})$ introduce the quantity

[TABLE]

which can be called output metric divergence of $\Phi$ with respect to $\sigma$.

Assume that the Hamiltonian $H_{B}$ of the system $B$ satisfies condition (23). Denote by $E_{X}(\rho)$ the energy $\mathrm{Tr}H_{X}\rho$ of a state $\rho$ in $\mathfrak{S}(\mathcal{H}_{X})$, $X=A,B$.

Proposition 4.  Let $\Phi:A\rightarrow B$ be a quantum channel and $\sigma$ a state in $\,\mathfrak{S}(\mathcal{H}_{B})$ with finite energy $E_{B}(\sigma)$. Let $\,\varepsilon=D(\Phi|\hskip 1.0pt\sigma)$ be the output metric divergence of $\,\Phi$ with respect to $\sigma$ defined in (40). If $\,E_{*}=\sup\limits_{E_{A}(\rho)\leq E}E_{B}(\Phi(\rho))\,$ is finite then

[TABLE]

*where $\,a=1/(2\varepsilon)$, $\widehat{F}_{H_{B}}$ is any upper bound for the function $F_{H_{B}}$ (defined in (24)) satisfying conditions (25) and $\,\kappa=\frac{1}{2}(1+E_{B}(\sigma)/E_{*})$.*121212$h_{2}(p)$ is the binary entropy, $g(p)=(1+p)h_{2}\!\left(\frac{p}{1+p}\right)=(p+1)\log(p+1)-p\log p$.

If $\,B$ is the $\,\ell$-mode quantum oscillator and $\,\widehat{F}_{H_{B}}=\widehat{F}_{\ell,\omega}$131313The function $\widehat{F}_{\ell,\omega}$ is defined in (35). then the above upper bound for $C_{\chi}(\Phi,H_{A},E)$ is tight (for large $E$ and optimal choice of $\,\sigma$).

Proof. The main assertion of the proposition directly follows from Proposition 2.2 and definition (39) of the Holevo capacity.

The last assertion follows from Example 3.2 below. $\square$

Example 9. Let $A=B$ be the $\,\ell$-mode quantum oscillator. Consider the channel

[TABLE]

where $\sigma$ is a given state with finite energy $E(\sigma)$ and $p\in[0,1]$.

By using joint convexity of the relative entropy, concavity of the von Neumann entropy and inequality (4) one can show that

[TABLE]

where $\gamma_{A}(E)$ is the Gibbs states of the system $A=B$ corresponding to the energy $E$.

In this case $D(\Phi^{\sigma}_{p}|\hskip 1.0pt\sigma)\leq(1-p)$ and $E_{*}=(1-p)E+pE(\sigma)$. Assume for simplicity that $E(\sigma)\leq E$. Then $E_{*}\leq E$ and Proposition 3.2 with $\,\widehat{F}_{H_{B}}=\widehat{F}_{\ell,\omega}$ gives the upper bound

[TABLE]

where $\,a=\frac{1}{2(1-p)}$, $\,\bar{p}=1-p$ and $\,f_{p}(t)=(1+t)/(1-(1-p)t)$. By Lemma 2.2 the right hand side of (42) is equal to

[TABLE]

Since $\widehat{F}_{\ell,\omega}(E)-H(\gamma_{A}(E))=o(1)$ as $E\rightarrow+\infty$, comparing this with (41) we see that the upper bound (42) is tight for large $E$. $\square$

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