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Physical Review A

Formal bounds of entanglement purification via entanglement swapping

Run-Ze Cong, Shuang Xu

Published Jan 6, 2025• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.111.012409

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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.
The nonlocal correlations between quantum states in an entangled system are essential to many quantum communications applications. A basic quantum operation, which permits the distribution of entanglement between two initially uncorrelated systems, is entanglement swapping. Here we present a rigorous formulation of entanglement swapping of any two partially mixed two-qubit states without limiting ourselves to any specific type of state or noise. Further, for two important classes of the input states, Bell diagonal and pure states, we describe how the concurrence of the final state is related to the concurrence of the initial states. First, we consider Bell diagonal states, and find bounds on the concurrence of the final state in terms of the concurrences of the initial states. These bounds are important for communications applications because polarization mode dispersion in fibers produces Bell diagonal states up to a local unitary rotation. Second, we show that swapping pure states occasionally results in a state of higher concurrence than either of the initial states. In addition, we find that two pure states are most likely to be capable of swapping to a state of increased concurrence when the two initial states have similar concurrences. Our analysis offers a completely general framework for investigating the behavior of any pair of two-qubit states when used for entanglement swapping.
We consider entanglement swapping, a key component of quantum network operations and entanglement distribution. Pure entangled states, which are the desired input to the swapping protocol, are typically mixed by environmental interactions, causing a reduction in their degree of entanglement. Thus an understanding of entanglement swapping with partially mixed states is of importance. Here we present a general analytical solution for entanglement swapping of arbitrary two-qubit states. Our result provides a comprehensive method for analyzing entanglement swapping in quantum networks. First, we show that the concurrence of a partially mixed state is conserved when this state is swapped with a Bell state. Then, we find upper and lower bounds on the concurrence of the state resulting from entanglement swapping for various classes of input states. Finally, we determine a general relationship between the ranks of the initial states and the rank of the final state after swapping.
We determine the preshared entanglement required for spatially separated parties, restricted to local operations and classical communication, to attain globally optimal probabilistic two-copy purification of arbitrary bipartite pure states under depolarizing noise. In every local dimension d≥2d\ge 2d≥2, one shared maximally entangled qubit pair suffices: local controlled-SWAP operations exactly reproduce the globally optimal successful transformation. Conversely, any finite-dimensional preshared resource state that attains the same benchmark, even under a positive-partial-transpose relaxation, must have entanglement of formation at least one ebit. For pure resources with one ebit, or for two-qubit resources including mixed states, attaining the benchmark is possible only for states with exactly two nonzero Schmidt weights, both equal to 1/21/21/2, namely those locally unitarily equivalent to a maximally entangled qubit pair. These findings provide a benchmark for evaluating the entanglement demands of noise management in quantum networks and modular quantum computers.
Entanglement swapping is gaining widespread attention due to its application in entanglement distribution among different parts of quantum appliances. We investigate the entanglement swapping for pure and noisy systems, and argue different entanglement quantifiers for quantum states. We explore the relationship between the entanglement of initial states and the average entanglement of final states in terms of concurrence and negativity. We find that if initial quantum states are maximally entangled and we make measurements in the Bell basis, then average concurrence and average negativity of final states give similar results. In this case, we simply obtain the average concurrence (average negativity) of the final states by taking the product of concurrences (negativities) of the initial states. However, the measurement in non-maximally entangled basis during entanglement swapping degrades the average swapped entanglement. Further, the product of the entanglement of the initial mixed states provides an upper bound to the average swapped entanglement of final states obtained after entanglement swapping. The negativity work well for weak entangled noisy states but concurrence gives better results for relatively strong entanglement regimes. We also discuss how successfully the output state can be used as a channel for the teleportation of an unknown qubit.
Entanglement purification is an indispensable ingredient in extended quantum communication networks and usually determines the efficiency and communication rate of quantum communication protocols. Different from all existing entanglement purification protocols (EPPs) where two or more copies of low quality mixed entangled states are selected from the same ensemble, here we describe a general and optimal EPP for arbitrary initial mixed states from different ensembles. We show that the successful operation of EPP may not obtain a higher fidelity mixed state, while the discarded source pair, which is usually regarded as a failure in existing EPPs, may have residual entanglement and can be reused to increase the yield of entanglement purification. We give the criterions of both the successful purification to obtain a higher fidelity mixed state and the existence of residual entanglement. Moreover, we reveal that entanglement purification procedure causes some entanglement loss. Finally, we provide an optimal approach to reduce the entanglement loss. This approach can also be used to increase the yield of entanglement purification. Our EPP may have potential application in long-distance quantum communications.
Four basis vectors of the Hilbert space of two qubits have the property that if three of them are product states, then the fourth one has to be a product state as well. We address the following situation: Consider a set of orthogonal vectors, each exhibiting a certain degree of entanglement. What is the bound on entanglement of the rest of the basis vectors to form a complete orthonormal basis? Specifically, we present an orthonormal basis, the $\ensuremath{\Xi}$ basis in the Hilbert space of two qubits, with one product state and three equally entangled states. The maximum of the so available entanglement is quantified. A close-to-optimal protocol is presented for entanglement purification via entanglement swapping of two-qubit states. It is based on a suitably chosen nonmaximally entangled basis and carried out in a single step without any ancillas. A similar application of the $\ensuremath{\Xi}$ basis is examined. In this latter case, all the involved entangled states have different and nonorthogonal Schmidt decompositions and, except for some possibly resulting states, none of them are maximally entangled. Entanglement of single pair purification is not conserved on average in this case.