Power and limitations of distributed quantum state purification

TL;DR

It is proved that, in the presence of depolarizing noise, no LOCC purification protocol starting from two copies can work blindly for all the states in three important sets: the set of all pure two-qubit states, the set of all two-qubit maximally entangled states, and the Bell basis.

Abstract

Quantum state purification protocols, which mitigate noise by converting multiple copies of noisy quantum states into fewer copies with a lower noise level, have applications in quantum communication and computation with imperfect devices. Here, we systematically study the task of state purification in distributed quantum systems, demanding that purification be achieved by local operations and classical communication (LOCC). We prove that, in the presence of depolarizing noise, no LOCC purification protocol starting from two copies can work blindly for all the states in three important sets: the set of all pure two-qubit states, the set of all two-qubit maximally entangled states, and the Bell basis. In stark contrast, we show that a targeted, single-state purification is always achievable in the presence of depolarizing noise, and we provide an explicit analytical LOCC protocol for every given two-qubit state. For arbitrary finite sets of pure states and arbitrary noise profiles, we develop an optimization-based algorithm that systematically designs LOCC purification protocols, and we demonstrate it through concrete examples. Overall, our results identify both fundamental limitations and practical noise reduction strategies for distributed quantum information processing.

Figures (11)

• Figure 1: Diagram of $n\rightarrow1$ purification protocol. (a) Global purification protocol with ${\cal E}_{A^nA'}\in {\rm CPTN}$. (b) Distributed purification protocol with ${\cal E}_{A^nB^nA'B'}\in {\rm LOCC}\cap {\rm CPTN}$.
• Figure 2: Diagram for the $2\rightarrow1$ analytical distributed purification protocol for noisy state ${\cal N}_{AB}^{\gamma,\gamma}(\psi)$. This purification protocol has two phases. In phase 1, the local unitaries map the noisy state ${\cal N}_{AB}^{\gamma,\gamma}(\psi)$ to ${\cal N}_{AB}^{\gamma,\gamma}(\psi_\alpha)$. Phase 2 can purify the noisy state ${\cal N}_{AB}^{\gamma,\gamma}(\psi_\alpha)$ for arbitrary $\alpha\in[0,1]$.
• Figure 3: Comparison of the performance of different $2\rightarrow 1$ distributed purification protocols. The black solid line is the PPT upper bound. The dotted blue line is achieved by the optimized purification protocol. The dash-dot red curve corresponds to the Phase 2 circuit shown in FIG. \ref{['fig:protocol_for_single_state']}. The dashed grey line refers to the fidelity without purification.
• Figure 4: Quantum circuit project quantum state to its symmetry subspace
• Figure 5: Quantum circuit for $2\rightarrow 1$ swap test gadget on arbitrary input noisy states ${\cal N}(\psi)$. The shaded area is the purification protocol ${\cal E}_{swap}$. The three-qubit gate between Hadamard gates is a controlled swap.

Theorems & Definitions (14)

• Definition 1
• Theorem 1
• Theorem 2
• Corollary 3
• Corollary 4
• Theorem 5
• Proposition 6
• Definition 2
• Definition 3
• Corollary 7: No-go theorem for the maximally entangled state set