Simultaneous superadditivity of the direct and complementary channel capacities

TL;DR

This work demonstrates that the coherent and private information of a channel and its complement can be simultaneously superadditive for arbitrarily many channel uses, and quantifies the limits of this effect.

Abstract

Quantum communication channels differ from their classical counterparts because their capacities can be superadditive. The principle of monogamy of entanglement suggests that superadditive improvements in the transmission capacity of a channel should reduce the amount of information loss to the environment. We challenge this intuition by demonstrating that the coherent and private information of a channel and its complement can be simultaneously superadditive for arbitrarily many channel uses. To quantify the limits of this effect, we consider the notion of max (resp. total) private information of a channel, which represents the maximum (resp. sum) of the private information of the channel itself and its complement, and study its relationship with the coherent information of the individual direct and complementary channels. For a varying number of channel uses, we show that these quantities can obey different interleaving sequences of inequalities.

Figures (4)

• Figure 1: Plot of the log of the lower bounds $f_{n,p,\alpha}(k)$ and $f^c_{n,p,\alpha}(k)$ on the difference quantities $\mathcal{Q}^{(k+1)}( \mathcal{N}_{n,p,d}) - \mathcal{P}^{(k)}_{\max}( \mathcal{N}_{n,p,d})$ and $\mathcal{Q}^{(k+1)}( \mathcal{N}^c_{n,p,d}) - \mathcal{P}^{(k)}( \mathcal{N}^c_{n,p,d})$, respectively (see Theorem \ref{['th:main2']}). Here, $n=100$, $p=0.4$, and $\alpha= 3$.
• Figure 2: Plot of the lower bound $Q_{max} = L(n+1)$ on $\mathcal{Q}^{(n+1)}(\mathcal{N}_{n,p,d})$ and the upper bound $\mathcal{U}_k = U'(k) \,\,\quad \text{if } k\leq k_0U"(k) \quad \text{otherwise}$ on $\mathcal{P}^{(k)}_{\rm{tot}}(\mathcal{N}_{n,p,d})$ for $p=0.09, n=100$, and $\alpha=3$. The lower and upper bound functions are defined in Eqs. \ref{['eq:Lk']},\ref{["eq:U'k"]},\ref{['eq:U"k']}. Clearly, $Q_{max}$ exceeds the total private capacity for at least $9$ uses of the channel.
• Figure 3: Visual depiction of how the Rocket channel $R_d$ can be used to transmit information with the help of pre-shared entanglement. Alice and Bob start with a pre-shared maximally entangled state (shown in red). Alice locally prepares another maximally entangled state (shown in purple) and sends half of each of the entangled states through $R_d$ to Bob as shown. Hence, only the top two dangling wires are in Alice's possession while the bottom four are with Bob. Since Bob knows which local random unitaries $U,V$ are applied during the transmission, he can use the pre-shared entanglement with Alice to undo the phase coupling operation as described. Here, $P^{\Gamma}$ is the partial transpose of $P$ with respect to the second subsystem. The two parties finally end up sharing one maximally entangled state (shown in orange), which can be used to send quantum information at rate $\log d$.
• Figure 4: Visual depiction of how the complementary Rocket channel $R^c_d$ can be used to transmit information with the help of pre-shared entanglement. Alice and Bob start with a pre-shared maximally entangled state (shown in red). Alice locally prepares another maximally entangled state (shown in purple) and sends half of each of the entangled states through $R^c_d$ to Bob as shown. Hence, only the bottom two dangling wires are in Alice's possession while the top four are with Bob. Since Bob knows which local random unitaries $U,V$ are applied during the transmission, he can use the pre-shared entanglement with Alice to undo the phase coupling operation as described. Here, $P^{\Gamma_1}$ is the partial transpose of $P$ with respect to the first subsystem. The two parties finally end up sharing one maximally entangled state (shown in orange), which can be used to send quantum information at rate $\log d$.

Theorems & Definitions (12)

• Theorem 1
• proof
• Theorem 2
• Lemma A.1
• Lemma B.1
• proof
• Theorem B.1
• proof
• Theorem B.2
• proof