The Channel Matrices for Determining the Entanglement Pattern of Multi-Partite State
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The Channel Matrices for Determining the Entanglement Pattern of Multi-Partite State R. Rahmawati 1 & A. Purwanto 1
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& B. A. Subagyo & B. D. Hatmoko
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Received: 11 December 2019 / Accepted: 29 April 2020/ # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
We propose the general reduced channel matrices of multi-partite state for determining the entanglement pattern of the state. The matrix elements of reduced channel matrices are nothing but the coefficients of the state so that the rank of channel matrices can be easier determined. The entanglement pattern can be obtained from the evaluation of the rank of reduced matrices in various level of the reduced state. We apply the method on three multi-qubit states as example. Keywords Quantum teleportation . Entanglement . Reduced channel matrix
1 Introduction The quantum teleportation, first proposed by Bennett et al. [1], is the transmission of quantum information from sender and receiver with the help of classical communication and previously shared quantum entanglement. After the first teleportation protocol by Bennett et al., several quantum teleportation protocols have proposed by many researchers. The controller was introduced between sender and receiver increasing the security of teleportation [2, 3]. Alice and Bob could act as both sender and receiver,
* A. Purwanto [email protected] R. Rahmawati [email protected] B. A. Subagyo [email protected] B. D. Hatmoko [email protected]
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Department of Physics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Surabaya 60111, Indonesia
International Journal of Theoretical Physics
was proposed in the bidirectional scheme [4–6]. Chen et al. [7] proposed the cyclic scheme where Alice sends her qubit to Bob, Bob himself sends his qubit to Charlie, and Charlie himself sends his qubit to Alice. The combination of cyclic and bidirectional schemes are used together to realize teleportation [8]. Those protocols give the consequences of using more qubits state as channels. For more qubit state, it is no longer about the entangled or not, but partially entangled or completely entangled [9]. In practice, it is not easy to determine rank of the density matrix of a multi-partite state via the reduce density matrix [9]. Hatmoko et al. [10] showed that reduced density matrix can be expressed as decomposition into the multiplication of the channel matrix by its Hermite conjugate. However, their work is restricted for four-qubit state. We generalize the method for determining the pattern of the n-partite and the higher dimensions entangled channel using the suitable channel matrix. Moreover, the entanglement plays an important role in quantum teleportation, exactly on the channel. In the first and simplest protocol [1] one-qubit information transmitted to Bob only need two-qubit channel state and must be entangled. When Alice sends two-qubit information to Bob, they require the four-qubit entangled channel.
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