Quantum Information and Computation

Entanglement purification with two-way classical communication

A.W. Leung, Peter W. Shor

Published Mar 1, 2008• 5 citations

Abstract

We present an entanglement purification protocol for Bell-diagonal mixed states and show that this protocol has improved yields over the recurrence methods and the method proposed by Maneva-Smolin. We then generalize this protocol to a family, and show that this family is also a generalization of the recurrence method, the modified recurrence method and the method proposed by Maneva-Smolin. We show that the yields of these protocols on the Werner state $\rho_F$ are higher than those of universal hashing for F less than 0.99999 and as F goes to 1.

DOI: 10.26421/qic8.3-4-6

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Quantum Information and Computation

Entanglement purification with two-way classical communication

A.W. Leung, Peter W. Shor

Published Mar 1, 2008• 5 citations

Publisher

Abstract

\rho_F

are higher than those of universal hashing for F less than 0.99999 and as F goes to 1.

DOI: 10.26421/qic8.3-4-6

Related papers

Adaptive Entanglement Purification Protocols with Two-way Classical Communication

Leung, Alan W., Peter W. Shor

We present a family of entanglement purification protocols that generalize four previous methods, namely the recurrence method, the modified recurrence method, and the two methods proposed by Maneva-Smolin and Leung-Shor. We will show that this family of protocols have improved yields over a wide range of initial fidelities F, and hence imply new lower bounds on the quantum capacity assisted by two-way classical communication of the quantum depolarizing channel. In particular, the yields of these protocols are higher than the yield of universal hashing for F less than 0.993 and as F goes to 1.We present a family of entanglement purification protocols that generalize four previous methods, namely the recurrence method, the modified recurrence method, and the two methods proposed by Maneva-Smolin and Leung-Shor. We will show that this family of protocols have improved yields over a wide range of initial fidelities F, and hence imply new lower bounds on the quantum capacity assisted by two-way classical communication of the quantum depolarizing channel. In particular, the yields of these protocols are higher than the yield of universal hashing for F less than 0.993 and as F goes to 1.

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Charles H. Bennett, David Peter Divincenzo, John A. Smolin +1 more

Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts ${\mathit{D}}_{1}$ (M) and ${\mathit{D}}_{2}$ (M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.Entanglement purification protocols (EPPs) and quantum error-correcting codes (QECCs) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbitrary quantum state |\ensuremath{\xi}〉 can be transmitted at some rate Q through a noisy channel \ensuremath{\chi} without degradation. We prove that an EPP involving one-way classical communication and acting on mixed state M^(\ensuremath{\chi}) (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel \ensuremath{\chi}) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts ${\mathit{D}}_{1}$(M) and ${\mathit{D}}_{2}$(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.

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Purification of multipartite entanglement by local operations

Ming Yang, Fei Yan, Zhuo‐Liang Cao

Multipartite entanglement purification is revisited by using the Local operations and classical communications(LOCCs). We demonstrate our idea by considering the tripartite case, i.e. the purification of tripartite entanglement. We express the general tripartite entangled states in a special representation of total spin operators $J_{123}^{2}$ and $% J_{12}^{2}$ of tripartite system with eigenvalues 15/4 and 2 respectively. This basis is a \textit{genuine basis} because it consists of of all the genuine entangled states of tripartite system. Our protocol is a recurrence one, and only two copies of the initial mixed tripartite entangled states are needed in each round. It is shown that if the initial fidelity is larger than a threshold 0.4, the purification process will succeed.The yield of the current protocol is higher than the previous multipartite entanglement purification protocols. As a by-product, we can get a bipartite pure Bell state when the purification protocol fails for W state. Our protocol also shows that there may be some special kind of tripartite entanglement which belongs to neither W-type entanglement nor GHZ-type entanglement.Multipartite entanglement purification is revisited by using the Local operations and classical communications(LOCCs). We demonstrate our idea by considering the tripartite case, i.e. the purification of tripartite entanglement. We express the general tripartite entangled states in a special representation of total spin operators $J_{123}^{2}$ and $% J_{12}^{2}$ of tripartite system with eigenvalues 15/4 and 2 respectively. This basis is a \textit{genuine basis} because it consists of of all the genuine entangled states of tripartite system. Our protocol is a recurrence one, and only two copies of the initial mixed tripartite entangled states are needed in each round. It is shown that if the initial fidelity is larger than a threshold 0.4, the purification process will succeed.The yield of the current protocol is higher than the previous multipartite entanglement purification protocols. As a by-product, we can get a bipartite pure Bell state when the purification protocol fails for W state. Our protocol also shows that there may be some special kind of tripartite entanglement which belongs to neither W-type entanglement nor GHZ-type entanglement.

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\rho_F

are higher than those of universal hashing for F less than 0.99999 and as F goes to 1.

{\mathit{D}}_{1}

(M) and

{\mathit{D}}_{2}

(M) that can be locally distilled from it by EPPs using one- and two-way classical communication, respectively, and give an exact expression for E(M) when M is Bell diagonal. While EPPs require classical communication, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way communication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single-error-correcting quantum block code. We prove that iff a QECC results in high fidelity for the case of no error then the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. \textcopyright{} 1996 The American Physical Society.

J_{123}^{2}

and

% J_{12}^{2}

of tripartite system with eigenvalues 15/4 and 2 respectively. This basis is a \textit{genuine basis} because it consists of of all the genuine entangled states of tripartite system. Our protocol is a recurrence one, and only two copies of the initial mixed tripartite entangled states are needed in each round. It is shown that if the initial fidelity is larger than a threshold 0.4, the purification process will succeed.The yield of the current protocol is higher than the previous multipartite entanglement purification protocols. As a by-product, we can get a bipartite pure Bell state when the purification protocol fails for W state. Our protocol also shows that there may be some special kind of tripartite entanglement which belongs to neither W-type entanglement nor GHZ-type entanglement.