Selective and Efficient Quantum Process Tomography with Single Photons

Christian Tomas Schmiegelow,1 Miguel Antonio Larotonda,2 and Juan Pablo Paz1

1Departamento de Fısica and IFIBA, FCEyN, UBA, Pabellon 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina2CEILAP, CITEDEF, J.B. de La Salle 4397, 1603 Villa Martelli, Buenos Aires, Argentina

(Received 29 December 2009; published 24 March 2010)

We present the results of the first photonic implementation of a new method for quantum process

tomography. The method {originally presented by A. Bendersky et al. [Phys. Rev. Lett. 100, 190403

(2008)]} enables the estimation of any element of the �-matrix that characterizes a quantum process using

resources that scale polynomially with the number of qubits. It is based on the idea of mapping the

estimation of any �-matrix element onto the average fidelity of a quantum channel and estimating the

latter by sampling randomly over a special set of states called a 2-design. With a heralded single photon

source we fully implement such algorithm and perform process tomography on a number of channels

affecting the polarization qubit. The method is compared with other existing ones, and its advantages are

discussed.

DOI: 10.1103/PhysRevLett.104.123601 PACS numbers: 42.50.Dv, 03.65.Wj, 03.67.Mn, 42.65.Lm

The complete characterization of a linear quantum pro-cess mapping initial states into final states (Eð�inÞ ¼ �out)is a very hard task. Actually, this well-known fact can beunderstood as follows: A general completely positive (CP)map can be fully characterized by the so-called �-matrix[1]. Choosing the basis Ea consisting of D2 operators, the

�-matrix representation for the map E is such that Eð�Þ ¼Pab�abEa�E

yb (D ¼ 2n for a system of n qubits). The

matrix �ab must be positive and Hermitian for the channelto be completely positive (CP) and Hermitian itself. More-over, for the map to be trace preserving, its �-matrix

should be such that the conditionP

ab�abEyaEb ¼ I is

satisfied. The matrix �ab is defined by D4 �D2 real pa-rameters, a number that scales exponentially with thenumber of qubits of the system. This implies that fullquantum process tomography (QPT) is unavoidably hard.Moreover, until recently, known methods to estimate ma-trix elements �ab were also inefficient since they requireresources (measured in terms of the number of repetitionsof each experiment, on the number of operations requiredto perform them, etc.) scaling exponentially with the num-ber of qubits [1–3]. Recently, we introduced an efficientstrategy enabling the estimation of any �ab element inves-ting for such purpose resources scaling polynomially withthe number of qubits of the system (and on the accuracyrequired for the estimation) [4]. In this Letter, we willpresent the first successful experimental implementationof such strategy using single photons and linear opticalelements. The experiment is not only an illustration of theuse of a general scheme, but also makes clear the advan-tages of the new tomographic method over itspredecessors.

It is convenient first to present the main ingredients ofthe only available method to perform selective and efficientquantum process tomography (SEQPT). For this purpose,it is best to first notice a general property of quantumchannels. Assuming only that the operator base Ea is or-

thonormal in the Hilbert Schmidt inner product, it can beshown that any element �ab is related to the average fi-delity of a modified channel Eab. Such channel depends onthe original E and on the operators Ea and Eb. It acts on any

state � as Eabð�Þ ¼ EðEya�EbÞ. Thus, we can show that [4]

�ab ¼ 1

D½ðDþ 1ÞFðEabÞ � �ab�: (1)

Here, the average fidelity FðEabÞ is computed integratingover all pure states j�i using the (unitarily invariant) Haarmeasure, i.e.,

FðEabÞ ¼Z

dj�ih�jEabðj�ih�jÞj�i: (2)

The method of SEQPT is based on the use of the aboveidentity and on the fact that the integral over the entireHilbert space can be exactly computed using finite resour-ces. It has been shown that the average fidelity can beexactly evaluated by computing the average over a finiteset of states that form a so-called 2-design [5]. In fact, if theset S ¼ fjc ji; j ¼ 1; . . . ; Kg is a 2-design, we have

FðEabÞ ¼ 1

K

Xj

hc jjEabðjc jihc jjÞjc ji: (3)

Although the exact computation of �ab using a 2-designwould involve performing a sum over an exponentiallylarge set, the estimation of such coefficients with finite(D-independent) precision is possible by randomly sam-pling over states jc ji. As described in [4] and discussed

with more detail below, the precision in the estimation

scales better than 1=ffiffiffiffiffiM

pwith the number of repetitions

M of the experiment.There are several known examples of 2-designs such as

the one formed by theDðDþ 1Þ states belonging to Dþ 1mutually unbiased bases [6] (two bases are mutually un-biased if and only if the absolute value of the overlap

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between states in such bases is 1=ffiffiffiffiD

p). Equation (3) sug-

gests an immediate way to characterize the quantum chan-nel: For every coefficient �ab, one needs to compute thesurvival probability of states jc ji when evolved over the

channel Eab and then average over the 2-design. For thestrategy to be viable, one needs not only to make sure thatthe channel can be implemented efficiently but also that thesum over the elements of the 2-design can be estimatedwith polynomial resources. The fact that these two taskscan be performed is the core of SEQPT. In particular, thediagonal elements �aa are average fidelities of the CPmapEaa, obtained by composing the original map E with the

unitary operation Ea: Eaað�Þ ¼ EðEya�EaÞ. Evaluating off

diagonal elements �ab requires a different strategy that atfirst sight seems to be rather different than the one used fordiagonal coefficients. However, the main point in thisstrategy, again, is to realize that any off diagonal coeffi-cient �ab is related through (1) with the average fidelity ofthe map Eab. This map is not CP but can be obtained as thedifference between two CP maps. For this reason, asshown in [4], the off diagonal coefficients can be obtainedas the mean value of an ancillary qubit conditioned to thesurvival of the state jc ji and averaged over the 2-design.

The complete algorithm is described in detail in Fig. 1. Infact, real and imaginary parts of �ab are obtained by condi-tionally measuring the mean values of �x and �y of an

ancillary qubit that interacts with the system withcontrolled-Ea;b operations.

Although SEQPT is the only available scheme to effi-ciently estimate any coefficient of the �-matrix, it hasnever been implemented in practice. Other methods havebeen demonstrated in different experimental setups (see,for example, [7–12]). We present here the results of the firstimplementation of SEQPT in a photonic quantum infor-mation processor. For this, we use a heralded single photonsource encoding two qubits in a single photon. We do thisby using both the polarization degree of freedom and themomentum (path) degree of freedom of a heralded photongenerated by parametric-down-conversion [13]. Themethod enables us to characterize unknown quantum chan-nels affecting the polarization qubit using the path qubit asan ancilla.

The full implementation is schematically shown inFig. 2. For the sake of clarity, we divide the descriptionof the method in four steps: (i) generation of initial states ofthe 2-design jc ji, (ii) application of controlled operations

between ancillary qubit and target qubit, (iii) the evolution

through the unknown process, and (iv) measurement in the�x or �y base for the ancilla (path) qubit conditioned on

the survival of the jc ji (polarization) states. With these

steps, we implement the algorithm of Fig. 1 with thephotonic setup described in Fig. 2. Stage (i) is rathersimple. For the case of a single qubit, it turns out that the2-design is formed by the DðDþ 1Þ ¼ 6 eigenstates of thethree Pauli operators (which define three mutually un-biased bases). For the polarization of photons, these are:two states with vertical-horizontal polarization, two withdiagonal (45–135 degrees) polarization, and two with cir-cular (right-left) polarization. The generation of such statesis done with the standard polarizer, half and quarter wave-plate configuration. Stage (ii) of the algorithm is concep-tually simple: the photon is split by the first nonpolarizingbeam splitter, which acts as the Hadamard gate in theancillary qubit, then any controlled operation on the polar-ization qubit can done by rotating independently the po-larization in each path using a quarter-half-quarter wave-plate configuration [13]. In order to apply the unknownprocess (iii), both paths are then made parallel and sentthough a zone where the unknown channel E is performedon the polarization qubit. Finally, (iv) to measure theexpectation values of�x and�y conditioned to the survival

of jc ji, we make both paths interfere with a relative phase

� at a second nonpolarizing beam splitter by tilting a glass

FIG. 1. The quantum algorithm for measuring Reð�abÞ for agiven channel E. The method requires an extra ancillary (clean)qubit. The imaginary part of �ab is estimated in the same way bymeasuring the polarization of the ancilla along the y axis.

FIG. 2 (color online). A continuous laser diode at 405 nm and40 mW generates frequency degenerate twin photons on BBOcrystal, cut for Type-II parametric-down-conversion. One photonis used as a herald while on the other, a polarization and pathqubit are encoded. We first prepare a state from the 2-design onthe polarization qubit with a polarizer, half and quarter waveplates; then we perform a Hadamard gate on the path with anonpolarizing 50:50 beam splitter. Next we perform the con-trolled operations Ea and Eb with a quarter-half-quarter waveplate sequence on each arm; both paths are then sent through theunknown channel E affecting the polarization qubit equally onboth paths. Finally, measurement of �x or �y in the path qubit is

done by interfering both paths at a second nonpolarizing beamsplitter with a phase � previously set by tilting a glass; projec-tion on the prepared polarized state is done by a quarter and halfwave plate followed by a polarizing beam splitter, and detectionis done by fiber coupled avalanche photodiodes. The interfer-ometer has an average visibility of 92%. Not shown in the figureis an intense beam counterpropagating in the interferometerwhich is monitored to actively control the position of one ofits mirrors with a piezoelectric disc attaining a stability of over�=30 over all measurements.

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in one of the paths and projecting into the polarizationstates with the inverse quarter, half wave plate and polar-izer configuration as in (i). With this scheme measure thequantities pabð�xðyÞ;�;�c j

Þ, which are the probabilities

of finding the ancilla in the � state of the xðyÞ baseconditioned on the survival of the input states jc ji for

each Eab. With these probabilities, we obtain all the neces-sary data to determine the matrix element �ab. A note mustbe made about diagonal elements: as mentioned above, inthis case, the scheme simplifies significantly because thereis no need for an ancilla qubit so one then can look at onlyone arm of the interferometer by blocking the other one.

The elements of �ab were detected for two differentnoisy processes affecting the polarization degree of free-

dom. The results are shown in Fig. 3. We must stress thatalthough we measured all the matrix elements character-izing both channels, our method determines any �ab inde-pendently and efficiently (i.e., as opposed to previousmethods [1], it is not necessary to fully characterize thechannel in order to determine a single �ab coefficient). Themeasured processes correspond to the identity channel(i.e., free propagation through air) and to a quarter waveplate at 0�. Those processes were also fully characterizedby means of the standard method of QPT (as explained inchapter 10 of [1]). The results obtained by both methodsare compared in Fig. 3 and turn out to be in very goodagreement. As a figure of merit to compare both schemes,we numerically calculate fidelity between the channelsdetermined by each method obtaining F ¼ 95, 1%�1:5% for the identity and F ¼ 96, 3%� 1:6% for thehalf wave plate at 0� [14]. The maximum fidelity attainableis mainly limited by interferometer visibility, which resultsin a not perfectly clean ancilla.SEQPT is a method that is suited to perform partial

process tomography selecting the relevant parameters onewants to estimate and investing polynomial resources forsuch estimation. It is interesting to compare the resourcesrequired to implement this new tomographic method andpreviously existing ones. To determine any single matrixelement �ab using SEQPT, we measured DðDþ 1Þ ¼ 6survival probabilities. On the contrary, obtaining a singlematrix element �ab using the standard QPT [1] requiresestimating D2 �D2 ¼ 16 transition probabilities. Thisimplies that even at the level of a single qubit, theSEQPT is more ‘‘efficient.’’ This comparison might seemsomewhat unfair since after such number of experiments,the standard method provides all the information requiredto estimate the full �ab matrix and not just a single elementof such matrix. However, when working with larger sys-tems, it is clear that one would never estimate the exactmatrix elements �ab by performing the average over theentire 2-design. In fact, this would always be exponentiallyhard (as the 2-design is a set containing an exponentiallylarge number of elements). Instead, the main virtue ofSEQPT is that it enables us to estimate any element withfixed accuracy by performing a number of measurementsthat only depends on such accuracy and is independent onthe number of qubits. As shown in [4], the error in theestimation of the average in Eq. (3) after M experiments

scales as 1=ffiffiffiffiffiM

p. However, it is interesting to get a tighter

bound for such error. In fact, if the average fidelity iscomputed by sampling over states jc ji which are never

prepared twice, then one can show that the error in the

average turns out to scale as �� /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1M ð1� M�1

K�1Þq

where

K ¼ DðDþ 1Þ is the number of elements in the 2-design.In fact, this equation tells us that the error vanishes as thenumber of experiments M ¼ K.Having the experimental data at hand, we can test the

behavior of the estimation error. Using the raw data, wesplit the 2-design in samples of different (variable) size and

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FIG. 3 (color online). Experimental results for SEQPT (a.1)and (a.2) and Standard QPT (b.1) and (b.2). First two rowsdisplay measured real (top) and imaginary (bottom) parts ofthe matrix �ab of the quantum process corresponding to freepropagation (identity channel). Similarly, the last two rowscorrespond to real (top) and imaginary (bottom) parts of thematrix �ab measured for a process corresponding to a QWPat 0�.

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computed the average over each of such samples. Theresults are shown in Fig. 4 where we display the behaviorof the estimation error for the matrix element �ab obtainedby sampling the 2-design in groups of increasing numberof states [the size of the sample, M, grows up to K ¼DðDþ 1Þ ¼ 6, which is the cardinal of the 2-design].These results are quite interesting and not expected:Although for a single qubit the sample space is rathersmall, the above mentioned bound turns out to be satisfiedrigorously for every instance of the experiment and notonly at the statistical level. We see from Fig. 4 that allpossible errors on all possible choices of sample partitionslie below the analytical bound. It is clear that this is notexpected for all possible random distributions, but the onesrealized in the experiment strictly satisfy the bound. In fact,it is not hard to imagine possible values for the results ofthe experiment that would violate the bound for certainsamples but hold to it at the statistical level. However, suchcases are not realized in the experiment which suggests thatit would be possible to find a tighter bound.

We conclude by noticing that looking at the quantumstate of the photon at the output of the final beam splitter inFig. 2 is quite revealing about the nature of SEQPT. In fact,it can be shown that if we prepare the initial state of thepolarization qubit to be jc ji, then the unnormalized output

state turns out to be described by the density matrixE½ðEa � EbÞkc jihc jjðEa � EbÞ� (the � signs are con-

trolled by a � ¼ 0, � phase with the glass in Fig. 2).This shows that the ancillary (path) qubit plays a rathersimple role in SEQPT: it is simply a tool to prepare aspecial initial state which is later sent through the channelE. This observation serves as a motivation for a simple, butilluminating, reformulation of the SEQPT algorithm. Infact, SEQPT is connected with a class of quantum algo-

rithms known as ‘‘deterministic quantum computationwith one clean qubit’’ (DQC1) [15,16]. However, as op-posed to DQC1, Fig. 1 shows that the measurement on theancilla can be moved to the initial stage of the algorithm(i.e., it can be performed before the channel E acts on thesystem). From this point of view, SEQPT can be describedas follows: The unnormalized state of the system condi-tioned on the detection of the �1 eigenvalue in the mea-surement of �x on the ancilla turns out to beðEa � EbÞjc jihc jjðEa � EbÞ. Thus, after the system

goes through the channel E, the probability of detectingthe state jc ji is simply the fidelity of the map E�ð�Þ ¼E½ðEa � EbÞ�ðEa � EbÞ�. The expectation value of �x iscomputed as the difference between the fidelities of themaps E�, which is nothing but the real part of �ab (analo-gously for the imaginary parts with� ¼ ��=2). This newinterpretation of SEQPT, suggested by the experimentindicates that the method could indeed be implementedwithout ancillary qubits at all. In any case, the presentexperiment shows the viability of SEQPT as a way tocharacterize a channel affecting polarization of single pho-tons and displays its advantages over other alternativeapproaches.The experiments were performed in the quantum optics

lab at CITEDEF. J. P. P. and M.A. L. are members ofCONICET. The authors acknowledge discussions withA. Bendersky, A. Hnilo, and R. Piegaia.

[1] M.A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).

[2] M. Mohseni and D.A. Lidar, Phys. Rev. Lett. 97, 170501(2006); Phys. Rev. A 75, 062331 (2007).

[3] G.M. D’Ariano and P. Lo Presti, Phys. Rev. Lett. 86, 4195(2001).

[4] A. Bendersky, F. Pastawski, and J. P. Paz, Phys. Rev. Lett.100, 190403 (2008); Phys. Rev. A 80, 032116 (2009).

[5] P. Delsarte, J. Goethals, and J. Seidel, GeometriaeDedicata 6, 363 (1977).

[6] A. Klappenecker and M. Rotteler, in Proc. Int. Symposiumon Information Theory (IEEE, Adelaide, Australia, 2005),p. 1740-1744.

[7] J. Emerson and M. Silva et al., Science 317, 1893 (2007).[8] W. Liu et al., Phys. Rev. A 77, 032328 (2008).[9] T. Monz et al., Phys. Rev. Lett. 102, 040501 (2009).[10] J. Altepeter et al., Phys. Rev. Lett. 90, 193601 (2003).[11] N. Boulant et al., Phys. Rev. A 67, 042322 (2003); M.

Howard et al., New J. Phys. 8, 33 (2006).[12] C. C. Lopez, B. Levi, D.G. Cory, Phys. Rev. A 79, 042328

(2009).[13] B. G. Englert, C. Kurtsiefer, and H. Weinfurter, Phys. Rev.

A 63, 032303 (2001).[14] The fidelity between channels is commonly defined as

F½E1; E2� ¼Rf ðE1ð�Þ; E2ð�Þ Þdjc i where � ¼ jc ihc j

and f½�1; �2� ¼ ðTrð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1

p�2

ffiffiffiffiffiffi�1

pp Þ Þ 2, see [10].[15] E. Knill and R. Laflamme, Phys. Rev. Lett. 81, 5672

(1998).[16] C. Miquel et al., Nature (London) 418, 59 (2002).

FIG. 4 (color online). Convergence for the �-matrix elementstheoretical (smooth line) and experimental (segmented lines).Shown are the errors in the estimation of every �ab for bothmeasured processes. Each curve shows the error made in theestimation for each �ab if only M measurements were made. Allpossible choices of the first M states are plotted, and they liebelow the theoretical bound. The insets show the errors for twospecific �ab elements. In (a), the error can stay close to thebound for some choices of the firstM measurements while in (b),is always almost null.

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