One-to-One Correspondence between Deterministic Port-Based Teleportation and Unitary Estimation

TL;DR

This work reveals a exact one-to-one correspondence between deterministic port-based teleportation (dPBT) with $N=n+1$ ports and $d$-dimensional unitary estimation with $n$ calls of the input unitary. By transforming covariant protocols between these tasks, it establishes $F_ ext{PBT}(n+1,d) = F_ ext{est}(n,d)$ and expresses the related matrices via a bridging map $R(n,d)$, yielding the asymptotic fidelity $F_ ext{PBT}(N,d) = 1- ext{Θ}(d^4/N^2)$ and the small-$n$ regime $F_ ext{est}(n,d) = (n+1)/d^2$ for $n\le d-1$. The results extend to a four-task equivalence among dPBT, unitary estimation, deterministic storage-and-retrieval, and deterministic parallel unitary inversion, enabling direct transfer of optimal strategies across these problems. The paper also proves that unitary inversion with $n\le d-1$ calls can be achieved optimally by estimation-based protocols even under general (including indefinite causal order) protocols, with fidelity $(n+1)/d^2$. This unification accelerates progress in dPBT and unitary estimation by allowing the wealth of known results in one domain to inform the other, and it refines the asymptotic scaling beyond previous bounds.

Abstract

Port-based teleportation is a variant of quantum teleportation, where the receiver can choose one of the ports in his part of the entangled state shared with the sender, but cannot apply other recovery operations. We show that the optimal fidelity of deterministic port-based teleportation (dPBT) using $N=n+1$ ports to teleport a $d$-dimensional state is equivalent to the optimal fidelity of $d$-dimensional unitary estimation using $n$ calls of the input unitary operation. From any given dPBT, we can explicitly construct the corresponding unitary estimation protocol achieving the same optimal fidelity, and vice versa. Using the obtained one-to-one correspondence between dPBT and unitary estimation, we derive the asymptotic optimal fidelity of port-based teleportation given by $1-O(d^4)N^{-2}\leq F \leq 1-Ω(d^4)N^{-2}$, which improves the previously known result given by $1-O(d^5)N^{-2} \leq F \leq 1-Ω(d^2) N^{-2}$. We also show that the optimal fidelity of unitary estimation for the case $n\leq d-1$ is $F = {n+1 \over d^2}$, and this fidelity is equal to the optimal fidelity of unitary inversion with $n\leq d-1$ calls of the input unitary operation even if we allow indefinite causal order among the calls.

Figures (2)

• Figure 1: This work shows the one-to-one correspondence between deterministic port-based teleportation (dPBT) and unitary estimation (see Thm. \ref{['thm:equivalence']}). Combining this result with Ref. bisio2010optimal and Ref. quintino2022deterministic, we also show the one-to-one correspondence with deterministic storage-and-retrieval (dSAR) of unitary operation and deterministic parallel unitary inversion. (a-1) In dPBT, Alice approximately teleports her unknown state $\rho$ by sending the measurement outcome $a$ of the joint measurement on the target state $\rho$ and half of a shared $2N$-qudit entangled state $\phi_\mathrm{PBT}$. Bob chooses the port $a$ of the other half to obtain a quantum state close to $\rho$. Its circuit description is shown in (a-2). By using the one-to-one correspondence, its asymptotic optimal fidelity is shown to be $F_\mathrm{PBT}(N,d) = 1-\Theta(d^4N^{-2})$ (see Cor. \ref{['cor:asymptotically_optimal_PBT']}). (b) In unitary estimation, $n$ calls of unitary operator $U$ are applied in parallel to the resource state $\phi_\mathrm{est}$, and the output state is measured in a positive operator-valued measure (POVM) measurement $\{M_{\hat{U}} \dd \hat{U}\}_{\hat{U}}$ to obtain the estimated unitary $\hat{U}$. By using the one-to-one correspondence, its optimal fidelity is shown to be $F_\mathrm{est}(n,d) = {n+1 \over d^2}$ for $n\leq d-1$ (see Cor. \ref{['cor:optimal_unitary_estimation_small_n']}). (c) In dSAR, $n$ calls of $U$ are applied into a resource state $\rho_\text{SAR}$ to obtain a program state $\psi_U$. In a later moment, the action of $U$ is retrieved by applying a quantum operation $\mathcal{D}$ on the joint system of the input state $\rho$ and the program state $\psi_U$. (d) In deterministic parallel unitary inversion, an operation $\mathcal{E}$ is applied before $n$ calls of an unknown unitary operator $U$, and an operation $\mathcal{D}$ is applied just after, in a way that the resulting composition is approximately $U^{-1}$. For $n\leq d-1$, the parallel protocol achieves the optimal fidelity even if we consider the most general protocol including the ones with indefinite causal order (see Thm. \ref{['thm:optimality_of_parallel_unitary_inversion']}).
• Figure 2: Explicit construction of a unitary estimation protocol from any given dPBT protocol and vice versa. (a) General protocol for dPBT. (b) A covariant protocol for dPBT is constructed by using Eqs. \ref{['eq:wmu']}, \ref{['eq:covariant_resource_state']} and \ref{['eq:srm']} to the general protocol (a) mozrzymas2018optimalleditzky2022optimality (see the details in Appendix \ref{['appendix_sec:covariant_PBT']}). (c) General adaptive protocol for unitary estimation. (d) A parallel covariant protocol for unitary estimation is constructed by using Eqs. \ref{['eq:def_of_probe_state']}, \ref{['eq:v_alpha']} and \ref{['eq:covariant_POVM']} to the general protocol (c) bisio2010optimal (see the details in Appendix \ref{['appendix_sec:covariant_unitary_estimation']}). This work shows a transformation between the covariant protocols for dPBT and unitary estimation given in Eqs. \ref{['eq:PBT_to_untiary_estimation']} and \ref{['eq:untiary_estimation_to_PBT']}. Combining this transformation with the constructions $\text{(a)} \to \text{(b)}$ and $\text{(c)} \to \text{(d)}$, we obtain a transformation between the sets of general protocols for unitary estimation and dPBT.

Theorems & Definitions (30)

• Theorem 1
• proof : Proof sketch
• Lemma 2
• proof : Proof sketch
• Corollary 3
• Corollary 4
• Theorem 5
• Lemma 6
• proof
• Lemma 7