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arXiv:2205.12091 · Physical Review A

Optimal two-qubit gates in recurrence protocols of entanglement purification

Francesco Preti, Tommaso Calarco, Juan Mauricio Torres, József Zsolt Bernád

Published Aug 22, 2022• 13 citations
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Abstract

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 $\text{SU}(4)$ 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.

DOI
10.1103/physreva.106.022422

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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.
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.
Entanglement purification allows the creation of qubit pairs of arbitrarily high fidelity with respect to a maximally entangled state, starting from a larger number of low-fidelity pairs. Purification requires quantum memory, a role for which electron spins are well suited. However, using existing recurrence protocols involving symmetric local two-qubit operations for spin qubits turns out to be rather unpractical. We present an efficient purification protocol requiring only a single pulsed Heisenberg- or XY-type exchange interaction between two qubit pairs. In contrast to known protocols, we allow for asymmetric bilateral two-qubit operations where the two communication parties operate differently on their qubits. In the optimal version of our protocol, the local two-qubit interactions in the case of Heisenberg exchange correspond to the swap\sqrt{swap}swap​ gate and its inverse.
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.
Entanglement purification describes a primitive in quantum information processing, where several copies of noisy quantum states are distilled into a few copies of nearly pure states of high quality via local operations and classical communication. Especially in the multiparticle case, the task of entanglement purification is complicated, as many inequivalent forms of pure state entanglement exist and purification protocols need to be tailored for different target states. In this paper we present optimized protocols for the purification of hypergraph states, which form a family of multiqubit states that are relevant from several perspectives. We start by reformulating an existing purification protocol in a graphical language. This allows for systematical optimization and we present improvements in three directions. First, one can optimize the sequences of the protocol with respect to the ordering of the parties. Second, one can use adaptive schemes, where the measurement results obtained within the protocol are used to modify the protocols. Finally, one can improve the protocol with respect to the efficiency, requiring fewer copies of noisy states to reach a certain target state.
This paper addresses the question of how to implement a desired two-qubit gate U using a given tunable two-qubit entangling interaction . We present a general method which is based on the K 1 AK 2 decomposition of unitary matrices ∊ SU (4) to calculate the smallest number of two-qubit gates U int ( t ) (based on ) and single-qubit rotations, and the explicit sequence of these operations that are required to implement U . We illustrate our protocol by calculating the implementation of (1) the transformation from standard basis to Bell basis, (2) the CNOT-gate and (3) the quantum Fourier transform for two kinds of interaction—Heisenberg exchange interaction and quantum inductive coupling—and discuss the relevance of our results for solid-state qubits.