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The Entanglement Measure of Multi-qubit Pure State in Quantum Error-correcting Code

Author: HuLiJuan
Tutor: GaoTing
School: Hebei Normal
Course: Applied Mathematics
Keywords: quantum state quantum entanglement concurrence reduced density matrix
CLC: O413.1
Type: Master's thesis
Year: 2010
Downloads: 55
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Abstract


The entanglement measure of the three common multi-qubit pure states in the quantumerror-correcting codes is started. Quantum entanglement is not only a key resource in quan-tum information processing and quantum computation, but also one of the most fascinatingfeatures of quantum theory. It is a most valuable research content. There are three commonquantum error-correcting codes, five-qubit quantum error-correcting code, seven-qubit quantumerror-correcting code-Steane and nine-qubit quantum error-correcting code-Shor. The five-qubitquantum error-correcting code is the smallest code that is capable of protecting against a singleerror. So it’s a much useful code.First, we introduce some useful background knowledge. Specifically, we describe the suc-cessfully entanglement measures, which we obtained the entanglement characterization of thetwo-body quantum systems and the three-body quantum systems. And it also explain the defi-nition of entanglement degree of multi-qubit pure state.In the second chapter, we calculate the multi-body entanglement degree of the previousthree common multi-qubit pure states of the quantum error-correcting code states. We givethe concurrences of the five-qubit quantum error-correcting code, seven-qubit quantum error-correcting code-Steane and nine-qubit quantum error-correcting code-Shor. Dure to the largenumber of the reduced-density operators, it causes great difficulty for us to calculate. So, wefirst started with a useful conclusion of Schmidt decomposition, which makes the calculationreduce the double. In the multi-body entanglement of the three multi-qubit pure states, the multi-body entanglement of the nine-qubit quantum error-correcting code is the most complicated.Because the number of the classification of the reduced-density operators is so many, we muststudy carefully with the special composed structure of the nine-qubit quantum error-correctingcode and make sure that the classification is accurate.In the three chapter, we give the higher-tangle of the three multi-qubit pure states, i.e.In the last chapter, we made the classification for high-dimensional two-body entanglementof the five-qubit quantum error-correcting code state, seven-qubit quantum error-correctingcode-Steane and nine-qubit quantum error-correcting code-Shor respectively. In this chapter,we must note the coefficient matrix of the quantum state and classification. These are the keysto success. Although the two sub-systems carried out classification according to the numberof particles contained, different sequencing of two subsystems cause different coefficient ma- trixes. Luckily, it don’t impact the final result. So, we only note the impact that caused differentparticles selecting in the sub-systems. Because the number of the selected particles is large, wemust carefully analyse to find out the laws. Among of them, the number of the classification ofnine-qubit quantum error-correcting code state is large. And the coefficient matrix is the mostcomplex. So, the writing of coefficient matrixe must be careful.The main conclusions of this paper are that the three multi-qubit pure states don’t havetwo-qubit entanglement of the real two-body. But they all have high-dimensional two-body en-tanglement, as well as truly multi-body entanglement. And three-body tangle of these threemulti-qubit pure states is one. But the degree of the multi-body entanglement don’t increasewith the number of particle increasing. The high-dimensional two-body entanglement of thefive-qubit quantum pure state has two classes. The high-dimensional two-body entanglementof the seven-qubit quantum pure state has three classes. And the high-dimensional two-bodyentanglement of the nine-qubit quantum pure state has four classes. In which there is a kindof entanglement, the degree is zero. That is to say, in such combination, there is no entan-glement between the two subsystems. So we can know the nine-qubit quantum pure state issemi-separable.

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CLC: > Mathematical sciences and chemical > Physics > Theoretical Physics > Quantum theory > Quantum mechanics ( wave mechanics,matrix mechanics )
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