[Paper Review] Square-root measurements and degradation of the resource state in port-based teleportation scheme

Michał Studziński, Marek Mozrzymas+1 more|arXiv (Cornell University)|May 31, 2021

Quantum Information and CryptographyComputer Science33 references7 citations

TL;DR

This paper introduces a recycling protocol for port-based teleportation (PBT) in arbitrary dimensions, analyzing degradation of the resource state after each teleportation round. By leveraging square-root measurements and group-theoretic symmetries via Schur-Weyl duality, the authors derive an explicit formula for recycling fidelity depending only on irreducible representation parameters, enabling precise quantification of resource state degradation for optimal PBT schemes across all d ≥ 2.

ABSTRACT

Port-based teleportation (PBT) is a protocol of quantum teleportation in which a receiver does not have to apply correction to the transmitted state. In this protocol two spatially separated parties can teleport an unknown quantum state only by exploiting joint measurements on number of shared $d-$dimensional maximally entangled states (resource state) together with a state to be teleported and one way classical communication. In this paper we analyse for the first time the recycling protocol for the deterministic PBT beyond the qubit case. In the recycling protocol the main idea is to re-use the remaining resource state after one or many rounds of PBT for further processes of teleportation. The key property is to learn how much the underlying resource state degrades after every round of the teleportation process. We measure this by evaluating quantum fidelity between respective resource states. To do so we first present analysis of the square-root measurements used by the sender in PBT by exploiting the symmetries of the system. In particular, we show how to effectively evaluate their square-roots and composition. These findings allow us to present the explicit formula for the recycling fidelity involving only group-theoretic parameters describing irreducible representations in the Schur-Weyl duality. For the first time, we also analyse the degradation of the resource state for the optimal PBT scheme and show its degradation for all $d\geq 2$. In the both versions, the qubit case is discussed separately resulting in compact expression for fidelity, depending only on the number of shared entangled pairs.

Motivation & Objective

• To investigate the feasibility of reusing entangled resource states in port-based teleportation (PBT) after each teleportation round.
• To quantify the degradation of the shared resource state after joint measurements in deterministic PBT, particularly in higher-dimensional systems (d ≥ 2).
• To develop a general framework for evaluating recycling fidelity beyond the qubit case using group-theoretic tools.
• To provide an explicit analytical expression for recycling fidelity in optimal PBT schemes using irreducible representations from Schur-Weyl duality.
• To extend previous qubit-only analyses to arbitrary local dimensions, enabling scalable and efficient PBT protocols with reusable resources.

Proposed method

• Analyzes square-root measurements used in PBT by exploiting symmetric structures in the system, particularly the action of the unitary group U(d) and its dual representation.
• Derives effective methods to compute square-roots and compositions of measurement operators using the symmetry of the Schur-Weyl duality decomposition.
• Expresses the recycling fidelity as a function of group-theoretic parameters—specifically, the irreducible representations of the symmetric group and unitary group under Schur-Weyl duality.
• Applies the formalism to both deterministic and optimal PBT schemes, computing the quantum fidelity between the original and degraded resource states after each round.
• Uses the transposition (SWAP) operation between ports to simulate the recycling process, isolating the remaining N−1 ports for subsequent rounds.
• Derives a compact, closed-form expression for recycling fidelity in the qubit case (d=2), depending only on the number of entangled pairs N.

Experimental results

Research questions

• RQ1 How does the resource state degrade after a single round of deterministic port-based teleportation in arbitrary dimension d?
• RQ2 Can the fidelity of the remaining entangled resource state be analytically quantified after reuse in a recycling protocol?
• RQ3 What is the role of square-root measurements in enabling the evaluation of state degradation during PBT?
• RQ4 How do the group-theoretic parameters from Schur-Weyl duality determine the fidelity of recycled resource states?
• RQ5 Does the degradation behavior differ significantly between optimal and non-optimal PBT schemes in higher dimensions?

Key findings

• The paper presents the first explicit formula for recycling fidelity in deterministic PBT beyond the qubit case, valid for all d ≥ 2.
• The recycling fidelity is expressed solely in terms of irreducible representation parameters from the Schur-Weyl duality, enabling exact analytical evaluation.
• For the qubit case (d=2), the recycling fidelity is derived as a compact expression depending only on the number of entangled pairs N.
• The authors confirm that resource state degradation occurs even in the optimal PBT scheme, with fidelity decreasing monotonically with each reuse round.
• The analysis reveals that the degradation rate is governed by the structure of the symmetric group and the representation theory of U(d), providing a unified framework for higher-dimensional PBT.
• The results demonstrate that resource reuse is feasible but comes with quantifiable fidelity loss, which scales with the number of teleportation rounds and system dimension d.