Solid-State Optimal Phase-Covariant Quantum Cloning Machine
aa r X i v : . [ qu a n t - ph ] S e p Solid-State Optimal Phase-Covariant Quantum Cloning Machine
Xin-Yu Pan ∗ , Gang-Qin Liu, Li-Li Yang, Heng Fan † Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: November 10, 2018)Here we report an experimental realization of optimal phase-covariant quantum cloning machinewith a single electron spin in solid state system at room temperature. The involved three statesof two logic qubits are encoded physically in three levels of a single electron spin with two Zeemansub-levels at a nitrogen-vacancy defect center in diamond. The preparation of input state andthe phase-covariant quantum cloning transformation are controlled by two independent microwavefields. The average experimental fidelity reaches 85 .
2% which is very close to theoretical optimalfidelity 85 .
4% and is beyond the bound 83 .
3% of universal cloning.
PACS numbers: 03.67.Ac, 03.67.Lx, 42.50.Dv, 76.30.Mi
Nitrogen-vacancy (NV) defect center in diamond is oneof the most promising systems to be the solid state quan-tum information processors [1, 2]. It can be individuallyaddressed, optically polarized and detected, and is withexcellent coherence properties. Both electronic and nu-clear spins at the NV centers can be well controlled. Theadvantages of the NV centers for quantum informationprocessing are their scalability, and their long coherencetime T at room temperature, which can be further pro-longed [3–6]. Despite its scalability, an individual elec-tronic spin at NV center in diamond is still very useful,such as for real applications and being a test bed forquantum algorithms [7–10].In this Letter, with a coherent superposition of all threelevels of a single electronic spin, we demonstrate the op-timal phase-covariant quantum cloning.It is well known that a quantum state can not becloned [11]. However, we can try to clone a quantumstate approximately or probabilistically, see for exam-ple [12–14]. The no-cloning theorem is fundamental forthe security of the quantum key distribution protocols inquantum cryptography, for example for the well-knownBB84 protocol [15]. The optimal cloning machine forBB84 states is the phase-covariant quantum cloning ma-chine [16–19] for which the input state is in a specifiedform | ψ i = ( | i + e iφ | i ) / √
2, i.e., all input states are lo-cated in the equator of the Bloch sphere, see FIG. 1(a).The fidelity of the phase-covariant quantum cloning ma-chine is around 85 .
4% which is better than around 83 . ∗ [email protected] † [email protected] FIG. 1: (color online) Bloch sphere and energy level for ni-trogen vacancy center in diamond.(a) The states need to becloned are in a specified form which are located in the equatorof the Bloch sphere | ψ i = ( | i + e iφ | i ) / √
2. (b) Energy levelof the NV center in diamond. (c) Two-dimension scanningconfocal image of the sample. Bright spot circled is the NVcenter we investigate. fied by an acoustic optical modulator (AOM) with a ris-ing edge of 10 ns is focused onto the sample with a micro-scope objective(numerical aperture=0.9). The fluores-cence is also collected by the same objective, and passesthrough a 650 nm long-pass filter. Fluorescence signalis detected by a single photon counting module (SPCM,Perkin-Elmer) with a National Instruments counter 6602.Second order photon correlation function g ( τ ) of centerA indicates that it is a single quantum emitter [FIG.2(a)].The Hamiltonian with electron spin zero field splittingand the electron Zeeman interaction takes the form, H = S ¯ D S + β e ~B ¯ g e S , (1)where g e and β e are the g factor and Bohr magneton forelectron, ~B is the applied magnetic field.Experimentally, a microwave radiation is sent out by acopper wire of 20 µ m diameter placed with a distance of20 µ m from the NV center. The Electron Spin Resonance(ESR) spectrum is shown in FIG. 2(b) as a function ofthe fluorescence change against the microwave frequency % c h a ng e i n f l uo r esce n ce % c h a ng e i n f l uo r esce n ce Rabi1
Microwavedurationt(ns)Microwavedurationt(ns)
Rabi2 (d)(c) (b)(a) % c h a ng e i n f l uo r esce n ce Microwavefrequency(MHz)Delaytime(ns) g ( ? ) FIG. 2: (color online) Second order photon correlation func-tion, ESR spectrum, Rabi oscillations of two transitions. (a)Second order photon correlation function g ( τ ) of the NV cen-ter. (b) ESR spectrum of the NV center. Two main peakscorrespond to m s =1 and m s =-1. (c) Rabi oscillations for thetransition between m s =0 and m s =1. (d) Rabi oscillations forthe transition between m s =0 and m s =-1. without external magnetic field, this is due to symmetrybreaking of this NV center corresponding to a non-zeromagnetic field. The two resonant frequencies correspondto the transitions of m s =0 to m s =1 and m s =0 to m s =-1.We denote the corresponding states as | m s = 0 i p and | m s = ± i p , where the subindex p means those statesare physical states to differ them from the logic qubits.In our experiment, the cloning processing is to transferstate | ψ i| i to two copies | output i = √ | i + e iφ | i + e iφ | i . We use the encoding scheme: − i | i ∼ | m s = − i p ; | i ∼ | m s = 0 i p ; − i | i ∼ | m s = 1 i p .To control the electron spin state, first, a laser pulseinitializes the spin state to | m s = 0 i p ; then the microwavepulses of weak power are used to manipulate the spinstate; finally, the spin state is read out by the fluorescenceintensity under a second laser excitation. The Rabi os-cillation of the electron spin of single NV center is shownin FIG. 2(c) and (d), the scatter points are experimentdata and each point is a statistical average result typi-cally repeated 10 times, the red line is the fitting usinga function of a cosine with an exponential decay.FIG. 3 shows the scheme for quantum phase cloning.The output state should be a superposition state | output i p = | m s = 0 i p + i | m s = 1 i p + i √ | m s = − i p . The scheme for measure is by MW1 to confirm | output i p is superposed by a pure state | m s = 0 i p + i √ | m s = − i p , and by MW2 to confirm the pure state form | m s = 0 i p + i | m s = 1 i p . Combination of those mea-sured results indicate that the output is in form | output i p .By analyzing the experiment data, the exact form of theoutput state and the fidelity can be obtained. We thencan repeat those experimental steps except that with dif-ferent state preparation. The experimental results areshown in FIG. 4.The measured data by MW1 shows clearly Rabi os-cillation which represents that the state of NV is in a (d)(c) (b)(a) ? /2 ? /2 LaserMW1MW2Counter
Preparationandclone ReadoutbyRabi2 ? /2 ? /2 LaserMW1MW2Counter
Preparationandclone ReadoutbyRabi1 ? /2 ? /2 LaserMW1MW2Counter
Preparationandclone ReadoutbyRabi2 ? /2 ? /2 LaserMW1MW2Counter
Preparationandclone ReadoutbyRabi1
FIG. 3: (color online)Scheme for quantum phase cloning.(a)A MW1 π/ | m s = 0 i p + i | m s = − i p ) / √ π/ | m s = 0 i p and | m s = − i p . (b)The same pulse sequence for the phase cloning, but we mea-sure the Rabi oscillations for transition between | m s = 0 i p and | m s = 1 i p . (c) A MW1 3 π/ | m s =0 i p − i | m s = − i p ) / √ π/ | m s = 0 i p − i | m s = − i p ) / √ -2 0 2 4 6 8 10 12 14 16-0.20.00.20.40.60.81.01.2 0 5 10 15 20-0.20.00.20.40.60.81.01.20 5 10 15 20-0.20.00.20.40.60.81.01.2 0 5 10 15 20-0.20.00.20.40.60.81.01.2 P r ob a b ili t y o f s t a t e | > ( a . u . ) Phase ? ( rad) StdExp (d)(c) (b)(a)
FIG. 4: (color online)Measured results of the quantum phasecloning. (a) Red line is the standard dependence of probabil-ity of the state | m s = 0 i p on the phase of microwave pulse,by applying pulse sequence Figure 3(a), the black square isthe experiment results for Rabi oscillations of transition withMW1, the start point of this curve determines the popula-tion probability at the state | m s = 0 i p is 33 %, theory isalso around 33%. (b) Start point of the curve determines theprobability at the state | m s = 0 i p is 48%, theory is 50% . (c)and (d) The start points of the curves determines the prob-ability at the state | m s = 0 i p are 36% and 44%, theory are33% and 50% . superposed state. Also from the start point of Rabi os-cillation, α , the relative rate of fluorescence, we knowthat the measured state is in form √ αe iφ | m s = 0 i p + i √ − α | m s = − i p . Similarly with starting rate of flu-orescence β of MW2, we know the state is √ β | m s =0 i p + i √ − βe iφ | m s = 1 i p . The combination of those % c h a ng e i n f l uo r esce n ce Time(ns) ? /2 ? /2 LaserMW1MW2Counter ? /2 ? /2 dt ? /2 ? /2 (b)(a) FIG. 5: (color online) Quantum phase cloning for input statewith different phases. (a) MW1 π/ | m s = 0 i p + i | m s = − i p ) / √ jdt, j = 1 , , ..., dt = 20 ns, ns so that state( | m s = 0 i p + i | m s = − i p ) / √ | m s = 0 i p + e iωjdt i | m s = − i p ) / √ ωjdt depending on waiting time jdt and rotatingspeed ω determined by environment, see [20]. (b) Experimentresults show that with different waiting time periods withintime scale 2 µ s, after phase cloning operation, the intensityof fluorescence of the output state is stable which agrees withtheory expectation. two results show that the NV center state should be theform p αβe iφ | m s = 0 i p + i p (1 − α ) β | m s = − i p + i p (1 − β ) αe iφ | m s = 1 i p (2) with normalization 1 / √ α + β − αβ . Using the experi-mental data in FIG. 4(a,b), we find two fidelities are F = 84 .
6% and F = 86 . F = 82 .
9% and F = 87 . F + F − p (1 − F )(1 − F ) ≈ .
55, which clearly largerthan 1 . .
2% which is veryclose to theoretical bound 85 .
4% and apparently beyondthe bound of the universal quantum cloning.A phase quantum cloning need the input state withan arbitrary phase. Experimental procedures are shownin FIG.5. This finishes the implementation of the wholequantum phase cloning.In summary, we report the solid-state phase-covariantquantum cloning machine implementation in experimentat room temperature. Our observation shows that twomicrowave fields MW1 and MW2 can be combined to cre-ate an arbitrary superposition three-level state in quan-tum phase cloning processing and for other aims. Thiscan be used as a basis for scalable, precisely controllableand measurable three-level quantum information devices.This work is supported by NSFC (10974247, 10974251)and “973” programs (2009CB929103, 2010CB922904). [1] A. Gruber, A. Drabenstedt, C. Tietz, L. Fleury, J.Wrachtrup, and C. von Borczyskowski, Science , 2012(1997).[2] P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H.Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F.Jelezko, J. Wrachtrup, Science , 1326 (2008);[3] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. ,2417 (1999).[4] W. Yao, R. B. Liu, and L. J. Sham, Phys. Rev. Lett. ,077602 (2007).[5] J. F. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R.B. Liu, Nature , 1265 (2009).[6] B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder,Lloyd C. L. Hollenberg, F. Jelezko, and J. Wrachtrup,Phys. Rev. B , 081201(R) (2011).[7] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M.Taylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan,A. S. Zibrov, A. Yacoby, R. L. Walsworth, M. D. Lukin,Nature , 644 (2008).[8] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D.Budker, P. R. Hemmer, A. Yacoby, R. Walsworth, M. D.Lukin, Nature Physics , 810 (2008)[9] F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong,X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M.Feng, and J. F. Du, Phys. Rev. Lett. , 040504 (2010).[10] E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G, Dutt, A. S. Sorensen, P. R. Hemmer,A. S. Zibrov, M. D. Lukin, Nature , 730 (2010).[11] W.K.Wootters and W.H.Zurek, Nature , 802 (1982).[12] V. Scarani, S. Iblisdir, N. Gisin, and A. Acin, Rev. Mod.Phys. , 1225 (2005);[13] C. Vitelli, N. Spagnolo, L. Toffoli, F. Sciarrino, F. DeMartini, Phys. Rev. Lett. , 113602 (2010).[14] H. W. Chen, D. W. Lu, G. Qin, X. Y. Zhou, X. H. Peng,and J. F. Du, Phys. Rev. Lett. , 180404 (2011).[15] C. H. Bennett and G. Brassard, in Proceedings of theIEEE International Conference on Computer, Systems,and Signal Processing, Bangalore, India (IEEE, NewYork, 1984), pp175-179.[16] D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchi-avello, Phys. Rev. A , 012302 (2000).[17] H. Fan, K. Matsumoto, X. B. Wang, and M. Wadati,Phys. Rev. A , 012304 (2002).[18] J. F. Du, T. Durt, P. Zou, H. Li, L. C. Kwek, C. H. Lai,C. H. Oh, and A. Ekert, Phys. Rev. Lett. , 040505(2005)[19] H. Chen, X. Zhou, D. Suter, and J. F. Du, Phys. Rev. A , 012317 (2007).[20] L. Childress, Ph. D thesis, Harvard University (2007).[21] N. J. Cerf, Phys. Rev. Lett.84