Continuousvariable quantum compressed sensing
Abstract
We significantly extend recently developed methods to faithfully reconstruct unknown quantum states that are approximately lowrank, using only a few measurement settings. Our new method is general enough to allow for measurements from a continuous family, and is also applicable to continuousvariable states. As a technical result, this work generalizes quantum compressed sensing to the situation where the measured observables are taken from a socalled tight frame (rather than an orthonormal basis) — hence covering most realistic measurement scenarios. As an application, we discuss the reconstruction of quantum states of light from homodyne detection and other types of measurements, and we present simulations that show the advantage of the proposed compressed sensing technique over present methods. Finally, we introduce a method to construct a certificate which guarantees the success of the reconstruction with no assumption on the state, and we show how slightly more measurements give rise to “universal” state reconstruction that is highly robust to noise.
1 Introduction
One of the most fundamental tasks in quantum mechanics is that of quantum state tomography, i.e., reliably reconstructing an unknown quantum state from measurements. Specifically in the context of quantum information processing in most experiments one has to eventually show what state had actually been prepared. Yet, surprisingly little attention has so far been devoted to the observation that standard methods of quantum state tomography scale very badly with the system size. Only quite recently, novel more efficient methods have been introduced which solve this problem in a more favorable way in the number of measurement settings that need to be performed [1, 2, 3, 4, 5, 6, 12]. This development is more timely than ever, given that the experimental progress with controlled quantum systems such as trapped ions is so rapid that traditional methods of state reconstruction will fail: E.g., ions can already be controlled in their quantum state [7]. Hence, further experimental progress appears severely challenged as long as ideas of reconstruction cannot keep up. Such new methods are based on ideas of quantum compressed sensing [1, 2, 6] — inspired by recent work on lowrank matrix completion [8, 9] — or on ideas of approximating unknown quantum states with matrixproduct states [4]. Indeed, using methods of quantum compressed sensing, one can reduce the number of measurement settings from in standard methods to for a quantum system with Hilbert space dimension , if the state is of rank . This is efficient in the sense that the number of measurements required is only slightly greater (by an factor) than the number of degrees of freedom in the unknown state.
These ideas have so far been tailored to the situation where observables are taken from an orthonormal operator basis, which is not always the natural situation at hand. In this paper, we introduce a theory of state reconstruction based on quantum compressed sensing that allows for continuous families of measurements, referred to as tight frames, which can be thought of overcomplete, nonorthogonal generalization of operator bases. These settings are particularly important in the context of continuousvariables, which are notably used to describe quantum optical systems beyond the singlephoton regime. These have drawn a considerable amount of research, both experimentally and theoretically, due to very desired features such as easy preparation and highly efficient detection. Note that when talking about a measurement, we always mean the estimation of an expectation value of an observable for which, of course, several repetition of some experimental procedure are necessary. In this paper we are mainly concerned with the number of distinct observables or measurement settings that are needed for tomography.^{1}^{1}1Other work addresses the number of copies of the unknown state that must be provided [10] — that is, the sample complexity of tomography.
In this work, we make significant progress towards a full theory of efficient state reconstruction via compressed sensing:

We introduce new incoherence properties for tight frames, that are sufficient to ensure efficient compressed sensing for lowrank states. This uses an extension of the “golfing” proof technique of [1, 2]. We give examples of tight frames that satisfy these properties. In addition, we show that, if one only wishes to reconstruct “typical” or “generic” lowrank states, there is a much larger class of tight frames that also lead to efficient compressed sensing.

We also describe a way to certify a successful reconstruction of the state, making our protocol unconditional and heralded. In this way, one does not need to make any a priori assumptions on the unknown state. Our method uses convex duality, and is different from other approaches to certification that focus mainly on pure states [1, 4, 11, 12]. Also, we discuss the robustness of the procedure under decoherence, imperfect measurements, and statistical noise. We show that, as long as all those effects are small, it is possible to certify that the reconstructed state is close to the true state.

We show that, using an incoherent tight frame, and a slightly larger number of measurements, one can achieve universal state reconstruction: a single fixed set of measurements can simultaneously distinguish among all possible lowrank states. This is a qualitatively stronger claim than those shown above, and it is obtained using a different technique, based on the “restricted isometry property” (RIP) [6, 13]. This implies strong error bounds, showing that our procedure for state reconstruction is robust to statistical noise, and that it works even when the true state is fullrank with rapidly decaying eigenvalues (in which case our procedure returns a lowrank approximation to the true state).^{2}^{2}2As a side note, the RIPbased analysis also shows that the compressed sensing state estimator is nearly minimaxoptimal [13], and it implies nearlyoptimal bounds on the sample complexity of lowrank quantum state tomography [10].

We show how our theory can be applied in realistic experimental scenarios, involving pointwise measurements of the Wigner function, and homodyne detection.

We demonstrate numerically that compressed sensing outperforms the naive approach to tomography not only in the asymptotic limit of large systems but also for system sizes commonly accessible in present day experiments.
This article is organized as follows: We start by introducing quantum compressed sensing in the general setting described by tight frames in Section 2. After discussing a suitable notion of efficiency, we show in Section 3 that efficient compressed sensing is possible if the tight frame fulfills certain incoherence properties. Section 4 is devoted to certified compressed sensing. We discuss how to certify the success of the reconstruction without prior assumptions on the tight frame, both in the ideal case and under the effects of errors. In Section 5 we show universal state reconstruction and error bounds. In Section 6, we investigate applications of the formalism to two common classes of quantum optical experiments; and in Section 7, numerical data, showing the efficiency for small systems, is presented.
2 Quantum compressed sensing
Consider a quantum system with Hilbert space dimension . In most cases of interest, is very large, but the states one wants to reconstruct are approximately lowrank, that is, they are wellapproximated by density matrices having rank . (Pure states correspond to the case where .) When dealing with continuousvariable systems, we will truncate the infinitedimensional Hilbert space and choose to be some large but finite cutoff. This is unavoidable, if one wants to do tomography as one cannot reconstruct a state that contains an infinite number of completely independent parameters. However, in most experimentally relevant situations, e.g., continuousvariable light modes with finite mean energy, all states can be arbitrarily well approximated by finitedimensional ones. We will elaborate on this claim when discussing other sources of errors such as decoherence or imperfect measurements.
Compressed sensing contains two key ideas. First, rather than measuring all degrees of freedom, it is sufficient to measure a randomly chosen subset of about degrees of freedom, provided these degrees of freedom satisfy certain incoherence properties. Secondly, one can reconstruct the state using an efficient algorithm. The obvious approach of searching for the lowestrank state compatible with the measurement results leads to a computationally intractable problem (generally NPhard). Instead, one can perform a convex relaxation, and minimize instead of the rank. Here stands for the Schatten norm: , , and are respectively the trace norm, Frobenius norm, and spectral (or operator) norm.
Let us denote the measured observables, i.e. Hermitian matrices, by , and suppose that we estimate their expectation values (by measuring many copies of the unknown state). Knowing these expectation values (for an unknown state ) is equivalent to knowing the value of the sampling operator , where we define
(1) 
where is the HilbertSchmidt scalar product. In all of our compressed sensing schemes, will be chosen independently at random from some distribution . The sampling operator is a linear superoperator on the dimensional real vector space of Hermitian matrices, or operators, . Such superoperators will always be denoted by capital script letters. Sometimes we will use the notation , multiplying the “matrix” by the “vector” ; this means the same thing as .
Let be the unknown state. In the ideal case, with perfect measurements and no statistical noise, we measure exactly. Then the procedure to reconstruct can be written as
(2) 
Note that a quantum state is a Hermitian matrix with the additional properties and . However, the reconstruction procedure (2) does not make use of this property and is, therefore, also applicable in more general settings, e.g. matrix completion. This problem can be stated as a semidefinite program (SDP) and, therefore, solved efficiently with many welldeveloped tools.
In the case of noisy data, we measure approximately, that is, we measure some such that , for some norm and tolerance that are chosen depending on the kind of noise that is expected. The constraint in (2) can then be replaced by , which implies .
We remark that equation (2) is the key to certifying our estimate for . Notice that if the solution of (2) is unique and fulfills , then it must be the case that . We will show later on how one can test the uniqueness of the solution , without assuming anything about . (This can be adapted to work with noisy data, without assuming anything about the noise.)
2.1 Measurements and tight frames
When we talk about a compressed sensing scheme, we mean any protocol based on the reconstruction procedure (2), with some choice of measurements described by the sampling operator (1). In Refs. [1, 2], the observables were required to be chosen uniformly at random from an operator basis. We substantially generalize these techniques, using the notion of a tight frame, which naturally captures many useful quantum measurements:
Definition 1 (Tight frame).
Let be a probability measure on some set , and for every , let be an observable, i.e., a Hermitian operator, and let be the (unnormalized) orthogonal projector which acts as . We say that is a tight frame if
(3) 
This can also be written as where is drawn according to . Because we deal with randomly drawn operators very often, will usually denote a random element of that has distribution . Note that we do not require that for all as it will be convenient in many applications. However, we do require a weaker normalization condition: which follows by taking the trace of (3).
We also define a generalized notion of a tight frame, where the sampling operator is not a sum of projectors; we will need this later to model homodyne detection on optical modes, where a single measurement setting provides more information than only one expectation value.
Definition 2 (Generalized tight frame).
Let be a probability measure on some set , and for every let be a positive operator. We say that forms a generalized tight frame if
(4) 
We note that the formalism can be also applied to 8port homodyne detection which corresponds, for a single mode, to projections on coherent states with .
2.2 Uniqueness of the solution to (2)
For to be the unique solution to (2), any deviation must be either tracenorm increasing, i.e., , or infeasible, i.e., . This is done by decomposing into a sum , and then showing that, with high probability, in the case where is large, must be infeasible, while in the case where is small, must be tracenorm increasing. Here, we denote by the real space of Hermitian matrices that send the kernel of on its image. In other words, the elements of are the Hermitian matrices whose restriction on and to the kernel of , i.e. where is the orthogonal projection on , is equal to . denotes the projection on this space .
Again, in the actual reconstruction, no assumptions have to be made concerning or . Theorem 1 gives a sufficient condition for uniqueness. The sign function of a Hermitian matrix is defined by applying the ordinary sign function to the matrix’ eigenvalues.
Theorem 1 (Uniqueness of the solution).
Proof: must be infeasible if which is the case if
(6) 
The righthand side is bounded as while the lefthand side fulfills
(7) 
Thus, (6) is satisfied if
(8) 
which, using definition (c), is equivalent to
(9) 
Using the pinching [19] and Hölder’s inequalities, as detailed in Ref. [2], yields
(10) 
The second term is equal to
(11) 
which is, according to (a) and (b), larger than
(12) 
Inserting this into (9) gives rise to condition (5) and concludes the proof. If all the elements in the tight frame fulfill we call it normalized and one can bound . In this case (5) in Theorem 1 can be weakened to
(13) 
2.3 Efficient quantum compressed sensing
Let be a state of dimension and rank . In the compressed sensing method of tomography, we choose observables randomly from some distribution, measure their expectation values with respect to , then solve (2) to obtain , which is our estimate of .
For a given state , there is some probability that the procedure may fail (i.e., it may return a solution that is not close to ). Note that this probability is taken with respect to the random choice of , and the random outcomes of the measurements. We say that the method succeeds with high probability if, for every lowrank state , the failure probability is small. Equivalently, the method succeeds with high probability if,
for every lowrank state , most choices of the observables can be used to successfully reconstruct .
Now, the basic question is: how large does have to be, to ensure that the method succeeds with high probability? A common situation is that the system under consideration consists of subsystems with local Hilbert space dimension ; then . Of course, no method of tomography can counter the exponential growth of the required number of measurements in . Thus, efficiency needs to be regarded relative to the measurements necessary for standard tomography. As even a pure state needs parameters to be described, this also is a lower bound to the number of observables that need to be measured. We allow for an additional polylogarithmic overhead and define efficiency as follows:
Definition 3 (Efficient quantum compressed sensing).
Compressed sensing for a state (with dimension and rank ) is regarded as efficient if: The number of measured observables satisfies , and the probability of failure satisfies .
If this is the case, can be made arbitrarily small by repeating the protocol and using a majority vote among the reconstructed states to get the final result. Then, the probability of failure decays exponentially in the number of repetitions.
Note that this is a very stringent definition of efficiency. One can also merely ask for any scaling of in . Of course, this weaker condition is easier to satisfy, as we shall see later on.
2.4 Sufficient conditions for efficiency
The general theory of quantum compressed sensing, which will be developed here, relies heavily on and significantly extends the analysis for the special case where the observables form an operator basis in Ref. [2]. The hypothesis for Theorem 1 is fulfilled if , , under the additional condition , which can be safely assumed to be true as we are only interested in the regime of . For normalized tight frames, the first condition can be weakened to . We show conditions to the tight frame under which those conditions are fulfilled with high probability.
For efficient compressed sensing to be possible, the observables need to fulfill certain incoherence properties. Roughly speaking, the observables are “incoherent” if they have small inner product with every possible state one wishes to reconstruct. For example, operator norm can be a measure of incoherence for reconstructing pure states, since . We distinguish two general cases (which we will define more precisely in the following sections):

“Fouriertype” compressed sensing, where almost all of the observables have small operator norm. In this case, efficient compressed sensing is possible for any lowrank state.

“NonFourier type” compressed sensing, where the observables may have large operator norm, but efficient compressed sensing is still possible for certain restricted classes of states, e.g., generic states.
2.5 Fouriertype efficient compressed sensing
The efficiency of a tomography protocol, as given in Definition 3, is a statement about a family of procedures acting on systems with growing dimension . We now give a sufficient condition for a family of tight frames to allow for efficient compressed sensing.
Theorem 2 (Fourier type).
Let be, for any , a tight frame. Let be any state with dimension and rank . Let . Set and let be the measure of this set. If
(14) 
efficient compressed sensing is possible for the family of states .
Here, the underlying “incoherence property” is the bound on the operator norm of the observables,
(15) 
which holds for “most” choices of . If there is no risk of confusion, we will omit the explicit dependencies on .
2.5.1 Perfect Fouriertype case
We have to first consider the case . Even though the proof in Ref. [2] can be applied with only minor changes, we state it in a way as complete and still nontechnical as possible where we focus on the asymptotic behavior and do not provide explicit constants. We need Lemma 5 from Ref. [2] which reads:
Lemma 1 (Large deviation bound for the projected sampling operator).
For all
(16) 
where is the oversampling factor which must fulfill for efficiency.
The tool to prove Lemma 1 and other bounds of this form is provided by the operatorBernstein inequality which was first given in Ref. [17] and which we state here without a proof.
Lemma 2 (OperatorBernstein inequality).
Let be i.i.d. Hermitian matrixvalued random variables with zero mean. Suppose there exist constants and such that , where the latter needs to be true for all realizations of the random variable. Define and . Then, for all
(17) 
The proof of Lemma 1 is given in Ref. [2] but we restate it here because it is quite instructive. Let be a random variable taking values in . We define random variables by and . Now and we have to estimate the maximum of and the norm of the variance of in order to apply Lemma 2. From the incoherence condition (15), we get by using the matrix Hölder inequality [19]
(18) 
This allows us to write
(19) 
and
(20) 
Here, and in the remainder, statements of the form (20) are meant to hold for all realization of the random variable as needed in the Operator Bernstein inequality. Inserting now (19) and (20) into Lemma 2 yields Lemma 1 which concludes the proof. Applying Lemma 1 for and choosing , the probability that can be made arbitrarily small.
Now we have to construct a certificate whose projection on is close to . This is done by an iterative process, called the golfing scheme [2]. The samples are grouped into groups which are indexed by and contain samples each. Let be the sampling operator of the th group and set , , , and .
Again, Lemma 1 can be used to show that with high probability (at the expense of a polylog growth of )
(21) 
and, therefore, from which we get
(22) 
while for the final inequality to hold it is enough to set . For the last remaining condition we need the subsequent statement:
Lemma 3 (Bound for the orthogonal projection).
Let and . Then
(23) 
Proof: Without loss of generality, consider and define the zeromean random variables which fulfill . Their variance is bounded by
(24) 
and their norm by
(25) 
Lemma 23 follows directly from using (24) and (25) in Lemma 2. Now we can bound
(26) 
Again, the probability of (26) not being true can be made as small as desired by choosing . Of course, this is also true for the total probability of failure which concludes the proof.
2.5.2 Imperfect Fouriertype case
We now show that the incoherence condition may be violated for some of the observables and adapt a technique used in Ref. [14]. Intuitively, when is small enough, we can just abort and restart the reconstruction procedure whenever we encounter a nonincoherent operator during our sampling process. The probability of this to happen is upper bounded by as obtained from (14) by a union bound over the measurements. This is equivalent to sampling only from the set . The conditional probability distribution on the observables does fulfill the approximate tightframe condition
(27) 
where where is the event that all of the chosen operators satisfy and its complement is denoted by . Let be the indicator function of . Then, This leads to
(28) 
With the help of Jensen’s inequality, we can simplify . Inserting this into (28) and rearranging, we get
(29) 
Our claim follows by taking which is always true by a union bound. We now have to justify why the tight frame condition (3) can be replaced by the approximate one in Eq. (27) in the proof of Lemma 1 and Lemma 23. We denote the probability measure which is conditioned on the event by .
Lemma 1 provides a bound to
We define the random variables and as in the proof of Lemma 1 and bound their variance as
(31)  
and their norm as . Using the operator Bernstein inequality yields
Lemma 4 (Large deviation bound for the projected sampling operator).
(32) 
for all .
where we have also used (2.5.2) to bound the second term in (27). Thus, up to an irrelevant constant factor, Lemma 4 replaces Lemma 1 wherever it is used.
To also replace Lemma 23, let , and note that
(33) 
The random variables are and where the variance is bounded by
(34)  
which gives, together with , and an application of the operatorBernstein inequality the subsequent statement.
Lemma 5 (Bound for the orthogonal projection).
Let and . Then
(35) 
Lemma 5 takes the place of Lemma 23 and, again, differs only by a constant factor in the exponent which concludes the proof of Theorem 2.
An example for a Fouriertype frame for which is given by the following situation. Here, with some probability, every Hermitian matrix with unit Frobenius norm is drawn in the measurement.
Example 1 (Tight frame containing all Hermitian matrices).
Any with can be viewed as a vector on the dimensional unit sphere. Therefore, on can define a rotationally invariant Haar measure on it. The tight frame formed by the Haar measure on all Hermitian matrices with fulfills Theorem 2. Therefore, it allows for efficient compressed sensing.
In order to satisfy Theorem 2, we have to show
(36) 
where . To see that this is true, we note that we are dealing with a normalized version of the extensively discussed Gaussian unitary ensemble (GUE) denoted by , . Now for all we have
(37) 
The first term can be bounded using a result from Ref. [16] yielding
(38) 
where are small constants while for the second term we use the properties of the distribution which are given the appendix. From this, we get
(39) 
We set and see that (36) is fulfilled for some constant when is large enough.
Product measurements are of great experimental importance: They describe the situation of addressing individual quantum systems, say, ions in an ion trap experiment or individual modes in an optical one. They are described by tight frames which are formed as tensor products of tight frames on the local systems. Given a tight frame which fulfills , one can obtain a tight frame on the dimensional Hilbert space by forming the fold tensor product. The strongest possible incoherence property we can obtain is . Unless , as for the Pauli matrices, grows too fast to allow for efficient compressed sensing for all states. This is even true if the incoherence condition may be violated on some set with .
2.6 NonFouriertype efficient compressed sensing
The conditions in Theorem 2 imply that efficient compressed sensing is possible for any lowrank state . This is a quite special situation and for Theorem 2 to be fulfilled, either a very special structure, like the one of the Pauli basis [1], or a large amount of randomness, like in the above example, is needed. As an example for a very different situation, consider the state together with the observables which corresponds to the sampling of single matrixentries (or the Hermitian combinations of two of them). Here, one needs to take attempts until one “hits” the single nonzero entry in the upperleft corner. This is not surprising because the operators in this basis fulfill . However, for most of the states, efficient compressed sensing is indeed possible in this basis. In Theorem 3, we give a sufficient condition for combinations of states and tight frames to work.
Theorem 3 (NonFouriertype efficient compressed sensing).
For a given tight frame , and a given rank state , denote by the set of observables for which at least one of the following conditions is not fulfilled:
(40)  
(41) 
If , efficient compressed sensing is possible for the state .
The golfing scheme works exactly like in the Fouriertype case, as does the proof of Lemma 1. However, Lemma 5 must be replaced by something else. Again, we use the technique of conditioning which means that we assume the incoherence condition to hold for all operators in the tight frame and the tight frame condition to be approximately true as in (27). First, we need some preparation.
Lemma 6 (Bound to the scalar product).
Let such that , , and
(42) 
for all . Then
(43) 
Proof: We consider the same same random variables as in the proof of Lemma 4 (note that we have again set ) and bound their variance as
(44) 
where we have used the incoherence property and
(45) 
To see that (45) holds, we start with
(46) 
where the first equality follows directly from the tight frame property, c.f. Ref. [2], while the second one stems from the definition of the conditional probability distribution . Rearranging and taking the norm yields
(47) 
which implies (45). Using (44) together with in Lemma 2, we obtain Lemma 6 which concludes the proof.
The above Lemma must by applied for , i.e., the operators occurring in the golfing scheme. By the second incoherence condition, (42) is fulfilled for . To ensure that incoherence is preserved during the golfing scheme, we must use a more complicated and technical argument than in Ref. [2] where a union bound over all elements of the operator basis was used which is clearly impossible in a tight frame with an infinite number of elements.
Lemma 7 (Replacing the union bound).
(48) 
where is the smallest number such that
(49) 
Proof: We fix an element from the tight frame and note that for
(50) 
The latter term is smaller than . To bound the former term, we define the random variable
(51) 
whose variance is bounded by
(52) 
and . Using once again the operator Bernstein inequality yields after squaring
(53) 
Eq. (53) says that the desired property is true with high probability for any fixed . To show that it is also true with high probability for most of the operators, we need a simple fact from probability theory, which is shown in the appendix.
Lemma 8 (Inverting probabilities).
Let and be two measure spaces and denote by a relation between the elements and . If
(54) 
then
(55) 
2.7 Reconstructing generic quantum states
In a next step, we investigate the reconstruction of random quantum states, that are sampled from probability measures that are invariant under the action of the unitary group by conjugation. We show examples of tight frames that satisfy the incoherence properties required in Theorem 3 to allow reconstruction of most quantum states.
Theorem 4 (Incoherence properties of generic states).
Let be a (family of) tight frame for which all operators fulfill , and pick a random rank quantum state , with a distribution that is invariant under the action of the unitary group. Then the probability that cannot be efficiently reconstructed by compressed sensing vanishes as .
Note that Theorem 4 holds for all unitarily invariant measures on the quantum states of rank regardless of the actual distribution of the eigenvalues.
Proof: We first show that for any fixed element of the tight frames, both incoherence properties are fulfilled with high probability. First, we turn to
(56) 
where is a unitary matrix which is chosen according to the Haar measure and we have fixed an element from the tight frame. We look at the th row of and note that . We write for the th column vector of and note that is just a random vector on a sphere . Thus, the squares of its coordinates are concentrated around , c.f. the appendix, and we get
(57) 
Using this in Eq. (56), inserting , and applying a union bound yields
(58) 
Employing again Lemma 8, this implies
(59) 
where is chosen according to the probability distribution of the tight frame. By allowing to grow polylogarithmically in , this probability vanishes polynomially in which means that it is violated to much only for a proportion of state vanishing polynomially as grows. Now we turn to the second nonFourier incoherence condition. Decomposing as a sum of projectors on orthogonal subspaces , we can write
(60) 
Using the concentration of measure on the sphere and yields
(61) 
which finally gives
(62) 
Since the additional factor of can be absorbed in , Theorem 4 follows from Eq. (62).
Tight frames for which this is the case include those where the rank of the operators does not grow with . The other extreme is given by the Pauli basis: From and it follows that . Colloquially speaking, a small spectral norm implies a large trace norm and vice versa. Thus, we have two classes of tight frames (Fourier likes ones and the ones with small norm) for which efficient compressed sensing is efficiently possible. Because they represent in some sense the two extreme cases (flat spectra vs. concentrated spectra), we have some reason to believe that this is indeed true for any tight frame.
3 Certification
3.1 Ideal case
Theorems 2 and 3 show that efficient compressed sensing is possible in a vast number of situations. They are stated in the asymptotic regime for clarity but could be furnished with reasonable prefactors for finite Hilbertspace dimension . However, when using compressed sensing in actual experiments, one encounters three main problems.

Firstly, the necessary number of measurements as calculated from the incoherence properties of the employed tight frame might still be too large to be feasible.

Secondly, repetition of the experiments to increase the probability of success to a satisfactory value may be expensive or difficult.

Thirdly, it is unknown how close to lowrank the state actually is. After all, no assumptions are made about the unknown input state.
The solutions to those problems is provided by certification. Instead of theoretically constructing some certificate based on with the help of the golfing scheme, we use the solution of the minimization problem to explicitly check whether the conditions for Theorem 1 are satisfied for . The candidate for the certificate can be calculated as
(63) 
where is obtained like but with replaced by and denotes the MoorePenrose pseudo inverse of . One can now check whether (5) is fulfilled. If the conditions for Theorem 1 are fulfilled and , the solution must be unique and equal to the state , i.e. tomography was successful.
3.2 Errors and noise
For compressed sensing to work in a realistic setting, the reconstruction procedure must be robust, i.e., small errors introduced by decoherence, errors stemming from imperfect measurements, and statistical noise due to the fact that every observable is only measured a finite number of times, should only lead to small errors in the reconstructed state. In addition, the Hilbert space might be infinitedimensional. When the mean energy, and therefore, the mean photon number , is finite, the error made by truncating the Hilbert space at photon number vanishes as
(64) 
which is shown in the appendix. This means that the expectation values with respect to the truncated state are close to the actually measured ones, i.e.,
(65) 
for all such that .
We assume that the observed data correspond to a matrix (not necessarily a state) with where is the lowrank, infinitedimensional state, i.e., the errors made by truncating to a finitedimensional Hilbert space are already included in , and where we denote by the projection to the image of the sampling operator. Such a tube condition is satisfied with very high probability for realistic error models like Gaussian noise [1, 18]. We relax the conditions in (2) to
(66) 
The solution of the SDP might not be of low rank. Because a lowrank state is needed for the construction of the certificate , we truncate to the largest eigenvalues (call this ) and obtain as above. As is in general not known, one has to perform the truncation of and the subsequent construction of the certificate for until a valid , as to be specified below, has been found. If this is not the case, the number of measurements was not enough and needs to be increased.
To provide an error bound, we denote the norm error made by the truncation of to rank by and obtain from the triangle inequality
(67) 
We calculate a candidate for a certificate as where is obtained from . If is valid, i.e., , and with , then the proof of Theorem 7 in Ref. [18] yields the robustness result
(68) 
By the equivalence of the norms, this also provides a norm bound at the expense of an additional factor .
Thus, with no further assumption than norm closeness of the observations to the state of interest it is possible to obtain a certified reconstruction which is also close to the state of interest. In this sense, quantum compressed sensing can achieve assumptionfree certified quantum state reconstruction in the presence of errors. This discussion applies to box errors, where each of the expectation values is assumed to be contained in a certain interval. The discussion of other error models will be the subject of forthcoming work.
4 Universal quantum compressed sensing
4.1 Universal quantum state reconstruction
The preceding discussion has focused on claims of the following form:
For every lowrank state , most choices of the observables can be used to successfully reconstruct .
In some situations, however, one can actually prove a much stronger statement, in which the order of the quantifiers is reversed:
Most choices of the observables will have the property that, for every lowrank state , the observables can be used to successfully reconstruct .
This is known as universal reconstruction; more simply, it says that a fixed set of observables can distinguish among all lowrank states simultaneously. Besides being of conceptual interest, universal reconstruction also implies stronger error bounds for reconstruction from noisy data.
Formally, we say that our method performs universal compressed sensing if