Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence

Abstract

Quantum information decoupling is a fundamental quantum information processing task, which also serves as a crucial tool in a diversity of topics in quantum physics. In this paper, we characterize the reliability function of catalytic quantum information decoupling, that is, the best exponential rate under which perfect decoupling is asymptotically approached. We have obtained the exact formula when the decoupling cost is below a critical value. In the situation of high cost, we provide meaningful upper and lower bounds. This result is then applied to quantum state merging, exploiting its inherent connection to decoupling. In addition, as technical tools, we derive the exact exponents for the smoothing of the conditional min-entropy and max-information, and we prove a novel bound for the convex-split lemma. Our results are given in terms of the sandwiched Rényi divergence, providing it with a new type of operational meaning in characterizing how fast the performance of quantum information tasks approaches the perfect.

Boundedness of the Upper Bound \(E_u(r)\)

Proposition 21

Let $\rho _{RA}\in \mathcal {S}(RA)$ be given. The function $E_u(r)=\sup \limits _{s\ge 0}\big \{s\big (r-\frac{1}{2}I_{1+s}(R:A)_\rho \big )\big \}$ is bounded in the interval $(-\infty , \frac{1}{2}I_\textrm{max}(R:A)_\rho )$.

Proof

We show that there is a constant C, such that for any $\epsilon >0$, $E_u\big (\frac{1}{2}I_\textrm{max}(R:A)_\rho -\epsilon \big )\le C$. Proposition 16 establishes that

$$\begin{aligned} E_u(r)=\lim _{n\rightarrow \infty }\frac{-1}{2n}\log {\text {Tr}}\left[ \mathcal {E}^n\big (\rho _{AB}^{\otimes n}\big ) \Big \{\mathcal {E}^n\big (\rho _{AB}^{\otimes n}\big )\ge 2^{n\cdot 2r} \rho _A^{\otimes n}\otimes \omega ^{(n)}_{B^n}\Big \} \right] , \end{aligned}$$

(A1)

where $\omega ^{(n)}_{B^n}$ is the universal symmetric state of Lemma 1 and $\mathcal {E}^n$ is the pinching map associated with $\rho _A^{\otimes n}\otimes \omega ^{(n)}_{B^n}$. Setting $M_A=\rho _A$ and letting $s\rightarrow \infty$ in Lemma 17, we get

$$\begin{aligned} \frac{1}{n}D_\textrm{max}\Big (\mathcal {E}^n(\rho _{AB}^{\otimes n})\big \Vert \rho _A^{\otimes n} \otimes \omega ^{(n)}_{B^n}\Big ) = I_\textrm{max}(A:B)_\rho +O\Big (\frac{\log n}{n}\Big ). \end{aligned}$$

This implies that, for arbitrary $\epsilon >0$, there exists a common eigenvector $|\varphi _n\rangle$ of $\mathcal {E}^n(\rho _{AB}^{\otimes n})$ and $\rho _A^{\otimes n}\otimes \omega ^{(n)}_{B^n}$, such that for n big enough,

$$\begin{aligned} \Big \{\mathcal {E}^n\big (\rho _{AB}^{\otimes n}\big )\ge 2^{n(I_\textrm{max}(A:B)_\rho -2\epsilon )}\rho _A^{\otimes n}\otimes \omega ^{(n)}_{B^n}\Big \} \ge | \varphi _n\rangle \!\langle \varphi _n | \end{aligned}$$

(A2)

and

$$\begin{aligned} \begin{aligned} \langle \varphi _n|\mathcal {E}^n\big (\rho _{AB}^{\otimes n}\big )|\varphi _n\rangle&\ge 2^{n(I_\textrm{max}(A:B)_\rho -2\epsilon )} \langle \varphi _n|\rho _A^{\otimes n}\otimes \omega ^{(n)}_{B^n}|\varphi _n\rangle \\&\ge 2^{n(I_\textrm{max}(A:B)_\rho -2\epsilon )} \lambda _\textrm{min}\big (\rho _A^{\otimes n}\otimes \omega ^{(n)}_{B^n}\big )\\&\ge 2^{n(I_\textrm{max}(A:B)_\rho -2\epsilon )} \big (\lambda _\textrm{min}(\rho _A)\big )^n\frac{1}{g_{n,|B|}\,|B|^n}. \end{aligned}\end{aligned}$$

(A3)

In Eq. (A3), we have used the fact $g_{n,|B|}\omega ^{(n)}_{B^n}\ge (\frac{\mathbbm {1}_B}{|B|})^{\otimes n}$ and $\lambda _\textrm{min}(X)$ denotes the minimal eigenvalue of X. Combining Eq. (A1), Eq. (A2) and Eq. (A3), we obtain

$$\begin{aligned} \begin{aligned} E_u\big (\frac{1}{2}I_\textrm{max}(A:B)_\rho -\epsilon \big )&\le \lim _{n\rightarrow \infty }\frac{-1}{2n}\log \langle \varphi _n|\mathcal {E}^n\big (\rho _{AB}^{\otimes n}\big )|\varphi _n\rangle \\&\le \frac{1}{2}\log \frac{|B|}{\lambda _\textrm{min}(\rho _A)} -\frac{1}{2}I_\textrm{max}(A:B)_\rho +\epsilon . \end{aligned}\end{aligned}$$

(A4)

At last, since $E_u(r)$ is monotonically increasing, we can choose $C=\frac{1}{2}\log \frac{|B|}{\lambda _\textrm{min}(\rho _A)} -\frac{1}{2}I_\textrm{max}(A:B)_\rho$. $\square$