Entanglement recycling in two-step port-based teleportation

Abstract

A protocol involving the repetitive (twofold, to be precise) application of PBT protocol to the same resource is studied. The quantities characterizing the resulting protocol, so-called \textit{two-step PBT}, namely \textit{enatnglement fidelity} and \textit{success probability} are provided for two scenarios, relying on application of pretty-good measurement, i.e. deterministic and probabilistic PBT with non-EPR resource. This results show that two-step PBT is an accurate protocol, provided the resource is sufficiently large. In particular, the deterministic two-step PBT obtains fidelity that is remarkably close to the optimal MPBT fidelity for teleportation of two quantum states. Additionally, the \textit{recycling fidelity}, i.e. the quantity characterizing the degradation of the resource state is calculated for repetitive application of probabilistic protocol, for both EPR and optimized resource, showing that entanglement recycling with two-step PBT is possible in the former case as well.

Figures (6)

• Figure 1: Two-step PBT protocol at a glance. Initially, Alice and Bob share an optimized resource state $\lvert{\Psi}\rangle_{AB} = (O_A \otimes I_B) \lvert{\Psi^+}\rangle_{AB}$ where registers $A = \lbrace A_1, \dotsc, A_N\rbrace$ and $B = \lbrace B_1, \dotsc, B_N\rbrace$ consist of $N$ qudits each. Alice's goal is to teleport qudit states $\rho_{1}=\mathop{\mathrm{Tr}}\nolimits_{\bar{A}_2}\rho_{\bar{A}}$ and $\rho_{2}=\mathop{\mathrm{Tr}}\nolimits_{\bar{A}_1}\rho_{\bar{A}}$ stored in her registers $\bar{A}_1$ and $\bar{A}_2$. First, Alice performs a POVM $\Pi$ on $\bar{A}_1 \cup A$. If the measurement outcome is $i \neq 0$(i.e. in the deterministic inexact scheme, or in the case of success in probabilistic exact), the first state ${\rho_1}$ is teleported to Bob's register $B_i$ (we depict the case when $i = N$). After discarding $\lbrace\bar{A}_1, A_i, B_i\rbrace$, Alice and Bob share a resource consisting of $N-1$ remaining registers on each side. In the second round, Alice performs a POVM $\widetilde{\Pi}$ on her registers $\bar{A}_2 \cup (\{A_1,\ldots,A_N\} \setminus A_i)$. Measuring the outcome $j \neq 0$ in this round indicates success (we depict the case when $j = N-1$). We call this protocol two-step PBT.
• Figure 2: Entanglement fidelity of the channel $\mathcal{N}$ can written as a tensor network contraction by bending and rearranging the wires and tensors from \ref{['fig:protocol']}.
• Figure 3: Numerical values of $p_{\mathrm{succ}}$ of two consecutive successful pPBT teleportations employing a recycled resource, and the conditional entanglement fidelity $F_e(\mathcal{N})/p_{\mathrm{succ}}$ of the corresponding teleportation channel $\mathcal{N}$ for $d \in \lbrace2,3,4\rbrace$.
• Figure 4: Comparison of deterministic two-step recycling PBT scheme with optimal deterministic multi PBT for two systems for $d \in \lbrace2,3\rbrace$.
• Figure 5: Conditional recycling fidelity for success (left) and failure (right) in standard pPBT scheme, for $d=2$ (blue dots) and $d=3$ (orange dots).

Theorems & Definitions (19)

• Theorem 1
• Theorem 2: \ref{['thm:two_step_psucc']}, informal
• Theorem 3: $F_{\mathrm{rec}}$ for EPR resource
• Theorem 4: $F_{\mathrm{rec}}$ for optimised resource
• Lemma 5: ram1992matrix
• Lemma 6
• proof
• Lemma 7
• Theorem 8
• Theorem 9