Two-photon tomography using on-chip quantum walks James G. Titchener,1 Alexander S. Solntsev,1 and Andrey A. Sukhorukov1 1Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, Australian Capital Territory 2601, Australia We present a conceptual approach to quantum tomography based on first expanding a quantum stateacrossextradegreesoffreedomandthenexploitingtheintroducedsparsitytoperformrecon- struction. Weformulateitsapplicationtophotoniccircuits,andshowthatmeasuredspatialphoton correlations at the output of a specially tailored discrete-continuous quantum-walk can enable full reconstruction of any two-photon spatially entangled and mixed state at the input. This approach doesnotrequireanytunableelements,soiswellsuitedforintegrationwithon-chipsuperconducting photon detectors. 6 1 PACSnumbers: 42.50.-p,42.82.Et,03.65.Wj 0 2 Akeybenefitofquantumsystemsisthattheirinforma- ing [7, 8] and could be applied for fast tomography of n tion capacity scales exponentially with the system size. near pure quantum states [9] and quantum process to- a J However this also presents a challenge, as fully charac- mography [10]. It has also been shown that knowing a 1 terizing such a system requires a correspondingly large quantum state is sparse can facilitate characterization 1 number of measurements. Many measurements are re- of three-photon quantum states just by measuring two- quired for two reasons. Firstly, the result of a single photon coincidences [11]. ] s quantum measurement is probabilistic, so to get infor- However the assumption that a system is sparse is not c mation about the underlying density matrix, the same validforarbitraryquantumsystems. Ourkeysuggestion i t measurementmustberepeatedmanytimesonneariden- is to first force the unknown system to become sparse, p tical states to calculate expectation values [1]. Secondly, andthenapplyacompressedsensing-likeapproachtoto- o . thedensitymatrixexistsinahighdimensionalspacethat mography. Weachievethisbyapplyingalineartransfor- s c cannotbefullyaccessedfromonesetofmeasuredexpec- mationtothesystemthatmapsittoanincreasednumber i tation values. Thus to characterize the density matrix ofmodes[Fig.1(a)],thusturningitintoasparsesystem. s y a variety of linear transformations are usually applied to Careful choice of this transformation allows imaging of h thesystemtomeasuremanydifferentexpectationvalues. thefullcomplexvalueddensitymatrixjustfrommeasur- p Generally to characterize a set of N qubits the measure- ingonesetofexpectationvaluesafterthetransformation. [ ment system is reconfigured to realize 4N different linear Thusthemixedquantumstatecanbefullycharacterized 1 transformations,andexpectationvaluesaremeasuredaf- with a static measurement setup. v ter each [2]. For example the density matrix of a pair of Hereweillustrateourmethodbyfocusingonperform- 4 polarizationentangledphotonscanbefoundusing16dif- ingtomographyofspatiallyentangledphotonpairs. The 4 ferent measurement settings realized by rotating a series transformation we will use to introduce sparsity is a hy- 3 2 of wave-plates and polarizers [3]. brid of discrete and continuous quantum-walks. In the 0 InthisLetterweshowthatreconfiguringthemeasure- past quantum-walks have been realized using atoms [4], 1. ment setup is actually not necessary for quantum state ions [12] and pairs of photons [13]. They have a wide 0 tomography. We demonstrate that one specially chosen range of potential applications, from helping to explain 6 staticmeasurementsettingissufficienttofullycharacter- energytransferinphotosynthesis[14]toprovidingaplat- 1 ize unknown density matrices. This insight means that formforuniversalquantumcomputation[15]andsolving : v the usually complex and error sensitive tomography pro- the boson sampling problem [16, 17]. Xi cessissimplifiedandnoiseintroducedwhenreconfiguring Specifically we consider hybrid quantum-walks within the measurement setup is avoided. This can be applied on-chip coupled waveguide arrays (WGA) [13]. It has r a to make quantum state tomography more accessible in previously been was shown that a WGA can be used to many different systems, from trapped atoms [4] to quan- perform interferometry [18–20], a classical analogue of tum states of light [5]. Furthermore, this could facilitate quantum state tomography. Light can be coupled into quantum tomography in situations where it was previ- a small number of waveguides, and the intensity mea- ously too complex to be practical. suredattheoutputofthewholeWGA.Thisallowsboth Our approach is inspired by compressed sensing. Con- the phase and amplitude of the input classical fields to ventionallyincompressedsensingitisassumedthatasig- be determined, just from intensity measurements. The nalissparseinsomebasis, andthisknowledgeallowsre- method also leverages sparsity (specifically, the knowl- constructionofthesignalfromfewermeasurementsthan edge that the input state was coupled into only a few suggestedbytheNyquist-Shannonsamplingtheorem[6]. selected waveguides) for full reconstruction of a complex This can be exploited for sub-wavelength optical imag- valued input field from only intensity measurements. 2 Wedeveloptheapproachtoallowreconstructionofthe two-photon correlationmeasurementsat theoutput. We two-photon density matrix from measurements of two- consider a mixed state comprised of two photons propa- photon correlations, employing a new class of WGA cir- gatingthroughawaveguidecircuit. Theneteffectofthe cuit realizing a hybrid of discrete and continuous quan- quantumwalkonasinglephotoncanbewrittenasalin- tum walks. These hybrid quantum walks map any input ear optical transformation, Uˆ [21]. This transformation state entangled across N ports to a sparse state at the will operate on the wavefunction of each photon coupled output M ports, where M > N [Fig. 1(a)]. This ap- into the circuit according to |ψ(cid:105)out = Uˆ |ψ(cid:105)in. For sim- proach is distinct from conventional compressed sensing plicity we assume that both photons undergo the same because, instead of assuming that the signal is sparse, unitarytransformation, butthefollowingcouldbeeasily we introduce sparsity to the signal in a controlled way generalized to the case where each photon undergoes a usingthehybridquantum-walk. Thereforethetechnique different transformation. is not limited to imaging sparse inputs, it is capable of The transformation operating on the joint wavefunc- reconstructing any mixed quantum state and thus forms tion of the two photons (Ψ = (cid:80) c |ψ (cid:105)(i)|ψ (cid:105)(i)) will i i 1 2 a complete system for quantum state tomography. read Ψout = Uˆ ⊗Uˆ Ψin. Now a statistical mixture of The ability to perform quantum state tomography in two photon wavefunctions is best described by the den- a static and 1-D quantum walk circuit is particularly sity matrix, ρˆ= (cid:80) p |Ψ (cid:105)(cid:104)Ψ |, where p is the proba- j j j j j promising when considered in the context of on-chip bility of the mixed quantum system being in state |Ψ (cid:105) j quantum photonics. Using normal tomography in an in- [1]. We see that under the transformation Uˆ the density tegrated photonic chip requires thermally tunable phase matrix changes as shifters to be reconfigured to perform different measure- ments [5, 21]. The latest technological advances enable ρˆ =Uˆ ⊗Uˆ ρˆ Uˆ ⊗Uˆ =Uˆ(4)ρˆ , (1) out in in on-chip integration of highly efficient superconducting whereUˆ(4) =Uˆ⊗Uˆ⊗Uˆ∗⊗Uˆ∗. Eq.(1)canbeexpressed photon detectors [22], however integration of multiple as a matrix equation as shown in Fig. 1(b). Here the in- tunable phase shifters operating at low temperature re- putandoutputdensitymatricesarewrittenasvectorsof mains an open problem. Our approach to tomography N4 and M4 elements respectively, N being the number wouldhavetheadvantageofbeingeasilyintegratedwith ofinputwaveguidesandM thenumberofoutputwaveg- on-chip detection schemes since it requires no tunability. uides. ThetransformationUˆ(4) canthenberewrittenas a M4×N4 matrix in a consistent way. Atthispointitisimportanttoconsiderhowtheoutput state will be measured. Typically in integrated quan- tum photonics the measured quantity will be the cor- relations in the arrival time of two photons at any of the output waveguides [5]. This will give the probability amplitudes, Γ , associated with observing one photon ij in waveguide i and the other in waveguide j. Mathe- matically, expectation values of these probability ampli- tudes are represented by applying the measurement op- erator |j(cid:105)|i(cid:105)(cid:104)i|(cid:104)j| to the density matrix [1], and accord- ingly Γ = (cid:104)i|(cid:104)j|ρˆ |j(cid:105)|i(cid:105). These amplitudes corre- ij out spond to the measurement of only a sub-set of the whole output density matrix. We illustrate this schematically in Fig. 1(b), where such measurable elements are high- lighted in yellow. Our goal is to use these measurements to reconstruct all the elements of the input density ma- trix. Accordingly, we reformulate Eq. (1) by excluding the unobservable elements of ρˆ , out FIG. 1. (a) A linear optical transformation operates on a Γ=MΓ ρˆ . (2) mixed two photon state, then correlations between the pho- in tonsaremeasured. (b)Linearmappingfrominputtooutput Here MΓ is a matrix containing only the rows of Uˆ(4) densitymatrixundertheunitarytransform. Highlightedrows that map the input state (ρˆ ) to correlations (Γ) in the andelementsshowthepartsofthetransformationassociated in with correlation measurements. output mixed state. Now the problem of quantum state tomography can To demonstrate our approach to tomography, we first be formulated as follows; can the input density matrix, introduce the mathematical formalism to propagate a ρˆ , be inferred from Eq. (2) given that Γ has been mea- in mixed quantum state through a quantum walk and take sured? To answer this question we first consider the di- 3 mensionality of ρˆ and Γ. There are generally M2 dif- in ferent correlations which can be measured, each being a real number. The input two-photon density matrix con- tains N4 elements, most of which are complex. However since it is Hermitian, it is fully defined by N4 real pa- rameters. For the tomography problem to be solvable, the amount of measured information should be equal to or exceed the number of unknowns, and accordingly we require M2 ≥ N4. Furthermore, if the two photons are indistinguishable, which we consider in the examples be- low, due to additional symmetries the condition reduces to M(M +1) N2(N +1)2 ≥ . (3) 2 4 This dimensionality requirement means that the trans- formationUˆ mustbeasparseone, mappingN inputsto M outputs with M >N. If the requirement is satisfied, thenitispossiblethatEq.(2)canbeinvertedtofindthe FIG.2. (a)Diagramofahybridquantumwalkfortwo-photon input density matrix. The inversion can be carried out tomography. (b)Realandimaginarypartsoftheinputmixed using the Moore-Penrose pseudoinverse [23], state. Each wavefunction in the mixture is represented by a different colored sphere, statistical weight of each wavefunc- ρˆ =(MΓ)−1Γ. (4) in tion is shown by the diameter of the sphere. (c) Simulated correlation measurements at the output. (d) Real and imag- Herethepseudoinverseisdefinedas(MΓ)−1 =VS−1U†, inary parts of the input density matrix, recovered using cor- where a singular value decomposition is performed as relation measurements shown in (c). MΓ = USV†. If Eq. (4) maps every possible set of cor- relations to a unique density matrix, then full quantum statetomographycanbeachievedsimplybyknowingthe walk can allow full tomography of a two-photon state correlations, Γ, and (MΓ)−1. Of course, (MΓ)−1, the in a planar device. mapping back from measured correlations to input den- An example of a linear optical circuit implementing sity matrix, is completely determined by Uˆ, the linear a suitable hybrid quantum walk is shown in Fig. 2(a). optical transform implemented on each photon before Here two input waveguides (N =2) are split into an ar- correlation measurement. This mapping must be care- rayoffour,realizingonestepofadiscretequantumwalk. fully chosen so that Eq. (4) produces a unique solution Followingthisthefourwaveguidesformacoupledwaveg- foranymeasuredcorrelations,makingquantumstateto- uide array (M =4), with dimensionless length L=3.76 mography possible. and coupling rate C = 1. We note that the waveguide We observe that continuous quantum walks in planar numbers satisfy the necessary condition in Eq. (3). This WGA’s do not lead to unique solutions of Eq. (4), and means the circuit has the potential to allow quantum thus cannot be used for tomography of mixed quantum state tomography just by taking correlation measure- states. Previous works with WGA’s have been focused ments at the output. on recovering classical amplitude profiles and mutual co- We use this circuit to demonstrate tomography of a herence functions of classical light fields [20, 24] or the mixed state comprised of 20 different pure states as density matrix of quantum states where the form of the shown in Fig. 2(b). This way of representing the mixed entangledpartofthewavefunctionisknownapriori[11]. state is used to demonstrate that the state is nontrivial, Furthermore it has been shown that recovery of the mu- containingmanyentangledtwo-photonpurestates. This tual coherence function requires nonlocal coupling in the representation of a mixed state was then used to calcu- WGA [24], which can be achieved in a 2-D WGA. This late the input density matrix, ρˆ , which is the typical in result extends to quantum state tomography also due (and most efficient) representation of such a state. We to the similarities between incoherent classical light and then propagate the density matrix through the hybrid mixed quantum states. The use of a 2-D WGA may WGA circuit, and model the output two-photon spatial be undesirable, especially for the goal of fully integrated correlations presented in Fig. 2(c). From the correlation quantum photonics on a planar chip. Thus we develop a measurementstheinputdensitymatrixisuniquelyrecov- special type of optical circuit, combining a discrete time ered using Eq. (4). The real and imaginary parts of the quantum walk with a continuous quantum walk. As we recovered input density matrix are shown in Fig. 2(d), demonstrate in the following, such a hybrid quantum and we have verified that they exactly match the input 4 state. the number of output waveguides can further reduce the condition number of the transformation. Although with more waveguides more single photon detectors are re- quired, and the complexity of the device increases. FIG. 4. (a) Diagram of linear optical circuit for tomogra- FIG. 3. (a) Mean error in the tomographic reconstruction phy of a two-photon mixed state coupled into three input of randomly generated mixed states using the optical circuit waveguides. (b)Simulatedoutputcorrelationmeasurements. shown. ThestandarddeviationofsimulatedGaussianerrorin (c) Real and (d) imaginary parts of the input density matrix thecorrelationmeasurementsisplottedalongthex-axis. (b) reconstructed from correlation measurements. The condition number of different hybrid quantum walking circuits. The x-axis shows increasing number of waveguides Tomography of states with more input degrees of free- in the WGA, while the y-axis shows increasing length of the dom is also possible. As an example, in Fig. 4(a) we array. demonstrate a static optical circuit, which can be used to recover the density matrix of any two-photon mixed Anyexperimentalimplementationsoftomographywill state coupled into three input waveguides. The circuit suffer from errors in the detection of photons, thus the is again a hybrid quantum walk, splitting three input tomographic technique must be robust. In Fig. 3(a) we waveguidesintosixthenallowingthemtoundergoacon- show the error tolerance of the optical circuit from Fig. tinuousquantumwalkinatenwaveguidearray. Accord- 2(a). Gaussian error is added to the correlation mea- ing to Eq. (3), eight output waveguides could provide surements, then tomography of a random mixed state is enough information to be measured via correlations in attempted. The x-axis shows increasing standard devi- order to determine the input density matrix. However ation of the Gaussian noise, while the y-axis shows the we chose to use ten output waveguides to increase the mean least squares error in the tomographic reconstruc- robustness of the tomography to errors. The output cor- tion, each point averaged over 10000 attempts at recon- relations are shown in Fig. 4(b). The reconstructed real structing random density matrices. This shows that ro- and imaginary parts of the input density matrix are pre- bust tomography of a two-photon state is possible in the sented in Figs. 4(c) and (d), respectively, and they ex- presence of errors. actly match the input state. More generally the stability of the technique is deter- In conclusion, we have introduced a method of per- minedbytheconditionnumber[23]ofthetransformation forming quantum state tomography that only requires MΓ. A high condition number means inversion will be a single set of expectation values to be measured in a impossible in realistic situations, because errors in the static system, in contrast to usual approaches with re- measurements will be highly amplified in the recovered configurable elements. We have developed hybrid quan- density matrix. In Fig. 3(b) the condition number of tum walk circuits for spatially entangled photon-pair to- the transformation is plotted against the number of out- mography, where the input state is reconstructed from put waveguides and the length of the WGA. We see that output correlation measurements. This could facilitate for the case of only four output waveguides discussed on-chip quantum state tomography with integrated su- above optimal choice of the length can give a sufficiently perconductingsingle-photondetectorsandnotunableel- low condition number to allow tomography. Increasing ements. 5 There are a diverse variety of other systems that Express 17, 23920 (2009). could also benefit from this type of tomography. They [8] A. Szameit et al., Nat. Mater. 11, 455 (2012). include nuclear magnetic resonance qubits [25], Bose- [9] D. Gross et al., Phys. Rev. Lett. 105, 150401 (2010). [10] C. H. Baldwin, A. Kalev, and I. H. Deutsch, Phys. Rev. Einstein [26] and exciton polariton condensates [27], A 90, 012110 (2014). atoms [4] and ions [28] in lattice potentials. Quantum [11] D. Oren et al., arXiv 1411.2238 (2014). walks have been demonstrated in a number of these sys- [12] H. Schmitz et al., Phys. Rev. Lett. 103, 090504 (2009). tems, making the correspondence to the two-photon to- [13] A. Peruzzo et al., Science 329, 1500 (2010). mography shown here quite direct. However quantum [14] G. S. Engel et al., Nature 446, 782 (2007). walks are not a fundamental requirement for this new [15] A. M. Childs, Phys. Rev. Lett. 102, 180501 (2009). approach to tomography. Our key concept is that a lin- [16] M. A. Broome et al., Science 339, 794 (2013). [17] M. Tillmann et al., Nature Photonics 7, 540 (2013). ear transformation can be applied to the system to in- [18] S. Minardi and T. Pertsch, Opt. Lett. 35, 3009 (2010). troducesparsity. Ourapproachshowshowtodesignthis [19] S. Minardi et al., Opt. Lett. 37, 3030 (2012). transformation to map every possible density matrix to [20] Y. Shechtman et al., Opt. Express 21, 24015 (2013). a unique set of measurable expectation values, thus en- [21] J. Carolan et al., Science 349, 711 (2015). abling quantum state tomography. [22] F. Najafi et al., Nat. Commun. 6, 5873 (2015). [23] W.H.Press,S.A.Teukolsky,W.T.Vetterling,andB.P. Flannery,NumericalRecipes: TheArtofScientificCom- puting,3rded.(CambridgeUniversityPress,Cambridge, 2007). [1] U. Fano, Rev. Mod. Phys. 29, 74 (1957). [24] S. Minardi, Phys. Rev. A 92, 013804 (2015). [2] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. [25] C.A.Ryan,M.Laforest,J.C.Boileau,andR.Laflamme, White, Phys. Rev. A 64, 052312 (2001). Phys. Rev. A 72, 062317 (2005). [3] A.G.White,D.F.V.James,P.H.Eberhard,andP.G. [26] O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 Kwiat, Phys. Rev. Lett. 83, 3103 (1999). (2006). [4] M. Karski et al., Science 325, 174 (2009). [27] T.Byrnes,N.Y.Kim,andY.Yamamoto,NaturePhysics [5] P. J. Shadbolt et al., Nature Photonics 6, 45 (2012). 10, 803 (2014). [6] D.L.Donoho,IEEETrans.Inf.Theory52,1289(2006). [28] F.Zahringeretal.,Phys.Rev.Lett.104,100503(2010). [7] S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, Opt.