Experimental realisation of generalised qubit measurements based on quantum walks
Yuan-yuan Zhao, Neng-kun Yu, Pawe Kurzynski, Guo-yong Xiang, Chuan-Feng Li, Guang-Can Guo
aa r X i v : . [ qu a n t - ph ] J a n Experimental realisation of generalised qubit measurements based on quantum walks
Yuan-yuan Zhao,
1, 2
Neng-kun Yu,
3, 4
Paweł Kurzyński,
5, 6
Guo-yong Xiang,
1, 2, ∗ Chuan-Feng Li,
1, 2 and Guang-Can Guo
1, 2 Key Laboratory of Quantum Information, University of Science and Technology of China,CAS, Hefei, 230026, People’s Republic of China Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China The Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada Department of Mathematics&Statistics, University of Guelph, Guelph, Ontario, Canada Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań , Poland (Dated: July 13, 2018)We report an experimental implementation of a single-qubit generalised measurement scenario(POVM) based on a quantum walk model. The qubit is encoded in a single-photon polarisation. Thephoton performs a quantum walk on an array of optical elements, where the polarisation-dependenttranslation is performed via birefringent beam displacers and a change of the polarisation is imple-mented with the help of wave-plates. We implement: (i) Trine-POVM, i.e., the POVM elementsuniformly distributed on an equatorial plane of the Bloch sphere; (ii) Symmetric-Informationally-Complete (SIC) POVM; and (iii) Unambiguous Discrimination of two non-orthogonal qubit states.
PACS numbers: 03.65.Ta, 03.67.Ac, 03.67.Lx, 42.25.Hz
I. INTRODUCTION
The basic unit of quantum information is a two-levelquantum system commonly known as a qubit. Qubitscan be implemented on physical objects such as polari-sation of photons or intrinsic angular momentum (spin / ) of quantum particles. Any quantum computationrelies on precise preparations, transformations and mea-surements of such systems. Before actual quantum com-puter is build, one has to master the ability to manipulatewith single qubits and learn how to readout informationencoded in them.The information readout from a quantum system isdone via a measurement. In most common scenario oneperforms the von Neumann measurement that projects astate of the qubit onto one of two perfectly distinguish-able (orthogonal) physical states of the system. Suchmeasurements are sharp in a sense that once the mea-surement is done, the outcome of the measurement isdetermined and any repetition of exactly the same mea-surement would yield the same outcome value.Physically, von Neumann measurements are realisedvia interaction of the system with the measurement appa-ratus. The pointer of the measurement apparatus is rep-resented via wave packed and the interaction causes thiswave packet to move either to the left or right, dependingon the value of the measured observable. In general, thisvalue might be undetermined and the pointer goes intosuperposition of being to the left and to the right fromits initial position. The actual collapse of wave functionis usually attributed to the observer who reads out the ∗ Electronic address: [email protected] measurement outcome from the pointer. The sharpnessof the measurement comes from the fact that the initialspread of the pointer’s state is assumed to be narrowand the translation caused by the interaction with themeasured observable is large enough to prevent overlapbetween the part of the wave packed that was shifted tothe right with the part that was shifted to the left.On the other hand, there are generalised measurementscenarios, the so called Positive-Operator Valued Mea-sure (POVM), in which one allows the principal system tointeract with an ancillary system, whose state is known,and later performs a von Neumann measurement on thejoint system. This effectively extends the dimensionalityof the Hilbert space and one can implement measure-ments of a quibit with more than two outcomes. As aresult, one gains a plethora of new possibilities. They al-low one, for example, to perform a tomography of qubitwith a single measurement setup [1, 2], or to discriminatebetween non-orthogonal quantum states [3, 4]. POVMswere implemented in laboratories using various setups[5–11].In [12] it was proposed that they can be implementedvia a discrete-time quantum walk which has been realisedin a laboratory using various physical systems [13–25].Quantum walks model an evolution of a particle in a dis-crete space. The particle moves either one step to the leftor right, depending on a state of a two-level system knownas a coin. Quantum walks were originally proposed as aninterference process resulting from a modified version ofa von Neumann measurement in which a pointer statedistribution is much broader than the shift of its posi-tion [26]. The position of the pointer plays the role of thequantum walker and the qubit that is measured plays therole of the coin. If the interaction between the pointerand the qubit occurs at the same time as the evolutionof the qubit, the measured value changes during the pro-cess and the pointer starts to move back and forth. Thismovement leads to an interference and the interferencepattern produced in this process can be interpreted asa POVM. In fact, in [12] it was shown that any POVMcan be implemented in such a way, provided a necessaryevolution is applied to the qubit.In this work we report an experimental implementa-tion of the above quantum walk POVM scenario. Here,we use the optical setup in which the qubit is encodedin a polarization state of a single photon and the posi-tion of the quantum walker is implemented on a pho-tonic path [25]. We construct setups realising (i) threePOVM elements symmetrically distributed on an equa-torial plane of the Bloch sphere (Trine-POVM); (ii)Symmetric-Informationally-Complete (SIC) POVM; and(iii) Unambiguous Discrimination of two non-orthogonalquantum states.
II. DISCRETE-TIME QUANTUM WALKS
Discrete-time quantum walks are quantum counter-parts of classical random walks in which a particle takesa random step to the left or right. In the classical casethe particle spreads in a diffusive manner (a standarddeviation of its position is proportional to √ t ) and aftermany steps a spatial probability distribution is Gaussian.In quantum case the system is described by two degreesof freedom | ψ i = | x, c i : the position x = . . . , − , , , . . . and a two-level internal degree of freedom known as acoin c = ← , → . The evolution is unitary and consists oftwo sub-operations U = CT , namely a unitary coin toss(that is a × unitary matrix acting only on the coindegree of freedom) C = (cid:18) cos θ e − iβ sin θ − e iβ sin θ cos θ (cid:19) (1)and a conditional translation operation T = X x | x + 1 , →ih x, → | + | x − , ←ih x, ← | . (2)Quantum walks with uniform (position-independent)coin operation spread ballistically (standard deviationproportional to t ) and its probability distribution differsform the classical Gaussian shape [29, 30]. On the otherhand, quantum walks with position-dependent coin op-eration C x = (cid:18) cos θ x e − iβ x sin θ x − e iβ x sin θ x cos θ x (cid:19) (3)can be used to observe localisation [31, 32], or to simulatephysical systems with non-homogenous interactions [33].In [12] it was shown that quantum walks with positionand time-dependent coin operations C x,t can be also usedto implement POVMs. III. POVM IMPLEMENTATION WITHQUANTUM WALKS
The probability distribution of a quantum walk, thatis initially localised at the origin x = 0 , depends on theinitial coin state α | →i + β | ←i and on the subsequentcoin operations. A single application of the operator T makes the particle to go into superposition α | , →i + β | − , ←i . In this case the position measurement wouldcorrespond to a von Neumann measurement of the coinin the basis {← , →} , where the result → corresponds tofinding the particle at x = 1 and ← to finding it at x = − . However, if one allows quantum walk to evolve formore than one step and the coin operation to change fromone step to another step, then the particle can spreadover many positions and the measurement of its locationat x may correspond to a measurement of some POVMelement E x . Indeed, a proper choice of coin operationscan lead to an arbitrary qubit POVM scenario [12].The POVM elements E i ( i = 1 , , . . . , n ) are nonnega-tive operators obeying n X i E i = . (4)They differ from standard von Neumann projectors Π i inthat they do not have to be orthonormal ( Π i Π j = δ i,j Π i ,but E i E j = δ i,j ) and that their number can be greaterthan the dimension of the system n > d . The follow-ing quantum walk algorithm, proposed in [12], generatesan arbitrary set of rank 1 POVM elements { E , . . . , E n } (rank 2 elements can be generated with a modified ver-sion of this algorithm).1. Initiate the quantum walk at position x = 0 withthe coin state corresponding to the qubit state onewants to measure.2. Set i := 1 .3. While i < n do the following: • Apply coin operation C (1) i at position x = 0 and identity elsewhere and then apply trans-lation operator T . • Apply coin operation C (2) i at position x = 1 , N OT = (cid:18) (cid:19) at position x = − and identity elsewhere andthen apply translation operator T . • i := i + 1 .Since in this work we are only interested in the measure-ment statistics, and not in the post-measurement states,we simplified the algorithm in [12] and omitted the laststep. The POVM elements that are generated dependsolely on the form of operators C (1) i and C (2) i . IV. EXPERIMENT
In our experiment, frequency-doubled femtosecondpulse (390 nm, 76 MHz repetition rate, 80 mw aver-age power) from a mode-locked Ti:sapphire laser pump atype-I beta-barium-borate (BBO, 6.0 × × , θ =29.9) crystal to produces the degenerate photon pairs.After being redirected by the mirrors (M1 and M2, asin the Fig. 1(a)) and the interference filters (IF, △ λ =3nm, λ =780 nm), the photon pairs generated in the spon-taneous parametric down-conversion (SPDC) process arecoupled into single-mode fibers separately. Single photonstate is prepared by triggering on one of these two pho-tons, and the coincidence counting rate collected by theavalanche photo-diodes (APD) are about × in oneminute. (b) CoincidenceDectection (a)
QWP1
IF HWP QWP BBO APDBDPBS trigger
M1M2pump
BD1 BD2 BD3 BD4
NOT NOTHWP1 HWP2 HWP3
BD4 BD5 BD6
NOTHWP4
HWP5QWP2
BD1 BD2 BD3
NOT NOTHWP1 HWP2 HWP3 (b)
CoincidenceDectectionCoincidenceDectection (a)
QWP1
IF HWP QWP BBO APDBDPBS trigger
M1M2pump
BD1 BD2 BD3 BD4
NOT NOTHWP1 HWP2 HWP3
BD4 BD5 BD6
NOTHWP4
HWP5QWP2
BD1 BD2 BD3
NOT NOTHWP1 HWP2 HWP3
FIG. 1: Experimental setup. (a) Experimental setup for con-structing the trine POVMs corresponding to | ψ i i and realiz-ing the unambiguous state discrimination. (b) Optical net-work for constructing SIC-POVMs. Initial coin states areprepared by passing the single photons through a polarizingbeam splitter (PBS), a half-wave plate (HWP) HWP1 and aquarter-wave plate (QWP) QWP1 in a specific configuration.The conditional position shifts are implemented by Beam Dis-placers (BDs) and the coin operators in different positions arerealized by wave plates with different angles (Table 1). Theindices in the figure denote the position of walker. One-dimensional discrete time quantum walk systemhas two degrees of freedom, x and c , x is the positionof the particle and c is the state of the coin. In ourexperiment they are encoded in the longitudinal spa-tial modes and polarizations | H i , | V i of the single pho-tons respectively. In this case, the conditional transla-tion operator as given by Eq.(2) is realized by the de-signed BD, that does not displace the vertical polarizedphotons ( | x, V i → | x − , V i ) but makes the horizon-tal polarized ones undergo a 4 mm lateral displacement( | x, H i → | x + 1 , H i ). A. Trine POVM
The experimental setup in Fig. 1(a) is used to con-struct the trine POVMs, | ψ i ih ψ i | ( i = 1 , , , | ψ i = | H i| ψ i = −
12 ( | H i − √ | V i ) | ψ i = −
12 ( | H i + √ | V i ) . (5)According to the settings of the coin operators, the opti-cal axes of BD1 and BD2 must be aligned, in other words,they form an interferometer. When rotating HWP1 andHWP3 by . ◦ , we observed that the interference visibil-ity of the interferometer was about . and the systemwas stable over 2.5 hours of timescale. After aligning,we begin to set the corresponding coin operators in eachstep. For the Trine POVMs, we have C (1)1 = 1 , C (2)1 = r (cid:18) √ −√ (cid:19) ,C (1)2 = r (cid:18) − (cid:19) , C (2)2 = 1 , (6)where C (2)1 and C (1)2 are realized by rotating HWP2 andHWP3 by . ◦ and . ◦ respectively. The initial trinecoin states | ψ i i are constructed by rotating HWP1 by ◦ , − ◦ and ◦ , while the anti-trine states | ¯ ψ i i areconstructed by rotating it by ◦ , ◦ and − ◦ . At last,every output port’s detection efficiency are calibrated sothat the differences among them are below .Fig. 2 shows that the results in our experiment agreewith theoretical predictions. The ratios / / / ( / / ) for the cases of | ψ i i ( | ¯ ψ i i , where h ψ i | ¯ ψ i i =0 ) are given in theory and the detailed numerical resultsof the probability distributions can be found in Table IIand III. To visualize that the setup has constructed theTrine POVMs, it is important to demonstrate that wecannot find states | ¯ ψ i in position 4, | ¯ ψ i in position 0and | ¯ ψ i in position 2. Fig. 2(b) and Table III show thatthe probabilities of these events are indeed very close tozero, with an average value of . . In addition, theresults for states | ψ i i also indicate the coefficients of thePOVM we constructed is , see Fig. 2(a). The errorsin our experiment mainly stem from the imperfect waveplates and the interferometers and the counting statisticsof the photons. (cid:3) FIG. 2: Results for trine POVMs. Histogram shows the prob-abilities of counting rates in position 0, 2 and 4 with inputstates ψ i (a) and ¯ ψ i (b), respectively. All results are normal-ized so that the sum of the counts in these three positions is1. The theoretical values are shown as the blue lines, whichare 2/3, 1/6, 1/6 for ψ i and 1/2, 1/2, 0 for ¯ ψ i (i=1, 2, 3);error bars are too small to identify. B. SIC POVM
The optical network in Fig. 1(b) construct the SICPOVMs, | ψ i ih ψ i | ( i = 1 , , , , | ψ i = | H i| ψ i = − √ | H i + r | V i| ψ i = − √ | H i + e i π r | V i| ψ i = − √ | H i + e − i π r | V i . (7)The coin operators C (1)1 = 1 , C (2)1 = 1 √ (cid:18) − (cid:19) ,C (1)2 = 1 √ (cid:18) − (cid:19) , C (2)2 = 1 √ (cid:18) √ −√ (cid:19) ,C (1)3 = 1 √ (cid:18) e − i π e i π e i π e − i π (cid:19) , C (2)3 = 1 (8)are realized by wave plates in various configurations (de-tails in Table I). C C C C C HW P ◦ ′ ◦ ′ ◦ ′ ◦ ′ −− QW P −− −− −− ◦ −− TABLE I: The configurations of the QWPs and HWPs torealize the coin operators for constructing the SIC-POVM.
For these settings, BD1 and BD2, BD3 and BD4 formtwo interferometers whose interference visibility are bothabove 0.993. The HWP1 and QWP1 with different anglesin front of a polarizing beam splitter (PBS) are used toproduce ψ i and the corresponding orthogonal states ¯ ψ i (Table VI and Table VII). As shown in Fig. 3, the resultsare also in good accordance with theoretical ratios : : : for ψ i and : : : 0 for ¯ ψ i . (cid:3) FIG. 3: Results for SIC POVMs. Histogram showing thenormalized probability of counting rate in position 0, 2, 4and 6 respectively with the input state ψ (a) and ¯ ψ i (b). Thetheoretical values are shown as the blue lines, which are givenby 1/2, 1/6, 1/6, 1/6 for states ψ i and 1/3, 1/3, 1/3, 0 forstates ¯ ψ ; error bars are too small to identify. C. Unambiguous state discrimination
For the unambiguous state discrimination of states, | ψ ± i = cos( θ/ | H i ± sin( θ/ | V i , we can use the same θ ( π /20) P − s u ccess ψ − ψ +Theory FIG. 4: Theoretical and experimental successful probabilityv.s. θ which is related to the state to be discrimination; errorbars are too small to identify. setup as in Fig. 1(a), with C (1)1 = 1 ,C (2)1 = q − (tan θ ) tan θ tan θ − q − (tan θ ) ,C (1)2 = r (cid:18) − (cid:19) , C (2)2 = 1 , (9)The state after four quantum walk steps becomes | ψ + i = √ cos θ | , H i + √ θ | , H i , (10)or | ψ − i = √ cos θ | , H i − √ θ | , H i . (11)Therefor, if the photon is detected at position x = 0 oneknows that the coin was definitely in the state | ψ − i . Ifit was detected at position x = 2 one knows that thecoin was definitely in the state | ψ + i . Finally, if it wasdetected at position x = 4 one gains no information andthe discrimination was unsuccessful.In our experiment, the input states ψ + and ψ − areprepared in π steps from ≤ θ ≤ π . The states for var-ious θ are prepared by rotating the HWP, placed beforethe polarizing beam splitter (PBS), and the coin oper-ator C (2)1 , C (1)2 and the NOT operator are realized by aHWP rotating in the angle of arcsin(tan θ ) , π and π respectively. From Fig. 4 we can see that the probabilityof the successful discrimination is increasing with θ . Formore details see Tables VIII and IX. V. CONCLUSIONS
We experimentally realised three generalised measure-ment scenarios for a qubit. These scenarios are based on a quantum walk model presented in [12]. Our resultsmatch the theoretical predictions. We believe that thesekind of experimental setups can be used in the future toimplement other types of generalised measurement sce-narios with multiple outcomes and rank 2 POVM ele-ments, and to study quantum walks with position andtime-dependent coin operations. Finally, we would liketo mention that similar results had been reported whilewe were working on this experiment [27, 28].
VI. ACKNOWLEDGEMENTS
The authors would like to thank Yongsheng Zhang forhelpful discussion. The work in USTC is supported byNational Fundamental Research Program (Grants No.2011CBA00200 and No. 2011CB9211200), National Nat-ural Science Foundation of China (Grants No. 61108009and No. 61222504). P.K. is supported by the NationalResearch Foundation and Ministry of Education in Sin-gapore.
VII. APPENDIX
The detailed results about ψ i , ¯ ψ i , ψ i and ¯ ψ i are shownin Table II, Table III, Table IV and Table V. Table VIand Table VII are the angles of the HWP1 and QWP1 forpreparing states ψ i and ¯ ψ i . The main parameters of un-ambiguous state discrimination and the detailed resultscan be found in Table VIII and Table IX. state P P P ψ . . . ψ . . . ψ . . . TABLE II: P , P and P are the nomarized probabilities of count-ing rate in position 0, 2 and 4. state P P P ¯ ψ . . . ψ . . . ψ . . . TABLE III: P , P and P are the nomarized probabilities ofcounting rate in position 0, 2 and 4. state P P P P ψ . . . . ψ . . . . ψ . . . . ψ . . . . TABLE IV:
The nomarized probabilities of counting rates for state ψ i in position 0, 2, 4 and 6. state P P P P ¯ ψ . . . . ψ . . . . ψ . . . . ψ . . . . TABLE V:
The nomarized probabilities of counting rates for state ¯ ψ i in position 0, 2, 4 and 6. ψ ψ ψ ψ HW P ◦ − ◦ ′ ◦ ′ ◦ QW P ◦ ◦ ′ − ◦ ′ − ◦ ′ TABLE VI: The angles of HWP1 and QWP1 used to preparethe states ψ i . ¯ ψ ¯ ψ ¯ ψ ¯ ψ HW P ◦ ◦ ′ − ◦ ′ ◦ QW P ◦ ◦ ′ − ◦ ′ − ◦ ′ TABLE VII: The angles of HWP1 and QWP1 used to preparethe states ¯ ψ i . θ θ / C (2)1 P theory P e π/
20 2 ◦ ′ ◦ ′ . . π/
10 4 ◦ ′ ◦ ′ . . π/
20 6 ◦ ′ ◦ ′ . . π/ ◦ ◦ ′ .
191 0 . π/ ◦ ′ ◦ ′ .
293 0 . π/
10 13 ◦ ′ ◦ ′ .
412 0 . π/
20 15 ◦ ′ ◦ ′ .
546 0 . π/ ◦ ◦ ′ .
691 0 . π/
20 20 ◦ ′ ◦ ′ .
844 0 . π/ ◦ ′ ◦ .
000 0 . TABLE VIII: θ , the angle related to the input state ψ + whichis prepared by HWP1 rotated to θ / ; C , the angle of HWP2 torealize the operator C . P theory and P e represent the theoreticaland the experimental successful probability. θ θ / C (2)1 P theory P e − π/ − ◦ ′ ◦ ′ . . − π/ − ◦ ′ ◦ ′ . . − π/ − ◦ ′ ◦ ′ . . − π/ − ◦ ◦ ′ .
191 0 . − π/ − ◦ ′ ◦ ′ .
293 0 . − π/ − ◦ ′ ◦ ′ .
412 0 . − π/ − ◦ ′ ◦ ′ .
546 0 . − π/ − ◦ ◦ ′ .
691 0 . − π/ − ◦ ′ ◦ ′ .
844 0 . − π/ − ◦ ′ ◦ .
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