基于两步测量方法及其最少观测次数的任意量子纯态估计
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摘要
量子状态层析所需要的完备观测次数d~2(d=2~n)随着状态的量子位数n的增加呈指数增长,这使得对高维量子态的层析变得十分困难.本文提出一种基于两步测量的量子态估计方法,可以对任意量子纯态的估计提供最少的观测次数.本文证明:当选择泡利观测算符,采用本文所提出的量子态估计方法对d=2n维希尔伯特空间中的任意n量子位纯态进行重构时,如果为本征态,那么所需最少观测次数memin仅为memin=n;对于包含l(2 6 l 6 d)个非零本征值的叠加态,重构所需最少观测次数msmin满足msmin=d+2l..3,此数目远小于压缩传感理论给出的量子态重构所需测量配置数目O(rd log d),以及目前已发表论文给出的纯态唯一确定所需最少观测次数4d..5.同时给出最少观测次数对应的最优观测算符集的构建方案,并通过仿真实验对本文所提出的量子态估计方法进行验证,实验中重构保真度均达到97%以上.
The number of complete observables required in quantum state tomography is d2(d = 2 n), which increases exponentially with the qubit number n of the quantum system, makes the reconstruction of the high dimensional quantum state become very difficult. In this paper, we propose a quantum two-step measurement method of the estimation of arbitrary quantum pure states with the minimum number of observables. We prove when choosing the observables of Pauli operators and the two steps measurement method proposed in this paper, the minimum number of observables required for the estimation of an n-qubit eigenstate is memin = n, and the minimum number of a superposition state consisting of l(2 6 l 6 d) nonzero eigenvalues satisfies msmin = d + 2 l.. 3. Either the number of eigenstate or super-position state is far less than the number of measurement configurations required by compressive sensing O(rd log d), and the minimum number of observables for pure states uniquely determination 4 d..5 in published papers up to now. We also give the method of selecting the corresponding observable sets, called the optimal observable set in this paper. Mathematical simulation experiments are carried out to validate the method of pure state reconstruction based on adaptive measurements.The fidelities in our experiments are all over 97%.
The number of complete observables required in quantum state tomography is d2(d = 2 n), which increases exponentially with the qubit number n of the quantum system, makes the reconstruction of the high dimensional quantum state become very difficult. In this paper, we propose a quantum two-step measurement method of the estimation of arbitrary quantum pure states with the minimum number of observables. We prove when choosing the observables of Pauli operators and the two steps measurement method proposed in this paper, the minimum number of observables required for the estimation of an n-qubit eigenstate is memin = n, and the minimum number of a superposition state consisting of l(2 6 l 6 d) nonzero eigenvalues satisfies msmin = d + 2 l.. 3. Either the number of eigenstate or super-position state is far less than the number of measurement configurations required by compressive sensing O(rd log d), and the minimum number of observables for pure states uniquely determination 4 d..5 in published papers up to now. We also give the method of selecting the corresponding observable sets, called the optimal observable set in this paper. Mathematical simulation experiments are carried out to validate the method of pure state reconstruction based on adaptive measurements.The fidelities in our experiments are all over 97%.
引文
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[5]D’ARIANO G M,PRESTI P L.Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation[J].Physical Review Letters,2001,86(19):4195–4198.
[6]CHEN J,DAWKINS H,JI Z.Uniqueness of quantum states compatible with given measurement results[J].Physical Review A,2013,88(1):012109.
[7]LI K,ZHANG J,CONG S.Fast reconstruction of high-qubit-number quantum states via low-rate measurements[J].Physical Review A,2017,96(1):012334.
[8]ZHENG K,LI K,CONG S.A reconstruction algorithm for compressive quantum tomography using various measurement sets[J].Scientific Reports,2016,6:38497.
[9]ZHANG J,LI K,CONG S.Efficient reconstruction of density matrices for high dimensional quantum state tomography[J].Signal Processing,2017,139:136–142.
[10]RIOFRIO C A,GROSS D,FLAMMIA S T.Experimental quantum compressed sensing for a seven-qubit system[J].Nature Communications,2017,8:15305.
[11]GROSS D.Recovering low-rank matrices from few coefficients in any basis[J].IEEE Transactions on Information Theory,2011,57(3):1548–1566.
[12]D’ARIANO G M,PARIS M G A,SACCHI M F.Quantum tomography[J].Advances in Imaging and Electron Physics,2003,128:206–309.
[13]FINKELSTEIN J.Pure-state informationally complete and“really”complete measurements[J].Physical Review A,2004,70(5):052107.
[14]PARIS M,REHACEK J.Quantum State Estimation[M].Berlin:Springer Publishing Company,Incorporated,2010.
[15]CARMELI C,HEINOSAARI T,SCHULTZ J.How many orthonormal bases are needed to distinguish all pure quantum states?[J].European Physical Journal D,2015,69(7):179–189.
[16]BUZEK V,DROBNY G,ADAM G.Reconstruction of quantum states of spin systems via the Jaynes principle ofmaximum entropy[J].Journal of Modern Optics,1997,44(11/12):2607–2627.
[17]HRADIL Z.Quantum-state estimation[J].Physical Review A,1997,55(3):1561–1564.
[18]OPATRNY T,WELSCH D G,VOGEL W.Least-squares inversion for density-matrix reconstruction[J].Physical Review A,1997,56(3):1788.
[2]OLIVEIRA I S,BONAGAMBA T J,SARTHOUR R S.NMR Quantum Information Processing[M].Amsterdan:Elsevier,2007.
[3]LI Z,YUNGMH,CHEN H.Solving quantum ground-state problems with nuclear magnetic resonance[J].Scientific Reports,2011,1(88):1–8.
[4]FANO U.Description of states in quantum mechanics by density matrix and operator techniques[J].Reviews Of Modern Physics,1957,29(1):74–93.
[5]D’ARIANO G M,PRESTI P L.Quantum tomography for measuring experimentally the matrix elements of an arbitrary quantum operation[J].Physical Review Letters,2001,86(19):4195–4198.
[6]CHEN J,DAWKINS H,JI Z.Uniqueness of quantum states compatible with given measurement results[J].Physical Review A,2013,88(1):012109.
[7]LI K,ZHANG J,CONG S.Fast reconstruction of high-qubit-number quantum states via low-rate measurements[J].Physical Review A,2017,96(1):012334.
[8]ZHENG K,LI K,CONG S.A reconstruction algorithm for compressive quantum tomography using various measurement sets[J].Scientific Reports,2016,6:38497.
[9]ZHANG J,LI K,CONG S.Efficient reconstruction of density matrices for high dimensional quantum state tomography[J].Signal Processing,2017,139:136–142.
[10]RIOFRIO C A,GROSS D,FLAMMIA S T.Experimental quantum compressed sensing for a seven-qubit system[J].Nature Communications,2017,8:15305.
[11]GROSS D.Recovering low-rank matrices from few coefficients in any basis[J].IEEE Transactions on Information Theory,2011,57(3):1548–1566.
[12]D’ARIANO G M,PARIS M G A,SACCHI M F.Quantum tomography[J].Advances in Imaging and Electron Physics,2003,128:206–309.
[13]FINKELSTEIN J.Pure-state informationally complete and“really”complete measurements[J].Physical Review A,2004,70(5):052107.
[14]PARIS M,REHACEK J.Quantum State Estimation[M].Berlin:Springer Publishing Company,Incorporated,2010.
[15]CARMELI C,HEINOSAARI T,SCHULTZ J.How many orthonormal bases are needed to distinguish all pure quantum states?[J].European Physical Journal D,2015,69(7):179–189.
[16]BUZEK V,DROBNY G,ADAM G.Reconstruction of quantum states of spin systems via the Jaynes principle ofmaximum entropy[J].Journal of Modern Optics,1997,44(11/12):2607–2627.
[17]HRADIL Z.Quantum-state estimation[J].Physical Review A,1997,55(3):1561–1564.
[18]OPATRNY T,WELSCH D G,VOGEL W.Least-squares inversion for density-matrix reconstruction[J].Physical Review A,1997,56(3):1788.