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arXiv:1007.3171 · Physical Review A

Optimal distillation of three-qubit W states

Ali Kerem Yildiz

Published Jul 19, 2010• 32 citations
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Abstract

Some of the asymmetric three-qubit $W$ states are used for perfect teleportation, superdense coding, and quantum-information splitting. We present the protocols for the optimal distillation of the asymmetric as well as the symmetric $W$ states from a single copy of any three-qubit $W$ class pure state.

DOI
10.1103/physreva.82.012317

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Some of the asymmetric three-qubit WWW states are used for perfect teleportation, superdense coding, and quantum-information splitting. We present the protocols for the optimal distillation of the asymmetric as well as the symmetric WWW states from a single copy of any three-qubit WWW class pure state.
True tripartite entanglement of the state of a system of three qubits can be classified on the basis of stochastic local operations and classical communications. Such states can be classified into two categories: GHZ states and WWW states. It is known that GHZ states can be used for teleportation and superdense coding, but the prototype WWW state cannot be. However, we show that there is a class of WWW states that can be used for perfect teleportation and superdense coding.
In [Phy. Rev. Lett., 98, 260501 (2007)] and previous papers, the W-like states were used in many quantum communication schemes, distillation, teleportation and superdense coding. So far, no one has compared the W-like states with the W state. We investigate the properties of three W-like states named as $|\vartheta ^{\prime }\rangle $, $|η^{\prime }\rangle $ and $|ξ^{\prime }\rangle $. We point out that $|\vartheta ^{\prime }\rangle $ (resp. $|η^{\prime }\rangle $ and $|ξ^{\prime }\rangle $) has the maximal von Neumann entanglement entropy (vNEE) S(ρA)=ln⁡2S(ρ_{A})=\ln 2S(ρA​)=ln2 (resp. S(ρB)=ln⁡2S(ρ_{B})=\ln 2S(ρB​)=ln2 and S(ρC)=ln⁡2S(ρ_{C})=\ln 2S(ρC​)=ln2) and the maximal tangle τA(BC)=1τ_{A(BC)}=1τA(BC)​=1 (resp. τB(AC)=1τ_{B(AC)}=1τB(AC)​=1 and τC(AB)=1τ_{C(AB)}=1τC(AB)​=1) while the W state does not. We also indicate that if some one of three qubits is traced out, then the remaining two qubits of $|\vartheta ^{\prime }\rangle $ (resp. $|η^{\prime }\rangle $ and $|ξ^{\prime }\rangle $) are more entangled than the two qubits of the W state by several entanglement measures. It means that the three W-like states have higher robustness against some particle loss than the W state. We show that $|\vartheta ^{\prime }\rangle $, $|η^{\prime }\rangle $ and $|ξ^{\prime }\rangle $ can be suitable for perfect teleportation and superdense coding but the W state cannot.
As we know, the states of triqubit systems have two important classes: GHZ-class and W-class. In this paper, the states of W-class are considered for teleportation and superdense coding, and are generalized to multi-particle systems. First we describe two transformations of the shared resources for teleportation and superdense coding, which allow many new protocols from some known ones for that. As an application of these transformations, we obtain a sufficient and necessary condition for a state of W-class being suitable for perfect teleportation and superdense coding. As another application, we find that state ∣W>123=1/2(∣100>123+∣010>123+sqrt2∣001>123)|W>_{123}={1/2}(|100>_{123}+|010>_{123}+\\sqrt{2}|001>_{123})∣W>123​=1/2(∣100>123​+∣010>123​+sqrt2∣001>123​) can be used to transmit three classical bits by sending two qubits, which was considered to be impossible by P. Agrawal and A. Pati [Phys. Rev. A to be published]. We generalize the states of W-class to multi-qubit systems and multi-particle systems with higher dimension. We propose two protocols for teleportation and superdense coding by using W-states of multi-qubit systems that generalize the protocols by using ∣W>123|W>_{123}∣W>123​ proposed by P. Agrawal and A. Pati. We obtain an optimal way to partition some W-states of multi-qubit systems into two subsystems, such that the entanglement between them achieves maximum value.
As we know, the states of triqubit systems have two important classes: GHZ-class and W-class. In this paper, the states of W-class are considered for teleportation and superdense coding, and they are generalized to multi-particle systems. First we describe two transformations on the shared resources for teleportation and superdense coding. With these transformations, we obtain a sufficient and necessary condition for a state of W-class being suitable for perfect teleportation and superdense coding. For the state which was thought to be not suitable for sending three classical bits by sending two qubits by Agrawal and Pati (2006 Phys. Rev. A 74 062320), we show that it may be used to fulfil that task, if entangled unitary operations on two qubits are allowed. We generalize the states of W-class to multi-qubit systems and multi-particle systems with higher dimension. We propose two protocols for teleportation and superdense coding by using W-states of multi-qubit systems that generalize the protocols by using | W ⟩ 123 proposed by Agrawal and Pati. We obtain an optimal way to partition some W-states of multi-qubit systems into two subsystems, such that the entanglement between them achieves maximum value.