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arXiv:2402.12287 · Physical Review A

Statistical evaluation and optimization of entanglement purification protocols

Francesco Preti, József Zsolt Bernád

Published Aug 26, 2024• 3 citations
arXivHTMLPDF

Abstract

Quantitative characterization of two-qubit entanglement purification protocols is introduced. Our approach is based on the concurrence and the hit-and-run algorithm applied to the convex set of all two-qubit states. We demonstrate that pioneering protocols are unable to improve the estimated initial average concurrence of almost uniformly sampled density matrices; however, as it is known, they still generate pairs of qubits in a state that is close to a Bell state. We also develop a more efficient protocol and investigate it numerically together with a recent proposal based on an entangling rank-2 projector. Furthermore, we present a class of variational purification protocols with continuous parameters and optimize their output concurrence. These optimized algorithms turn out to surpass former proposals and our protocol by means of not wasting too many entangled states.

DOI
10.1103/physreva.110.022619

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Quantitative characterization of two-qubit entanglement purification protocols is introduced. Our approach is based on the concurrence and the hit-and-run algorithm applied to the convex set of all two-qubit states. We demonstrate that pioneering protocols are unable to improve the estimated initial average concurrence of almost uniformly sampled density matrices; however, as it is known, they still generate pairs of qubits in a state that is close to a Bell state. We also develop a more efficient protocol and investigate it numerically together with a recent proposal based on an entangling rank-2 projector. Furthermore, we present a class of variational purification protocols with continuous parameters and optimize their output concurrence. These optimized algorithms turn out to surpass former proposals and our protocol by means of not wasting too many entangled states.
We present the convergence study of a recurrence entanglement purification protocol using arbitrary two-qubit initial states. The protocol is based on a rank 2 projector in the Bell basis which serves as a two-qubit operation replacing the usual controlled-not gate. We show that the whole space of two-qubit density matrices is mapped onto an invariant subspace characterized by seven real parameters. By analyzing this type of density matrices we are able to find general conditions for entanglement purification in the form of two inequalities between pairs of diagonal elements and pairs of coherences. We show that purifiable initial states do not necessary require a fidelity larger than one half with respect to any maximally entangled pure state. Furthermore, we find a family of states parametrized by their concurrence that can be perfectly converted into a Bell state in just one step of the protocol with probability proportional to the square of the concurrence.
We propose and investigate a method to optimize recurrence entanglement purification protocols. The approach is based on a numerical search in the whole set of SU(4)\text{SU}(4)SU(4) matrices with the aid of a quasi-Newton algorithm. Our method evaluates average concurrences where the probabilistic occurrence of mixed entangled states is also taken into account. We show for certain families of states that optimal protocols are not necessarily achieved by bilaterally applied controlled-not gates. As we discover several optimal solutions, the proposed method offers some flexibility in experimental implementations of entanglement purification protocols and interesting perspectives in quantum information processing.
Bipartite entanglement purification is the conversion of copies of weakly entangled pairs shared between two separated parties into a smaller number of strongly entangled shared pairs using only local operations and classical communication. Choosing between different entanglement purification protocols generally involves weighing up a trade-off between the ratio of strongly entangled pairs produced to weakly entangled pairs consumed, which we call the rate of the protocol, and the degree of the entanglement of the strongly entangled pairs, typically measured by the fidelity of those pairs to maximally entangled states. By randomly choosing a protocol according to a probability distribution over a list of protocols for each pair we want to produce, we can achieve rates and fidelities not achieved by any of the original protocols. Here, we show how to choose this distribution to maximize the rate at which we produce qubit pairs with a given fidelity to a Bell state or, equivalently, to maximize the fidelity to a Bell state of the qubit pairs produced at a given rate. We investigate both the asymptotic case, where the number of initial pairs goes to infinity, and the finite-size regime, where protocols are restricted to a finite number of weakly entangled pairs.
Abstract Entanglement between distant quantum systems is a critical resource for implementing quantum communication. This property is affected by external agents and can be restored by employing efficient entanglement purification protocols. In this work, we propose an entanglement purification protocol based on two entangling two-qubit operations that replace the usual controlled-NOT (CNOT) gate. These operations arise from a generalised quantum measurement and can be understood as measurement operators in a positive operator-valued measure. Furthermore, two variants of the core protocol are introduced and their performances are studied in terms of the overall success probability of reaching a Bell state and the number of purifiable states. Based on rank-two states, we can obtain analytical expressions for the success probability that we extend and refine using numerical calculations to the case of maximally entangled states. We also consider more general rank-three states to show that our procedure is in general more convenient compared to purification protocols based on Bell diagonal states. Finally, we test the protocols using initial random states. In all cases, we find a better performance and a larger amount of purifiable states using our schemes compared to the CNOT-based protocol.
We investigate novel protocols for entanglement purification of qubit Bell pairs. Employing genetic algorithms for the design of the purification circuit, we obtain shorter circuits achieving higher success rates and better final fidelities than what is currently available in the literature. We provide a software tool for analytical and numerical study of the generated purification circuits, under customizable error models. These new purification protocols pave the way to practical implementations of modular quantum computers and quantum repeaters. Our approach is particularly attentive to the effects of finite resources and imperfect local operations - phenomena neglected in the usual asymptotic approach to the problem. The choice of the building blocks permitted in the construction of the circuits is based on a thorough enumeration of the local Clifford operations that act as permutations on the basis of Bell states.
We study the entanglement purification via entanglement swapping in terms of optimal output state concurrence. A strict formula that governs the process is first given analytically for the case where both input entangled pairs are in arbitrary pure states. Then, it is generalized, with strong numerical evidence, to the case with one of the entangled pairs being in an arbitrary mixed state. The formula subsequently gives a formal upper bound of entanglement purification via entanglement swapping, as well as how the maximum entanglement purification can be attained by engineering the pure state entangled pair. Moreover, the formalism of the upper bound in turn suggests that entanglement swapping is most suited for reducing the influence of pure dissipative noise, which can be applied to extend the lifespan of stationary qubits in quantum communications.