Abstract
The bilateral controlled NOT (CNOT) operation plays a key role in standard entanglement purification process, but the CNOT operation may not be the optimal joint operation in the sense that the output entanglement is maximized. In this paper, the CNOT operations in both the Schmidt-projection based entanglement concentration and the entanglement purification schemes are replaced with a general joint unitary operation, and the optimal matching relations between the entangling power of the joint unitary operation and the non-maximal entangled channel are found for optimizing the entanglement in- crement or the output entanglement. The result is somewhat counter-intuitive for entanglement concentration. The output entanglement is maximized when the entangling power of the joint unitary operation and the quantum channel satisfy certain relation. There exist a variety of joint operations with non-maximal entangling power that can induce a maximal output entanglement, which will greatly broaden the set of the potential joint operations in entanglement concentration. In addition, the entanglement increment in purification process is maximized only by the joint unitary operations (including CNOT) with maximal entangling power.
Highlights
Entanglement plays an important role in quantum information processing
The bilateral controlled NOT (CNOT) operation in standard entanglement concentration process is replaced with a general bilateral bipartite operation, which is elaborated in the Methods section
By replacing the CNOT operation with a general joint operation, we studied the role of entangling power of the joint unitary operation in entanglement concentration and entanglement purification and found the matching relations between the entangling power of the joint unitary operation and the parameter of the quantum channel so that the entanglement of the output state or the entanglement increment is maximized
Summary
Entanglement plays an important role in quantum information processing. It is a prerequisite in quantum teleportation[1,2], entanglement swapping[3], quantum cryptography[4] and so on. These results show that the complicated CNOT operation is not a necessity in entanglement concentration and purification, and the bilateral coupling between the two copies of the states to be concentrated or purified is the core part of an entanglement concentration or purification process The role of this kind of coupling has been demonstrated[9,10], and the optimal output fidelity is achieved with a general coupling rather than a CNOT-like coupling. This result only applies to the phase-error single-photon entangled mixed states, and whether there exist the similar results for the purification of general mixed entangled states(such as Werner state) or the concentration of unknown non-maximally entangled pure states is not clear. We revisited the entanglement purification and the Schmidt-projection based entanglement concentration schemes with the CNOT operation being replaced with a general joint unitary operation, and the optimal matching relations between the entangling power of the joint unitary operation and the non-maximal entangled channel are found for optimizing the entanglement increment and output entanglement in entanglement purification and concentration
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