Random unitary maps for quantum state reconstruction
Seth T. Merkel, Carlos A. Riofrio, Steven T. Flammia, Ivan H. Deutsch
aa r X i v : . [ qu a n t - ph ] D ec Random unitary maps for quantum state reconstruction
Seth T. Merkel, Carlos A. Riofr´ıo, Steven T. Flammia, and Ivan H. Deutsch Institute for Quantum Computing, Waterloo, ON N2L 3G1, Canada Center for Quantum Information and Control (CQuIC), Department of Physics and Astronomy,University of New Mexico, Albuquerque, NM, 87131, USA Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: December 9, 2009)We study the possibility of performing quantum state reconstruction from a measurement recordthat is obtained as a sequence of expectation values of a Hermitian operator evolving under repeatedapplication of a single random unitary map, U . We show that while this single-parameter orbitin operator space is not informationally complete, it can be used to yield surprisingly high-fidelityreconstruction. For a d -dimensional Hilbert space with the initial observable in su ( d ), the mea-surement record lacks information about a matrix subspace of dimension ≥ d − d −
1. We determine the conditions on U such that the bound is saturated, and showthey are achieved by almost all pseudorandom unitary matrices. When we further impose the con-straint that the physical density matrix must be positive, we obtain even higher fidelity than thatpredicted from the missing subspace. With prior knowledge that the state is pure, the reconstructionwill be perfect (in the limit of vanishing noise) and for arbitrary mixed states, the fidelity is over0.96, even for small d , and reaching F > .
99 for d >
9. We also study the implementation of thisprotocol based on the relationship between random matrices and quantum chaos. We show that theFloquet operator of the quantum kicked top provides a means of generating the required type ofmeasurement record, with implications on the relationship between quantum chaos and informationgain.
PACS numbers: 32.80.Qk,42.50.-p,02.30.Yy
I. INTRODUCTION
Quantum state reconstruction (QSR, or quantum to-mography) is a fundamental tool in quantum informationscience that has been carried out in a variety of systemsand in a variety of protocols [1, 2]. The essential proce-dure of QSR is to use the statistics of the measurementresults on an ensemble of identical systems to make a bestestimate of the prepared state ρ . This can be achieved,e.g., through a series of strong projective measurementsof a set of Hermitian observables [3] or through contin-uous weak measurement of a time-series of observables[4, 5]. High fidelity QSR typically requires an “informa-tionally complete” measurement record. One can obtaininformational completeness by measuring the expecta-tion values of a set of Hermitian operators that spanan operator basis for ρ , or some more general opera-tor “frame” [6, 7]. Restricting our attention to a Hilbertspace of finite dimension d , and fixing the normalizationof ρ , the set of Hermitian operators must form a ba-sis for the Lie algebra su ( d ). Laboratory realization ofsuch a record is intimately tied to controllability , i.e., theability to reconfigure the apparatus in such a way as togenerate arbitrary unitary maps. In the continuous mea-surement context, when the system is controllable it ispossible to choose control fields for the system such that,when viewed in the Heisenberg picture, the observablesevolve over the span of the algebra. While the necessity ofinformation completeness is rigorous if one requires highfidelity for the reconstruction of all arbitrary states, in anumber of situations this condition can be substantially relaxed. Examples include schemes that are designed toachieve high performance of the reconstruction on aver-age [8], or over only some restricted set of the state space[9]. For these protocols, the performance of a restrictedset of measurements is often nearly as good as for aninformationally complete set of measurements, and yetrequire dramatically fewer measurement resources.In this paper, we study another example of informa-tionally incomplete measurements that nonetheless canbe used in a high-fidelity QSR — measurement of a time-series of operators generated by a single-parameter ran-dom evolution. Consider continuous weak measurementof an observable, O , through a meter that couples to anensemble of N identical systems. The members of the en-semble undergo identical, separable time evolution in awell chosen manner. Assuming the subsytems remain in aproduct state, or equivalently that a mean-field approx-imation is valid, we can write our measurement recordquite generally as M ( t ) = N hOi ( t ) + δM ( t ) , (1)where, δM ( t ) describes the deviation from the meanvalue arising from state-dependent quantum uncertaintyand noise in the detection system. For unitary evolution, hOi ( t ) = Tr (cid:0) U † ( t ) O U ( t ) ρ (cid:1) . The QSR problem is toretrodict the initial state of the system ρ from the sig-nal M ( t ). We can simplify the analysis of this problemby considering a discrete set of measurements at intervals∆ t , (cid:8) O n ≡ U † ( n ∆ t ) O U ( n ∆ t ) (cid:9) . The ultimate fidelity ofthe QSR will be limited by the finite signal-to-noise ratio.While the choice of unitary evolution necessary to deter-mine an arbitrary ρ from M n is not unique, a necessaryand sufficient condition is that the set {O n } be infor-mationally complete. A good strategy is to choose thedynamics such that for each n , U ( n ∆ t ) is a random ma-trix, chosen from an appropriate Haar measure. In thatcase our measurement record is not only provably infor-mationally complete but is also unbiased over time. Sup-pose, however, we choose U ( n ∆ t ) = ( U ) n , where U isa fixed unitary matrix. In this case, the observable series O n traces out a single orbit in operator space; we call thisa one-parameter measurement record. As we will show,the record is not informationally complete, but neverthe-less can lead to high fidelity QSR for all states but a setof small measure if U is a random unitary, especiallyfor large dimensional spaces. These results elucidate theconnection between random evolution and informationgain at the quantum level.The remainder of the article is organized as follows.In Sec. II we show that the measurement operators gen-erated from a single parameter trajectory cannot spanthe entirety of the operator algebra, su ( d ), but that theoperators that lie outside of the subspace of the mea-surement record are a vanishingly small fraction in thelimit d → ∞ . Next, we study the performance of QSRusing the weak measurement protocol [4] for these incom-plete measurement records. We show that even at small d , when one includes the physical constraint of positivity of the density matrix, QSR performs surprising well foralmost all quantum states, well beyond that expected ifone had solely considered the vector space geometry ofthe Lie algebra. Finally, in Sec. III, we connect theseabstract results to physical realizations using the unitaryFloquet maps of the quantum kicked top [10] whose asso-ciated classical dynamics is chaotic. As quantum chaosis associated with pseudorandom matrix statistics, thisprotocol provides intriguing new signatures of quantumchaos in QSR. II. ONE-PARAMETER MEASUREMENTRECORDS
In this section, we study whether or not informa-tion completeness for QSR is achievable from a one-parameter measurement record. The one-parameter or-bit in operator space is defined by the time-series O n =( U † ) n O ( U ) n , where O is a Hermitian operator and U is a fixed unitary matrix. We will restrict the observable O to have zero trace since the component proportionalto the identity gives no useful information in QSR. Wethus ask, is it possible to reconstruct a generic quan-tum state ρ if one can measure the expectation valuesof all of the observables in the time series? To answerthis, we consider A ≡ span {O n } , and determine the sizethe orthocomplement subspace with respect to the traceinner product, A ⊥ ; operators in this set are not mea-sured in the time-series. Such missing information ren-ders the measurement incomplete, and thus incompatiblewith perfect QSR, no matter what signal-to-noise ratio is available in the laboratory.To find the dimension of A , consider the sub-space of operators that are preserved under conjuga-tion by U , G ≡ n g ∈ su ( d ) (cid:12)(cid:12)(cid:12) U gU † = g o . Let B = { g ∈ G | Tr( g O ) = 0 } . It thus follows that B ⊆ A ⊥ since ∀ g ∈ B Tr( O n g ) = Tr (cid:16) ( U † ) n O ( U ) n g (cid:17) = Tr( O g ) = 0 . (2)As the two spaces are orthogonal, dim A + dim B ≤ dim( su ( d )) = d −
1. Now, if U has nondegener-ate eigenvalues, G will be isomorphic to the the largestcommuting subalgebra of su ( d ) (the Cartan subalgebra),but for degenerate U , G will contain additional ele-ments. Since the Cartan subalgebra has dimension d − G ≥ d −
1. By definition, B is obtained from G byprojecting out one direction in operator space, and thusdim B = dim G − ≥ d −
2. It follows thatdim
A ≤ dim( su ( d )) − dim B ≤ d − d + 1 . (3)This is the first principal result – a one-parameter mea-surement record is not informationally complete sincedim A ⊥ > d > U and O required to saturate bound in Eq. (3).Since U is a unitary matrix it is always diagonalizableas U = d X j =1 e − iφ j | j ih j | , (4)and in this basis O n has the representation O n = d X j,k =1 e − in ( φ j − φ k ) h k | O | j i | k ih j | . (5)The diagonal component has no n -dependence, so it isuseful to rewrite O n as O n = d X j =1 h j | O | j i | j ih j | + d X j = k e − in ( φ j − φ k ) h k | O | j i | k ih j | . (6)To show that that A is spanned by d − d + 1 linearlyindependent matrices we must have that d − d X n =0 a n O n = 0 iff a n = 0 ∀ n. (7)We can write this condition out explicitly using Eq. (6)as d − d X n =0 a n d X j =1 h j | O | j i | j ih j | + d X j = k d − d X n =0 a n e − in ( φ j − φ k ) h k | O | j i | k ih j | = 0 . (8)The system of equations is underconstrained if either h j | O | j i = 0 for all j or h j | O | k i = 0 for any j = k .Assuming this is not the case, the condition for lineardependence is given by a set of linear equations on a n ofthe form x x · · · x d − d x x · · · x d − d ... ... ... . . . ...1 x d − d x d − d · · · x d − dd − d | {z } V a a . . .a d − d = 0 . (9)Here we have written x = 1 and x m = e − i ( φ j − φ k ) , forsome indexing of the pairs ( j, k ) to 1 ≤ m ≤ d − d . Thecondition for linear independence is simply det V = 0.Expressed as above, one can see that V is an instanceof a Vandermonde matrix, whose determinant is easy toevaluate through the formula [11]det V = Y ≤ j 1. For very large Hilbert spacedimensions, the implication is that all but a vanishingfraction of the information regarding measurements onthe quantum system is contained in this type of record.The reconstruction fidelity is a fairly complicated nonlin-ear function of the measurement record, F ( ρ est , ρ ) = (cid:20) Tr (cid:18)q √ ρ est ρ √ ρ est (cid:19)(cid:21) . (12)It is not clear that this fidelity will be directly relatedto the fraction of operator space spanned by our set ofobservables. Surprisingly, the situation is in fact morefavorable than this na¨ıve assumption. In the next sec-tion we will see that merely requiring the reconstructeddensity matrix be positive provides a powerful constraint,allowing us to use a one-parameter measurement recordinduced by the orbit of a single pseudorandom unitarymatrix to perform very high fidelity reconstructions evenfor small dimensional Hilbert spaces, for all but a verysmall subset of states. III. DENSITY MATRIX RECONSTRUCTIONFROM AN INCOMPLETE MEASUREMENT The QSR protocol we consider was first proposed bySilberfarb et al. [4] and implemented by Smith et al. [5]. In this scenario one has access to an ensemble of N identically prepared systems all initialized to the samestate, ρ . The system is weakly measured yielding therecord given in Eq. (1). For sufficiently weak coupling,the deviation of the measurement result from the quan-tum expectation value is dominated by the noise on thedetector (e.g., shot noise of a laser probe) rather thanthe quantum fluctuations of measurement outcomes in-trinsic to the state (known as projection noise). In thiscase, there is negligible backaction on the quantum stateduring the course of the measurement, and the ensembleremains factorized. We treat the detector noise as Gaus-sian white noise defined as δM ( t ) = σW ( t ), where W ( t )is a Weiner process and σ determines the noise variance.We examine a stroboscopic time-series of the measure-ment record, where the observables evolve according tothe one-parameter trajectory discussed in the previoussection. At discrete times t = n ∆ t , M n = N Tr( O n ρ ) + σW n , (13)where O n and ρ are Heisenberg operators. The prob-lem is thus reduced to one of stochastic estimation of ρ given { M n } . To accomplish this, the density op-erator is expanded in a basis of Hermitian matrices, { E α } , so ρ = P α r α E α + I/d . We take ρ to be unittrace, explicitly removing the identity from the opera-tor basis. Defining a rectangular matrix with elements˜ O nα = Tr( O n E α ), the measurement time-series given inEq. (13) can be expressed as the vector M = N ˜ O r + σ W .Because the fluctuations around the mean are Gaussiandistributed, the maximum-likelihood-estimate of the un-known parameters { r α } is the least-squared fit, given bythe vector r ML = 1 N ( ˜ O T ˜ O ) − ˜ O T M . (14)If the measurement record is informationally complete,which occurs when the covariance matrix C = ˜ O T ˜ O hasfull rank d − 1, and in the absence of measurement noise,the maximum-likely-estimate, ρ ML = P α r ML α E α + I/d ,is exactly ρ . If the set of observables {O n } is not infor-mationally complete, then C is not full rank and we mustreplace the inverse in Eq. (14) with the Moore-Penrosepseudo-inverse, i.e., inverting only over the space in whichthe covariance matrix has support. In this case, a statewith support on the null space of C will yield a maximumlikelihood estimate with sub-unit reconstruction fidelity.In the presence of measurement noise, or when themeasurement record is incomplete, the estimate ρ ML canhave negative eigenvalues which are unphysical. To ob-tain a better reconstruction of the state, we seek thephysical density matrix “closest” to ρ ML . We use thecovariance matrix C as a cost function or metric to mea-sure the distance between ρ ML and new estimate ¯ ρ bydefining k r ML − ¯ r k = ( r ML − ¯ r ) T C ( r ML − ¯ r ) . (15)Technically speaking, this quantity is not a norm butrather a seminorm, meaning that there exist some vectors v such that k v k = 0 but v = 0. This cost penalizesus for taking displacements in directions that increasethe variance in the measurement uncertainty. We thusneed an ¯ r that minimizes the distance k r ML − ¯ r k whilerespecting the constraint P α ¯ r α E α + I/d ≥ 0. Whilethere is generally no analytic solution to this problem, theoptimization is a semidefinite program which is efficientlysolvable numerically [21, 22].The condition of positivity is a powerful constraint thatdescribes correlations between observables that can liealong orthogonal directions in operator space. For exam-ple, in the case of a 2-level quantum system, if h σ z i = 1,positivity implies h σ x i = h σ y i = 0, fully specifying thestate from a single expectation value. In the context ofnoisy measuremets, the positivity constraint allows usto perform high fidelity QSR in the face of uncertaintyby enforcing consistency conditions on our measurementoutcomes. When we consider incomplete measurementrecords, positivity can place bounds on the means of ob-servables which otherwise would be completely undeter-mined. This can greatly increase the fidelity of QSR. In-tuitively, while many vectors ¯ r might minimize Eq. (15),only very few of these are also compatible with positivity. FIG. 1: (Color online) Numerical simulations the QSR pro-tocol for pure states as function of n th expectation valuemeasured in the time-series, and for different dimensions ofthe Hilbert space, d . Each data point represents the aver-age reconstruction fidelity of 100 pure states drawn from theFubini-Study measure, additionally averaged over measure-ment records derived from ten different Harr-random unitarypropagators. As we show below, in the context of a one-parameter mea-surement record generated by a single random matrix,the requirement of positivity provides substantial addi-tional information leading to very high fidelity of QSR,well beyond what one would na¨ıvely predict.We are now prepared to quantitatively analyze the per-formance of our QSR protocol in the case of the one-parameter measurement record arising from an incom-plete set of observables that satisfy Eq. (11). As oursystem, we consider a system with spin J described bya Hilbert space of dimension d = 2 J + 1. We will fix O = J z and select a random unitary matrix U from theHaar measure on SU ( d ). Such a random matrix will al-most always satisfy the constraints of Eq. (11), except fora set of measure zero. As our goal is to determine how theinformation missing in a subspace of observables impactsthe QSR fidelity, we will simplify the analysis by assum-ing that noise on the measurement is vanishingly small.We study the performance of different classes of states,randomly chosen by an appropriate measure. For eachset of states, we will look at the average fidelity betweenthe initial and reconstructed states, hFi = R d ρ F (¯ ρ, ρ ),where d ρ is a measure on the space of density operators.The simplest case to analyze is when we have prior in-formation that ρ is a pure state, | ψ i . Figure 1 shows theaverage fidelity as one sequentially measures the expecta-tion value of the n th observable in the series, for differentdimensions of the Hilbert space d . Averages are takenfor 10 choices of random unitary matrices, each of whichis averaged over 100 random pure states distributed onthe Fubini-Study measure [23]. Two striking features areseen in these plots: (i) unit fidelity is achieved for any d even though the record was said to be informationallyincomplete; (ii) the protocol reconstructs the state wellbefore we measure all d − d + 1 independent observ-ables. The inclusion of positivity dramatically improvesthe reconstruction fidelity for pure states. In fact, a one-parameter measurement record generated by a random U can be used to reconstruct almost all pure states per-fectly in the absence of noise.The performance of the QSR protocol can be under-stood given the prior information we have assumed. Apure state is specified by 2 d − d − d +1 expectation values. Thus, it shouldcome as no surprise that the measurement record con-tains enough information to reconstruct the state. Infact, in this case one can use positivity to explicitly re-cover the missing information exactly from the measure-ment record, without resorting to the numerical semidef-inite program discussed above. In general, the missinginformation is associated with matrices that commutewith U . Thus, when expressed in the eigenbasis of U ,only the diagonal matrix elements of the density opera-tor might not be estimated. A necessary (but not gen-erally sufficient) condition for a matrix ρ to be positivesemidefinite is that its matrix elements must satisfy thefollowing set of inequalities: ρ ii ρ jj − | ρ ij | ≥ 0, i.e. allof the 2 × ρ ii = ( | ρ ij || ρ ik | ) / | ρ jk | . A special case is if all of the off-diagonal matrix elements of ρ are zero. Then the statemust be one of the eigenvectors of U , but such stateslie in a set of measure zero. Measurements that are in-formationally complete solely for pure states are calledPSI-compete [24, 25].While one can easily explain the high-fidelity perfor-mance of the QSR protocol in the case of a pure state, formixed states, this is far from clear, and the power of thepositivity constraint comes fully to the fore. For mixedstates, and d > 2, the average fidelity of our reconstruc-tion will never reach unity because some density matri-ces cannot be reconstructed from the information in ourincomplete measurement record. For example, some con-vex combinations of eigenstates of U are indistinguish-able from the maximally mixed state, even though thefidelity between the two can be very small. Nonetheless,as we see below, the one-parameter measurement recordgenerated by a single random unitary still performs verywell on average, even for generic mixed states.Figure 2 shows the average fidelity for two choices ofmeasures on density matrices, the Bures measure and theHilbert-Schmidt measure [23, 26], with states sampledaccording to the construction provided by Osipov, Som-mers and ˙Zyczkowski [27]. For both distributions we lookat a long-time limit of the time-series, here 10( d − d + 1)measurement steps, and plot the average fidelity as afunction of the dimension of the Hilbert space rather FIG. 2: (Color online) Fidelity of the QSR protocol for mixedstates as a function of the dimension of Hilbert space d . Eachpoint represents the average reconstruction fidelity from ameasurement record of length 10( d − d + 1) (a long-timelimit) generated from a different random unitary propagatorfrom the Harr measure. We average over 200 density matri-ces drawn from the Bures (blue crosses) or Hilbert-Schmidt(green x’s) measures. For each dimension we show the averagefidelities from twenty of these measurement procedures. than n . In the limiting case of negligible noise on themeasurement, we have already extracted all the possibleinformation about the state after d − d +1 measurements.In practice, increasing the measurement record serves tosmear out the information over the measured observables,leading to a more uniform distribution for the non-zeroeigenvalues of C , which is numerically favorable. As seenin these two plots, on average, the one-parameter mea-surement records perform surprisingly well. In all casesthe mean fidelity is greater than 0 . 96 with a minimumaround d = 3 or d = 4. After this dip, the minimum ofthe fidelity looks to be monotonically increasing with thesize of the Hilbert space. Additionally, the particular in-stantiation of the random unitary map appears to makevery little difference (less than 0.01 fidelity), with theresidual difference decreasing as the dimension increases. IV. EXAMPLE: QUANTUM KICKED-TOP In Sec. II we discussed that the conditions given inEq. (11) can be satisfied by a pseudorandom unitary ma-trix, instead of a true random matrix sampled from theHaar measure. One such class of matrices are the Flo-quet maps associated with “quantum chaos”, i.e., peri-odic maps whose classical dynamical description shows aglobally chaotic phase space. An example is the quantumkicked-top [10], a system that recently has been realizedin a cold atomic ensemble [28]. In this section we explorehow our QSR protocol performs in this context, providinga possible route to laboratory studies, and novel signa-tures of chaos in quantum information.The standard quantum kicked top (QKT) dynamicsconsists of a constant quadratic twisting of a spin (“top”),punctuated by a periodic train of delta-kicks of the spinaround an orthogonal axis. The Floquet operator forthis perodic map is typically written as the product ofnoncommuting unitary matrices, U QKT = e − iφJ z /J e − iθJ x . (16)The parameters θ and φ represent the angles of linear andnonlinear rotation respectively. The dynamics exhibit aclassically chaotic phase space for an appropriate choiceof these parameters [10]. The connection between chaosin this system and random matrices has been well stud-ied, particularly, the relationship between the level statis-tics of the Floquet eigenvalues, chaos, and symmetry.Floquet maps associated with global chaos are randommatrices that divide into different classes. If the Floquetoperator is time-reversal invariant, the level statistics arethat of the circular orthogonal ensemble (COE); with-out additional symmetry they are members of the cir-cular unitary ensemble (CUE). The latter group is U ( d )or SU ( d ) depending on the context. The measurementrecords generated from matrices chosen from either theCOE or CUE will satisfy the eigenvalue conditions de-scribed in Eq. (11) almost surely. The QKT is knownto have a time-reversal symmetry and classically chaoticdynamics. It still does not have COE statistics in the full d = 2 J + 1 Hilbert space, however, due to an additionalsymmetry. The QKT map is invariant under a π -rotationabout the x -axis, leading to a parity symmetry. This sys-tem therefore has a doubly degenerate eigenspectrum,breaking the conditions in Eq. (11). While such Flo-quet operators generate a measurement record that havemuch less information relative to an arbitrary state, wecan perform high-fidelity QSR in for states restricted toa subspace defined by the additional symmetry, here, thestates that have even parity under reflection around the x -axis. To do this we require that our initial operator O is also symmetric under reflection, e.g., O = J x .We present examples of this type of reconstruction inFig. 3. Here we look at the QKT dynamics for a spin J = 3 particle (a d = 7 dimensional Hilbert space). Wechoose the parameters φ = 7 and θ = 0 . C has rank 19. This agrees with our predic-tions, since this space has a 4-fold degenerate − × 4) componentsfrom each of the two off-diagonal blocks. For this dimen-sion d − d + 1 − 24 = 19. FIG. 3: (Color online) Reconstruction fidelity for a varietyof states versus n . The initial measurement observable is J x ,and each subsequent observable whose expectation value wemeasure is obtained by evolving under the Floquet map of aquantum kicked top, Eq. (16). For generic random states inthe whole Hilbert space (pure or mixed), the reconstructionpreforms poorly. Density matrices that are invariant with re-spect to π -rotation around the x-axis, such as the cat state,an eigenstate of the parity operator, or an incoherent mix-ture of odd-parity J x eigenstates, are reconstructed with highfidelity.FIG. 4: (Color online) Same as Fig. 3 when the evolution isgiven by the double kicked top. All states, pure or mixed,asymptote to fidelities near unity, with the pure states reach-ing their maxima quicker than the mixed states. We can examine the effects of a pseudorandom unitarythat satisfies Eq. (11) on the whole space if we look at the“double kicked top” where we alternate kicking about x and y . Here the Floquet operator has the form U = e − iφJ z /J e − iθ x J x e − iφ ′ J z /J e − iθ y J y . (17)If we choose φ = φ ′ = 6, θ x = π/ θ y = 0 . J z we can choose O = J z in order to satisfy the first condition of Eq. (11).In Fig. 4 we show the QSR performance for a time-series generated by the double kicked top in a Hilbertspace of a spin-3 particle. Here the fidelity asymptotesto unity for all of our choices of states, as we would ex-pect from the simulations in Sec. III. The reconstruc-tion reaches its asymptotic fidelity when we have made d − d + 1 = 43 measurements. As we saw previously,the pure states require less measurements to reach theirasymptotic value and the QSR can be perfect in the ab-sence of noise. V. SUMMARY AND OUTLOOK We have studied measurement records that are derivedby stroboscopically measuring the expectation values ofa single observable of a system that is evolving under therepeated application of a single unitary map. We haveshown that this record never contains complete informa-tion about the quantum state, however for unitary mapschosen randomly or pseudorandomly, only a vanishingfraction of the information is missing. When combinedwith the constraint of positivity, this incomplete mea-surement record led to a protocol for quantum state re-construction that had high-fidelity performance for typ-ical mixed and pure quantum states. For pure states wecan achieve unit fidelity reconstruction (in the absenceof noise) and for mixed states the fidelity is greater than0.99 for d > su ( d ) andare positive. Our conjecture is that both the increas-ing average fidelity and the decreasing dependence onthe sampled unitary can be a explained based on con-centration of measure in analogy with the work in [9].Essentially, most randomly sampled states have very lit-tle support on a the subspace that we do not measure.Related work in the field of matrix completion has shownthat such “incoherence” between states and (incomplete)measurements can provably lead to high-fidelity state re-constructions especially when the states in question arelow rank [29] or near to low-rank states [9]. Irrespectiveof a rigorous proof, it is an empirical fact that our pro-tocol works well with typical states and typical unitarymaps. We expect that as the dimension of the Hilbertspace increases, almost all of the states and unitary evo-lutions that are sampled will be very close to a typicalvalue, resulting in high QSR fidelity. Acknowledgments We thank Andrew Scott for helpful discussions. 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