Entanglement recycling in two-step port-based teleportation

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.

💡 Research Summary

The paper investigates a novel “two‑step” port‑based teleportation (PBT) protocol in which the same entangled resource—composed of N maximally entangled “ports”—is reused for two successive teleportations. The authors formalize the protocol, quantify its performance, and analyze the degradation of the resource after the first use.

In the first step, Alice performs a joint measurement on her message register and all N ports. Using a pretty‑good measurement (PGM) or its probabilistic variant, she obtains an outcome i (i≠0 for success). The state is then teleported to Bob’s i‑th port, and a SWAP operation removes the used port from further use, leaving N‑1 ports for the second step. In the second step Alice repeats the same type of measurement on the remaining ports and teleports a second quantum state. The overall process defines a two‑step quantum channel 𝒩_N.

Two figures of merit are introduced: (i) the entanglement fidelity F_ent(N), which measures the average fidelity of the joint output state with respect to the ideal output, and (ii) the average success probability p_succ(N), which accounts for the fact that the protocol may be probabilistic (p_succ<1). The authors derive closed‑form expressions for both quantities when the PGM is employed.

Theorem 1 shows that the (unnormalized) entanglement fidelity can be written as
 F_ent(N)= (1/d⁴) vᵀ M v,
where M is the familiar “teleportation matrix” from earlier PBT literature and the vector v encodes the choice of the resource state on Alice’s side. In the deterministic (inexact) regime the channel is trace‑preserving, so no normalization is needed; optimizing the resource state simply amounts to maximizing the Rayleigh quotient vᵀ M v, i.e., taking the largest eigenvalue of M.

Theorem 2 (informal version of Theorem 9) states that the average success probability of the two‑step probabilistic PBT coincides with the optimal multi‑port probabilistic scheme. Consequently, reusing the same resource does not degrade the overall success probability compared with the best known multi‑port protocols.

Beyond these performance metrics, the paper introduces a “recycling fidelity” F_rec(N,d) to quantify how much the resource state deteriorates after the first teleportation. F_rec is defined as the weighted average overlap between the post‑teleportation resource (together with the half of the EPR pair that has been teleported) and an ideal resource that would be perfectly suited for a second PBT round. The authors decompose F_rec into contributions from the failure and success branches of the first step. In the asymptotic limit N→∞ the failure probability vanishes, so the success branch dominates.

Theorem 3 proves that when the standard EPR resource (N copies of |Φ⁺⟩) is used, the degradation after one probabilistic PBT round is negligible for large N; thus the same EPR resource can be recycled for a second round with only a tiny loss in fidelity. This demonstrates that two‑step PBT works not only for the PGM but also for other POVMs that are optimal for the EPR case.

Theorem 4 shows a contrasting behavior for an optimized (non‑EPR) resource. After one probabilistic PBT round the recycling fidelity does not improve with increasing N; in fact it remains significantly lower than unity, indicating that the resource has been substantially altered and is no longer suitable for a second high‑fidelity teleportation. This highlights a fundamental trade‑off: resources that are optimal for a single teleportation may be poor candidates for reuse.

The technical backbone of the analysis relies on representation theory of the partially transposed permutation algebra and mixed Schur–Weyl duality. By exploiting the structure of the algebra A_{d,N,2} and its generators (partial transpositions and contractions), the authors obtain explicit forms for the matrix M and for the eigenvectors that define the optimal resource. The PGM’s optimality for one‑step PBT (proved in earlier works) is leveraged to extend the results to the two‑step scenario.

Overall, the paper delivers three major contributions:

1. Exact performance formulas for entanglement fidelity and success probability of two‑step PBT with PGM measurements, showing that the protocol attains essentially the same performance as the best known multi‑port schemes.

2. A rigorous analysis of resource degradation, introducing the recycling fidelity and demonstrating that EPR resources can be recycled efficiently, whereas optimized resources cannot.

3. A clear methodological framework based on algebraic representation theory that simplifies the otherwise cumbersome calculations typical of PBT analyses.

These results have practical implications for quantum networks where entangled resources are scarce. In scenarios where many teleportations are required, using standard EPR ports and reusing them after each use may be preferable to employing highly optimized but single‑use resources. Future work suggested includes extending the analysis to more than two steps, investigating noisy channels, and implementing the protocol in experimental platforms.