Detecting Entanglement of Unknown Quantum States with Random Measurements

In quantum information theory, the reliable and effective detection of entanglement is of paramount importance. However, given an unknown state, assessing its entanglement is a challenging task. To attack this problem, we investigate the use of random local measurements, from which entanglement witnesses are then constructed via semidefinite programming methods. We propose a scheme of successively increasing the number of measurements until the presence of entanglement can be unambiguously concluded, and investigate its performance in various examples.


I. INTRODUCTION
Entanglement is a valuable resource for accomplishing quantum information tasks with a performance exceeding what can be optimally achieved classically. Thus, the unambiguous detection of entanglement is a fundamental problem in quantum information processing. However, even though in the limit of high Hilbert space dimension, a randomly chosen quantum state is almost certainly close to maximally entangled, this is in general a hard task [1].
Nevertheless, important insights have been gained for the case that the state to be analysed is at least partially known. Given some prior knowledge of the density matrix (obtained, for example, by assuming that the given state is close to a target state that has been imperfectly prepared), few local measurements suffice to detect entanglement [2]. In a different vein, local measurements along randomly chosen directions may be used to certify nonclassical correlations by means of a Bell-inequality violation, if the state distributed between the different parties is known, thus obviating the need for a shared reference frame [3][4][5].
How can entanglement be detected in the case in which one does not have any knowledge of the given state? A method to characterize entanglement in such a case is given by quantum tomography [6], i.e., reconstructing the full density matrix from a pre-chosen set of measurements.
However, this method has several drawbacks: first of all, it is enormously resource-intensive, with the number of measurements that need to be performed scaling exponentially with the number of particles of the system. Furthermore, recent results have shown that the most commonly used tomography schemes may be subject to systematic errors that can lead to overestimating the entanglement contained in a quantum state [7].
Additionally, if one is merely interested in the question of whether a given state is entangled, full tomography yields an excess of information; thus, it may be feasible * Electronic address: szangolies@thphy.uni-duesseldorf.de to extract the required knowledge using fewer measurements.
In this article, we investigate the possibility of detecting the entanglement of an unknown state using random measurements. The intuition behind this is that if one is given an unknown, i.e. random, state, there is no canonical way to choose measurements tailored to optimally determining the entanglement of this state.
The method of choice for entanglement detection we will be concerned with is that of so-called entanglement witnesses [8][9][10]. An entanglement witness is a Hermitian operator W such that for all separable states σ, and for at least one entangled state ρ. Thus, a negative expectation value as a result of measuring W certifies the presence of entanglement. Note that the absolute of a negative expectation value for W also leads to a lower bound on an entanglement measure [11].
For simplicity, we will in the following mainly concentrate on a certain class of entanglement witnesses, the so-called decomposable witnesses [8,12]. These are witness operators W that can be written in the form Here, the superscript T A denotes the partial transposition operation with respect to subsystem A, and Q and P are both positive semidefinite operators, that is, they have no negative eigenvalues. This is the most general witness capable of detecting states with nonpositive partial transpose: tr(ρW ) < 0 implies ρ T A 0, since tr(ρW ) = tr(P ρ) + tr(Qρ T A ), which can only be smaller than zero if ρ T A is not positive. Nonpositivity of the partial transpose is a sufficient criterion for entanglement in all dimensions, and is even necessary in case dim(ρ) ≤ 2×3 [8], which yields the famous PPT-criterion for entanglement detection.
The paper is organized as follows. In Section II, we investigate the possibility of detecting entanglement using a randomly chosen witness operator. Then, in Section III, we focus on constructing witnesses from random local measurements. To this end, in Section III A we employ a decomposition of witness operators, and carry out optimization by means of a semidefinite program in order to find the best witness constructible from a given set of measurement settings and outcomes. Then, in Section III B, we propose a scheme that successively increases the number of measurements, until the semidefinite program certifies the presence of entanglement. In Section IV, we test this scheme on various experimentally relevant examples. Section V is devoted to investigating the statistical robustness of our method, and in Section VI, we conclude.

II. RANDOMLY GENERATED ENTANGLEMENT WITNESSES
As a first attempt towards addressing the problem of unknown-state entanglement detection, one might, reasoning that since there is no way to single out a particular witness, simply consider using witnesses that themselves are (Haar-)randomly drawn from the set of all Hermitian operators acting on H AB = H A ⊗ H B . Operationally, this corresponds to carrying out a random global measurement on the system. The first step, then, is to identify witness operators among all possible measurements. This can be done by checking two criteria which both have to be fulfilled: First, one checks whether the operator is indefinite. If it is not, then it cannot possibly be a witness, as its expectation value would have the same sign on separable and entangled states alike. Second, one checks whether the operator's partial transpose is positive: Since for any separable state σ, σ T A is again a valid (separable) density operator, we can find σ such that σ T A = σ, and thus, if meaning that the expectation value of W is positive on all separable states. Hence, as W is indefinite by the previous requirement, it must detect at least one entangled state. Not all entanglement witnesses can be found in this way. For the remainder, an optimization procedure that minimizes the expectation value of the operator on the set of separable states, based on the one presented in Ref. [13], will be used. Positivity of this minimal expectation value then certifies the operator as an entanglement witness, provided a global minimum is attained. To certify that this occurs with a high degree of confidence, the optimization is performed 10 3 times with different initial comditions.
In order to test the witnesses obtained in this way, we generate random states, and attempt to detect them, using randomly chosen operators. Random (mixed) states are produced as follows. First, a random unitary matrix can be generated by orthogonalizing a matrix with uniformly random, bounded complex entries us-ing the Gram-Schmidt method (which guarantees Haarrandomness [14]). A random k × n−dimensional pure state is then obtained by generating a random unitary and extracting one of its column vectors. Then, the k−dimensional environment is traced out, yielding an n−dimensional mixed state randomly distributed according to [15] where the step function Θ enforces positivity, and the Dirac δ guarantees the normalization of the density matrix thus obtained. The use of this prescription is motivated by the fact that mixed states, in quantum theory, always arise from interactions with an environment, together with the assumption that the total (pure) system-environment state is distributed according to the unitarily invariant (Haar-) measure [15].
Additionally, random maximally entangled states can be generated by starting from a given d-dimensional maximally entangled state, |Ψ + = 1 √ d i |i ⊗d , and transforming it with random local unitaries U i [15], i.e.
where U A and U B are random unitaries acting on the local Hilbert spaces H A and H B , thus effecting a random transformation leaving the amount of entanglement constant. However, the results of carrying out this investigation are discouraging: Generating random Hermitian operators by generating a diagonal matrix D with uniformly random real entries sampled from a bounded interval, and then forming where U is a random unitary matrix, we find that even for a two-qubit Hilbert space, witnesses are rare among randomly chosen operators. Indeed, among 10 5 candidate operators, only a fraction of 1.73 ± 0.05% were identified as witnesses by the positivity of their partial transpose, with a further 1.34 ± 0.04% being found to be witnesses by means of the overlap-minimization algorithm. Here and in the remainder of this section, the specified errors are due to finite-size statistical effects. Even further discouraging is the fact that typically, witnesses being chosen according to this prescription have a very small likelihood of actually detecting the state one is presently interested in-testing witnesses found according to both criteria on a set of random states, it was found that only a fraction of (9.3 ± 0.7) · 10 −6 of states was detected by witnesses having positive partial transpose, while a further (13±1)·10 −6 were detected by witnesses found using the overlap-minimization approach. Even in the case of maximally entangled states, only (1.094 ± 0.007) · 10 −2 , respectively (1.092 ± 0.008) · 10 −2 , of all states were detected using these methods. Fortunately, as we will show in the remainder of the paper, this is not yet grounds to give up on the project of detecting entanglement using random measurements. In the following section we will introduce the method of constructing entanglement witnesses from random measurements carried out on the local Hilbert spaces.

III. ENTANGLEMENT WITNESSES FROM RANDOM LOCAL MEASUREMENTS
From this point on, we will only consider decomposable witnesses, that is, witnesses of the form given in Eq. (3). The reason for this is, as we will see, that such witnesses can be efficiently constructed using a semidefinite program. Hence, from now on, the term 'witness' should always be understood to mean 'decomposable witness', unless specified otherwise.

A. Constructing Witnesses
Any Hermitian operator O in B(H AB ), the space of linear bounded operators acting on the total Hilbert space H AB , can be decomposed into local Hermitian operators with c ij ∈ R. Thus, finding a witness operator W , given local measurements A i and B j , amounts to finding coefficients c ij such that for some entangled ρ, and for all separable ρ. This translates to the following optimization problem: Here, c is the vector of the coefficients c ij , and m = ( A i ⊗ B j ) is the vector of the expectation values of A i ⊗ B j . Both vectors are obtained by mapping the indices i and j to a single index α. The semidefinite program (SDP) thus formulated calculates the operator W , such that W has the lowest possible expectation value given the experimentally obtained data, while being a normalized decomposable entanglement witness. If this expectation value now is negative, it can be unambiguously concluded that ρ is entangled.
This raises the question of how many measurements need to be performed in order to be able to certify the presence of entanglement-or, in other words, how many measurements one needs minimally to be able to construct an entanglement witness for an arbitrary state.
An upper bound for this number is given by the number of measurements in a tomographically complete set: since randomly drawn measurement directions are linearly independent, such a set forms a basis for the Hilbert-Schmidt space of operators, and thus, any operator can be written as a linear combination of them.
For a system of dimension d = d A · d B , the density matrix ρ contains d 2 − 1 independent parameters, each of which needs to be fixed by a suitable measurement. However, in total, (d 2 An immediate advantage of the method proposed above is that it can be used to devise a scheme which always terminates with the successful detection of entanglement, if entanglement that can be detected using the PPT-criterion is indeed present. This can be achieved by successively adding measurement directions on Alice's and Bob's local systems, then running the semidefinite program, until entanglement has been detected.
In order to better illustrate this strategy, we order the measurements as follows.
In this ordering, the measurements in the top row and leftmost column need not be performed, but instead come 'for free' upon performance of the measurements in the second row and second column.
Different ways of traversing these measurements now suggest themselves. Strategies may differ in the experimental complexity they necessitate: We clearly want to minimize the number of measurements that need to be carried out in order to detect a given state; however, at the same time, resetting measurement apparata is itself an investment of time resources and may also introduce a source of errors. If, for instance, we choose to first implement all measurements of the form A 1 ⊗ B i , that is, leave the apparatus at A fixed, and only change the setting at B, we have to re-set the apparatus at B in order to measure the observables A 2 ⊗ B i , A 3 ⊗ B i , and so on; but this is in general only imperfectly possible.
Among the strategies to traverse Table I An intrinsic advantage is provided by strategy 2: since every new round conveys a maximum of new information, we may expect it to perform, on average, better than the other strategies; in fact, since one needs at least two distinct local measurements to detect entanglement [16], this is the only strategy that enables entanglement detection already in the second round. Thus, in the reminder of this paper, we will use this strategy.

IV. DETECTING NPT-ENTANGLEMENT OF UNKNOWN STATES
In the previous section, we have established a method to detect entanglement of states about which nothing is known except the dimension, and hence, which can be assumed to be drawn randomly. In order to now put this method to the test, we will simulate its performance on states chosen randomly as described above.
To quantify entanglement, we resort to the negativity, a well-known entanglement measure that was introduced in Ref. [18]. This is also motivated by the fact that we detect states exactly if their partial transpose is nonpositive, that is, their negativity is nonzero. Furthermore, the expectation value of (optimal) decomposable witnesses is equal to the negativity [17].
The negativity is defined as where ||X|| 1 = tr( √ XX † ) denotes the trace norm, and the ω i are the eigenvalues of ρ T A . Thus, the negativity of ρ is the absolute sum of the negative eigenvalues of its partial transposition [18].
The distribution of negativities is in general such that it is very unlikely to randomly draw a highly entangled state, and, even though the bulk of states gets more entangled in higher dimensions, this gets worse with increased system size.
Our analysis concerns the performance of our method in detecting quantum states uniformly randomly chosen as described above. As a figure of merit, we use the number of measurements needed until a given entangled state is detected; thus, our algorithm first computes the expectation values of a certain number of random measurements A i and B j given the random state ρ, and then executes the semidefinite program to see whether the state's entanglement has been detected. If that is not the case, additional measurements are simulated one at a time, where each additional measurement is composed from local measurements in the form A i ⊗B j , and chosen according to the scheme discussed in Section III B. The algorithm is run until entanglement is detected. The results, for a system of two qubits, are shown in Figure 1, while Figure 2 analogously shows the same analysis for two qutrits. As one would expect, the method becomes more effective for more strongly entangled states, since these lie closer to the border of the set of all quantum states, and thus, less information is needed to distinguish between them and separable states. Compared with the naive method discussed in Section II, we also observe a marked increase in efficiency: where the mere random measuring of entanglement witnesses detected even highly entangled states with infeasibly low probability, here, highly entangled states are detected after performing only a few measurements.
FIG. 2: 3 · 10 4 runs of the algorithm with 3 × 3-dimensional states drawn randomly. The data is normalized with respect to the binning of the negativity values, such that for each value of the negativity, it sums to one. For 3 × 3-dimensional systems, the gain in efficiency with more highly entangled states is even more pronounced. Although few states of truly high entanglement content are produced by our random method, even moderately entangled states are typically detected after only around 20 measurements, whereas full tomography would need 64 measurements in general.
Indeed, as shown in Figure 3, for maximally entangled 3 × 3-dimensional states, one can typically detect their entanglement after performing only 10 ± 3 measurements (where the error is due to finite size statistics).

V. STATISTICAL ANALYSIS
Up to this point, we have concentrated on the ideal case, that is, used the exact expectation values A i ⊗ B j in order to construct witness operators. However, in a real experiment, we do not have access to these values, but only to experimental data subject to, at the very least, statistical fluctuations. These fluctuations introduce an uncertainty into the experimentally obtained expectation values. Given some dichotomic operator M , if we perform N experiments, obtaining n + times +1 and n − times the value −1, we can calculate the approximate expectation value where the bar denotes the experimentally obtained mean value. The value n + now carries a statistical error ∆n + = N p(1 − p), where p is the probability of obtaining the outcome +1 in a single measurement, since it is binomially distributed. Using standard error propagation, the resulting error of M is Of course, without knowing the state ρ, we do not have access to the probability p. However, as p(1 − p) assumes its maximum of 1 4 at p = 1 2 , we can use the worst-case approximation According to the second strategy defined in Section III, the witness is of the form with M i = A i ⊗ B i . Here, we have the fortunate case that we need not worry about the covariance between different observables, as all are chosen independently. It would now be tempting to simply proceed using, again, error propagation, obtaining the formula (16) However, doing so would neglect the fact that the coefficients c i are not independent of the mean values M i , since they are, in fact, derived from them.
This difficulty can be overcome by dividing the data into two sets, using one to derive, via the semidefinite program, the coefficients c i , and then evaluating this witness using the expectation values generated by the other set of data (cf. [19]). That way, the coefficients are independent of the errors in the expectation values of the second set, and thus, can be assumed to be simple constants, and Eq. 16 can be used to finally obtain In case every observable is measured an equal number of times, i.e. N i = N for all i, this further simplifies to This estimate can be used to facilitate the decision if, in an experiment, one should rather consider adding new measurements, or performing additional repetitions of the measurements already made: Especially in the case of an observed expectation value close to 0, a very small error is necessary to conclude the presence of entanglement; however, since additional measurements will tend to drive the expectation value closer to the state's negativity, adding another measurement instead may be more advantageous.
Consider, to this end, the example in Figure 4: There, for a single detection of a low-negativity (N = 0.0163) two qubit state, the maximum of the 3σ-error interval is plotted as a function of the number N of measurement repetitions. As can be seen, while a detection, that is, a certification of tr(W ρ) < 0 with 3σ-confidence, is possible for 6 global measurements, doing so requires a high number of repetitions for each individual measurement; however, adding another measurement, this number drops drastically.

VI. CONCLUSIONS
We have proposed a new method to detect the entanglement of quantum states, about which nothing is known except the dimension, using random local measurements. From these measurements, a witness operator is constructed by means of a semidefinite optimization procedure, minimizing the expectation value in a way that ensures that the operator is a valid witness at all times. Thus, once this expectation value becomes negative, one can unambiguously conclude that the state must be entangled.
Our main objective was to investigate the feasibility of this method in an experimental context, and in particular, compare it to existing methods of investigating unknown states, such as for instance local tomography. To this end, we have carried out numerical simulations, investigating the performance of our method using states drawn uniformly at random.
We found a significant reduction in the number of measurements one needs to perform in order to certify the presence of entanglement, as compared to doing full tomography, yet still without introducing any additional assumptions about the state. For instance, in the case of a 3 × 3-dimensional system, maximally entangled states may be detected after only 10 ± 3 random measurements, as compared to the 64 measurements needed for full tomography.
A question of special relevance to the experimental implementation of this method is the statistics needed in order to certify entanglement to some given degree of confidence. In order to facilitate this, we derived an upper bound on the error of the witness expectation value, and provided a method to gauge the statistical fluctuations during an experiment, by binning the data into independent sets, and cross-checking the witness constructed from one set using data from the other [19]. Simulation of this procedure showed a good robustness of the method to statistical errors.
With these promising results in mind, its usefulness in actual experimental practise needs to be demonstrated. Furthermore, one could widen the theoretical scope of our considerations towards higher-dimensional systems, as well as multipartite ones.