Extracting the physical sector of quantum states
Dmitri Mogilevtsev, Yong-Siah Teo, Jaroslav Rehacek, Zdenek Hradil, Johannes Tiedau, Regina Kruse, Georg Harder, Christine Silberhorn, Luis L. Sánchez-Soto
aa r X i v : . [ qu a n t - ph ] A ug Extracting the physical sector of quantum states
D Mogilevtsev , , Y S Teo , J ˇReh´aˇcek , Z Hradil , J Tiedau ,R Kruse , G Harder , C Silberhorn , , L L Sanchez-Soto , Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Santo Andr´e, SP,09210-170 Brazil B. I. Stepanov Institute of Physics, National Academy of Science of Belarus, NezavisimostiAvenue 68, Minsk 220072, Belarus BK21 Frontier Physics Research Division, Seoul National University, 08826 Seoul, SouthKorea Department of Optics, Palack´y University, 17. listopadu 12, 77146 Olomouc, CzechRepublic Department Physik, Universit¨at Paderborn, Warburger Straße 100, 33098 Paderborn,Germany Max-Planck-Institut f¨ur die Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany Departamento de ´Optica, Facultad de F´ısica, Universidad Complutense, 28040 Madrid,Spain
Abstract.
The physical nature of any quantum source guarantees the existence of an effectiveHilbert space of finite dimension, the physical sector, in which its state is completelycharacterized with arbitrarily high accuracy. The extraction of this sector is essential for statetomography. We show that the physical sector of a state, defined in some pre-chosen basis, canbe systematically retrieved with a procedure using only data collected from a set of commutingquantum measurement outcomes, with no other assumptions about the source. We demonstratethe versatility and efficiency of the physical-sector extraction by applying it to simulated andexperimental data for quantum light sources, as well as quantum systems of finite dimensions.PACS numbers: 03.65.Ud, 03.65.Wj, 03.67.-a
Submitted to:
New J. Phys.
1. Introduction
The physical laws of quantum mechanics ensure that all experimental observations can bedescribed in an effective
Hilbert space of finite dimension, to which we shall refer as the physical sector of the state. The systematic extraction of this physical sector is crucial forreliable quantum state tomography.Photonic sources constitute an archetypical example where such an extraction isindispensable. Theoretically, the states describing these sources reside in an infinite-dimensional Hilbert space. Nonetheless, the elements of the associated density matrices decayto zero for sufficiently large photon numbers, so that there always exists a finite-dimensionalphysical sector that contains the state with sufficient accuracy. Reliable state tomography canthus be performed once this physical sector is correctly extracted.Experiments on estimates of the correct physical sector have been carried out [1, 2].One common strategy is to make an educated guess about the state (such as Gaussianity [3] xtracting the physical sector of quantum states r , so that specialized rank- r compressed-sensing measurementscan be employed to uniquely characterize the state with much fewer measurement settings.Very generally, educated guesses of certain properties of the state requires additional physicalverifications. Algorithms for statistical model selection, such as the Akaike [11–13] orSchwarz criteria [14,15] or the likelihood sieve [16,17], have also been developed to estimatethe physical sector. These algorithms provide another practical solution to reducing thecomplexity of the tomography problem. In the presence of the positivity constraint [18, 19],their application to quantum states becomes more sophisticated, as the procedures for derivingstopping criteria that supplies the final appropriate model subspace for the unknown state areintricate.On the other hand, finite-dimensional systems represent another example for whicha systematic physical-sector extraction becomes important. In the context of quantuminformation, ongoing developments in dimension-witness testing [20–24] offer somesolutions to finding the minimal dimension of a black box required to justify the given setof measurement data in a device-independent way. Searching for dimension witnesses ofarbitrary dimensions is still challenging [23].In reference [25], we showed that, when the measurement device is calibrated, onecan systematically extract the physical sector (that is, both the Hilbert-space support anddimension) and simultaneously reconstruct any unknown state directly from the measurementdata without any assumption about the state. In this paper, we introduce an even moreefficient procedure that extracts the physical sector of any state from the data without statereconstruction and provide the pseudocode. This procedure requires nothing more than dataobtained from a set of commuting measurements. As in [25], the extraction of the physicalsector does not depend on any other assumptions or calibration details about the source. Byconstruction, this procedure has a linear complexity in the dimension of the physical sector. Toshowcase its versatility, we apply it to simulated and experimental data for photonic sourcesand systems of finite dimensions. In this way, we offer a deterministic solution to the problemof extracting the correct physical sector for any quantum state in measurement-calibratedsituations.
2. Physical sectors and commuting measurements
The concept of physical sectors and their relations to commuting measurements is probablybest understood with a concrete example. Let us consider, in the Fock basis, a quantum stateof light described by the density operator ρ b = . ∗ . ∗ · · ·∗ ∗ ∗ ∗ · · · . ∗ . ∗ · · ·∗ ∗ ∗ ∗ · · · ... ... ... ... . . . , (2.1)where ∗ denotes elements of its density matrix that are so tiny that treating them to bezero incurs very small truncation errors. If all ∗ = ρ is the pure state | i h | describedby | i ∝ | α i + |− α i , with the coherent state of amplitude α = . xtracting the physical sector of quantum states ∗ elements are tiny, the state ρ is essentially fully characterized bya 3-dimensional sector, such that elements beyond this sector supply almost no contributionto ρ . This forms a truncated Hilbert subspace where tomography can be carried out reliably.This subspace is given by H sub = span {| i , | i , | i} . However from (2.1), we realize thatthis subspace is not the smallest one that supports ρ . The smallest subspace H phys = span {| i , | i} is in fact spanned by only two basis kets. This defines the 2-dimensional physical sector .In general, the physical sector H phys is defined to be the smallest Hilbert subspace thatfully supports a given state with a truncation error smaller than some tiny ε in some basis.Evidently, the choice of basis affects the description of H phys . If one already knows that ρ is close to | i h | , then choosing | i as part of a basis gives a 1-dimensional H phys . Suchknowledge is of course absent when ρ is unknown. In such a practical scenario in quantumoptics, we may adopt the most common Fock basis for representing ρ and H phys . Whendealing with general quantum systems, the basis that is most natural in typical experimentsmay be chosen, such as the Pauli computational basis for qubit systems. Let us revisit the example in (2.1). Because of the positivity constraint imposed on ρ ,whenever a diagonal element is ∗ , then elements in the row and column that intersect thiselement are all ∗ . Also, if a diagonal element is not ∗ , then it is obvious that H phys isspanned by the basis ket for this diagonal element. For this example, the 2-dimensional H phys completely characterizes ρ with the 2 = ρ , ρ , Re ( ρ ) and Im ( ρ ) .It follows that knowing the location of significant diagonal elements are all we need toascertain H phys . For this purpose the only necessary tool is a set of commuting measurementoutcomes with their common eigenbasis being the pre-chosen basis for H phys . After themeasurement data are performed with these commuting outcomes, all one needs to dois perform an extraction procedure on the data to obtain H phys . This procedure wouldproceed to test a growing set of basis kets until it informs that the current set spans H phys that fully supports the data. We note here that the extraction works for any other sort ofgeneralized measurements in principle, although we shall consider commuting measurementsin subsequent discussions since they are the simplest kind necessary for extracting physicalsectors in large Hilbert-space dimensions.
3. The extraction of the physical sector
In some pre-chosen basis, the physical-sector extraction procedure (PSEP) iteratively checkswhether its data are supported by the cumulative sequence of H sub with truncation errorsmaller than some tiny ε . PSEP starts deciding whether, say, H sub = span {| n i , | n i} ofthe smallest dimension d = H phys . Otherwise, PSEP continues and decides if H sub = span {| n i , | n i , | n i} adequately supports the data, and so on until finally PSEP assigns a d phys -dimensional H sub = H phys with some statistical reliability. In each iterative step, there are three objectivesto be met: (Ci) PSEP must decide if the data are supported with H sub spanned by some set of basis ketsor not. (Cii) PSEP must report the reliability of the statement “ H sub supports ρ with truncation errorless than ε ”. xtracting the physical sector of quantum states (Ciii) PSEP must ensure that the final accepted set of basis kets span H phys , the smallest H sub that supports ρ .In what follows, we show that all these objectives can be fulfilled with only the informationencoded in the measurement data. We proceed by first listing a few notations. In an experiment, a set of measured commutingoutcomes are described by positive operators ∑ j Π j =
1. They give measurement probabilities p j = tr ( ρΠ j ) according to the Born rule. Each commuting outcome, in the commoneigenstates | n i h n | that are also used to represent the physical sector, can be written as Π j = ∑ l c jl | l i h l | (3.1)with positive weights c jl that characterize the outcome.To decide whether the p j s are supported with some Hilbert subspace H sub , the easiestway is to introduce Hermitian decision observablesW sub = ∑ j y j Π j (3.2)for real parameters y j . The decision observable for testing H sub , along with its y j s, satisfiesthe defining property, h n | W sub | n i = (cid:26) | n i ∈ H sub , a n > ρ is completely supported in H sub , then theexpectation value h W sub i = ∑ j y j p j = H sub = H phys ). Quantum systems of finite dimensions possess statesof this kind. In quantum optics however, ρ is not completely supported in any subspace,but possesses decaying density-matrix elements with increasing photon numbers [such as theexample in (2.1)]. A laser source, for instance, cannot produce light of an infinite intensity.Furthermore, the Born probabilities p j are never measured. Instead, the data consist of relativefrequencies f j that estimate the probabilities with statistical fluctuation. Therefore, if wedefine the decision random variable (RV) w sub = ∑ j y j f j (3.4)that estimates h W sub i , then PSEP may assign H sub = H phys with a truncation error defined by | w sub | that is smaller than ε . The decision RV w sub is an unbiased RV in that the data average of w sub is the true value h W sub i that PSEP achieves to estimate ( E [ w sub ] = h W sub i ). This means that in the limit of largenumber of measured detection events N for the data { f j } , w sub approaches its expected value E [ w sub ] , which in turn tends to zero in the limit H sub → H phys . This limiting behavior invitesus to understand the truncation error | w sub | using the well-known Hoeffding inequality [26],which states that α ≡ Pr {| w sub | ≥ ε } ≤ − N ε ∑ j y j ! . (3.5) xtracting the physical sector of quantum states α of having a truncation errorgreater than or equal to ε , which is the significance level of the hypothesis that w sub = E [ w sub ] for all conceivable future data [27]. With N ≥ − ( α / ) ε ∑ j y j , (3.6)we are assured with α significance that the main factor for a nonzero | w sub | comes frominsufficient support from H sub since statistical fluctuation is heavily suppressed.One can obtain the more experimentally-friendly inequality [26] α ≤ B sub = (cid:18) − | w sub | ∆ (cid:19) (3.7)in terms of the variance ∆ of w sub , where we take ε ≈ | w sub | as a sensible guide to thetruncation-error threshold. For N ≫
1, the 1 / N scaling of ∆ allows the quantity B sub toprovide an indication on the reliability of the statement “ H sub supports ρ with truncationerror less than ε ” with a reasonable statistical estimate for ∆ from the data. If (3.7) holds for H sub and some pre-chosen α , then the assignment H phys = H sub is made. Quite generally, w sub and ∆ reveal the influence of both statistical and systematic errors [28]. Therefore, byconstruction, for sufficiently large N , H sub eventually converges to the unique H phys at α significance with increasing size of the basis set for properly chosen H sub . The choice of H sub at each iterative step of PSEP must be made so that the final extracted support is indeed H phys , the smallest support for ρ . To ensure that H phys is really extracted, and not some other larger H sub that also supportsthe data, we once more return to the example in (2.1). For that pure state, in the Fock basis,the H sub that supports the state is effectively 3-dimensional, whereas H phys is effectively2 dimensional. With sufficiently large number of detection events N , if one naively carriesout PSEP starting from H sub = span {| i} , PSEP would recognize that H sub cannot supportthe data, continue to test the next larger subspace H sub = span {| i , | i} , where it wouldagain conclude insufficient support. Only after the third step will PSEP accept H sub = span {| i , | i , | i} as the support at some fixed α significance. However, H sub = H phys .In order to efficiently extract H phys , we need only one additional clue from the data ,that is the relative size of the diagonal elements of ρ . We emphasize here that we are not interested in the precise values of the diagonal elements, but only a very rough estimateof their relative ratios to guide PSEP. With this clue, we can then apply PSEP using theappropriately ordered sequence of basis kets to most efficiently terminate PSEP and obtain thesmallest possible support for the data. For the pure-state example, the decreasing magnitudeof the diagonal elements gives the order {| i , | i} . For any arbitrary set of commuting Π j s,given the measurement matrix C of coefficients c jl , sorting the column C − f , defined by theMoore-Penrose pseudoinverse C − of C , in descending order suffices to guide PSEP‡. Thissorting permits the efficient completion of PSEP in O ( d phys ) steps without doing quantumtomography. Other sorting algorithms are, of course, possible without any information aboutthe diagonal-element estimates. One can perform other tests on different permutations of basiskets within the extracted Hilbert-subspace support, although the number of steps required tocomplete PSEP would be larger than O ( d phys ) . ‡ This is not tomography for the photon number distribution, but merely a very rough estimate on the relative ratiosof diagonal elements, since C − f is not positive. xtracting the physical sector of quantum states An astute reader would have already noticed that it is the H phys within the field-of-view (FOV)of the data that can be reliably extracted. The FOV is affected by three factors: the degreeof linear independence of the measured outcomes, the choice of some very large subspaceto apply PSEP whose dimension does not exceed this degree of linear independence, and theaccuracy of the data (the value of N ). In real experiments, the number of linearly independentoutcomes measured is always finite. With the corresponding finite data set, there exists a largesubspace for extracting H phys , in which the decision observables W sub always satisfy (3.3) forany H sub . For sufficiently large N , the collected data will capture all significant features of H phys within this data FOV.Indeed, if the source is truly a black box, then defining the data FOV can be tricky. Trueblack boxes are, however, atypical in a practical tomography experiment since it is usuallythe observer who prepares the state of the source and can therefore be confident that the stateprepared should not deviate too far from the target state as long as the setup is reasonably well-controlled. The data FOV should therefore be guided by this common sense. On the otherhand, the extraction of H phys in device-independent cryptography, where both the source andmeasurement are completely untrusted for arbitrary quantum systems, is still an open problem.We note here that the measurement in (3.1) may incorporate realistic imperfections,such as noise, finite detection efficiency, that are faced in a number of realistic schemes.For instance, the commuting diagonal outcomes may represent on/off detectors of varyingefficiencies, or incorporate thermal noise [29, 30]. All such measurements are presumed to becalibratable, as non-calibrated measurements require other methods to probe the source. Asan example, suppose that the measurement is inefficient but still trustworthy enough for theobserver to describe its outcomes by the set { η j Π j } with unknown inefficiencies η j < η j = η j ( T , . . . , T l ) for l that is typically much less thanthe total number of outcomes in practical experiments. Then the straightforward practice is tofirst calibrate all T j s before using them to subsequently carry out PSEP for other sources. Onemay also choose to calibrate T j already during the sorting stage by “solving” the linear system t = C − f ′ , where f ′ j = f j / η j is now linear in the data f j and nonlinear in T j . The estimationof T j falls under parameter tomography that is beyond the scope of this discussion, whichfocuses on the idea of locating physical sectors and not the exact values of density matrices.
4. The pseudocode for physical-sector extraction
Suppose we have a set of commuting measurement data { f j } that form the column f , aswell as the associated outcomes Π j of some eigenbasis {| i , | i , | i , . . . } that is adopted torepresent H phys . For some pre-chosen basis and α significance, the pseudocode for PSEP ispresented as follows: STEP
1. Compute the measurement matrix C and sort C − f in descending order to obtain theordered index i . Then, define the ordered sequence of basis kets {| n i i , | n i i , | n i i , . . . } . STEP
2. Set k = H sub = span {| n i i} . STEP
3. Construct W sub by solving the linear system of equations in equation (3.3) for the y j s. STEP
4. Compute w sub , ∆ and hence B sub . For typical multinomial data, ∆ = ∑ jk y j y k ( δ j , k p j − p j p k ) / N . STEP
5. Increase k by one and include | n i k i in H sub . xtracting the physical sector of quantum states STEP
6. Repeat
STEP B sub ≥ α . Finally, report H phys = H sub and α andproceed to perform quantum-state tomography in H phys .
5. Results
To illustrate PSEP, we consider the state in (2.1) and ρ = | i h | + | i h | + | i h | .Simulated data are generated with a random set of commuting measurement outcomes. Theextracted physical sectors are shown in figure 1.Data statistical fluctuation may be further minimized by averaging B sub over manydifferent sets of commuting outcomes. Moreover, one can detect additional systematic errorsthat are not attributed to truncation artifacts by inspecting the corresponding histograms forerrors larger than the statistical fluctuation.We next proceed to experimentally validate PSEP by measuring photon-click events ofa time-multiplexed detector (TMD). We use a fiber-integrated setup to generate and measurea mixture of coherent states, as depicted in figure 2(a). Coherent states are produced bya pulsed diode laser with 35 ps pulses at 200 kHz and a wavelength of 1550 nm. Thesepulses are then modulated with a telecom Mach-Zehnder amplitude modulator, driven with asquare-wave signal at 230 kHz. This produces pseudorandom pulse patterns with two fixedamplitudes. After passing through fiber-attenuators, the state is measured with an eight-binTMD [31, 32] with a bin separation of 125 ns and two superconducting nanowire detectors.We record statistics of all possible 2 bin configurations, which corresponds to a total of 256 n B n Figure 1.
Physical sectors extracted with PSEP from simulated data of N = detectionevents for (a) the pure state in (2.1) (black solid curve represents its photon-numberdistribution) and (b) the mixed state ρ = | i h | + | i h | + | i h | . 2000 random setsof 40 commuting measurement outcomes were used to calculate the average B sub in everyiterative step k . The (blue) histogram plots B sub for the default ordering of the basis kets labeledwith n = , , ... . The physical sector H phys (yellow region) is revealed after completingPSEP with respect to a 5% significance level ( α = .
05) (red solid line). xtracting the physical sector of quantum states n B n n B n n B n (a) qubit ( π ,π, π ) (b) qutrit ( π , π , π ) (c) ququart ( π , π , π ) Figure 2.
Schematic diagram of (a) the experimental setup to measure a mixture of coherentstates and (b) the result of PSEP on the data for a mixture of two coherent states of meanphoton numbers 9.043 and 36. Panel (a) describes coherent states from a pulsed laser passthrough an amplitude modulator (AM), which switches between two values of attenuation.Neutral density (ND) filters further attenuate the light to the single photon level. The timemultiplexing detector (TMD) consists of three fiber couplers, delay lines and superconductingnanowire single photon detectors (SPD). The physical sector in panel (b) is extracted fromdata of N = . × detection events. 5000 different sets of 60 outcomes out of the measured256 were used to calculate the average B sub in every iterative step. Other figure specificationsfollow those of figure 1. TMD outcomes.To characterize the TMD outcomes for the measurement, we perform standard detectortomography, using well calibrated coherent probe states [33, 34]. The setup is similar to theprevious one, but we replace the modulator by a controllable variable attenuator. We calibratethe attenuation with respect to a power meter at the laser output. This allows us to produce aset of 150 probe states with a power separation of 0.2 dB.TMD data of a statistical mixture of two coherent states are collected and PSEP issubsequently performed on these data. The accuracy of the extracted physical sector isultimately sensitive to experimental imperfections. In this case, these imperfections areminimized owing to the state-of-the-art superconductor technology, the fruit of which isa histogram that is as clean as it gets in an experimental setting. Figure 2(b) providesconvincing evidence of the feasibility and practical performance of the technique, where realdata statistical fluctuation is present. This physical sector may subsequently be taken as theobjective starting point for a more detailed investigation of the quantum signal with tools fortomography and diagnostics.
To analyze another aspect of PSEP, in this section, we apply it to quantum systems offinite dimensions with discrete-variable commuting measurement outcomes. As a specificexample, we consider the arrangement in reference [22], which uses single photons toencode the information simultaneously in horizontal ( H ) and vertical ( V ) polarizations, andin two spatial modes ( a and b ). We define four basis states: | i ≡ | H , a i , | i ≡ | V , a i , | i ≡ | H , b i , and | i ≡ | V , b i . On passing through three suitably oriented half-wave platesat angles θ , θ , and θ , the state of such hybrid systems can be converted to the pure state ρ = | θ , θ , θ i h θ , θ , θ | , defined by | θ , θ , θ i = sin ( θ ) sin ( θ ) | i − sin ( θ ) cos ( θ ) | i + cos ( θ ) cos ( θ ) | i + cos ( θ ) sin ( θ ) | i . (5.1)Thus, by adjusting the orientation angles of the wave plates, one could produce qubits,qutrits or ququarts from such a hybrid source. Here, we show that PSEP can rapidly extract H phys by inspecting only the data measured from a set of commuting quantum measurements. xtracting the physical sector of quantum states Laser
AM TMD ND a) SPD1SPD2
TMD b) Figure 3.
PSEP for hybrid quantum systems of finite dimensions that potentially generateseither (a) a qubit state, (b) a qutrit state, (c) or a ququart state according to equation (5.1).With N = . × detection events, all three physical sectors (yellow region) are correctlyextracted. For the ququart, the slightly higher reordered B H sub bar at n = N ) is a manifestation of the favorable sensitivity of the procedure to specificquantum-state features, not just the overall physical sector. Figure specifications follow thoseof figure 1. Figure 3 presents the plots for a qubit, qutrit and ququart system characterized by the different( θ , θ , θ ) configurations.We have thus shown that in the typical experimental scenarios where the measurementsetup is reasonably-well calibrated, and hence trusted, H phys can be systematically extractedwithin the subspace spanned by the measurement outcomes. This allows an observer tolater probe the details of the unknown but trusted quantum source using only the data athand. Notice that the relevant basis states, labeled by n , form a basis for the commutingmeasurement on the black box. As such, this procedure is not a bootstrapping instruction.Rather, it systematically identifies the correct H phys without any other ad hoc assertions aboutthe source. In this way, we turn PSEP into an efficient deterministic dimension tester withcomplexity O ( d phys ) , as we have already learnt from section 3.3.
6. Conclusions
We have formulated a systematic procedure to extract the physical sector, the smallest Hilbert-subspace support, of an unknown quantum state using only the measurement data and nothingelse. This is possible because information about the physical sector is always entirely encodedin the data. This extraction requires only few efficient iterative steps of the order of thephysical-sector dimension.We demonstrated the validity and versatility of the procedure with simulated andexperimental data from quantum light sources, as well as finite-dimensional quantum systems.The results support the clear message that, for well-calibrated measurement devices, thephysical sector can always be systematically extracted and verified with statistical tools, inwhich quantum-state tomography can be performed accurately. No a priori assumptions aboutthe source, which require additional testing, are necessary. The proposed method should serve xtracting the physical sector of quantum states
Acknowledgments
D. M. acknowledges support from the National Academy of Sciences of Belarus throughthe program “Convergence”, the European Commission through the SUPERTWIN project(Contract No. 686731), and by FAPESP (Grant No. 2014/21188-0). Y. S. T. acknowledgessupport from the BK21 Plus Program (Grant 21A20131111123) funded by the Ministry ofEducation (MOE, Korea) and National Research Foundation of Korea (NRF). J. ˇR and Z. H.acknowledge support from the Grant Agency of the Czech Republic (Grant No. 15-03194S),and the IGA Project of Palack´y University (Grant No. PRF 2016-005). J. T., R. K., G. H., andCh. S. acknowledge the European Commission through the QCumber project (Contract No.665148). Finally, L. L. S. S. acknowledges the Spanish MINECO (Grant FIS2015-67963-P). xtracting the physical sector of quantum states [1] ˇReh´aˇcek J and Paris M G A 2004 Lecture Notes in Physics – Quantum State Estimation vol 649 (BerlinHeidelberg: Springer)[2] Lvovsky A I and Raymer M G 2009 Continuous-variable optical quantum-state tomography
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