量子通信中的纠缠特性研究

Size: px

Start display at page:

Download "量子通信中的纠缠特性研究"

彩苍
7 years ago
Views:

1 南京大学本科生毕业论文 ( 设计 ) 中文摘要毕业论文题目 : 量子通信中的纠缠特性研究电子科学与工程学院电子信息科学与技术专业 009 级本科生姓名 : 李超凡指导教师 ( 姓名职称 ): 刘建才副教授摘要 : 量子纠缠现象自上个世纪 30 年代被提出以来, 一直是物理学研究的热门问题, 这一现象集中体现了许多不同于经典世界的特性, 典型的就是非定域性, 即对一对量子纠缠系统中的一个进行测量会即时地影响另一个量子系统的状态最初, 非定域性被用来否定量子力学的完备性, 一些人认为应该存在符合定域性假设的理论来描述量子力学所预言的现象但是, 在 0 世纪的 60 年代至 80 年代, 理论及实验的进展最终确认不可能存在满足定域性假设的理论来描述量子力学, 量子纠缠的非定域特性却成为了量子信息领域的重要资源随着量子远程传态等量子通信方案的提出并被实验证实, 量子纠缠在量子信息中的运用变得愈加重要本文介绍量子信息中的一些基本概念, 并着重以量子传态为背景, 探讨了量子纠缠与量子测量之间的关系, 阐明了量子纠缠对于量子测量的影响同时, 利用一个具体案例说明了量子测量也可以影响量子纠缠, 即所谓纠缠交换现象, 并说明了纠缠交换在理论上可以由量子传态导出关键词 : 量子通信 ; 量子纠缠 ; 量子传态 ; 超密编码 ; 纠缠交换

2 南京大学本科生毕业论文 ( 设计 ) 英文摘要 THESIS:Entanglement Research in Quantum Teleportation DEPARTMENT :Department of Electronic Science and Engineering SPECIALIZATION: Electronic and Information Science UNDERGRADUATE: Li, Chaofan MENTOR: Assoc.Prof. Liu, Jiancai ABSTRACT : Since it was proposed in 30s of 0 th century, quantum entanglement was always a topic of research within physics community. This phenomenon reflects many strange properties--which even don t exist in classical world--especially the non-locality, referring to the phenomenon that measuring one quantum system of an entangled pair will instantaneously change the quantum state of the other one. When it was first proposed, the non-locality of quantum entanglement was used as an example to demonstrate that the theory of quantum mechanics was incomplete. And, some physicists believed that there should be an theory not violating the local realism hypothesis to describe the phenomenon predicted by quantum mechanics. However, to the 80s of 0 th century, many experiments had proved that it is impossible for a theory not violating local realism to describe those strange phenomenon. And physicists gradually recognized the non-locality of entanglement as a resource for quantum information research. It was increasingly important as some applications of entanglement like

3 quantum teleportation was proposed and proved by experiments. This thesis introduces some basic concepts of quantum information and then discusses the relationship between quantum entanglement and quantum measurement within the context of quantum teleportation. It demonstrate how the quantum entanglement influence the quantum measurement. And with the aid of an example of entanglement swapping of photons published recently, this thesis illustrates the effects of the quantum measurement on the quantum entanglement, i.e. Entanglement swapping. And then it shows that the quantum swapping can be regarded as the teleportation of mixed states and the quantum swapping of photons that don t coexist as the teleportation known states. KEY WORDS:Quantum communication ; Quantum entanglement ; Teleportation;Superdense coding;entanglement Swapping

5 A A B AB A A B A I U a a A B AB A B A A A B A U U I a a a aa

7 A B A B A B C C ψ C ψ C B A A C B A A ψ C A

8 ψ ψ a 0 + b ab ψ ψ ψ a + b 1 ϕ ψ ϕ ψ

9 {M m } mm p(m) ψ M mm m ψ p(m)1 p(m) ψ ( M mm m ) ψ 1 m m m ψ M mm m I m M m ψ» ψ M mm m ψ Q ˆQ ˆQ{M m } ˆQ m mm mm m ψ M m M m ψ ψ M mm m ψ ψ M mm mm m M m ψ ψ M mm m ψ ψ M mm m ψ M m ψ ψ M mm m ψ ψ M m Mm ψ ψ M mm m ψ M m ψ» ψ M m M m ψ

10 M m» ψ M m M m ψ M m» ψ M mm m ψ M m M m M m i i m i m i m {M m } {M m } i m { i m } ψ i m i m ψ» ψ im i m ψ i m ψ i m ψ i m i m im ψ i m ψ 1 ψ i m {i m } ψ a 0 + b 1 {M 0 0 0, M } 0 ψ 0 0 ψ» ψ 0 0 ψ 0 ψ 0 ψ {M 0, M 1 } 0 ψ ψ

11 4 AB A B (a a 1 1 ) (b b 1 1 ) a 0 b a 0 b a 1 b a 1 b 1 11 A B AB a 0b 0 + a 0b 1 + a 1b 0 + a 1b 1 a 0 (b 0 + b 1) + a 1 (b 0 + b 1) AB Φ + β 00 Ψ + β 01 Φ β 10 Ψ β

12 U I U U ψ U U ψ A B CCAB

13 CNOT AB A a 1 0 +a 1 B b b 1 AB a 1 b a 1 b 01 + a b a b 11 CNOT AB CNOT ( a 1 b 1 ) a b a 1 b a 1 b a b a b a 1 b 1 a 1 b a b a b 1 a 1 b a 1 b 01 + a b 10 + a b 1 11 AB AB AB A A 1 B

14 H ±1 H h 0 h 0 h 1 h 1 h 0 h 1 h h ψ 1 ψ 1 0 ψ 1 h 0 h 1 0 1

15 ψ ψ ψ ψ 0 ψ ψ 1 ψ 1 ψ ψ 1 ψ CNOT ( ψ 1 ψ CNOT [ 1 ( )] ( ) ( ) A B

16 B C XY Z X X A B Y Z B C A A α a 0 + b B C ab A B C BC 1 ( B C B C ) 0 1 BC 1 ( ) ABC (a 0 + b 1 )( ) 1 (a 001 a b 101 b 110 ) 1 ( 00 a 1 01 a b 1 11 b 0 )

17 ABC 1 [( Φ+ + Φ )a 1 ( Ψ + + Ψ )a 0 + ( Ψ + Ψ )b 1 ( Φ + Φ )b 0 ] 1 [ Φ+ (a 1 b 0 ) + Φ (a 1 + b 0 ) + Ψ + ( a 0 + b 1 ) + Ψ ( a 0 b 1 )] ABC BC ABC ABC AB C AB C α ab AB {M 00 Φ + Φ +, M 01 Ψ + Ψ +, M 10 Φ Φ, M 11 Ψ Ψ } AB AB C C AB Φ C a 1 + b 0 XY A α X A B AB A B A B AB

18 {M 0 0 0, M } A B X X CNOT Φ + CNOT CNOT Φ CNOT CNOT Ψ + CNOT CNOT Ψ CNOT H H Φ + 00 Ψ + 01 Φ 10 Ψ 11 β

19 Φ Φ + Ψ + σ σ (σ 1 I) Φ + Ψ (σ 3 I) Φ + Φ 00 11

20 Y Zσ 1 σ 3 ψ ψ ψ {M 0 0 0, M } (1/ ) 1/ ψ p(0) 0 p(1) /

21 ψ ρ ρ ψ ψ ψ i p(i)ρ ρ i p(i) ψ i ψ i {M 1, M,..., M m }m ψ i m

22 i p(m) i p(m i)p(i) p(i) ψ i M mm m ψ i i (M mm m p(i) ψ i ψ i ) (M mm m ρ) i AB A ρ A B ( AB AB ) B ( A A B B ) A A ( B B ) A A B B A A A AB A A AB AB A B A B (ρ)1(ρ )1 (ρ) 1 (ρ) ( i p(i) ψ i ψ i ) i p(i)( ψ i ψ i ) i p(i) ψ i ψ i i p(i) 1 ρ ( ψ ψ ) ψ ψ ψ ψ ψ ψ ρ (ρ ) (ρ) 1

23 AB i p i A i B i { A i }{ B i } B C Ψ AB B C B C C B B AB ab 1 0 C AB 1 [ Φ+ (a b) + Φ (a + b) + Ψ + ( a + b) + Ψ ( a b)] {M 00 Φ + Φ +, M 10 Φ Φ, M 01 Ψ + Ψ +, M 11

24 Ψ Ψ } p(00) BA M 00M 00 AB BA Φ + Ψ + AB a b Φ + Φ + Φ + Φ + 4 a b 4 p(01) BA M 01M 01 AB BA Ψ + Ψ + AB a + b Ψ + Ψ + Ψ + Ψ + a + b 4 4 p(10) BA M 10M 10 AB BA Φ Φ a + b AB Φ Φ Φ Φ a + b 4 4 p(11) BA M 11M 11 AB BA Ψ Ψ AB a b Ψ Ψ Ψ Ψ a b 4 4 p(00)p(01)p(10)p(11) ψ ψ ψ 1 ψ 1 ψ ψ ψ 1 ψ ψ B C C B ψ 1 ψ ψ 1 ψ a 00 + b 11 ab a + b 1 ρ 1 (a 00 + b 11 )(a 00 + b 11 ) a b ab ba a 0 ò ïa 0 0 b 0 b a 0 0 ab ba 0 0 b

25 ρ 1 ρ 1 (ρ 1 ) a 0 0 ab ( a 0 ) ( ab ) ( ) ( ) ( ) ba 0 0 b ba 0 0 b a 0 0 b (ρ 1) (ρ 1) ( a 0 ) 0 b a 4 + b 4 1(ρ )1 ψ 1 ψ 0 a 1 b {M 0 h + h +, M 1 h h } h ± 0 ± 1 m(m mm m ρ) h ± h ± 0 p(0) ( h + h + a 0 ) 0 b ( a 0 ) b 1 ( a b a b 1

26 ψ 1 ψ a 00 + b 11 h ± h ± ab h ± h ± ψ a 0 + b 1 0 1/ab p(0) ψ h + h + h + h + ψ ψ h + h + ψ a + b ψ 1 ψ a 00 + b 11 ψ a 0 + b 1 ABC ( Ψ + + Φ + + Ψ + Φ )/ ψ 1 ψ {M 0 h + h +, M 1 h h } ψ 1 ψ h ± ψ 1 ψ a h+ + h 0 + b h+ h 1 h+ (a 0 + b 1 ) + h (a 0 b 1 ) {M 0 h + h +, M 1 h h } ψ 1 ψ ψ h+ + h {M 0 h + h +, M 1 h h } ψ 1 ψ ψ a 0 + b 1 {M 0 0 0, M } ψ 1 ψ ψ a 0 +b 1 a b {M i i i } ψ i p i i AB i p i i ψ i p i i p i 1 {M i i i }{M i i i }

27 ψ i p i i AB ρ ij ij p i p j i ψ i j ψ j p i p j i j ψ i ψ j AA ρ A B ( ij p i p j i j ψ i ψ j ij ij p i p j( ψ i ψ j ) i j p i p j(ij) i j (ij)tr( ψ i ψ j ){M i i i } ψ i p i i k p(k) ( k k k k ρ A ) ( k k ρ A ) ij p i p j(ij)( k k i j ) j j p k (kk)( k j ) p k (kk) j k p k (ij)i j (ij) ( ψ i ψ i ) 1 M k k k i p i i p(k) i p i p i i k k k k i p k AB i p i i

28 A A A A ABCA a 0 + b C 0 1

29 [1] Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical review, 47(10):777, [] William K Wootters and Wojciech H Zurek. A single quantum cannot be cloned. Nature, 99(5886):80 803, 198. [3] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. Teleporting an unknown quantum state via dual classical and einsteinpodolsky-rosen channels. Phys. Rev. Lett., 70: , Mar [4] Jaewoo Joo, Young-Jai Park, Sangchul Oh, and Jaewan Kim. Quantum teleportation via a w state. New Journal of Physics, 5(1):136, 003. [5] Manzoor Ikram, Shi-Yao Zhu, and M Suhail Zubairy. Quantum teleportation of an entangled state. Physical Review A, 6():9415, 000. [6] Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. Cambridge university press, 010. [7] Akira Furusawa, Jens Lykke Sørensen, Samuel L Braunstein, Christopher A Fuchs, H Jeff Kimble, and Eugene S Polzik. Unconditional quantum teleportation. Science, 8(5389): , [8] Lev Vaidman. Teleportation of quantum states. PHYSICAL REVIEW A, 49(), [9] John Von Neumann. Mathematical foundations of quantum mechanics. Princeton University Press,

30 [10] David Jeffery Griffiths. Introduction To Quantum Mechanics, /E. Pearson Education India, 005. [11] Xiaoqin Li, Yanwen Wu, Duncan Steel, D Gammon, TH Stievater, DS Katzer, D Park, C Piermarocchi, and LJ Sham. An all-optical quantum gate in a semiconductor quantum dot. Science, 301(5634): , 003. [1] Ferdinand Schmidt-Kaler, Hartmut Häffner, Mark Riebe, Stephan Gulde, Gavin PT Lancaster, Thomas Deuschle, Christoph Becher, Christian F Roos, Jürgen Eschner, and Rainer Blatt. Realization of the cirac zoller controlled-not quantum gate. Nature, 4(6930): , 003. [13] Charles H Bennett and Stephen J Wiesner. Communication via one-and two-particle operators on einstein-podolsky-rosen states. Physical review letters, 69(0): , 199. [14] E Megidish, A Halevy, T Shacham, T Dvir, L Dovrat, and HS Eisenberg. Entanglement swapping between photons that have never coexisted. Physical Review Letters, 110(1): 10403, 013. #include <lapacke.h> #include <math.h> #include <complex.h> #define min(a,b) ((a)>(b)?(b):(a)) int schmidt_decomposition( int da, int db, /*The dimension of the first and the second *quantum state. */ lapack_complex_double* v, /*the amplitute of every states, *index is the just the state as binary value. *For example,

31 *.8* 01> +.6* 11> *is represented as * [0,0.8, 0,0.6] * * 00B 01B 10B 11B */ */ */ */ double* sv, /*the coefficients of the schmidt decompsition lapack_complex_double* ia, /*the first Schmidt base lapack_complex_double* ib){ /*the second Schmidt base register int i,j; lapack_complex_double ibt[db*min(da,db)]; double superb[min(da,db) -1]; if(lapacke_zgesvd(lapack_row_major, 'A', 'A', da, db, v, db, sv, ia, min(da,db), ibt, db, superb) > 0){ } return(-1); /*ib conjugate transpose of ibt */ for(i0;i<da;i++){ for(j0;j<db;j++){ ib[da*j+i] conj(ibt[db*i+j]);}} return 0; } #include <lapacke.h> #include "ruby.h" #include "complex.h" #define min(a,b) ((a)>(b)?(b):(a)) extern int schmidt_decomposition( int da, int db, lapack_complex_double * v,

32 double *sv, lapack_complex_double *ia, lapack_complex_double * ib); VALUE cqti ; static VALUE sd(value self, VALUE da_v, VALUE db_v, VALUE v_v){ int da, db; long len; register int i,j; if(fixnum_p(da_v)&&fixnum_p(db_v)){ da FIXINT(da_v); db FIXINT(db_v); }else{ rb_raise(rb_eargerror, "The first two arguments must be integers"); } Check_Type(v_v, T_ARRAY); len RARRAY_LEN(v_v) ; } if(len! da * db){ rb_raise(rb_eargerror, "Length of the array must equal the product of the first two arguments");} lapack_complex_double v[da*db],ia[da*min(da,db)],ib[db*min(da,db)]; double sv[min(da,db)]; for(i0;i<len;i++){ VALUE ary_i rb_ary_entry(v_v, i); switch(type(ary_i)){ case(t_complex): v[i]numdbl(rcomplex(ary_i)->real) +NUMDBL(RCOMPLEX(ary_i)->imag)*I; break; case(t_float): case(t_fixnum): case(t_rational): v[i] NUMDBL(ary_i); break; default: rb_raise(rb_eargerror, "Contains elements of illegal type within the array"); break;

33 } } schmidt_decomposition(da,db,v,sv,ia,ib); VALUE retary, rethash[min(da,db)], va[min(da,db)], vb[min(da,db)]; for(j0;j<min(da,db);j++){ rethash[j] rb_hash_new(); va[j] rb_ary_new(da); for(i0;i<da;i++){ rb_ary_store(va[j], i, rb_complex(rb_float_new(creal(ia[i*da+j ])), rb_float_new(cimag(ia[i*da+j]))));} rb_hash_aset(rethash[j], IDSYM(rb_intern("ia")), va[j]); vb[j] rb_ary_new(db); for(i0;i<db;i++){ rb_ary_store(vb[j], i, rb_complex(rb_float_new(creal(ib[i*db+j ])), rb_float_new(cimag(ib[i*db+j]))));} rb_hash_aset(rethash[j], IDSYM(rb_intern("ib")), vb[j]); rb_hash_aset(rethash[j], IDSYM(rb_intern("sv")), rb_float_new(sv[j ])); } retary rb_ary_new4(min(da,db), rethash); return retary; void Init_qti(){ // cqti rb_define_class("qti", rb_cobject); rb_define_method(rb_cobject, "sd", sd, 3);//like a normal function, because everything is the child of Object class } irb(main):00:0> require './qti.so' irb(main):015:0> sd(,,[3**(-1.0/),3**(-1.0/), 0,3**(-1.0/)]) > [{:ia>[( i), ( i)], :ib>[( i), ( i)], :sv> }, {:ia>[( i), ( i)],

34 :ib>[( i), ( i)], :sv> }]

国家实验教学示范中心

国家实验教学示范中心附件 2: 辽宁省实验教学示范中心建设单位申请书学校名称 : 中心名称 : 渤海大学物理实验教学中心类别代码 : 1 中心主任姓名 : 高亚军联系电话 ( 手机 ): 13941687355 中心网址 : www.bhu.edu.cn/page/depart/slxy/syzx/index.html