Optimal Quantum Information Transmission Under a Continuous-Variable Erasure Channel

Abstract

Quantum capacity gives the fundamental limit of information transmission through a channel. However, evaluating the quantum capacities of a continuous-variable bosonic quantum channel, as well as finding an optimal code to achieve the optimal information transmission rate, is in general challenging. In this work, we derive the quantum capacity and entanglement-assisted quantum capacity of the bosonic continuous-variable erasure channel when subject to energy constraints. We then construct random codes based on scrambling information within the typical subspace of the encoding state and prove that these codes are asymptotically optimal up to a constant gap. Finally, using our random coding scheme we design a bosonic variation of the Hayden-Preskill protocol and find that information recovery depends on the ratio between the input and output modes. This is in contrast with the conventional discrete-variable scenario which requires only a fixed number of additional output qudits.

Figures (2)

• Figure 1: Random coding scheme for entanglement transmission. Alice begins with $K$ copies of TMSVs, keeping one mode per pair in the reference $R$ and placing the other in $A_1$. She introduces an additional $N-K$ modes in $A_2$ and applies a random unitary $U$ on $A=A_1A_2$. Each mode is then sent through the single-mode CV erasure channel $\Lambda_p$. Bob uses the classical flags to retain the $\approx(1-p)N$ unerased modes ($B$), while the erased modes go to the environment ($E$). In the standard scheme, $A_2$ consists of random coherent states, whereas in the entanglement-assisted scheme (dashed orange) Alice and Bob instead share $N-K$ copies of TMSVs across $A_2\bar{B}$, with Bob accessing $\bar{B}$ during decoding. When considering the black hole toy model, $A_2$ corresponds with the initial black hole, $A_1$ is the information Alice throws in, $U$ is the scrambling dynamics of the black hole, $B$ is the emitted Hawking radiation and $E$ is the residual black hole.
• Figure 2: (a) The quantum capacities as a function of erasure probability. (b) The maximum achievable rate of the random codes in the (i) standard and (ii) entanglement-assisted cases for up to $\bar{n} = 10$. The dotted curves are the associated capacities. (c) The maximum erasure probability before no information transmission is possible, $p_*$ against the energy constraint $\bar{n}$ for our codes. They asymptotically approach $p_*=0.5$ and $p_* = 1$ in the standard and entanglement-assisted cases. (d) The constant functions (i) $c(p, q)$ and (ii) $c_{\textrm{EA}}(p, q)$ as a function of $p$ for different values of $q$. The $q = 0.368$ and $q = 0.271$ curves in (i) and (ii) are the values of $q$ for which $c$ is maximised, and $q_{\textrm{optm}}$ curve is the value of $q$ that maximises the rate (i.e., the black dotted curves in (d)(i) and (d)(ii) correspond to the energy-independent gap in the plots (b)(i) and (b)(ii) respectively).