Numerical estimation schemes for quantum tomography

In general, however, the measurement data obtained from complexquantum systems are informationally incomplete and, as a rule, do not yield a unique state estimator.. Finally, we also pro

NATIONAL UNIVERSITY OF SINGAPORE

DOCTORAL THESISsubmitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy in Science

Teo Yong Siah

Numerical Estimation Schemes for Quantum Tomography

Thesis Advisor: Berthold-Georg ENGLERT

Department of Physics/

Centre for Quantum Technologies/

NUS Graduate School for Integrative Sciences and

Engineering

2012/2013

Numerical Estimation Schemes for Quantum Tomography

A survey of novel numerical techniques for quantum estimation

Teo Yong Siah

The author would like to express his gratitude to his Ph.D thesis visor Prof Berthold-Georg Englert, a Principal Investigator at the Centrefor Quantum Technologies, National University of Singapore, for his patientguidance The most part of this dissertation involves work that was done

super-in collaboration with Prof Jaroslav ˇReh´aˇcek and Prof Zdenˇek Hradil fromthe Department of Optics at Palack´y University in Olomouc, Czech Republic.The author would also like to thank Zhu Huangjun, Thiang Guo Chuan and

Ng Hui Khoon for the many insightful discussions Finally, the author thanksthe NUS Graduate School for Integrative Sciences and Engineering and theCentre for Quantum Technologies for their support

Y S Teo

1.1 Introduction 1

1.2 Preliminaries of quantum state estimation 5

1.2.1 Estimation theory 5

1.2.2 Uncertainties in quantum estimation 15

1.3 Informationally complete quantum state estimation 20

1.3.1 Steepest-ascent (direct-gradient) algorithm 20

1.3.2 Conjugate-gradient algorithm 24

1.4 Informationally incomplete quantum state estimation 33

1.4.1 General iterative scheme 34

1.4.2 Qubit tomography 39

1.4.3 Two-qubit tomography 40

1.4.4 Imperfect measurements 41

1.4.5 A new perspective 50

1.5 Hedged quantum state estimation – a comparison 82

1.5.1 The hedged likelihood functional 84

1.5.2 The HML algorithm 86

1.5.3 Informationally incomplete measurements 88

1.6 Chapter summary 89

2 Two-qubit Entanglement Detection with State Estimation 93 2.1 Witness bases measurement 93

2.2 Properties of two-qubit informationally complete witness bases 100 2.2.1 Construction 100

2.2.2 Local unitary equivalence 105

2.2.3 A summary 107

2.3 Adaptive witness bases measurement with state estimation 107

3 Quantum Process Estimation 113 3.1 Introduction 113

3.2 Preliminaries of quantum process estimation 116

3.3 The iterative algorithm 118

3.4 Adaptive strategies 125

3.4.1 Optimization over a fixed set of linearly independent input states 126

3.4.2 Optimization over the Hilbert space 128

3.4.3 A combination of both adaptive strategies 137

3.4.4 Fixed measurement resources 139

3.5 Chapter summary 141

Contents v

C Formula for Computing the Non-classicality Depth 153

One statistically meaningful technique to estimate the unknown quantumstate based on a set of informationally complete measurement data is themaximum-likelihood method (ML) This technique yields a unique MLestimator for a given complete set of data An iterative algorithm wasproposed by Jaroslav ˇReh´aˇcek et al to search for a positive estimatorthat maximizes the likelihood functional We first show that this algorithmcoincides with the steepest-ascent technique and develop a new algorithmbased on the conjugate-gradient method that can be more efficient than thesteepest-ascent version We inspect the performance of this new algorithmwith Monte Carlo numerical simulations

In general, however, the measurement data obtained from complexquantum systems are informationally incomplete and, as a rule, do not yield

a unique state estimator We establish an estimation scheme where both thelikelihood and the von Neumann entropy functionals are maximized in order

to systematically select the most-likely estimator with the largest entropy,that is, the least-bias maximum-likelihood and maximum-entropy estimator(MLME), consistent with a given set of measurement data This is equivalent

to the joint consideration of our partial knowledge and of our ignorance aboutthe source to reconstruct its identity The MLME technique is then applied

to both experimental and simulation data

Next, we take a look at a recent proposal by R Blume-Kohout — thehedged maximum-likelihood method — for quantum state estimation andderive an iterative scheme (HML) to look for the estimator that maximizes

the hedged likelihood functional We then report some interesting features

of these HML estimators in the context of informationally incomplete surements and compare them with the MLME estimators using numericalsimulations

mea-Entanglement detection via witness measurements is a useful technique

to check if an unknown quantum state is an entangled one The MLMEalgorithm can also be used to increase the efficiency of entanglement detec-tion, using the data obtained from measuring sets of witness bases This isbetter than the conventional witness measurement strategy in which only theexpectation value of each witness is estimated and used to infer the existence

of entanglement in the unknown quantum state In our proposed strategies,all information from the collected data is used to detect entanglement andwhen this fails, state estimation can be performed to estimate the unknownstate Adaptive strategies to measure these witness bases will also bepresented

Finally, we also propose a similar algorithm, as in quantum state mation, for incomplete quantum process estimation based on the combinedprinciples of maximum-likelihood and maximum-entropy, to yield a uniqueestimator for an unknown quantum process when one has a set of informa-tionally incomplete data We apply this iterative algorithm adaptively tovarious situations in order to minimize the amount of measurement resourcesrequired to estimate the unknown quantum process with incomplete data

esti-List of Tables

1.1 Table of the average number of iterations and average duration

to complete one full run of the respective iterative schemes forthe four-qubit GHZ and W states The POM for the sim-ulations consists of the tensor products of four single-qubittetrahedron outcomes The above illustrates that on average,ML-CG I, which is ML-CG with fixed ǫk, performs better, interms of the average duration of a full run, than the regular di-rect and conjugate-gradient schemes with ǫkoptimization, eventhough the average number of iterations can sometimes be sig-nificantly reduced using the optimized schemes The additionaltime taken for the type II algorithms is mainly due to the heavymatrix evaluations in the line search procedure 32

2.1 Signatures of the relevant projectors in witness basis ment set-up for polarization qubits (0 b=v, 1 b=h), detected by theset-up of Fig 2.1, with no wave plates in the input ports As aconsequence of the Hong–Ou–Mandel effect [HOM87] (Chung

measure-Ki Hong, Zhe-Yu Ou and Leonard Mandel), the cases (1,0,0,1)and (0,1,1,0) do not occur 98

2.2 The six witness bases of the kind depicted in Fig 2.1 that enablefull tomography of the two-qubit state The second and thirdcolumns list the unitary operators U1wp and U2wp that describethe effect of the wave plates WPs 1 and WPs 2, respectively, onthe polarization of the incoming photons The fourth columnstates the three two-qubit operators whose expectation valuesare determined when the eigenstate basis of the correspondingwitness is measured 99

2.3 The results of local unitary equivalences for sets in Classes 2

to 6 Class 1 contains only three sets which are mutually lated by the qubit Clifford transformation that permutes thequbit Weyl operators The value under the column “1-V1”,for instance, gives the number of 1-V1 transformations that areperformed on a fixed reference set in each of the families thatfalls in the class For example, the first row says that for eachfamily out of 17 in Class 2, including the reference set, thereexists a total of 16 sets with 15 of them generated by apply-ing various types of n-V1 transformations on the reference set

re-in the family Families with the configuration (4,6,4,1,0,0), forinstance, are due to the fact that two witness bases in the ref-erence set of every family, having the same u1,u2 settings, areunaffected by the V1transformations and so there are 41

= 4

1-V1 transformations, 42

= 6 2-V1 transformations, 43

= 4 3-V1transformations and 44

= 1 4-V1 transformations Every ily in Class 4 has half of the 32 sets equivalent to the other halfvia an overall V1 transformation on the entire set For instance,sets that are generated by the 1-V1 and 5-V1 transformations

fam-on the reference set in a particular family are related via anoverall V1 transformation and so on Half the set generated bythe 3-V1 transformations on the reference set is equivalent tothe other half generated by the same type of transformations inthe same manner The entries under the last column includesthe reference set in each family There are 1392 informationallycomplete sets of witness bases out of these five classes Togetherwith the three sets in Class 3, there is a total of 1395 sets 106

experi-1.2 A total of 1500 random two-qubit full-rank mixed state weresimulated with eight thousand detection copies over fifty exper-imental runs By fixing the precision ε = 10−7, the scatter plots

of the average number of iterative steps over the experimentalruns for ML-DG with fixed ǫk = ǫ (ML-DG I) (Red), ML-DGwith optimized ǫk (ML-DG II) (Blue) and ML-CG (Green) in-dicate an expected trend For all the randomly chosen states,ML-CG outperforms ML-DG II with an average improvement

of about 55% On average, ML-CG requires about 95% lessnumber of iterative steps than ML-DG I for the same precision 31

1.3 Here is the corresponding plot of the average duration of onecomplete run of each of the three schemes: ML-DG I(Red),ML-DG II (Blue) and ML-CG (Green) The average improve-ment on which ML-CG outperforms ML-DG II, in terms ofthe average duration of one complete run, is about 65% Thecorresponding improvement of ML-CG over ML-DG I is about75% 31

1.4 Two-qubit tomography using joint trine POMs consisting ofnine outcomes A Monte Carlo simulation is performed withthe number of detection copies N = 106 on a random truestate described by a real statistical operator The verticalaxis represents the real matrix elements for both the trueand reconstructed statistical operators in the computationalbasis The horizontal axes respectively represents the rowand column labels of the matrices The trace-class distance

Dtr = tr{|ˆρMLME− ρtrue|}/2 is 0.158 41

1.5 Two-qubit tomography using a random two-qubit POM sisting of nine full-rank outcomes A Monte Carlo simulation

con-is performed with N = 106 on a random true state represented

by a complex positive matrix of unit trace The vertical axis

in each of (a) and (b) represents the real matrix elements ofthe respective true and reconstructed statistical operators insome computational basis and that in each of (c) and (d) rep-resents the respective imaginary matrix elements In this case

Dtr = 0.206 42

1.6 A simulated experiment on a random state, in which 5000qubits were measured using a random imperfect two-outcomePOM The plot markers, which are indicated by dots, representthe entropies of the MLME estimators generated by the naivescheme starting from random states in the uniform distributionwith respect to the Hilbert-Schmidt measure 103 such estima-tors were computed The thick solid line represents the entropy

of the MLME estimator generated by Scheme B 49

List of Figures xiii

1.7 A comparison of two different schemes for a fixed random complete POM with 103 random qubit true states distributeduniformly with respect to the Hilbert-Schmidt measure Fiftyexperiments were simulated for every true state, with N = 5000for each experiment, and the average trace-class distance Dtravg

in-was computed The entire simulation in-was done with a set ofrandomly generated, informationally incomplete POM consist-ing of three outcomes A POM outcome was discarded to sim-ulate the situation in which two functioning detectors out ofthe three are registering the qubits The plot markers denoted

by “+” represent reconstructed states using Scheme A, andthose denoted by “” represent the reconstructed states usingScheme B The missing probabilities estimated by the recon-structed states using the Scheme B are typically closer to themissing frequencies that would be measured if the discardeddetector was functioning compared to those estimated by thereconstructed states using Scheme A About 80% of the totalnumber of true states respond better under the second scheme 50

1.8 Schematic diagrams of I(λ, ρ) on the space of statistical ators The maximally-mixed state resides at the center of thesquare base which represents the Hilbert space At the extremalpoints of λ, I(λ = 0; ρ) = log(L({nj}; ρ))/N, with a convexplateau at the maximal value, andI(λ → ∞; ρ) = λS(ρ) Plot(c) shows the functional with an appropriate choice of value for

oper-λ for MLME An additional hill-like structure resulting fromS(ρ) is introduced over the plateau, so that the estimator withthe largest entropy can be selected from the convex set of MLestimators within the plateau 53

1.9 A simulation on quantum tomography on a randomly generatedmixed state of light in the five–dimensional Fock space In thisplot, the number of copies of quantum systems measured isfixed at N = 104 A choice of 20 quadrature eigenstates made

up of four different ϑ settings, with five x values corresponding

to each setting, which are projected onto this space was usedand state estimators are constructed for different values of λ

As λ decreases, both the entropy and likelihood functionalsapproach their respective optimal values obtained from MLME(i.e when λ → 0) When λ is zero, there is a convex set ofestimators giving the optimal likelihood value For very large

λ values, the estimators approach the maximally-mixed stateand hence S(ρ) approaches the maximal value log 5 60

1.10 A simulation on quantum tomography on a randomly generatedmixed state ρtrueof light in the 20–dimensional Fock space with

a slightly positive W00= 0.141 Dtr and W00 respectively note the trace-class distance between the reconstructed estima-tor and the true state and the Wigner functional at the phasespace origin, both averaged over 50 experiments with N = 104.The same set of 20 quadrature eigenstates as in Fig 1.9,projected onto this space was used and this set of measure-ments is informationally complete in the two-, three-, and four-dimensional Fock subspaces (shaded region) The values W00

de-and Dtrwere obtained by ML [SMBF93,OTBG06,NNNH+06]

in subspaces of dimensions two to four, and by the MLMEscheme in dimensions greater than four The plot shows astrong dependence ofW00and Dtr on the subspace dimension

In this case, it is obvious that the negativity of W00 inferred

by a reconstruction in a subspace too small is just an artifact

of the truncations Also, Dtr decreases as the reconstructionsubspace increases in dimension This demonstrates the ad-vantages of the MLME scheme over the ML method 63

1.11 A schematic diagram representing the time-multiplexed setupwith K + 1 output ports The Tjs are the respective trans-mission probabilities for the jth beam splitter The overallefficiency for, say, the kth port is given by ˜ηk = ηk(1− Tk+

TK+1δk,K+1)Qk−1

j=1Tj 65

List of Figures xv

1.12 Density plots of the Wigner functions, in phase space, of ous statistical operators for (a) the true state (20-dimensionalstationary state of a laser, µ = 4) with ˜τ ≈ 0.394, (b) the5-dimensional ML estimator with ˜τ ≈ 0.921 and (c) the 11-dimensional MLME estimator with ˜τ ≈ 0.489 Here, brighterregions indicate the locations of larger Wigner function values,and vice versa The statistical operator for (b) is obtained us-ing ML by assuming a 5-dimensional subspace in which thedisplaced POM outcomes are informationally complete Thestatistical operator for (c) is obtained by assuming a largersubspace of dimension 11 using MLME Numerous artificialnon-classical features of the ML estimator, a signature of itshighly oscillatory Wigner function, are manifested as an abnor-mally large value of ˜τ , an inevitable byproduct of state-spacetruncation One can see that with MLME, extraneous artifacts

vari-of the Wigner function resulted from such a truncation can belargely removed 68

1.13 Density plots of the Wigner functions, in phase space, of variousstatistical operators for (a) the true state (ρα′, α′= 5), (b) the8-dimensional ML estimator, (c) the 10-dimensional and (d)15-dimensional MLME estimators In this case, the Wignerfunction of the ML estimator differs greatly from that of thetrue state, an example of misleading information obtained viastate-space truncation A transition in the structure of theWigner function occurs at Dsub= 10, with the MLME estima-tor for Dsub = 15 giving a more accurate estimated picture ofthe Wigner function of the true state 69

1.14 Schematic diagram of the diffraction patterns of an incominglight beam that is obtained from a SH wave front sensor Thelight beam is transformed by an array of microlenses (aper-tures) A CCD camera is placed at the rear focal plane of thearray The measurement data consist of the measured inten-sities of the beam The intensity at the jth pixel, located atposition xj, behind the kth microlens aperture is denoted by

Ik(xj) 72

1.15 Experimental set-up involving a single-mode fiber (SMF), aspatial light modulator (SLM), an aperture stop (A) and aShack-Hartmann (SH) sensor 76

1.16 CCD image for the state ρtruecoh The relevant part of the SHreadout used for the beam reconstruction is shown Contribu-tions from the individual SH apertures are indicated by brightspots, with each spot made up of multiple pixels Note thatthe two void regions correspond to the phase singularities ofthe state ρsupcoh This hints that ρtrue

coh ≈ ρsupcoh 80

1.17 MLME state estimation from informationally incomplete datafor Dsub = 9 The real (left) and imaginary (right) parts ofthe reconstructed coherence operator ˆρMLMEcoh are shown Thereconstruction subspace is spanned by the modes LGl, with

l = 0, 1, , 8 In this case, 56 out of 91 independent outcomes,required for complete characterization of ρtruecoh, are not acces-sible, yet the MLME estimator ˆρMLME

coh is close to ρsupcoh, with afidelity of 92% 81

1.18 Average fidelities, computed over 50 random choices of tational bases, of the estimators for different dimensions Dsubofthe reconstruction subspace The unfilled (filled) circular plotmarkers correspond to informationally complete (incomplete)tomography, respectively 82

compu-1.19 A numerical comparison between HML and MLME A total of

500 random true states ρtrue are generated for each POM Forevery true state, a total of 100 experiments for a fixed N = 500were simulated and the average trace-class distance Davgtr wasplotted In each plot, for almost all the random states, theestimators ˆρHML (+) and ˆρMLME () almost coincide on average 90

List of Figures xvii

2.1 A linear-optics set-up that offers an experimental tion of the optimal witness of Eq (2.1.4) for polarization qubits.Two photons that are indistinguishable by their spatial-spectralproperties are simultaneously incident on a half-transparentmirror, photon 1 from the left and photon 2 from the right.They carry one polarization qubit each, with their unknowntwo-qubit state to be analyzed After being transmittedthrough, or reflected off, the half-transparent mirror, the pho-tons are detected behind polarizing beam splitters that reflectvertically polarized photons and transmit horizontally polarizedones The four detectors LH, LV, RV, and RH must be able todiscriminate between one-photon and two-photon events Thefour eigenstates of the family of entanglement witnesses are dis-tinguished by different signatures; see Table 2.1 By letting thephotons pass through polarization changing wave plates in theinput ports, labeled by WPs 1 and WPs 2, one realizes otherwitnesses that differ from the witness of Eq (2.1.4) by localunitary transformations 97

implementa-2.2 A simulation on the measurement of the set of six tionally complete two-qubit entanglement witness bases for 104random two-qubit pure states, as well as full-rank mixed states,with the measurements done for one state at a time 109

informa-3.1 Numerical simulations on the two-qubit (d = 22) and qubit (d = 23) quantum channels where Di = Do = d Theprojectors of symmetric informationally complete POMs (SICPOMs) are chosen as the linearly independent input states forall the simulations (L = d2) For the measurements, informa-tionally complete POMs consisting of tensor products of qubitSIC POMs are used (M = d2) Each qubit SIC POM consists

three-of a set three-of pure states whose Bloch vectors form the “legs three-of atetrahedron” in the Bloch sphere For the two-qubit channels,

N = 104 and an average over 50 experiments is taken to pute the trace-class distances For the three-qubit channel, themeasurement data are generated without statistical noise Forunitary channels, one can see that the MLME algorithm canstill give fairly accurate estimations with a smaller number ofinput states than that of a linearly independent set Numeri-cal simulations of arbitrary two-qubit and three-qubit unitarychannels suggest that the number is approximately d2/2 forSIC POM input states, above which there is insignificant to-mographic improvement 124

com-3.2 A comparison of three incomplete QPT schemes: the adaptive MLME scheme, the adaptive MLME scheme and theadaptive MPL-MLME scheme Monte Carlo simulations arecarried out on two different types of imperfect cnot gates de-scribed in the text Here, N = 104 and an average over 50experiments is taken to compute the trace-class distances Forboth the non-adaptive as well as the adaptive MLME schemes,the 16 linearly independent input states are chosen to be tensorproducts of projectors of the kets |0i, |1i, (|0i + |1i)/√2 and(|0i+|1i i)/√2 For all schemes, the POM outcomes are chosen

non-to be the tensor products of qubit SIC POMs The non-tomographicperformance of the adaptive MPL-MLME scheme is the bestout of the three The plots show that the tomographic effi-ciency can be further improved by optimizing the input statesover the Hilbert space instead of restricting to a fixed set oflinearly independent input states, albeit the small difference intomographic performance between the two adaptive schemesfor some quantum processes 134

List of Figures xix

3.3 The dependence of the size of the likelihood plateau (∆) andthe normalized log-likelihood maximum on the number of in-put states The respective performances of the non-adaptiveMLME scheme, the adaptive MLME scheme and the adap-tive MPL-MLME scheme are computed based on noiselessmeasurement data for an imperfect cnot gate with ǫ = 0.1.For both the non-adaptive MLME scheme and the adaptiveMLME scheme, the 16 linearly independent input states arechosen to be tensor products of projectors of the kets |0i, |1i,(|0i + |1i)/√2 and (|0i + |1i i)/√2 For all schemes, the POMoutcomes are chosen to be the tensor products of qubit SICPOMs From the plot, the rate of decrease of ∆ is the great-est with the adaptive MPL-MLME scheme The increase inthe normalized log-likelihood maxima with the adaptive MPL-MLME scheme may also be interpreted as greater maximuminformation gain after measurements using the optimal inputstates as compared to the other schemes 136

3.4 A comparison of three incomplete QPT schemes: the adaptive MLME scheme, the adaptive MLME scheme and acombination of the adaptive MPL-MLME scheme and the adap-tive MLME scheme (hybrid scheme) Monte Carlo simulationsare carried out on the imperfect cnot gate with ǫ = 0.1.Here, N = 104 and an average over 50 experiments is taken tocompute the trace-class distances For both the non-adaptive

non-as well non-as the adaptive MLME schemes, the default set of

16 linearly independent input states are chosen to be tensorproducts of projectors of the kets |0i, |1i, (|0i + |1i)/√2 and(|0i + |1i i)/√2 For all schemes, a set of 16 randomly gener-ated positive operators, which are all linearly independent ofone another, are used to form the POM For this POM, the av-erage repetition frequency of the adaptive MPL-MLME scheme

is very high after four input states are used The first inputstate for all schemes is chosen to be the same separable state

ρ(1)i =|00i h00| For the third scheme, the second to the fourthinput states (shaded region) are optimized using the adaptiveMPL-MLME strategy and the subsequent input states are cho-sen via the adaptive MLME strategy using the default set ofinput states which excludes |00i h00| The plot shows that theoverall performance of the combined strategy is better than theadaptive MLME strategy alone 138

3.5 Numerical simulation on the imperfect two-qubit cnot gatewith random noise for fixed LN = 104 An average over 50experiments is taken to compute the trace-class distances Theadaptive MPL-MLME strategy is used when the number ofinput states L is less than 16 140

List of Symbols

←1 unit dyadic 16

1H identity operator inH 118

1K identity operator inK 118kOk2 2-norm of an operator O 6

C qubit Clifford unitary operator 99

C(ρtrue, ˆρ) cost functional of ˆρ for ρtrue 5C(ˆρ) average cost functional of ˆρ 6

CH-S(ρtrue− ˆρ) covariance between ρtrue and ˆρ 15

D

average over all possible D 6

∂~r gradient operator with respect to ~r 16

∆ operator variance of a convex set of ˆEMLs (QPT) 137

D Lagrange functional 34

D dimension of the Hilbert space 6

D(α) displacement operator for a given α 65

Dtr trace-class distance 41(dρ) prior 6

Dsub dimension of the truncated Hilbert space 66(dτd) integration measure for the D-dimensional Hilbert space 6

← symbol for dyadic 15

E Choi-Jami´o lkowski operator for a quantum process 116

ˆ

EML ML process estimator 133

EML operator centroid of a convex set of ˆEMLs 135ˆ

EMLME MLME process estimator 120

Eprior prior information about Etrue 115

ǫ, ǫk small positive step size in an iterative algorithm 22

ε precision for terminating an iterative algorithm 22

η overall detection efficiency 43

Etrue Choi-Jami´o lkowski operator for the true process 114

F Fisher’s information dyadic 17

F frame superoperator 12

List of Symbols xxiii

fj measured frequency of outcome Πj 10

Γj D-dimensional trace-orthonormal basis operators 13

γk+1 Polak-Ribi`ere criterion evaluated in the kth step 27

G (G′) sum of all the POM outcomes 43

G, H positive operators that parametrize a quadratic form 6

ˆ symbol for an estimator 2

Hn(x) degree-n Hermite polynomial in x 149

H, H′ Hilbert space of the input states (QPT) 113

~hk( ~Hk) conjugate direction vector operators for the kth step 24

hk(x) impulse response function 72

h, v horizontal and vertical polarizations of photons 98

I identity superoperator 12I(λ; ρ) information functional of ρ 51I(λ; E) information functional of E 119

Ik(xj) intensity of a beam, at the jth pixel, on the focal plane of

the kth microlens aperture 75

Ikprop(x) intensity of a beam, at position x, after propagating from

the kth microlens aperture 74

Jµ(y) Bessel function in y of order µ 154

| i , h | kets and bras respectively 7

K Hilbert space of the output states (QPT) 113

Km Kraus operators 113

L total number of input states (QPT) 114

λj, λ, µ Lagrange multipliers 34

Λ Lagrange operator 121

Lνn(y) degree-n associated Laguerre polynomials in y of order ν67

LGl Laguerre-Gaussian mode of order l 78L(D; ρ), L({nj}; ρ) likelihood functional of ρ (perfect measurements) 6L({nlm}; E) likelihood functional of E (perfect measurements) 118

L′({nlm}; E) likelihood functional of E (imperfect measurements) 122log ˜L({νlm}; E, ρ) projected log-likelihood functional (QPT) 129

L′({nj}; ρ) likelihood functional of ρ (imperfect measurements) 43

LH({nj}; ρ) hedged likelihood functional of ρ 83

Lz orbital angular momentum operator in the z direction 78

M total number of POM outcomes (QPT) 114

EMPL, ˆρMPL MPL process and state estimators (QPT) 131

|ni Fock states 58

nj number of occurrences of outcome Πj 9

nlm number of occurrences of outcome Πm with ρ(l)i (QPT)118

List of Symbols xxv

Nports number of output ports in TMD detection 64

N measured total number of copies of quantum systems 3

Ntrue true total number of copies 43

n>0 number of positive eigenvalues of M 55

P parity operator 62

P (α) Glauber-Sudarshan P function of α 60

P momentum quadrature 58

~

∂ two-component gradient operator 22

Πj outcomes of a probability operator measurement (POM) 1

Π′j outcomes of an imperfect POM 57

Πwitj outcomes of a witness basis 108

Πk(xj) outcomes describing the SH detections 76

plm probability of an outcome Πm with ρ(l)

i (QPT) 118

mplm = 1/L 118

pj probability of an outcome Πj 10p(D∩ ρ) probability of having D and ρ simultaneously 6p(D|ρ) conditional probability of obtaining D given ρ 6

ˆj estimated probabilities 2

ptruej true probabilities 14

| iprod product ket 94π(ρ) prior probability distribution of ρ 6

˜lm, ˜νm, ˜pm projected quantites (QPT) 129

ψ(x) complex amplitude of a light beam, at position x, from the

source 71

ψprop(k) (x) complex amplitude of a propagated light beam, at position

x, from the kth aperture 72

ψk′(x) complex amplitude of a light beam, at position x on the

focal plane of the kth aperture 73

Q(α) HusimiQ function for a given α 67R(x, p, τ), R(α, τ) function of α and τ, with 0 ≤ τ ≤ 1, for computing ˜τ 67

ρ state or statistical operator 5

ρ2ph two-photon state 100

ρcoh coherence operator 74

ˆMLMEcoh MLME estimator for ρtrue

coh 80

ρtruecoh true coherence operator describing a light beam 75

ρi, Di input state and its dimension (QPT) 113

ρ(l)i lth input state (QPT) 114

ρo, Do output state and its dimension (QPT) 113

ρ(l)o lth output state (QPT) 118

ˆB, ˆρB(H) Bayesian estimator, with and without bias H 8

ρent entangled state 94

ˆ state estimator 2

ˆI,λ state estimator that maximizes I(λ; ρ) for fixed λ 51

ˆHML HML state estimator 85

ˆME ME state estimator 35

List of Symbols xxvii

ˆML ML state estimator 9

ˆMLME MLME state estimator 35

ρsep separable state 94

ρss stationary state of a laser 66

ρtrue true state obtained from measuring infinite copies 3

jfjΠj/pj 20

rjtrue coefficients of ρtrueexpressed in terms of Γjs 15

~rtrue column of rtruej 15S(E) von Neumann entropy of E 116

ˆ

σ linear-inversion estimator 13

Ψ , Ψ

superkets and superbras repectively for the operator Ψ 10

S({fj}|{pj}) relative entropy 52S(ρ) (Smax) von Neumann entropy of ρ (maximum value) 33

˜

τ non-classicality depth 67

Θj dual operator of the outcome Πj 12

tl partial transpose on the lth subsystem 94

Tj transmission probabilities 65

~t column of coefficients for ˆρ expressed in terms of Γjs 15

~tML column of coefficients for ˆρML expressed in terms of Γjs 18

Uk unitary response operator for the kth microlens 75

U1wp, U2wp unitary transformations effected by wave plates 99

W00 Wigner functional at the phase space origin 62W(x, p), W(α) Wigner function in phase space 61

a set of positive operators Πj that compose a probability operator ment (POM) After that, the measurement data obtained are used to inferthe quantum state of the source Such a procedure of state inference, whichshall be our main focus in this dissertation, is also known as quantum stateestimation.

measure-The central idea of quantum state estimation is to attribute a well-definedobjective true state to each measured quantum system that is emitted fromthe source, making a connection with the frequentist’s definition of classicalestimation An observer, after measuring a finite number of copies, will obtain

a state estimator that is generally different from that obtained by another

ob-server, after measuring his own copies in a different way This is not surprisingsince the quantum state of the source directly reflects the amount of informa-tion an observer gains after measuring his copies [CFS02] As the number ofcopies approaches infinity, different estimation procedures ultimately lead tothe same true quantum state of the source if the measurements completelycharacterize the source However, such an idealized situation is never achiev-able in any laboratory setting, as one can only perform measurements on finitecopies of quantum systems As a result, the state estimator obtained will bedifferent from the true state and depends on the details of the estimationprocedure To make statistical predictions, the corresponding operator ˆρ de-scribing this estimator must be a statistical operator, which is positive Thiswill ensure that the estimated probability ˆpj = tr{ˆρΠj} for an outcome Πj ofany set of POM is positive We shall denote all estimated quantities with a

“hat” symbol

The frequentist’s notion of quantum state estimation, described above, isfundamentally different from the Bayesian point of view [P ˇR04, CFS02], inwhich there is no objective true state of the source to be characterized Rather,the quantum state of a given source is treated purely as knowledge that is to

be updated by the measurement data obtained from finite copies, subjected tosome prior information about the distribution of statistical operators In thelatter viewpoint, the quantum state of the source is naturally regarded as asubjective reality that is based on the measurements performed by an observer,rather than a definite state that is associated to the source Unfortunately, due

to its technical difficulty, a feasible Bayesian estimation scheme for quantumstates is presently undeveloped

There are two popular methods for the frequentist’s version of quantum

1.1 Introduction 3

state estimation: Bayesian state estimation∗ and maximum-likelihood tion (ML) The Bayesian state estimation method [SBC01,BKH06, BK10b]constructs a state estimator from an integral average over all possible quan-tum states to estimate the unknown true state The likelihood functional,which yields the likelihood of obtaining a particular sequence of measurementdetections given a quantum state, serves as a weight for the average This ap-proach includes all the neighboring states near the maximum of the likelihoodfunctional as possible guesses for the unknown ρtrue These neighboring statesare given especially significant weight when N , the measured total number ofcopies, is small, in which case the likelihood functional is only broadly peaked

estima-at the maximum However, the integral average unavoidably depends on howone measures volumes in the state space, and there is no universal and unam-biguous method for that The ML method [Fis22,Hel76,P ˇR04,RHKL07ˇ ], onthe other hand, simply chooses the estimator as the statistical operator thatmaximizes the likelihood functional For a sufficiently large number of copies,both methods give the same estimator since the likelihood functional peaksvery strongly at the maximum

When the measurement outcomes form an informationally complete set,the measurement data obtained will contain maximal information about thesource Thus, a unique state estimator can be inferred with ML Unfortu-nately, in tomography experiments performed on complex quantum systemswith many degrees of freedom, it is not possible to implement such an informa-tionally complete set of measurement outcomes As a result, some informationabout the source will be missing and its quantum state cannot be completelycharacterized For example, if a source produces a mode of light that is de-scribed by an infinite-dimensional statistical operator ρtrue, then no matter

∗ Not to be confused with the Bayesian view of quantum estimation as discussed ously.

previ-how ingeniously a measurement scheme is designed to probe incoming tons prepared by this source, an infinite amount of information about themode of light will always remain unknown The ML estimator obtained fromthese informationally incomplete data is no longer unique and there will ingeneral be infinitely many other ML estimators that are consistent with thedata.

pho-The standard approach to this problem is to apply an ad hoc truncation

on the Hilbert space and perform the state reconstruction in a particular space This results in a smaller number of unknown parameters that can then

sub-be uniquely determined by the measurement scheme Since the truncation islargely based on the observer’s intuition about the expected result, that isthe true state that describes an infinite number of copies of such quantumsystems, this cannot be a truly objective method [RMH08ˇ ] A more objectivealternative is to consider the largest possible reconstruction subspace that iscompatible with any existing prior knowledge about the source For example,

if an observer has prior knowledge about the range of the energy spectrum

a given light source can have, he should consider the largest possible struction subspace that contains quantum states describing the source in thisrange of energies This inevitably introduces more unknown parameters thatcannot be uniquely determined by the measurements and one should selectthe state estimator in this subspace that is least biased

recon-In Refs [TZE+11] and [TSE+12], we reported an iterative algorithm(MLME) to estimate unknown quantum states from incomplete measurementdata by maximizing the likelihood and von Neumann entropy functionals Theapplication of this algorithm was illustrated with simulations and experimen-tal data and we concluded that, together with a more objective Hilbert spacetruncation, this approach can serve as a reliable and statistically meaningful

1.2 Preliminaries of quantum state estimation 5

quantum state estimation with incomplete data

In this first chapter, we will discuss, at great lengths, the principles ofquantum state estimation and establish some novel algorithms using variousnumerical methods

optimiza-is minimized based on the measurement data D These measurement data arecollected in an experiment carried out on the unknown source producing mul-tiple copies N of quantum systems, each prepared in the state ρtrue The datacollection is usually done with a probability operator measurement (POM)such thatP

jΠj = 1

Since ρtrue is always unknown, in order to obtain a generically reliableestimator, the objective functional to be minimized has to be independent of

ρ ≡ ρtrue There are many kinds of such objective functionals we can use

A typical kind of objective functional, which we will consider here as themain example, is one that accounts for all possible experimental data D onecan obtain in an experiment This allows us to find the estimator that is,

in this sense, a universally optimal estimator for the cost functional that isindependent of the data To this end, we introduce the average cost functional

C(ˆρ) =X

D

Z(dτd) p(D∩ ρ)C(ρ, ˆρ) , (1.2.1)

where (dτd) is a pre-chosen integration measure for the D-dimensional Hilbertspace and p(D∩ ρ) is the probability of having D and the state ρ simulta-neously The summation notation P

D refers to an average over all possible

D The statistical identity p(D∩ ρ) = p(D|ρ)π(ρ) separates p(D ∩ ρ) into

a product of a conditional probability distribution and a prior probabilitydistribution π(ρ) of all possible states ρ The conditional probability p(D|ρ),which involves the data, is defined in terms of the likelihood functional L(D; ρ)inasmuch as

p(D|ρ) = R L(D; ρ)

(dτd)π(ρ)L(D; ρ). (1.2.2)The functionalL(D; ρ) gives the likelihood of a state ρ yielding the measure-ment data D The prior probability distribution π(ρ), on the other hand,reflects the prior knowledge one has about the source One can define theprior (dρ)≡ (dτd)π(ρ) After inserting all the necessary elements, the objec-tive functional is given by

C(ˆρ) =X

D

R(dρ)L(D; ρ)C(ρ, ˆρ)R

G

ρ+H 1+tr{H}− ˆρo

defined by the positive operators G and H, can be used as the cost functionaland this quantifies a “distance” between ρ and ˆρ Here kGk2 refers to theoperator 2-norm of G defined as

kGk2 = max

|yi6=0

phy| G†G|yip

1.2 Preliminaries of quantum state estimation 7

This is equal to the largest eigenvalue of G≥ 0, since for any ket |yi,

phy| G2|yiphy|yi

=

str

In the derivation, the fact that 0≤ G =Pj|gji gjhgj| is exploited

To show that C1(ρ, ˆρ) is indeed bounded from above by 1, we note that

tr

ρ+H 1+tr{H}− ˆρ2G

ˆ2 #kGk2

In establishing the first inequality, the simple identity tr{ρG} ≤largest eigenvalue of G =kGk2 is used This general quadratic form C1(ρ, ˆρ)

has a unique minimum as long as G≥ 0 Such a functional gives non-zero costfor ˆρ6= ρ and the special case G = 1, H = 0 yields the familiar square of thenormalized Hilbert-Schmidt distance (David Hilbert and Erhard Schmidt).

An extreme case of such a cost functional is given by

+

ρ+H 1+tr{H}

ρ+H 1+tr{H}

+

ρ+H 1+tr{H}



GiR

1 + tr{H}. (1.2.10)Since minimizing C1 requires that δC1(ˆρ) = 0, we thus have ˆρ = ˆρB(H).The statistical operator ˆρB(H) is known as the Bayesian estimator (ThomasBayes) of ρtrue for a given operator H A common variant of the Bayesianestimator [SBC01, BKH06, BK10b] is defined as ˆρB = ˆρB(0) In general,the integral average strongly depends on the definition of (dρ), which has no

quantum state estimation with incomplete data

In this first chapter, we will discuss, at great lengths, the principles ofquantum state estimation. .. form an informationally complete set,the measurement data obtained will contain maximal information about thesource Thus, a unique state estimator can be inferred with ML Unfortu-nately, in tomography. .. the source Unfortunately, due

to its technical difficulty, a feasible Bayesian estimation scheme for quantumstates is presently undeveloped

There are two popular methods for the frequentist’s