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Fourier Transform Quantum State Tomography Mohammadreza Mohammadi1, Agata M. Branćzyk1 and Daniel F. V. James1 1CQIQC and IOS, Department of Physics, University of Toronto, 60 Saint George St., Toronto, Ontario M5S 1A7, Canada∗ (Dated: January 18, 2013) We propose a technique for performing quantum state tomography of photonic polarization- encodedmulti-qubitstates. Ourmethodusesasinglerotating waveplate,apolarizingbeamsplitter andtwophoton-countingdetectorsperphotonmode. Asthewaveplaterotates,thephotoncounters measure a pseudo-continuous signal which is then Fourier transformed. The density matrix of the state is reconstructed using the relationship between the Fourier coefficients of the signal and the Stokes’ parameters that represent the state. The experimental complexity, i.e. different wave plate rotation frequencies, scales linearly with the number of qubits. 3 1 PACSnumbers: 03.65.Wj,42.50.-p,42.50.Ex 0 2 Quantum state preparation is an essential ingredient from the Fourier coefficients of this signal. The experi- n in the realization of quantum technologies such as quan- mentalcomplexityofthismethodscaleslinearlywiththe a J tum computing [1], quantum cryptography [2] and other number of qubits in terms of the number of settings re- quantum information protocols [3]. A crucial aspect of quired (i.e. wave plate rotation frequencies) rather than 7 1 reliablestatepreparationistheabilitytoaccuratelychar- exponentially, as is the case with QST that uses discrete acterize the state of a quantum system. To this end, measurement settings. Similar techniques that rely on ] quantumstatetomography(QST)allowsthereconstruc- rotating wave-plates are used in classical optics to deter- h tionofastate’sdensitymatrixfrommeasurementstatis- mine the polarization state of the electromagnetic field p ticsaccumulatedthroughrepeatedindependentmeasure- [31]. In the context of non-classical light, Fourier spec- - t ments of multiple identically-prepared systems [4–6]. troscopyhasbeenusedtocharacterizethejointspectrum n a In linear-optics, where quantum information is en- of photons [32]. u The remainder of this paper is organised as follows. coded in the polarization of a single photon, different q In Section I, we give a brief review of QST of multi- measurement settings are realized with a combination of [ qubitstates. InSectionII,weintroducingourschemefor linear optical elements such as wave plates, beam split- 2 ters and polarizing beam splitters, followed by photon Fouriertransformtomography(FTT)andprovideexam- v counting. QST was first accomplished in such systems ples for one and two qubits. In Section III, we provide 9 concluding remarks by White et al. [7], where the measurement settings 9 corresponded directly to the Stokes’ parameters used to 3 1 characterize the polarization state of the classical elec- I. QUANTUM STATE TOMOGRAPHY . tromagnetic field [8]. Later it was suggested that an 1 over-complete symmetric six-measurement set [9] or an 1 informationally-complete symmetric four-measurement Tomography is the process of constructing a represen- 2 1 set [9–13] be used for improved performance. Other ex- tation of an object by imaging it in different sections. In : tensions, such as those considering optimal experimental quantum state tomography, we aim to construct a rep- v design under realistic technical constraints [14, 15], or resentation of a quantum state ρˆfrom different measure- i X modifications due to inaccessible information [16–22] or mentoutcomes. Ann-qubitsystemisspecifiedby4n−1 r preferable measurements choices [9, 11, 12, 23–30] have real parameters. We therefore require at least this many a also been considered. outcomes of linearly independent measurements to spec- ify ρˆ. To date, all implementations of QST of photonic The probability of obtaining measurement outcome j, polarization-encoded qubits have utilized either multiple given a measurement operator Mˆ , is given by wave-platesand/ormultiplebeamsplittersperqubit. We j propose a technique that uses only one wave plate and n p =(cid:104)Mˆ (cid:105)=Tr[ρˆMˆ ]= j , (1) onepolarizingbeamsplitter(PBS)perqubitmode. Each j j j N j modemisincidentonasinglewaveplaterotatingatfre- quencyΩm followedbyapolarizingbeamsplitter(PBS). where nj is the number of counts and Nj is a constant PhotoncountersattheoutputportsofthePBSmeasure dependentonthedetectorefficiencyanddurationofdata apseudo-continuoussignalandthestateisreconstructed collection. In a polarization-encoded linear optical system, any projective measurement can be realized with a quarter-wave plate, a half-wave plate and a polarizing beam splitter, as shown in FIG. 1 a). A popular choice ∗Electronicaddress: [email protected] corresponds to the three Pauli operators. 2 a) c) II. FOURIER TRANSFORM TOMOGRAPHY H H HWP V V In this section, we show how the quantum state of a QWP PBS WP PBS multi-qubit system can be represented by a single joint- H co probabilitysignalandhowthemeasurementofthissignal WP PBS V circuitincidence enIanbloesurthperorpecoosnasl,triudcetnitoincaolfctohpeieqsuoafnttuhme ssttaattee.are pre- b) H H pared and subsequently pass through a series of optical elements. Foramulti-photonstate,eachphotonmodem V V is incident on a single wave plate rotating at frequency WP PBS WP PBS Ω followed by a polarizing beam splitter (PBS). Pho- m FIG. 1: (Color online) Schematic diagrams of: a) typical toncountersattheoutputportsofthePBScontinuously QST set-up which uses a combination of quarter- and half- measure the intensity, which can be processed to recover waveplatestoperformarbitrary-basismeasurements;b)FTT Stokes’ parameters. A schematic of this setup is shown set-upwhichusesonerotatingwaveplate;c)multi-qubitFTT in in FIG 1 b) for a single qubit and 1 c) for multiple which uses one wave-plate per qubit mode m, rotating at qubits. For multiple qubits, the signal measured is a frequency Ωm. “coincidence intensity” corresponding to the joint probability of detecting photons at each PBS. The time-dependent single-qubit projection-valued We can always write the density matrix of an n-qubit measure(PVM)associatedwiththeprobabilityofdetect- system in terms of Hermitian operators σˆ ing a photon in the horizontal or vertical output modes i of each PBS is given by {MˆH(t),MˆV(t)} where m m ρˆ= 1 (cid:88)3 S σˆ ⊗···⊗σˆ , (2) Mˆma(t)=Uˆm†|a(cid:105)(cid:104)a|Uˆm, (4) 2n i1,...,in i1 in i1,...,in=0 for a=H,V, where m labels the qubit mode and (cid:18) (cid:19) (cid:18) (cid:19) β β wσˆhereaσˆre0 =th|eHP(cid:105)a(cid:104)Huli|+op|Ver(cid:105)a(cid:104)tVor|si:stσˆhei=de|nHti(cid:105)ty(cid:104)Vo|p+era|tVo(cid:105)r(cid:104)aHn|d, Uˆm(t)=cos 2 σˆ0−isin 2 (cid:126)vm(t)·(cid:126)σ (5) 1−3 1 σˆ = i(|V(cid:105)(cid:104)H| −|H(cid:105)(cid:104)V|) and σˆ = |H(cid:105)(cid:104)H| −|V(cid:105)(cid:104)V|. 2 3 is the unitary operator associated with a wave plate in The coefficients S = Tr[ρˆ(σˆ ⊗ ··· ⊗ σˆ )] com- i1,...,in i1 in modem. Uˆ (t)rotatestheoperators|a(cid:105)(cid:104)a| ontheBloch pletely characterize the state. S are normalized m i1,...,in sphere by an angle β, about the vector generalizations of the classical parameters introduced by Stokes in 1852 [8], and will hereafter be simply referred (cid:126)v (t)=cos(ω t)(cid:126)k+sin(ω t)(cid:126)i, (6) to as Stokes’ parameters. m m m CombiningEquations(1)and(2),wefindalinearrela- where(cid:126)kand(cid:126)iareunitvectorsinEuclidianspace(defined tionship between Stokes’ parameters and the probability by the axes in FIG 2) and(cid:126)v·(cid:126)σ =v σˆ +v σˆ +v σˆ . As 1 1 2 2 3 3 p : j the wave plate rotates about the beam-axis at frequency Ω in real space, (cid:126)v (t) rotates about the y-axis in Eu- m m p = 1 (cid:88)3 S Tr[σˆ ⊗···⊗σˆ Mˆ ], (3) cthliedifaanstspaxaicseoafttfhreeqwuaevnecyplωatme=is a2lΩigmn.edWaeta0ssduemgreeetshatot j 2n i1,...,in i1 in j i1,...,in=0 the horizontal as defined by the polarization of the photons. Aphasefactorcanbeincludedin(cid:126)v (t)toaccount m for different initial alignment of the wave plate. The re- whereMˆj actsontheentiremulti-qubitsystem. Bymak- sultingprojectorMˆH(t)tracesoutafigure-8pathonthe ing 4n−1 linearly independent measurements, it is pos- m Bloch sphere, as shown in FIG 2. The retardance of the sible to solve for Stokes’ parameters and reconstruct the wave plate determines the size of the figure-8. density matrix according to Equation (2). This can be To characterize an n-qubit state, one measures a joint achieved through a variety of methods, including simple probability of detecting a photon in the H mode of each linear inversion, least-squares estimation or the popular PBS. This is given by maximum likelihood estimation method [33]. Alterna- tively, one can look to a growing number of exciting new 3 1 (cid:88) techniques such as the forced purity routine [34], Bay- p (t)= S χ ...χ , (7) n 2n i1...,in 1,i1 n,in sean mean estimation [35], compressed sensing [36], von i1,...,in=0 Neumann entropy maximization [37], hedged maximum likelihood estimation [38], minimax estimation [39], and where techniques that focus on reconstructing the state with χ :=Tr[σˆ MˆH(t)], (8) reliable error bars [40] and confidence regions [41]. m,i i m 3 probabilities. Without loss of generality, we restrict 0 < ω < ··· < 1 ω . For two qubits, ω = rω where r > 1. If r is n 2 1 an irrational number, the signal does not have a finite period. If r is a rational number, we can write r = p/q, wherepandq areintegers. Inthiscase,theperiodofthe two-qubit signal is given by 2πq T(r)= , (10) ω gcd(p,q) 1 where gcd(p,q) is the greatest common denominator of p and q. For n > 2, the period of the signal can be determined via recursion. A shorter period is favourable from an experimental perspective which, for a constant ω , occurs when r is an integer. The lowest integer that 1 ensures sufficient Fourier coefficients to solve for Stokes’ FIG. 2: (Color online) Path traced out by Mˆ(H)(t), defined parameters is r =5. m inEquation(4),for: β =π/4(green,dotted);β =π/2(blue, Inpracticepn willnotbeacontinuousfunctionoftime dashed); and β =11π/15 (orange, solid). but rather a discretized approximation. The discretized signal will be divided into time bins, with N coincidence counts in each bin. The number of time bins per pe- and therefore riod, N, must be at least the Nyquist rate, i.e. twice the highest frequency contained within the signal, to avoid χm,0 =1 (9a) aliasing. χm,1 =s2sin(2ωmt) (9b) The discrete probability in bin τj will be given by χ =2cssin(ω t) (9c) m,2 m n χm,3 =c2+s2cos(2ωmt) (9d) pn(τj)= HN...H , (11) where c=cos(β/2) and s=sin(β/2). where N =(cid:80) n , n is the number of Notethatthechoiceofanalyzingthesignalfrommode qubitsandn k1,...,k2nis=tHh,Venukm1,.b..,ekr2nofcoincidencecounts H rather than mode V is arbitrary and typically both for a given pkro1,j.e..c,kt2onr Mˆk1(t)⊗···⊗Mˆk2n(t). modes will need to be measured to ensure normalised m m In principle, β can take on any value other than an integer multiple of π. However in practice, some values 500 will be more susceptible to noise than others. We use theequallyweightedvariance(EWV)[42],whichassesses 400 the noise immunity of the wave-plate, to show that β ≈ 11π/15 is most immune to noise, as defined in [42]. A N 300 plot of the EWV is shown in FIG 3. Such a retardance V W canbeachievedwithanoff-the-shelfwaveplatedesigned E 200 for a wave length different to that of the experiment. In the remainder of this section, we provide specific 100 examples for one- and two-qubit states. 0 0 Π Π 3Π Π 5Π 3Π 7Π Π 8 4 8 2 8 4 8 Β A. Example: one qubit FIG.3: (Coloronline)Theequallyweightedvariance(EWV) [42] assesses the noise immunity of the wave-plate. Here we For a single qubit, the signal is given by plotEWV×N foroneperiodasafunctionofβ(orange,solid), where N is the total number of bins per period. A smaller 3 EWV is associated with better immunity to noise. The best 1(cid:88) p (t)= S χ (12) noise immunity occurs at β ≈ 2.27 ≈ 11π/15 for EWV ≈ 1 2 i 1,i 41.7/N. The black, dotted line shows EWVopt = 40/N, at- i=0 S S tainable by optimal tomographic schemes such as those that = 0 + 1s2sin(2ω t)+S cssin(ω t) measure the Pauli matrices or the SIC-POVM [10, 11, 13]. 2 2 1 2 1 (13) ETWheVbqluwep,=da8s4h/eNd.liNneotsehotwhasttthheeEEWWVVfoisrdaimQWenPsio(nβle=ss.π/2), + S23 (cid:0)c2+s2cos(2ω1t)(cid:1) . 4 1.0 1.0 a) b) 0.8 0.8 0.6 0.6 t t 1 2 p0.4 p0.4 0.2 0.2 0.0 0.0 0 Π Π 3Π 2Π 5Π 3Π 7Π 4Π 0 Π Π 3Π 2Π 5Π 3Π 7Π 4Π 2 2 2 2 2 2 2 2 Ω Ω 1 1 1.0 0.4 0.4 0.4 0.8 0.2 0.2 0.2 af 0.6 bf 0.0 af 0.0 bf 0.0 0.4 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.0 0 1 2 3 4 0 1 2 3 4 0 2 4 6 8 10121416 0 2 4 6 8 10121416 f f f f FIG.4: Theprobabilitysignalp ,asafunctionofrevolutionsofthefirstwaveplate,andFouriercoefficientsa andb forthe n f f state: a)ρˆ giveninEquation(18);andb)|ψ(cid:105) giveninEquation(23). Thelabelf denotesthetermintheFourierseriesand 1 2 corresponds to the subscript in Equations (14) and (22) This can be written as parameter d, such that a p1(t)≡ 20 +b1sin(ω1t) (14) ρˆ1 =d|ψ(cid:105)1(cid:104)ψ|1+(1−d)σˆz|ψ(cid:105)1(cid:104)ψ|1σˆz. (18) +a cos(2ω t)+b sin(2ω t) , Specifically, let’s consider 2 1 2 1 1 where the Fourier coefficients are given by |ψ(cid:105) = √ (|H(cid:105)+e−iπ/4|V(cid:105)) (19) 1 2 a =S +S c2; (15a) 0 0 3 and d = 0.1. A retardance of β = 11π/15 produces the b =S cs; (15b) 1 2 signalshowninFIG4a). PerformingafastFouriertrans- a = S3s2; (15c) form (FFT) of the discretized signal yields the Fourier 2 2 coefficientsinFIG4a). Thecoefficientsa andb corre- f f b = S1s2, (15d) spond to the real and imaginary parts of the list gener- 2 2 ated by the FFT respectively. Inserting the coefficients, a = 1, b = 0.210, a = 0 and b = −0.236 into the 0 1 2 2 where c = cos(β/2) and s = sin(β/2). Linear inversion density matrix in Equation (17) gives of Equations (15) gives the Stokes’ parameters in terms of the Fourier coefficients: (cid:18) 0.5 −0.283−0.283i(cid:19) ρ = , (20) 1 −0.283+0.283i 0.5 2a c2 S =a − 2 ; (16a) 0 0 s2 which corresponds to the density operator in Equation S = 2b2 ; (16b) (18) for d=0.1. 1 s2 b S = 1 ; (16c) 2 cs B. Example: two qubits 2a S = 2 . (16d) 3 s2 Fortwoqubits,thejointprobabilityofdetectingapho- ton in the horizontal output ports of each PBS is given Substitution into Equation (2), gives the density matrix by in terms of the Fourier coefficients: 3 ρ1 = (cid:18) 12sb(22a0++i22bca1s2) 12(a0sb+22 −2ai22)bc1s− 2sa22 (cid:19) . (17) p2(t)= i1(cid:88),i2=0Si41,i2χ1,i1χ2,i2 (21) As an example, consider a single-qubit state |ψ(cid:105)1 that = a20 +(cid:88)(cid:0)afcos(ωf(cid:48)t)+bfsin(ωf(cid:48)t)(cid:1) , (22) has experienced depolarizing noise, characterized by the f=1 5 where in the second line, we have written the signal in Waveplateretardancesofβ =11π/15andafrequency terms of its Fourier coefficients. The extent of the sum- ratio r = ω /ω = 5 produces the signal shown in FIG 2 1 mationdependsonthespecificchoiceofrelativefrequen- 4. The explicit expression for p (t) for this choice of 2 cies, and ω(cid:48) are functions of ω and ω from the set measurement settings can be found in Appendix A. Per- f 1 2 {ω ,ω ,2ω ,2ω ,ω ±ω ,ω ±2ω ,2ω ±ω ,2ω ±2ω }. formingafastFouriertransform(FFT)ofthediscretized 1 2 1 2 1 2 1 2 1 2 1 2 The elements of this set are not necessarily in order of signal yields the Fourier coefficients in FIG 4 b). size, and to relate them to ω(cid:48) one needs to consider ex- f plicit values for ω1 and ω2. LinearinversionoftheexpressionsfortheFouriercoef- As an example, consider the two-qubit state ficientsintermsoftheStokes’parameters,giveninEqua- tions (A3), followed by substitution into Equation (2), 1 |ψ(cid:105) = √ (|H(cid:105)|V(cid:105)+|R(cid:105)|L(cid:105)) . (23) along with the Fourier coefficients determined from the 2 2 signalinFIG4b),givesthereconstructeddensitymatrix  0.125 0.25+0.125i −0.125i 0.125  0.25−0.125i 0.625 −0.125−0.25i 0.25−0.125i ρ =   , (24) 2  0.125i −0.125+0.25i 0.125 0.125i  0.125 0.25+0.125i −0.125i 0.125 which corresponds to the density operator ρˆ =|ψ(cid:105) (cid:104)ψ| (USRA) and the DARPA (QuBE) program. 2 2 2 where |ψ(cid:105) is defined in Equation (23). In general, given 2 separable qubits, p factorizes into a product of p for 2 1 each qubit. Appendix A: Probability signal and Fourier III. SUMMARY & CONCLUDING REMARKS coefficients for two-qubit state We presented a scheme for performing quantum state tomography of photonic polarization-encoded multi- In this appendix, we give the probability signal for qubit states. The scheme is simpler than standard to- the specific two-qubit example described in Section IIB. mographicprotocolsinthatonlyonewaveplateandone WealsoprovideexpressionsfortheFouriercoefficientsin polarizing beam splitter is required per photon mode. terms of the Stokes’ parameters, as well as the inverted In this scheme, photon-counting detectors measure a expressions for the Stokes’ parameters in terms of the pseudo-continuous time-dependent joint probability as Fourier coefficients. the wave plates rotate at frequency Ω . The Fourier co- m The signal probability for the state efficients of the signal give the Stokes’ parameters which describe the state. For a single qubit, the optimal wave plate retardance is β ≈11π/15. 1 |ψ(cid:105) = √ (|H(cid:105)|V(cid:105)+|R(cid:105)|L(cid:105)) , (A1) This technique reduces the number of required optical 2 2 elementsandtheexperimentalcomplexityscaleslinearly with the number of qubits, in terms of the number of settingsrequired(waveplaterotationfrequencies)rather with a frequency ratio r =ω2/ω1 =5, is than exponentially, as is the case with QST that uses discrete measurement settings. a An open question is whether the representation of a p (t)= 0 +b sin(ω t)+b sin(2ω t)+b sin(3ω t) 2 2 1 1 2 1 3 1 quantum state as a continuous signal will provide intu- +b sin(5ω t)+b sin(7ω t)+b sin(8ω t) itivemeansforestablishingcertainpropertiesofthestate 5 1 7 1 8 1 such as its entanglement. +b9sin(9ω1t)+b10sin(10ω1t)+b11sin(11ω1t) +b sin(12ω t)+a cos(2ω t)+a cos(3ω t) 12 1 2 1 3 1 +a cos(8ω t)+a cos(4ω t)+a cos(6ω t) 8 1 4 1 6 1 IV. ACKNOWLEDGEMENTS +a cos(7ω t)+a cos(9ω t)+a cos(10ω t) 7 1 9 1 10 1 +a cos(11ω t)+a cos(12ω t) , The authors thank Dylan Mahler and Paul Kwiat for 11 1 12 1 (A2) helpful discussions. This work was funded by NSERC 6 where the Fourier coefficients are Substituting the Stokes’ parameters, along with the Fourier coefficients in FIG 4 b), into Equation (2), gives a0 = (cid:0)c2(cid:0)c2S3,3+S0,3+S3,0(cid:1)+S0,0(cid:1)/2 (A3a) the reconstructed density matrix in Equation (24) which a =s2(cid:0)c2S +S (cid:1)/4 (A3b) corresponds to the density operator ρˆ2 =|ψ(cid:105)2(cid:104)ψ|2 where 2 3,3 3,0 |ψ(cid:105) is defined in Equation (A1). a = −a =cs3S /4 (A3c) 2 3 7 1,2 a = −a =c2s2S /2 (A3d) 4 6 2,2 a =s4(S +S )/8 (A3e) 8 1,1 3,3 a = =−a =cs3S /4 (A3f) 9 11 2,1 a =s2(cid:0)c2S +S (cid:1)/4 (A3g) 10 3,3 0,3 a =s4(S −S )/8, (A3h) 12 3,3 1,1 and b =cs(cid:0)c2S +S (cid:1)/2 (A4a) 1 2,3 2,0 b =s2(cid:0)c2S +S (cid:1)/2 (A4b) 2 1,3 1,0 b =b =cs3S /4 (A4c) 3 7 3,2 b =cs(cid:0)c2S +S (cid:1)/2 (A4d) 5 3,2 0,2 b =s4(S −S )/8 (A4e) 8 3,1 1,3 b = −b =−cs3S /4 (A4f) 9 11 2,3 b =s2(cid:0)c2S +S (cid:1)/4 (A4g) 10 3,1 0,1 b =s4(S +S )/8, (A4h) 12 1,3 3,1 where c = cos(β/2) and s = sin(β/2). 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