Reusability of Quantum Catalysts

TL;DR

The paper addresses the finite operational lifetime of embezzling quantum catalysts in quantum information tasks, focusing on entanglement distillation and quantum teleportation. It develops a quantitative framework and analyzes two canonical catalysis schemes, CSLA and ESA, deriving explicit bounds on the maximum number of reusable rounds $r_{CS}$ and $r_E$ under a prescribed fidelity-improvement threshold $\varepsilon$. By extending the framework to catalytic teleportation, the authors show how reuse limits govern sustained high-fidelity state transfer in quantum communication. The work advances a resource-theory perspective where catalysts are evolving agents subject to degradation, offering practical guidelines for designing sustainable catalytic protocols in quantum networks.

Abstract

Quantum catalysts enable transformations that otherwise would be forbidden, offering a pathway to surpass conventional limits in quantum information processing. Among them, embezzling catalysts stand out for achieving near-perfect performance while tolerating only minimal disturbance, bridging the gap between ideal and practical catalysis. Yet, this superior capability comes at a cost: Each use slightly degrades the catalyst, leading to an inevitable accumulation of imperfection. This gradual decay defines their most distinctive property -- reusability -- which, despite its fundamental importance, remains largely unexplored. Here, we establish a quantitative framework to characterize the operational lifetime of embezzling catalysts, focusing on their role in entanglement distillation and extending the analysis to quantum teleportation. We show that the catalytic advantage inevitably diminishes with repeated use, deriving bounds on the maximum effective reuse rounds for a desired performance gain. Our results uncover the finite reusability of catalysts in quantum processes and point toward sustainable strategies for quantum communication.

Figures (8)

• Figure 1: Schematic of Catalytic Entanglement Distillation. (a) shows the standard protocol without auxiliary entangled resources, where a single noisy state $\rho_{AB}$ is purified solely through a LOCC operation $\Lambda$. (b) depicts a single-round catalytic distillation protocol assisted by an embezzling state $\tau_{CC'}$, which provides auxiliary entanglement while remaining nearly unchanged. (c) extends this to a multi-round catalytic distillation scheme, in which the catalyst is reused across successive rounds. For each round $i = 1, 2, \dots, r$, the operation $\Lambda^{\text{Cat}}_i$ is identical to that in (b) and acts on the subsystem $A_i C C' B_i$, with each round starting from an identical input state $\rho_{AB}$.
• Figure 2: Embezzling-State–Assisted (ESA) Distillation. The protocol begins by replacing the main system's state $\rho$ with $\ketbra{11}{11}$ and then applying the same unitary operation $U$ to the subsystems $AC$ and $BC'$, respectively. The embezzling state $\tau^{E}$ (see Eq. \ref{['eq:emb-state']}) is prepared on the ancillary systems $C$ and $C'$. The unitary $U$ is defined through its action on the computational basis as $U\ket{ij} = \ket{kl}$, where the indices satisfy $l = \lceil ((i-1)M + j)/d \rceil$ and $k = (i-1)M + j - (l-1)d$.
• Figure 3: Convex-Split–Lemma–Assisted (CSLA) Distillation. Representation of two rounds of the catalytic protocol $\Lambda^{CS}$ defined in Eq. \ref{['eq:lmdCS']} for the case $n=4$. Indices $1$–$3$ label the catalytic subsystems, while $4$ and $5$ denote the main systems. Green cubes represent the input states $\rho$, and yellow spheres indicate the components of the embezzling catalyst $\tau^{CS} = \tau_3 \otimes \tau_2 \otimes \tau_1$. (a) In the first round, $\Lambda^{CS}_1$ acts on the main system $4$ and catalytic subsystems $1$–$3$, producing an equal mixture over four outcomes, with the reduced state $\rho^{CS,1}_4 = (\rho + 3\tau)/4$. (b) In the second round, $\Lambda^{CS}_2$ acts on the main system $5$ and the same catalytic subsystems. Each input branch generates four new outcomes, yielding a uniform mixture over $16$ states. The reduced state on system $5$ is $\rho^{CS,2}_5 = (7\rho + 9\tau)/16$, while system $4$ retains $\rho^{CS,2}_4 = [4(\rho + 3\tau)]/16=\rho^{CS,1}_4$, reflecting contributions carried over from the first round.
• Figure 4: Reusability of CSLA Catalysts. Numerical characterization of the catalytic performance and reusability of the CSLA catalyst $\tau^{CS}$ in entanglement distillation. Constructed through the convex-split-lemma as $\tau^{CS} = \tau^{\otimes (n-1)}$ (see Lem. \ref{['lem:cs']}), this catalyst demonstrates sustained fidelity enhancement over successive rounds, revealing the operational lifetime of catalytic resources. Figures (a) and (b) show the fidelity $F(\rho^{CS,r})$ as a function of the distillation round $r$ and the parameter $n$ specifying the CSLA catalyst $\tau^{CS}= \tau^{\otimes (n-1)}$. The yellow plane represents the fidelity of the original noisy state $\rho$, and the green plane indicates the performance threshold defining the boundary of effective catalyst reusability. Figures (c) and (d) depict the dependence of the maximum effective reuse rounds $r_{CS}$ (see Eq. \ref{['eq:drCS']}) on the fidelity improvement threshold $\epsilon$ and the catalyst size parameter $n$. The observed scaling trend delineates the fundamental trade-off between enhanced distillation accuracy, increasing catalytic dimensionality, and the gradual loss of reusability inherent to the CSLA catalyst.
• Figure 5: Reusability of ESA Catalysts. The catalyst $\tau^{E}$ is constructed from an embezzling state with Schmidt rank $M$ (see Lem. \ref{['lem:E']}). (a) Entanglement fidelity $F(\rho^{E,r}_r)$ as a function of the number of catalytic entanglement distillation rounds $r$ for different target dimensions $d$. The fidelity exhibits a gradual decay with repeated use, converging to the limit $1/d$ when the number of rounds exceeds $\lceil\log_d M\rceil$. (b) Dependence of the maximum effective reuse rounds $r_E$ on the catalyst's Schmidt rank $M$, evaluated for $d=2$ and a fidelity-gain threshold $\epsilon=0.05$. The scaling highlights the extended operational lifetime afforded by catalysts with more entanglement.

Theorems & Definitions (9)

• Definition 2.1: Embezzling Catalysts Datta_2023
• Lemma 3.1: CSLA Distillation Xing2024Teleportation
• Lemma 3.2: ESA Distillation Xing2024Teleportation
• Theorem 3.3: Reusability of CSLA Catalyst
• proof
• Theorem 3.4: Reusability of ESA Catalyst
• proof
• Corollary 4.1: CSLA Teleportation
• Corollary 4.2: ESA Teleportation