Reverse Coherent Information

In this Letter we define a family of entanglement distribution protocols assisted by feedback classical communication that gives an operational interpretation to reverse coherent information, i.e., the symmetric counterpart of the well-known coherent information. This leads to the definition of a new entanglement distribution capacity that exceeds the unassisted capacity for some interesting channels.

Figure 1

Relations between the quantum communication and entanglement distribution capacities. We first start by two general remarks: (I) Being able to send a noiseless qubit is a stronger resource than distributing units of entanglement ($e$-bits): ${\mathcal{E}}_x\ge {\mathcal{Q}}_x$ for all $x$. (II) Increasing the complexity of the assistance cannot decrease the capacity: $\mathcal{X}\le {\mathcal{X}}_{\leftarrow }\le {\mathcal{X}}_{\leftrightarrow }$. The following remarks concern their corresponding number on the figure. (1) The equality $\mathcal{E}=\mathcal{Q}={\mathcal{Q}}_{\to }$ was shown in 12. (2) Any entanglement distribution protocol with free forward classical communication can be transformed into a quantum communication protocol by appending teleportation to it. (3) Results from combining 1 and 2. (4) Combining 1, 3 and II. (5) It is easy to prove that ${\mathcal{E}}_{\leftarrow }={\mathcal{Q}}_{\leftrightarrow }$ for the erasure channel [13, 14]. In 14 it was shown that the erasure channel satisfies the strict inequality ${\mathcal{Q}}_{\leftarrow }<{\mathcal{Q}}_{\leftrightarrow }$, which gives ${\mathcal{E}}_{\leftarrow }\ne {\mathcal{Q}}_{\leftarrow }$.

Figure 2

The first round of the entanglement distribution protocol assisted by classical feedback consists of three steps: (i) Alice sends qubit $A_1$ through the quantum channel $\Lambda$; (ii) Bob applies an incomplete quantum measurement ${\mathcal{B}}_1$ over his received qubit $B_1$ and communicates the outcome $b_1$ to Alice; (iii) Alice applies a global quantum operation ${\mathcal{A}}_1^{b_1}$ over the joint system of $R$ and the remaining $n-1$ qubits $A_i$. The next rounds are straightforward extensions of the first one.

Figure 3

A simplification of the general entanglement distribution protocol assisted by classical feedback (Fig. 2) limits the protocol to a last single round of processing. After Alice has sent all her qubits ($A_1{A}_2\dots A_n$) through the quantum channel $\Lambda$, Bob applies a collective incomplete measurement $\mathcal{B}$ among all the qubits $B_1{B}_2\dots B_n$ and communicates the classical outcome $b$ to Alice. Finally, conditioned on the message $b$, Alice applies the quantum operation ${\mathcal{A}}^b$ on system $R$.

Figure 4

(a) Comparison of $\mathcal{E}({\mathcal{D}}_{\eta })$ (dashed line) and ${\mathcal{E}}_R({\mathcal{D}}_{\eta })$ (solid line) as functions of the damping parameter $\eta$ for the amplitude damping channel, together with the capacity ${\mathcal{E}}^{(1)}({\mathcal{D}}_{\eta })={\mathcal{E}}_R({\mathcal{D}}_{\eta })$ of the generalized amplitude damping channel with a maximally mixed environment ($\alpha =1/2$) (dotted line). (b) Optimal input population $p$ achieving the previous capacities.

Figure 5

Quantum circuit corresponding to the Stinespring’s dilation of the generalized amplitude damping channel ${\mathcal{D}}_{(\eta ,\alpha )}$. Alice’s input state ${\rho }_{\mathrm{in}}$ and half of an entangled state $|{\Psi }_{\alpha }⟩$ interact through the relaxation operation $U_{\mathrm{RO}}$ composed of two CNOT gates and a controlled rotation around the $y$ axis of the Bloch sphere [${cos}^2(\gamma /2)=\eta$].

Figure 6

(a) Tolerable thermal noise of the generalized amplitude damping channel ${\mathcal{D}}_{(\eta ,\alpha )}$ (minimum $\alpha$ such that the capacity is zero) as a function of the damping parameter $\eta$ for: ${\mathcal{E}}^{(1)}({\mathcal{D}}_{(\eta ,\alpha )})$ (dashed line), and ${\mathcal{E}}_R({\mathcal{D}}_{(\eta ,\alpha )})$ (solid line). (b) Input population $p$ achieving the curves of (a).