光场的熵压缩性质及量子非局域性
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摘要
量子光学是研究光场的量子统计特性及采用全量子理论分析光与物质相互作用的量子特征的学科。实验上已证实，量子光场存在着三种非经典效应：光子反聚束效应，亚泊淞分布和压缩态。具有压缩效应的量子态，由于其某个物理量的量子涨落低于标准量子涨落，使得它在光通讯等光学领域有着广阔的应用前景。在大量文献中，一般是从海森堡测不准关系出发，用光场的某一正交分量的均方差是否小于真空极限判断光场的该正交分量是否出现压缩效应。那么是否有更灵敏的方法用以判断光场的压缩效应，本文尝试用信息熵来衡量光场的量子涨落。我们建立了信息熵的概念，并从熵的测不准关系出发，类比方差压缩，研究真空态与粒子数态所形成的叠加态光场的熵压缩性质。
量子信息学是以量子力学为基础的量子信息理论，是量子力学与信息科学相结合的产物。量子信息学包括量子密码、量子通信、量子计算机等几个方面，近年来在理论和实验上都取得了重大的突破。其中量子密码就是密码术与量子力学结合的产物，是基于非局域性的量子相关效应所形成的量子密码传送方案。而量子力学自诞生以来，关于量子力学的思想基础和基本问题的争论，从来就没有停止过。本文中研究了量子力学的基本特性之一——非局域性；回顾了EPR佯谬，对Bell不等式定理及其相关的推广不等式进行概述，并利用Bell-CHSH不等式研究两个原子所构成的量子纠缠态的量子非局域性。
Quantum optics is a subject on studying quantum statistical and analyzing the quantum character of light interacting with matter. It has been proved by experiment that quantum states of cavity field can exhibit three non-classical features, i.e. photo antibunching effect sub-Poissonian statistics distribution and squeezing states. In the squeezing states, because the quantum fluctuation of its one quadrature is below that of the standard level, the squeezing states have comprehensive applications in optical communication systems, and so on. In many literatures, people commonly begin with the Heisenberg uncertainty principle, verdict that the light field is squeezing while the quantum fluctuation of its orthogonal variant below that of the vacuum's. Does there exist a more sensitive measure to judge the squeezing effect of the light field, we try to do it by the information entropy. In this article, we built the concept of the information entropy, and starting on the entropy uncertainty relations, comparing w
ith the Heisenberg uncertainty relations, studied the entropy squeezing effect of the superposition of the vacuum state and the Fock states.
Quantum information which based on the quantum mechanics is the result of combining quantum mechanics and information science. It mainly includes quantum cryptography quantum communication quantum computer, and so on. In recent years, it has been made great breakthrough at the aspect of theory and experiment. For example quantum cryptography is a code teleportation scheme which combining cryptography with quantum mechanics and based on the quantum nonlocality. But the debating on the quantum mechanics' essence problem is never stopping when it was born. This thesis studied the quantum nonlocality which is one of the quantum mechanics' Characters. Firstly we
summarized the EPR paradox, and Bell inequality theorem as well as its generalized Bell type inequalities. Then using the CHSH inequality we studied the quantum nonlocality of two spin-1/2 's entangled quantum states.
量子信息学是以量子力学为基础的量子信息理论，是量子力学与信息科学相结合的产物。量子信息学包括量子密码、量子通信、量子计算机等几个方面，近年来在理论和实验上都取得了重大的突破。其中量子密码就是密码术与量子力学结合的产物，是基于非局域性的量子相关效应所形成的量子密码传送方案。而量子力学自诞生以来，关于量子力学的思想基础和基本问题的争论，从来就没有停止过。本文中研究了量子力学的基本特性之一——非局域性；回顾了EPR佯谬，对Bell不等式定理及其相关的推广不等式进行概述，并利用Bell-CHSH不等式研究两个原子所构成的量子纠缠态的量子非局域性。
Quantum optics is a subject on studying quantum statistical and analyzing the quantum character of light interacting with matter. It has been proved by experiment that quantum states of cavity field can exhibit three non-classical features, i.e. photo antibunching effect sub-Poissonian statistics distribution and squeezing states. In the squeezing states, because the quantum fluctuation of its one quadrature is below that of the standard level, the squeezing states have comprehensive applications in optical communication systems, and so on. In many literatures, people commonly begin with the Heisenberg uncertainty principle, verdict that the light field is squeezing while the quantum fluctuation of its orthogonal variant below that of the vacuum's. Does there exist a more sensitive measure to judge the squeezing effect of the light field, we try to do it by the information entropy. In this article, we built the concept of the information entropy, and starting on the entropy uncertainty relations, comparing w
ith the Heisenberg uncertainty relations, studied the entropy squeezing effect of the superposition of the vacuum state and the Fock states.
Quantum information which based on the quantum mechanics is the result of combining quantum mechanics and information science. It mainly includes quantum cryptography quantum communication quantum computer, and so on. In recent years, it has been made great breakthrough at the aspect of theory and experiment. For example quantum cryptography is a code teleportation scheme which combining cryptography with quantum mechanics and based on the quantum nonlocality. But the debating on the quantum mechanics' essence problem is never stopping when it was born. This thesis studied the quantum nonlocality which is one of the quantum mechanics' Characters. Firstly we
summarized the EPR paradox, and Bell inequality theorem as well as its generalized Bell type inequalities. Then using the CHSH inequality we studied the quantum nonlocality of two spin-1/2 's entangled quantum states.
引文
1．彭金生，李高翔．近代量子光学导论[M]．科学出版社，1996：165．
2．郭光灿．量子光学[M]．北京：高等教育出版社，1990：534．
3．吴龙泉，李洪才．关于几种叠加态的压缩性质的讨论[J]．福建师范大学学报(自然科学版)，2002,18(1)：24-28．
4．Li H C, Wu L Q. Generation of Superpositions of Squeezed States Via Raman Interaction [J]. Opt.Comm.2001,197:97-101.
5．李洪才，吴龙泉．在腔场与可移动的镜子系统中对镜子位置的测量[J]．光子学报，2002，31(10)：1169-1173．
6．李承祖，黄明球等．量子通信和量子计算[M]．湖南．长沙：国防科技大学出版社，2000：3．
7．Buzek V, Keitel H, Knight P L. Sampling Entropies and Operational Phase-space Measurement[J]. Phy. Rev. (A), 1995,51 (3):2575-2593.
8．Shannon C E, Weaver W, The Mathematical Theory of Communication.[M]. Urbana: The University of Illinois Press, 1963,81-96.
9．Vaccaro J A, Orlowski A. Phase Properties of Kerr Media Via Variance and Entropy as of Uncertainty[J]. Phys. Rev. (A), 1995,51 (5): 4172-4180.
10．闫珂柱，孔祥和，夏云杰．奇偶相干态的Wehrl熵和shannon熵[J]．光学学报，1998，18(6)：717-721．
11．于国臣，屈爱存，王连水．叠加相干态的Wehrl熵和shannon熵[J]．量子电子学报，2000，17(1)：26-30．
12．Uffink J B M, Hilgevoord J. Uncertainty Principle and Uncertainty Relation [J]. Found.Phys., 1975,15(9):925-944.
13．Beckner W. Inequalities in Fourier Analysis[J]. Ann.Math.,1975, 102(2): 159-182.
14．Bialyncki-Birula I, Mycielski J. Uncertainty Relations for Information Entropy in Wave Mechanics [J].Commun.Math.Phys., 1975,44(2):
129-132.
15. 方卯发，陈菊梅．熵测不准关系与光场的熵压缩[J]．光学学报，2001，21(1)：8-11．
16. Orlowski A. Information Entropy and Squeezing of Quantum Fluctuations[J].Phys. Rev. (A), 1997,56(4):2545-2548.
17. Jorge S R. Position-momentum Entropic Uncertainty Relation and Complementarity in Single-slit and Double-slit Experiments [J]. Phys. Rev. (A), 1998,57(3): 1519-1525.
18. Michael J W H. Universal Geometric Approach to Uncertainty, Entropy, and Information [J]. Phys. Rev. (A),1999,59(4):2602-2615.
19. H. Everett. Ⅲ .Ph.D.thesis. The Many-Worlds Interpretation of Quantum Mechanics[M],Princeton University Press.Princeton.1973.
20. Wodkiesicz K., Knight P L, Buckle S J et al.. Squeezing and Superposition States[J]. Phys. Rev. (A), 1987.35(6):2567-2577.
21. 牛万青．Bell定理的研究[M]．硕士论文．2002．5．
22. Einstein A, et al. Can Quantum-Mechanical Description of Physical Reality be Considered Complete? [J]. Phys. Rev.(47),1935:777
23. Bohm D. Quantum Theory [M]. New York: Prentice Hall, 1951.
24. Bohm D. A Suggested Interpretation of Quantum Theory in Terms of "Hidden Variable" [J]. Phys. Rev. (85), 1952:1-2.; Causality and Chance in Modern Physics [M]. New York: Harper, 1957.
25. Bell J S. On the Einstein-Podolsky-Rosen Paradox [J]. Physics, 1, 1964:195-200.
26. Aspect A, et al. Experimental Test of Bell's Inequalities Using Time -Varying Analyzers [J]. Phys. Rev. Lett (49), 1982:1804.
27. Clauser J F, Home M A, Shimony A. Proposed Experiment to Test Local Hidden-variable theories [J]. Phys. Rev. Lett, 1969,23(15):880-884
28. 张永德．量子测量和量子计算简述．曾谨言，裴寿镛主编．量子力学新进展[M]，第一辑．北京大学出版社，2000，7．
29. 李承祖．黄明球等．量子通信和量子计算[M]．湖南·长沙：国防科技大学出版社，2000：94．
30. 张永德。量子力学[M]。北京：科学出版社，2002。
31. D．Bohm，量子理论[M]。候德彭译，商务印书馆，1982。
32. S.L. Braunstein, A,Mann, and M.Reven. Maximal Violation of Bell Inequalities for Mixed States [J].Phys. Rev. Lett (68),1992:3259-3261.
33. A. Peres. Finite Violation of a Bell Inequality for Arbitrarily Large Spin [J].Phys.Rev.A, 1992,46(7):4413-4414.
34. N.Gisin, A.Peres. Maximal Violation of Bell's Inequality for Arbitrarily Large Spin [J].Phys.Lett.A, 1992,162:15-17.
35. N. Mermin. Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States[J]. Phys.Rev.Lett, 1990,65(15): 1838-1840.
36. M.Ardehali. Bell Inequalities with a Magnitude of Violation that Grows Exponentially with the Number of Particles[J].Phys.Rev.A, 1992,46(9):5375-5377.
37. N.Gisin and H.Bechmann-Pasquinucci. Bell Inequality, Bell States and Maximally Entangled States for n Qubits [J]. quant-ph/9804045.
38. A.Cabello. Bell's Inequality for n Spin-s Particles [J].quant-ph/0202126.
39. A.Cabello. Multiparty Multilevel GHZ States[J].quant-ph/0007065.
40. J. S. Bell. Speakable and Unspeakable in Quantum Mechanics[M].Cambridge 期内University Press, Cambridge. England. 1987.
41. K.Banaszk, and K.Wodkiewicz. Nonlocality of EPR State in the Winger representation [J]. Phys.Rev.A,1998,58(6):4345-4347. K.Banaszk, and K.Wodkiewicz. Testing Quantum Nonlocality in Phase Space [J].Phys.Rev.Lett, 1999,82(10):2009-2013.
42. Chen Z B, Pan J W, Hou G. Maximal Violation of Bell's Inequalities for Continuous Variable Systems[J].Phys.Rev.Lett,2002.88(4):040406.
43. P.Grangier. Nature.2001(409):774.
44. G.Weihs, et al. Viloation of Bell's Inequality under Strict Einstein Locality Conditions [J]. quant-ph/9810080.
45. A.Lamas-Linares, et al. Stimulated Emission of Polarization-entangled Photons [J].quant-ph/01100048.
46. J.C.Howel, et al. Experimental Violation of a Spin-1 Bell Inequality Using Maximally Entangled [J].Phys.Rev.Lett,2002(88):030401.
47. 周正威．博士学位论文，2001．4。
48. Chen Z B, Zhang Y D. GHZ nonlocality for continuous-variable systems [J]. Phys.Rev(A),202,65(4):044102.
49. Jeong H, Lee J, Kim M S. Dynamics of Nonlocality for a Two-mode Squeezed State in a Thermal Environment[J].Phys. Rev, (A),2000.61 (5):052101.
50. 吴强、朱国骏、张永德等，纠缠薛定谔猫态的非局域性及其在热库中的演化[J]．光学学报，2002，22(12)：1409-1412．
2．郭光灿．量子光学[M]．北京：高等教育出版社，1990：534．
3．吴龙泉，李洪才．关于几种叠加态的压缩性质的讨论[J]．福建师范大学学报(自然科学版)，2002,18(1)：24-28．
4．Li H C, Wu L Q. Generation of Superpositions of Squeezed States Via Raman Interaction [J]. Opt.Comm.2001,197:97-101.
5．李洪才，吴龙泉．在腔场与可移动的镜子系统中对镜子位置的测量[J]．光子学报，2002，31(10)：1169-1173．
6．李承祖，黄明球等．量子通信和量子计算[M]．湖南．长沙：国防科技大学出版社，2000：3．
7．Buzek V, Keitel H, Knight P L. Sampling Entropies and Operational Phase-space Measurement[J]. Phy. Rev. (A), 1995,51 (3):2575-2593.
8．Shannon C E, Weaver W, The Mathematical Theory of Communication.[M]. Urbana: The University of Illinois Press, 1963,81-96.
9．Vaccaro J A, Orlowski A. Phase Properties of Kerr Media Via Variance and Entropy as of Uncertainty[J]. Phys. Rev. (A), 1995,51 (5): 4172-4180.
10．闫珂柱，孔祥和，夏云杰．奇偶相干态的Wehrl熵和shannon熵[J]．光学学报，1998，18(6)：717-721．
11．于国臣，屈爱存，王连水．叠加相干态的Wehrl熵和shannon熵[J]．量子电子学报，2000，17(1)：26-30．
12．Uffink J B M, Hilgevoord J. Uncertainty Principle and Uncertainty Relation [J]. Found.Phys., 1975,15(9):925-944.
13．Beckner W. Inequalities in Fourier Analysis[J]. Ann.Math.,1975, 102(2): 159-182.
14．Bialyncki-Birula I, Mycielski J. Uncertainty Relations for Information Entropy in Wave Mechanics [J].Commun.Math.Phys., 1975,44(2):
129-132.
15. 方卯发，陈菊梅．熵测不准关系与光场的熵压缩[J]．光学学报，2001，21(1)：8-11．
16. Orlowski A. Information Entropy and Squeezing of Quantum Fluctuations[J].Phys. Rev. (A), 1997,56(4):2545-2548.
17. Jorge S R. Position-momentum Entropic Uncertainty Relation and Complementarity in Single-slit and Double-slit Experiments [J]. Phys. Rev. (A), 1998,57(3): 1519-1525.
18. Michael J W H. Universal Geometric Approach to Uncertainty, Entropy, and Information [J]. Phys. Rev. (A),1999,59(4):2602-2615.
19. H. Everett. Ⅲ .Ph.D.thesis. The Many-Worlds Interpretation of Quantum Mechanics[M],Princeton University Press.Princeton.1973.
20. Wodkiesicz K., Knight P L, Buckle S J et al.. Squeezing and Superposition States[J]. Phys. Rev. (A), 1987.35(6):2567-2577.
21. 牛万青．Bell定理的研究[M]．硕士论文．2002．5．
22. Einstein A, et al. Can Quantum-Mechanical Description of Physical Reality be Considered Complete? [J]. Phys. Rev.(47),1935:777
23. Bohm D. Quantum Theory [M]. New York: Prentice Hall, 1951.
24. Bohm D. A Suggested Interpretation of Quantum Theory in Terms of "Hidden Variable" [J]. Phys. Rev. (85), 1952:1-2.; Causality and Chance in Modern Physics [M]. New York: Harper, 1957.
25. Bell J S. On the Einstein-Podolsky-Rosen Paradox [J]. Physics, 1, 1964:195-200.
26. Aspect A, et al. Experimental Test of Bell's Inequalities Using Time -Varying Analyzers [J]. Phys. Rev. Lett (49), 1982:1804.
27. Clauser J F, Home M A, Shimony A. Proposed Experiment to Test Local Hidden-variable theories [J]. Phys. Rev. Lett, 1969,23(15):880-884
28. 张永德．量子测量和量子计算简述．曾谨言，裴寿镛主编．量子力学新进展[M]，第一辑．北京大学出版社，2000，7．
29. 李承祖．黄明球等．量子通信和量子计算[M]．湖南·长沙：国防科技大学出版社，2000：94．
30. 张永德。量子力学[M]。北京：科学出版社，2002。
31. D．Bohm，量子理论[M]。候德彭译，商务印书馆，1982。
32. S.L. Braunstein, A,Mann, and M.Reven. Maximal Violation of Bell Inequalities for Mixed States [J].Phys. Rev. Lett (68),1992:3259-3261.
33. A. Peres. Finite Violation of a Bell Inequality for Arbitrarily Large Spin [J].Phys.Rev.A, 1992,46(7):4413-4414.
34. N.Gisin, A.Peres. Maximal Violation of Bell's Inequality for Arbitrarily Large Spin [J].Phys.Lett.A, 1992,162:15-17.
35. N. Mermin. Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States[J]. Phys.Rev.Lett, 1990,65(15): 1838-1840.
36. M.Ardehali. Bell Inequalities with a Magnitude of Violation that Grows Exponentially with the Number of Particles[J].Phys.Rev.A, 1992,46(9):5375-5377.
37. N.Gisin and H.Bechmann-Pasquinucci. Bell Inequality, Bell States and Maximally Entangled States for n Qubits [J]. quant-ph/9804045.
38. A.Cabello. Bell's Inequality for n Spin-s Particles [J].quant-ph/0202126.
39. A.Cabello. Multiparty Multilevel GHZ States[J].quant-ph/0007065.
40. J. S. Bell. Speakable and Unspeakable in Quantum Mechanics[M].Cambridge 期内University Press, Cambridge. England. 1987.
41. K.Banaszk, and K.Wodkiewicz. Nonlocality of EPR State in the Winger representation [J]. Phys.Rev.A,1998,58(6):4345-4347. K.Banaszk, and K.Wodkiewicz. Testing Quantum Nonlocality in Phase Space [J].Phys.Rev.Lett, 1999,82(10):2009-2013.
42. Chen Z B, Pan J W, Hou G. Maximal Violation of Bell's Inequalities for Continuous Variable Systems[J].Phys.Rev.Lett,2002.88(4):040406.
43. P.Grangier. Nature.2001(409):774.
44. G.Weihs, et al. Viloation of Bell's Inequality under Strict Einstein Locality Conditions [J]. quant-ph/9810080.
45. A.Lamas-Linares, et al. Stimulated Emission of Polarization-entangled Photons [J].quant-ph/01100048.
46. J.C.Howel, et al. Experimental Violation of a Spin-1 Bell Inequality Using Maximally Entangled [J].Phys.Rev.Lett,2002(88):030401.
47. 周正威．博士学位论文，2001．4。
48. Chen Z B, Zhang Y D. GHZ nonlocality for continuous-variable systems [J]. Phys.Rev(A),202,65(4):044102.
49. Jeong H, Lee J, Kim M S. Dynamics of Nonlocality for a Two-mode Squeezed State in a Thermal Environment[J].Phys. Rev, (A),2000.61 (5):052101.
50. 吴强、朱国骏、张永德等，纠缠薛定谔猫态的非局域性及其在热库中的演化[J]．光学学报，2002，22(12)：1409-1412．