Which-way experiment with an internal degree of freedom

Transcription

1 Which-way experiment with an internal degree of freedom Konrad Banaszek Michał Karpiński Czesław Radzewicz University of Warsaw Paweł Horodecki Technical University of Gdańsk National Quantum Information Centre in Gdańsk

2 Mach-Zehnder interferometer Á p + (Á) p (Á) p (Á) = 1 2 (1 cos Á) Á

3 1 p 2 (j0i Q + j1i Q ) Which way? Á je ini i E 1 p 2 (j0i Q je 0 i E + j1i Q je 1 i E )

4 Minimum-error measurement je 0 i E je 1 i E p ¹p ¹p p Guess: upper path lower path Maximum distinguishability for Equiprobable input states Symmetric errors p ¹p D = q 1 jhe 0 je 1 ij 2 p + ¹p = 1

5 Visibility Á p + (Á) p (Á) ^% 0 Q = 1 2 Ã 1 V V 1! p (Á) 1 2 jv j = jhe 1 je 0 ij Á

6 Trade-off B.-G. Englert, Phys. Rev. Lett. 77, 2154 (1996) Which-way information D = q 1 jhe 0 je 1 ij 2 Interference visibility jv j = jhe 1 je 0 ij Bound D 2 + jv j 2 1

7 Spin Á d. For polarization-independent beam splitters and detectors we expect V = 1 ) D = 0

8 Projective spin filtering Fractional visibility Conjecture: V º at the n th pair of output ports: p º (Á) = 1 2 [pº Re(e iá V º )] D 2 + d 1 X º=0 jv º j 12 A 1

9 Example jvi jhi ^K ^K 0 Á 0 ^K 1 ^I ^I ^X ^X ^Y ^Y ^Z ^Z ^K 1 jªi = 1 p 2 (j0i Q + j1i Q ) jãi S ³ jªihªj = 1 2³ j0iq h0j 1 2 ^I S + j1i Q h1j 1 2 ^I S +(j0i Q h1j + j1i Q h0j) 1 2 (jãi S hãj)t

10 Channel properties For any choice of projective filters, X º jv º j 1 2 ) No stringent bound on distinguishability Alternative realization of the channel: 1 ³ p j0iq + j1i 2 Q ³ Ãh jhi S + Ã v jvi S jeini i E! jª 0 i QSE = 1 ³ 2 hj0i Q Ã h jhi S je 1 i E + Ã h jvi S je 2 i E +Ã v jhi S je 3 i E + Ã v jvi S je 4 i E +j1i Q ³ Ãh jhi S je 1 i E + Ã v jhi S je 2 i E +Ã h jvi S je 3 i E + Ã v jvi S je 4 i E i

11 ^% 0(i) E Environment states = 2Tr S ³Qhijª 0 i QSE hª 0 jii Q i = 0; 1 Input polarisation ^% 0(0) E ^% 0(1) E jhi S hhj jvi S hvj ³ 1 2 je1 i E he 1 j +je 2 i E he 2 j ³ 1 2 je3 i E he 3 j +je 4 i E he 4 j ³ 1 2 je1 i E he 1 j +je 3 i E he 3 j ³ 1 2 je2 i E he 2 j +je 4 i E he 4 j ³ 1 2 jhis hhj +jvi S hvj 1 4 4X i=1 je i i E he i j 1 4 4X i=1 je i i E he i j

12 Channel As the interaction with the environment does not transfer the particle between the arms of the interferometer: ³ jii Q hjj ^¾ S = jiiq hjj ij (^¾ S ); i; j = 0; 1 00 ; 11 spin channels for individual arms 01 characterizes coherence between arms (not necessarily a positive map!) Example: 00 (^¾) = 11 (^¾) = 1 d ^I Tr^¾ 01 (^¾) = 10 (^¾) = 1 d^¾t

13 Result K. Banaszek, P. Horodecki, M. Karpiński, and C. Radzewicz, Nature Commun. 4, 2594 (2013) Trade-off: D 2 + V 2 G 1 where generalized visibility is defined as: V G = d (I 01 ) ³ (^I q q ^% 0 )j + ih + j(^i ^% 1 ) ^% 0 ; ^% 1 j + i d d conditional input spin states in individual arms normalized maximally entangled state jj jj trace norm

14 Examples 1) No noise: 01 = I Independently of spin preparation, V G = q Tr^% 0 Tr^% 1 = 1 2) 01 (^¾) = (Tr^¾)^¾ 0 ; where Tr^¾ 0 = 1: The generalized visibility is given by fidelity: V G = jj p^% 0 p^%1 jj = Trq p^%0^% 1 p^%0 and reaches one iff ^% 0 = ^% 1 3) 01 (^¾) = ^¾ T =d V G = 1 d jjp^% 0 jj jj p^% 1 jj q Tr^% 0 Tr^% 1 = 1 For a pure preparation. Equality holds iff ^% 0 = ^% 1 = 1 d ^I V G = 1 d

15 Choi-Jamiołkowski isomorphism Maximally entangled state of two replicas of the system: j + i QSQ 0 S 0 QS Q 0 S 0 ³ ^ QSQ 0 S 0 = (I ) j + i QS 0 QS 0 h + j

16 Purification Purification: ^ QSQ 0 S 0 = Tr E ³ j iqsqs 0 E h j Because the particle is not transferred between paths: j i QSQ 0 S 0 E = p 1 ³ j00iqq 0j 2 0 i SS 0 E +j11i QQ 0 j 1 i SS 0 E Equivalent channel representation: (^%) Q 0 S 0 = 2dTr QSE³ j iqsq 0 S 0 E h j^%t QS

17 Environment states ^% 0(i) E = dtr SS 0 ³ (^%i ) T S j ii SS 0 E h ij ; i = 0; 1 Environment states can be purified ^% 0(i) E = Tr SS ³jE 0 (i) i SS 0 E he(i) j using je i i SS 0 E = ^U (i) SS 0 q d(^% i ) T S j ii SS 0 E ; where ^U (i) SS 0 are arbitrary unitaries.

18 Distiguishability bound Because adding subsystems can only increase distinguishability, D = 1 2 jj^%0(0) E q ^%0(1) E jj 1 jhe 0 je 1 ij 2 The strongest bound is obtained by maximizing over scalar products V G = = d max ^U (0) SS 0; ^U (1) SS 0 jhe 0 je 1 ij (I 01 ) ³ (^I q ^% 0 )j + ih + j(^i q ^% 1 )

19 Fractional visibility measurement jã ¹ 0 i S jã ¹ 1 i S jâ º 0 i S jâ º 1 i S Á p ¹º (Á) = 1 2 [p¹º + Re(V ¹º e iá )] Fractional visibility for preparation V ¹º = hâ º 0 j 01 ¹ and filter º ³ jã ¹ 0 ihã¹ 1 j jâ º 1 i

20 Estimating generalized visibility For any we have jj ^Ajj Tr( ^U ^A). Consequently, V G d ^U y ^U ^I q Trnh ( ^% T q 1 ^I)^U( ^% T 0 ^I) i (I 01 ) ³ j + ih + j o Fractional visibilities can be written as: V ¹º = dtr nh³jã ¹ 0 ihã¹ j T 1 jâ º 1 ihâ º i 0 j (I 01 ) ³ j + ih + j o Take a set of complex satisfying X ¹º ¹º ³ ¹ ¹º jã 0 ihã¹ j T 1 jâ º 1 ihâ º 0 j = ( q ^% T 1 ^I) ^U( q ^% T 0 ^I); V G X ¹º ¹º V ¹º ^U y ^U ^I

21 Projective filtering Let us take º = e iá º The operator ^U = X º satisfies e iá º ³ jã 0 ihã 1 j T jâ º 1 ihâ º 0 j ^U y ^U ^I, hence V G X º e iá º V º Maximization over arbitrary phases V G X º Á º jv º j yields

22 Experiment K. Banaszek, P. Horodecki, M. Karpiński, and C. Radzewicz, Nature Commun. 4, 2594 (2013) Preparations Filters ¹ = hh; hv; vh; vv º = hh; hv; vh; vv V ¹º Theoretical detection probability p ¹º = 1 2

23 Fractional visibilities ¹ = vv ¹ = hh

24 Fractional visibilities cont d ¹ = vh ¹ = hv

25 X ¹º Mixed input polarization ³ ¹ ¹º jã 0 ihã¹ j T 1 jâ º 1 ihâ º 0 q^% j = ( T q 1 ^I)^U( ^% T 0 ^I) ^% 0 = ^% 1 = 1 2 ^I Take hh;hh = 1 2 eiµ 1 hv;vh = 1 2 eiµ 2 vh;hv = 1 2 eiµ 3 vv;vv = 1 2 eiµ 4 and zero otherwise V G 12 e iµ 1V hh;hh + e iµ 2V hv;vh V G 12 +e iµ 3V vh;hv + e iµ 4V vv;vv Maximization over phases yields ³ jv hh;hh j + jv hv;vh j +jv vh;hv j + jv vv;vv j

26 Experimental data Uncertainty Pure preparation: V G 0:580(6) at most. Completely mixed preparation: V G 12 ³ jv hh;hh j + jv hv;vh j + jv vh;hv j + jv vv;vv j = 0:960(6) from just four measurements of fractional visibilities!

27 Conclusions Which-way information can be protected by introducing noise in the internal degree of freedom Stringent bounds on the amount of which-way information can be obtained from few visibility measurements Prepare-and-measure quantum cryptography with composite quantum systems: noise may be needed to ensure security A non-trivial set of preparations and measurements may be required to verify security

28 Choi-Jamiołkowski isomorphism j + i QSQ 0 S 0 QS Q 0 S 0 ³ ^ QSQ 0 S 0 = (I ) j + i QS 0 QS 0 h + j

29 Bipartite state Q Q Alice S S Bob ^ QSQ 0 S 0 = 3 4 j +i QQ 0h + j ^% (+) SS j i QQ 0h j ^% ( ) SS 0 Path (qubit) states: j i QQ 0 = 1 p 2 (j00i QQ 0 j11i QQ 0) Measurements in the 0/1 basis are random and perfectly correlated Cryptographic key?

30 Spin states Spin states: Q Q S S Alice SS 0 = 1 3 (^I jª i SS 0hª j); Singlet state Bob ^% ( ) SS ^ 0 = jª QSQ 0 S 0 = 3 i SS 0hª j jª i = p 1 (j01i j10i) 4 j +i QQ 0h + j ^% (+) 2 SS j i QQ 0h j ^% ( ) SS 0 ^% (+) States ^% (+) SS 0 and ^% ( ) SS 0 are orthogonal but they cannot be distinguished unambiguously by local operations and classical communication Key obtained from qubits is secure QQ 0 Distillable entanglement, bounded by log-negativity E D log is strictly less than the key rate!

31 Private states General theory: K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, Phys. Rev. Lett. 94, (2005); IEEE Trans. Inf. Theory 55, 1898 (2009) Q Q Alice S S QQ 0 - key subsystem SS 0 - shield subsystem Bob

32 Density matrix ^ QSQ 0 S 0 = 0 1 C A QQ' key security

33 State reconstruction K. Dobek. M. Karpiński, R. Demkowicz-Dobrzański, K. Banaszek, and P. Horodecki, Phys. Rev. Lett. 106, (2011) Distillable entanglement E D 0:581(4) Key rate (cqq scenario) K 0:690(7)