Uniform finite-dimensional approximation of basic capacities of energy-constrained channels

Abstract

We consider energy-constrained infinite-dimensional quantum channels from a given system (satisfying a certain condition) to any other systems. We show that dealing with basic capacities of these channels we may assume (accepting arbitrarily small error $\varepsilon$) that all channels have the same finite-dimensional input space—the subspace corresponding to the $m(\varepsilon )$ minimal eigenvalues of the input Hamiltonian. We also show that for the class of energy-limited channels (mapping energy-bounded states to energy-bounded states) the above result is valid with substantially smaller dimension $m(\varepsilon )$. The uniform finite-dimensional approximation allows us to prove the uniform continuity of the basic capacities on the set of all quantum channels with respect to the strong (pointwise) convergence topology. For all the capacities, we obtain continuity bounds depending only on the input energy bound and the energy-constrained diamond-norm distance between quantum channels (generating the strong convergence on the set of quantum channels).

Appendix

The following lemma is the QCMI version of Lemma 7 in [5].¹⁵

Lemma 7

Let $\,{\Phi }$ and $\,{\Psi }$ be quantum channels from A to B satisfying condition (44), C, D any systems, $n\in {\mathbb {N}}$ and $\,\rho$ a state in $\,S(\mathcal {H}^{\otimes n}_{A}\otimes \mathcal {H}_{CD})$ such that $\,\sum _{k=1}^n\mathrm {Tr}H_A\rho _{A_k}\le nE<+\infty$. If $\,\frac{1}{2}\Vert {\Phi }-{\Psi }\Vert _{\diamond }^E\le \varepsilon$ then¹⁶

$$\begin{aligned}&(1/n)\left| I(B^n:D|C)_{{\Phi }^{\otimes n}\otimes \mathrm {Id}_{CD}(\rho )}-I(B^n:D|C)_{{\Psi }^{\otimes n}\otimes \mathrm {Id}_{CD}(\rho )}\right| \nonumber \\&\quad \le \displaystyle (4t+2r(t,\varepsilon ))\widehat{F}_{H_{B}}\left( \frac{E_p}{t}\right) +2g\left( r(t,\varepsilon )\right) +4h_2(t)+ \frac{2}{p}\widehat{F}_{H_{B}}(E_p) \end{aligned}$$

(52)

for any $\,p\ge 1$ and $\,t\in (0,\frac{1}{2}]$, where $E_p=\alpha p E+E_c$, $r(t,\varepsilon )=\frac{\varepsilon +t/2}{1-t}$, and $\,\widehat{F}_{H_B}$ is any function with properties (45), (46) and (47).

If $\,\mathrm {Tr}H_A\rho _{A_k}\le E\,$ for all $\,k=\overline{1,n}\,$ then (52) holds with $p=1$ without the last term in the right-hand side.

Proof

Denote by ${\Delta }^n({\Phi },{\Psi },\rho )$ the left-hand side of (52). By the proof of Proposition 12B in [18] (based on the Leung–Smith telescopic method), we have

$$\begin{aligned} n{\Delta }^n({\Phi },{\Psi },\rho )\le \sum _{k=1}^{n}|I(B_k:D|X)_{\sigma _k}-I(B_k:D|X)_{\sigma _{k-1}}|, \end{aligned}$$

where $X=B_1\ldots B_{k-1}B_{k+1}\ldots B_nC$ and $\sigma _k={\Phi }^{\otimes k}\otimes {\Psi }^{\otimes (n-k)}\otimes \mathrm {Id}_{CD}(\rho )$, $k=0,1,\ldots ,n$. The proof of Proposition 12B in [18] also implies that

$$\begin{aligned} \Vert \sigma _k-\sigma _{k-1}\Vert _1\le \sup \left\{ \Vert ({\Phi }-{\Psi })\otimes \mathrm {Id}_R(\omega )\Vert _1\,|\;\omega _{A}=\rho _{A_k}\right\} \le \Vert {\Phi }-{\Psi }\Vert _{\diamond }^{x_k}, \end{aligned}$$

(53)

where $x_k=\mathrm {Tr}H_A\rho _{A_k}$.

Since $[\sigma _k]_{B_k}={\Phi }(\rho _{A_k})$ and $[\sigma _{k-1}]_{B_k}={\Psi }(\rho _{A_k})$, we have

$$\begin{aligned} \mathrm {Tr}H_B[\sigma _k]_{B_k}, \mathrm {Tr}H_B[\sigma _{k-1}]_{B_k}\le \alpha x_k + E_c. \end{aligned}$$

(54)

Let $N_1$ be the set of indexes k for which $x_k\le p E$ and $N_2=\{1,..,n\}\setminus N_1$. Thus,

$$\begin{aligned} n{\Delta }^n({\Phi },{\Psi },\rho )\le \sum _{k\in N_1}D_k+\sum _{k\in N_2}D_k,\quad D_k=|I(B_k:D|X)_{\sigma _k}-I(B_k:D|X)_{\sigma _{k-1}}|. \end{aligned}$$

For each $k\in N_1$ Proposition 5 in [18] along with (53) and (54) implies

for any $\,t_k\in (0,\frac{1}{2\varepsilon _k}]$, where $\varepsilon _k=\frac{1}{2}\Vert {\Phi }-{\Psi }\Vert ^{x_k}_{\diamond }$. By choosing free parameters $t_k$ such that $\,\varepsilon _kt_k=t\,$ for all $k\in N_1$ we obtain

where $n_1=\sharp (N_1)$ and $\bar{\varepsilon }_1\doteq n_1^{-1}\sum _{k\in N_1}\varepsilon _k$. The last inequality follows from monotonicity of the function $\widehat{F}_{H_B}$ (since $x_k\le pE$ for all $k\in N_1$) and concavity of the function g(x).

By using monotonicity and concavity of the function $E\mapsto \Vert {\Phi }\Vert ^E_{\diamond }$ (proved in [5]) it is easy to show that $\bar{\varepsilon }_1\le \frac{1}{2}\Vert {\Phi }-{\Psi }\Vert _{\diamond }^E\le \varepsilon$. So, by monotonicity of g(x) we have

$$\begin{aligned} n^{-1}\sum _{k\in N_1}D_k\le \left( 4t+\frac{2\varepsilon +t}{1-t}\right) \widehat{F}_{H_B}\left( \frac{\alpha pE +E_c}{t}\right) +2g\left( \frac{\varepsilon +t/2}{1-t}\right) +4h_2(t). \end{aligned}$$

For each $k\in N_2$ upper bound (18), the nonnegativity of QCMI and inequalities (54) imply that

$$\begin{aligned} D_k\le 2\max \{H([\sigma _k]_{B_k}), H([\sigma _{k-1}]_{B_k})\}\le 2\widehat{F}_{H_B}(\alpha x_k + E_c). \end{aligned}$$

So, by concavity of $\widehat{F}_{H_B}$ we have

$$\begin{aligned} \sum _{k\in N_2}D_k\le 2\sum _{k\in N_2}\widehat{F}_{H_B}(\alpha x_k + E_c)\le 2n_2\widehat{F}_{H_B}(\alpha X_2 + E_c), \end{aligned}$$

where $n_2=\sharp (N_2)$ and $X_2=n_2^{-1}\sum _{k\in N_2}x_2$. Since $\sum _{k\in N_2}x_k\le nE$ and $x_k> pE$ for all $k\in N_2$, we have $X_2\le nE/n_2$ and $n_2/n\le 1/p$. By using monotonicity of $\widehat{F}_{H_B}$ and applying Lemma 1 to the concave nonnegative function $x\mapsto \widehat{F}_{H_B}(\alpha x + E_c)$ on $\mathbb {R}_+$ we obtain

$$\begin{aligned} n^{-1}\sum _{k\in N_2}D_k\le 2(n_2/n)\widehat{F}_{H_B}(\alpha (n/n_2) E + E_c)\le (2/p)\widehat{F}_{H_B}(\alpha p E + E_c). \end{aligned}$$

This and the above estimate for $n^{-1}\sum _{k\in N_1}D_k$ imply (52).

The last assertion of the lemma follows from the above arguments with $p=1$, since in this case the set $N_2$ is empty. $\square$