Reliable Simulation of Quantum Channels: the Error Exponent

TL;DR

We study the reliability function $E^{\mathrm{sim}}(\mathcal{N},r)$ for reverse Shannon simulation of a quantum channel $\mathcal{N}_{A\rightarrow B}$ under channel purified distance, deriving both exponential achievability and one-shot converse bounds expressed via the sandwiched Rényi information $I_{1+s}(\mathcal{N})$. A de Finetti/symmetrization framework yields a finite-blocklength achievability bound, and the two bounds coincide for rates $r$ up to the critical threshold $R_{\mathrm{critical}}$, giving the exact low-rate exponent and connecting it to $I_{1+s}(\mathcal{N})$. The results provide an operational interpretation of $I_{\alpha}(\mathcal{N})$ for $\alpha\in(1,2]$ in the direct error exponent and extend the Quantum Reverse Shannon Theorem to finite blocklength and rate regimes relevant for practical quantum communication scenarios. These findings quantify how quickly perfect channel simulation is approached and have implications for entanglement-assisted quantum communication protocols.

Abstract

The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel's entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel's sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.

Figures (2)

• Figure 1: Reliability function of quantum channel simulation. $E_l(r):=\frac{1}{2} \max_{0 \leq s \leq 1}\{s(r-I_{1+s}(\mathcal{N}))\}$ is the lower bound of Eq. (\ref{['eq:reliability-l']}). $E_u(r):=\frac{1}{2} \sup_{s \geq 0}\{s(r-I_{1+s}(\mathcal{N}))\}$ is the upper bound of Eq. (\ref{['eq:reliability-u']}). The two bounds are equal in the interval $[0,R_\text{critical}]$, giving the exact reliability function. Above the critical value $R_\text{critical}$, the upper bound $E_u(r)$ increases faster and it diverges to infinity when $r>I_\text{max}(\mathcal{N}):=\lim\limits_{\alpha\rightarrow\infty}I_\alpha(\mathcal{N})$, while the lower bound $E_l(r)$ becomes linear and reaches $\log d-\frac{1}{2}I_2(\mathcal{N})$ at $r=2\log d$ with $d:=\min\{|A|,|B|\}$.
• Figure 2: Critical value and mutual information of the qubit depolarizing channel. Upper curve is $R_{\rm{critical}}(\mathcal{N}^{(p)})$, and lower curve is $I(\mathcal{N}^{(p)})$.

Theorems & Definitions (20)

• Definition 1: reverse Shannon simulation
• Definition 2: performance function
• Definition 3: reliability function
• Definition 4
• Definition 5
• Lemma 6
• Lemma 7
• Lemma 8
• Theorem 9
• Theorem 10