Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem

TL;DR

The paper proves that the entanglement-assisted classical capacity of a quantum channel is CE(N) = max_rho [ H(rho) + H(N(rho)) - H((N ⊗ I) Φ_rho) ], establishing achievability and converse via typical subspace arguments and Holevo bounds. It then evaluates CE for Gaussian bosonic channels under energy constraints and for the qubit amplitude-damping channel, illustrating when entanglement boosts capacity and how optimal inputs behave. A classical reverse Shannon theorem is presented, showing that any DMC of capacity C can be simulated by C bits of forward communication given shared randomness, with exact asymptotic fidelity; this classical result motivates the quantum reverse Shannon conjecture (QRSC). The QRSC posits that, with unlimited entanglement, all quantum channels of equal C_E are asymptotically equivalent in capacity, a simplification that would dramatically unify quantum channel behavior and has implications for feedback in quantum communication. Together, these results deepen understanding of how entanglement and ancillary resources shape quantum channel capacities and channel simulability.

Abstract

The entanglement-assisted classical capacity of a noisy quantum channel is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel, assuming that the sender and receiver have access to the resource of shared quantum entanglement, which may be used up by the communication protocol. We show that this capacity is given by an expression parallel to that for the capacity of a purely classical channel: i.e., the maximum, over channel inputs $ρ$, of the entropy of the channel input plus the entropy of the channel output minus their joint entropy, the latter being defined as the entropy of an entangled purification of $ρ$ after half of it has passed through the channel. We calculate entanglement-assisted capacities for two interesting quantum channels, the qubit amplitude damping channel and the bosonic channel with amplification/attenuation and Gaussian noise. We discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel, alone or in the presence of ancillary resources such as prior entanglement. In the classical analog of entanglement assisted communication--communication over a discrete memoryless channel (DMC) between parties who share prior random information--we show that one parameter is sufficient, i.e., that in the presence of prior shared random information, all DMC's of equal capacity can simulate one another with unit asymptotic efficiency.

Figures (7)

• Figure 1: A quantum system Q in mixed state $\rho$ is sent through the noisy channel ${\cal N}$, which may be viewed as a unitary interaction $U$ with an environment E. Meanwhile a purifying reference system R is sent through the identity channel ${\cal I}$. The final joint state of RQ has the same entropy as the final state ${\cal E}(\rho)$ of the environment.
• Figure 2: In Lemma \ref{['show-unitary']}, $A$ is the input space for the original map ${\cal N}$. $A \cup A'$ is the input space for the map ${\cal N}'$. The output space for both maps is $B$. The space $R$ is a reference system used to purify states in $A$ and $A'$.
• Figure 3: For Lemma \ref{['concavity-lemma']}, $A$ is a Hilbert space we send through the channel ${\hat{\cal N}}$, and $B$ is the output space. This mapping $\hat{\cal N}$ can be made unitary by adding an environment space $E$. We let $R$ be a reference system which purifies the systems $\rho_0$ and $\rho_1$ in $A$, and $C_1$ and $C_2$ be two qubits purifying $AR$ as described in the text.
• Figure 4: This figure shows the curves given by the ratio of capacities $C_E/C_{\rm{Shan}}$ for the quantum Gaussian channel with noise $N$ and the nine combinations of values: amplification/attenuation parameter $k=0.1$, $1$, or $3$; and signal strength $S=0.1$, $1$, or $10$. The dotted curves have $S=0.1$; the solid curves have $S=1$; and the dashed curves have $S=10$. Within each set, the curves have the values $k=0.1$, $k=1$, and $k=3$ from bottom to top.
• Figure 5: The solid curves show the ratio of capacities $C_E/C_{\mathrm{Shan}}$ for the quantum Gaussian channel with signal strength $S$, amplification/attenuation paramter $k=1$ and noise $N=0.1$, $0.3$, $1$, $3$, and $10$ (from bottom to top). The dashed curve is the limit of the solid curves as $N$ goes to $\infty$; namely, $C_E/C_{\mathrm{Shan}} = (S+1)\log(1+1/S)$. These curves approach $\infty$ as $S$ goes to $0$, and approach $1$ as $S$ goes to $\infty$.

Theorems & Definitions (8)

• Theorem 1
• Lemma 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 2: Classical Reverse Shannon Theorem
• Corollary 1: Efficient simulation of one noisy channel by another
• Corollary 2: Efficient simulation of noisy channels on constrained sources