任意階最大量子糾纏態的純化

Quantum n-level qudit purification scheme

羅信斌(Hsin-Pin Lo)

指導教授 ： 徐立義

中原大學/理學院/應用物理研究所/碩士(2006年)

https://doi.org/10.6840/cycu200600447

摘要

摘要
在一九九零年代早期, 作為一跨學科的新興領域, 量
子資訊科學有著驚人的成長。這是由於人們預見量子資
訊科學將在計算、通訊、及密碼理論中將帶來革命性
的進步。其中, 量子糾纏扮演著重要的角色。許多量子
通訊協定, 像是稠密編碼(dense coding)、量子遙傳
(quantum teleportation)、量子金鑰(quantum key
distribution) 等等, 都必須經由量子糾纏方能實現。
若距離遙遠的愛麗絲以及鮑伯, 想要實現上述的量子
通訊協定, 兩人須擁有預先準備多的最大量子糾纏態。
但是, 最大量子糾纏態並不容易保存, 非常容易因為與
環境之間的交互作用而變成混合態(mixed state)。在
本篇論文中, 我們將討論如何將因環境所造成的混合
態加以純化(purify) 成原本預備的最大量子糾纏態。
我們的純化過程可分為三步驟: 首先對愛麗絲以及鮑
伯兩人所擁有的最大量子糾纏態分別作哈達瑪轉換以
及逆哈達瑪轉換。接下來,將轉換後的結果以Controlled-Not 邏輯閘作用。最後測量目標位元對, 如果測量結果相同就保留控制位元對到下次純化; 如果結果不同就將控制位元對以及目標位元對都捨棄。如此重複純化, 直到將所有最大量子糾纏態作用完畢, 得到純的最大量子糾纏態。

我們將其純化過程進行數值模擬。其結果簡述如下: 我
們分別對n 階的混合態, 其中n 等於3、4、5、6、7進行
模擬。經過我們的純化過程後, 都可以得到我們所需要
的最大量子糾纏態, 且其機率為一。而當純化的階數增
加時, 所需要純化的初始機率會下降。理論指出: 對n
階的混合態而言, 若初始機率大於1/n, 則此混合態為
不可分離的(inseparable)。我們的結果顯示: 只要此
混合態為不可分離的, 即可純化出最大糾纏態。另外,
我們還討論一個特殊的狀況, 找最小可以得到純化的
初始機率。發現經過一些初始值的選取後, 就可以把純
化的初始值降低, 以利我們得到純化的結果。這樣的結
果也可以適用到任意階最大量子糾纏態的純化中。

關鍵字

量子資訊

並列摘要

Abstract
Due to the promising potential causing revolutionary advance involving computation, communication and cryptography, quantum information science (QIS), as an emerging and cross-disciplinary subject has been rapidly expand and began its explosive growth since the early to mid 1990’s. Therein, quantum entanglement plays an important role in QIS. Now, entanglement is regarded as a key resource in quantum communication protocols, such as quantum key distribution, dense coding and teleportation. In the physical realization of the above protocols, an ensemble of maximally entangled states should be prepared and shared in advance for the two distant parties. However, an essential obstacle comes form decoherence.
The interaction between quantum bits
(qubits) and the environment generally disturbs
and even destroys the entanglement. Therefore,
entanglement purification on the unknown mixed
states is necessary before the implementation of
any entanglement-based quantum communication protocol.
In this thesis, a scheme of how to purify n-level
bipartite maximally-entangled states is proposed.
This scheme consists of three steps in each round. At the first purification step, one partite performs the local Hadamard transformation and the other party the corresponding inverse transformation.
At the second step, the bilateral controllednot
operations are performed. At the last step,
the target pairs are measured, if the outcome is
coincided, the controlled pairs should be kept for the next round; otherwise, the controlled pairs and target pairs both should be discarded. These three steps should be performed repeatly and finally maximally entangled states can be accessed. We also numerically simulate the proposed purification scheme. In brief, the results show that the proposed scheme can purify the n-level bipartite maximally-entangled states, where n can be 3, 4, 5, 6 and 7, if the initial fidelity is larger than 1/n. Moreover, the final fidelity of the intended maximally entangled states can reach 1 after about ten rounds. On the other hand, the theory indicates that the n-level bipartite mixed states are inseparable if the fidelity of some maximally entangled state is larger than 1/n. Therefore, our simulation implies that the purification of quantum entanglement can be achieved for any inseparable mixed states. In addition, some specific cases are discussed. Especially, in the n=3 case, the proposed purification scheme can fail in some special initial fidelity distributions.
The criteria of successful purification can be derived analytically.

並列關鍵字

quantum information

參考文獻

[1] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).

[2] P. Benoiff, Phys. Rev. Lett. 48, 1581 (1982).

[4] C. H. Bennett, IBM J. Res. Dev. 17, 525(1973).

[6] D. Bouwmeester, J. W. Pan, K. Mattle, M. Eible,

H. Weinfurter and A. Zeilinger, Nature. , 390,