Practical methods for witnessing genuine multi-qubit entanglement in the vicinity of symmetric states

We present general numerical methods to construct witness operators for entanglement detection and estimation of the fidelity. Our methods are applied to detecting entanglement in the vicinity of a six-qubit Dicke state with three excitations and also to further entangled symmetric states. All our witnesses are designed to keep the measurement effort small. We present also general results on the efficient local decomposition of permutationally invariant operators, which makes it possible to measure projectors to symmetric states efficiently


I. INTRODUCTION
Entanglement plays a central role in quantum mechanics and in quantum information processing applications [1].
Moreover, it appears also as the main goal in today's quantum physics experiments aiming to create various quantum states [2]. For example, entanglement has been realized with photonic systems using parametric down-conversion and conditional detection [3,4,5,6,7,8,9], with trapped cold ions [10,11,12], in cold atomic ensembles [13], in cold atoms in optical lattices [14] and in diamond between the electron and nuclear spins [15]. These experiments aimed at creating entangled states. Entanglement makes it possible for some quantum algorithms (e.g., prime factoring, search in a database) to outperform their classical counterparts. Entangled particles are needed for quantum teleportation and other quantum communication protocols. Moreover, the creation of large entangled states might lead to new insights about how a classical macro-world emerges from a quantum micro-world.
In a multi-qubit experiment, typically the full density matrix is not known, and only few measurements can be made, yet one would still like to ensure that the prepared state is entangled. One possibility is applying entanglement witnesses [16,17]. These are observables that have a positive expectation value for separable states, while for some entangled states their expectation value is negative. Since these witness operators are multi-qubit operators, they typically cannot be measured directly and must be decomposed into the sum of locally measurable operators, which are just products of single-qubit operators [4,18,19].
For many quantum states, like the Greenberger-Horne-Zeilinger (GHZ, [20]) states and the cluster states [21] such a decomposition of projector-based witness operators seems to be very difficult: The number of terms in a decomposition to a sum of products of Pauli matrices increases rapidly with the number of qubits. However, practically useful entanglement witnesses with two measurement settings can be constructed for such states [5,22]. It also turned out that there are decompositions of the projector for GHZ and W-states in which the increase with the number of qubits is linear [23].
However, optimal decomposition of an operator is a very difficult, unsolved problem. Moreover, in general, it is still a difficult task to construct efficient entanglement witnesses for a given quantum state. For that, typically we need to obtain the maximum of some operators for product states. In most of the cases we would like to detect genuine multipartite entanglement. For that, we need to obtain the maximum of these operators for biseparable states, which is again a very hard problem.
In this paper our goal is to design witnesses that make it possible to detect genuine multipartite entanglement with few measurements, and also to estimate the fidelity of an experimentally prepared state with respect to the target state. Here three strategies are applied to find an experimentally realizable witness. (i) The first strategy is based on measuring the projector-based witness for the detection of genuine multipartite entanglement. |Ψ is the target state of the experiment. For reducing experimental effort, the aim is to find an efficient decomposition of the projector. (ii) The second strategy is to find a witness that needs much fewer measurements than the projector witness, but the price for that might be a lower robustness against noise. The search for such a witness can be simplified if we look for a witness W such that for some α > 0. Such a witness is guaranteed to detect genuine multi-qubit entanglement. The advantage of this approach is that the expectation value of W can be used to find a lower bound on the fidelity. (iii) The third strategy is to find a witness independent from the projector witness. In this case one has to find an easily measurable operator whose expectation value takes its maximum for the target state. Then, one has to find the maximum of this operator for biseparable states. Any state that has an operator expectation value larger than that is genuine multipartite entangled.
For the optimization of entanglement witnesses for small experimental effort and large robustness to noise, we use semidefinite programming [24,25,26,27,28,29]. Our methods can efficiently be used for multi-qubit systems with up to about 10 qubits. This is important, since there are many situations where semidefinite programming could help theoretically, but in practice the calculations cannot be carried out even for systems of modest size.
We use our methods to design witnesses detecting entanglement in the vicinity of symmetric Dicke states. An N -qubit symmetric Dicke state with m excitations is defined as [30,31] where k P k (.) denotes summation over all distinct permutations of the spins. |D (1) N is the well known N -qubit W state. The witnesses we will introduce in the following have already been used in the photonic experiment described in [32], aiming to observe a |D state [32,33]. We show that genuine multi-qubit entanglement can be detected and the fidelity with respect to the above highly entangled state can efficiently be estimated with two and three measurement settings, respectively. As a byproduct, we will also derive an upper bound for the number of settings needed to measure any permutationally invariant operator. We show that such operators can efficiently be measured even for large systems.
The structure of our paper is as follows. In section 2, we present the basic methods for constructing witnesses. In section 3, we use these methods for constructing witnesses to detect entanglement in the vicinity of a six-qubit symmetric Dicke state with three excitations. In section 4, we present witnesses for states obtained from the above state by measuring some of the qubits. In Appendix A, we summarize the tasks that can be solved by semidefinite programming, when looking for suitable entanglement witnesses. In Appendix B, we summarize some of the relevant numerical routines of the QUBIT4MATLAB 3.0 program package [34]. In Appendix C, we present entanglement conditions for systems with 5 − 10 qubits that will be relevant in future experiments.

II. BASIC DEFINITIONS AND GENERAL METHODS
A multi-qubit quantum state is entangled if it cannot be written as a convex combination of product states. However, in a multi-qubit experiment we would like to detect genuine multi-qubit entanglement [35]: The presence of such entanglement indicates that all the qubits are entangled with each other, not only some of them. We will now need the following definitions: Definition 1. A pure multi-qubit quantum state is called biseparable if it can be written as the tensor product of two, possibly entangled, multi-qubit states A mixed state is called biseparable, if it can be obtained by mixing pure biseparable states. If a state is not biseparable then it is called genuine multi-partite entangled. In this paper we will consider witness operators that detect genuine multipartite entanglement. Definition 2. While an entanglement witness is an observable, typically it cannot be measured directly. This is because in most experiments only local measurements are possible. At each qubit k we are able to measure a single-qubit operator M k , which we can do simultaneously at all the qubits. If we repeat such measurements, then we obtain the expectation values of 2 N − 1 multi-qubit operators. For example, for N = 3 these are M 1 ⊗ 1 1 ⊗ 1 1, 1 The set of single-qubit operators measured is called measurement setting [4] and it can be given as {M 1 , M 2 , M 3 , ..., M N }. When we consider an entanglement condition, it is important to know, how many measurement settings are needed for its evaluation.

Definition 3.
Many experiments aim at preparing some, typically pure quantum state ̺. An entanglement witness is then designed to detect the entanglement of this state. However, in real experiments such a state is never produced perfectly, and the realized state is mixed with noise as given by the following formula where p noise is the ratio of noise and ̺ noise is the noise. If we consider white noise then ̺ noise = 1 1/2 N . The noise tolerance of a witness W is characterized by the largest p noise for which we still have Tr(W̺ noisy ) < 0.
In this paper, we will consider three possibilities for detecting genuine multi-qubit entanglement, explained in the following subsections. Later, we will use these ideas to construct various entanglement witnesses.

A. Projector witness
A witness detecting genuine multi-qubit entanglement in the vicinity of a pure state |Ψ can be constructed with the projector as where λ is the maximum of the Schmidt coefficients for |Ψ , when all bipartitions are considered [4]. For the states considered in this paper, projector-based witnesses are given by [4,12,37] W (P ) These witnesses must be decomposed into the sum of locally measurable terms. For this decomposition, the following observations will turn out to be very important. Observation 1. A permutationally invariant operator A can always be decomposed as [45] where a n are single qubit operators, and such a decomposition can straightforwardly be obtained.
Proof. Any permutationally invariant multi-qubit operator A can be decomposed as where B n,m are single qubit operators, c n are constants, and P k are the full set of operators permuting the qubits. For odd N, we can use the identity Substituting (11) into (10), we obtain a decomposition of the form (9). Equation (11) can be proved by carrying out the summation and expanding the brackets. Due to the s 1 s 2 s 3 · · · s N = +1 condition, the coefficient of B n,1 ⊗ B n,2 ⊗ B n,3 ⊗ ... ⊗ B n,N is 1. The coefficient of terms like B n,1 ⊗ B n,1 ⊗ B n,3 ⊗ ... ⊗ B n,N , that is, terms containing one of the variables more than once is zero. For even N, a similar proof can be carried out using [49] Next, we give two examples for the application of (11) and (12) for the decomposition of simple expressions where σ k are the Pauli spin matrices. While the first example does not reduce the number of settings needed, the second example reduces the number of settings from 6 to 4. Next, we present a method to get efficient decompositions for permutationally invariant operators. Observation 2. Any N -qubit permutationally invariant operator A can be measured with at most local measurement settings, using (11) and (12). Proof. We have to decompose first A into the sum of Pauli group elements as where c ijm are some constants. Then, such a decomposition can be transformed into another one of the form (9), using (11) and (12). All of the settings needed are of the form {a, a, a, ..., a} where a = n x σ x + n y σ y + n z σ z , n k are integer and 1 ≤ k |n k | ≤ N. Simple counting leads to an upper bound L N for the number of settings given in (15).
Here we considered that (n x , n y , n z ) and (−n x , −n y , −n z ) describe the same setting. An even better bound can be obtained using that (n x , n y , n z ) and (cn x , cn y , cn z ) for some c = 0 represent the same setting. An algorithm based on this leads to the bounds L ′ N = 9, 25, 49, 97, 145, 241, 337, 481, 625 for N = 2, 3, ..., 10 qubits, respectively. For the projector |D |, the decomposition to Pauli group elements contain only terms in which each Pauli matrix appears an even number of times. Hence, all of the settings needed are of the form {a, a, a, ..., a} where a = 2n x σ x + 2n y σ y + 2n z σ z , n k are integer and 1 ≤ k |n k | ≤ N/2. For this reason, L N/2 and L ′ N/2 are upper bounds for the number of settings needed to measure this operator.
Let us discuss the consequences of Observations 1 and 2. They essentially state that the number of settings needed to measure a permutationally invariant operator scales only polynomially with the number of qubits. This is important since for operators that are not permutationally invariant, the scaling is known to be exponential [36]. Moreover, even if we can measure only correlation terms of the form a ⊗N , we can measure any permutationally invariant operator.

B. Witnesses based on the projector witness
We can construct witnesses that are easier to measure than the projector witness, but they are still based on the projector witness. We use the idea mentioned in the introduction. If W (P ) is the projector witness and (2) is fulfilled for some α > 0, then W is also a witness. This is because W has a negative expectation value only for states for which W (P ) also has a negative expectation value. The advantage of obtaining witnesses this way is that we can have a lower bound on the fidelity from the expectation value of the witness as We will look for such witnesses numerically, such that the noise tolerance of the witness be the largest possible. This search can be simplified by the following observation. Observation 3. Since we would like to construct a witness detecting genuine multiqubit entanglement in the vicinity of a permutationally invariant state, it is enough to consider witness operators that are also permutationally invariant. Proof. Let us consider a witness operator that detects entanglement in the vicinity of a permutationally invariant state ̺ and its expectation value takes its minimum for ̺. Then, based on (5), the witness W detects entanglement if For a permutationally invariant state ̺, we have ̺ = 1 NP k P k ̺P k , where N P is the number of different permutation operators P k . We assume that the same holds also for ̺ noise . Let us define the permutationally invariant operator where B is the set of biseparable states. Hence, W ′ is a witness detecting genuine multipartite entanglement. Since we have Tr(W̺) = Tr(W ′ ̺), and Tr(W̺ noise ) = Tr(W ′ ̺ noise ), the robustness to noise of W ′ is identical to that of W. Hence, it is sufficient to look for witnesses that are permutationally invariant. We will first consider measuring the {σ x , σ x , σ x , σ x , σ x , σ x } and {σ y , σ y , σ y , σ y , σ y , σ y } settings, where σ l are the Pauli spin matrices. This we call the two-setting case. Then we will consider measuring also the {σ z , σ z , σ z , σ z , σ z , σ z } setting, which we call the three-setting case. Due to Observation 3, we consider only permutationally invariant witnesses. Such witnesses can be written as where the summation is over all distinct permutations, and α 0 and α ln are some constants. We will consider a simpler but equivalent formulation where c 0 , c ln are the coefficients of the linear combination defining the witness and J l are the components of the total angular momentum given as Here σ (k) l denotes a Pauli spin matrix acting on qubit (k). Finally, if we consider detecting entanglement in the vicinity of |D (N/2) N states, then further simplifications can be made. For this state and also for the completely mixed state all odd moments of J l have a zero expectation value. For any witness of the form (21), the maximum for biseparable states does not change if we flip the sign of c ln for all odd n. Hence, following from an argument similar to the one in Observation 3 concerning permutational symmetry, it is enough to consider only even powers of J l in our witnesses.

C. Witnesses independent from the projector witness.
In general, we can also design witnesses without any relation to the projector witness. We can use an easily measurable operator M to make a witness of the form where c is some constant. To make sure that (23) is a witness for genuine multipartite entanglement, i.e, W is positive on all biseparable states, we have to set c to where B is the set of biseparable states. The optimization needed for (24) can be done analytically. For example, for the |D state a witness has been presented that detects genuine four-qubit entanglement by measuring second moments of angular momentum operators [37]. However, analytical calculations become exceedingly difficult as the number of qubits increases.
The optimization can also be done numerically, but one cannot be sure that simple numerical optimization finds the global maximum. (See Appendix B for a reference to such a MATLAB program.) Semidefinite programming is known to find the global optimum, but the optimization task (24) cannot be solved directly by semidefinite programming. Instead of looking for the maximum for biseparable states, using semidefinite programming, we can look for the maximum for states that have a positive partial transpose [24,38]. (See Appendices A and B.) This way we can obtain for which c ′ ≥ c. The first maximization is over all bipartitions I. Thus, when putting c ′ into the place of c in (23), we obtain a witness that detects only genuine multipartite entanglement. In many cases simple numerics show that c = c ′ . In this case our witnesses are optimal in the sense that some biseparable state gives a zero expectation value for these witnesses. Finally, let us discuss how to find the operator M in (23) for a two-or a three-setting witness, in particular, for detecting entanglement in the vicinity of |D (N/2) N . Based on section II B, we have to look for an operator that contains only even powers of J l . Hence, the general form of a two-setting witness with moments up to second order is where c DN is a constant [39]. The coefficients of J 2 x and J 2 y could still be different, however, this would not lead to witnesses with a better robustness to noise.
For other symmetric Dicke states, based on similar arguments, a general form of a witness containing moments of J l up to second order such that it takes its minimum for |D where c q and q are constants. For the witnesses described in this section, the optimization process is more timeconsuming than for the witnesses related to the projector witness. Because of that we presented witnesses of the above type that are constructed only with the first and second moments of the angular momentum operators, and thus contain few free parameters.

III. WITNESSES FOR A SIX-QUBIT DICKE STATE WITH THREE EXCITATIONS
In this section, we will consider entanglement detection close to a six-qubit symmetric Dicke state with three excitations, denoted as |D . There are several proposals for creating Dicke states in various physical systems [40,41,42,43].
A. Witnesses based on the projector witness

Two-setting witness
Let us consider the two-setting case and define first the optimization problem we want to solve. We would like to look for the witness W with the largest noise tolerance that fulfills the following requirements: For the two-setting case we set {B k } = {1 1, J 2 x , J 2 y , J 4 x , J 4 y , J 6 x , J 6 y }. The second condition makes sure that W is also a witness detecting genuine multipartite entanglement.
Note that any optimization algorithm can be used for looking for W. Even if we do not find the global optimum, that is, the witness with the largest possible robustness to white noise, W is still a witness detecting genuine multipartite which tolerates white noise if p noise < 0.1391. Straightforward calculation shows that W

Three-setting witness
Similarly we can look for the optimal witness for the three-setting case. The result is White noise is tolerated if p noise < 0.2735. It is easy to check that W is a witness as W (P 3) D(6,3) − 2.5W (P ) ≥ 0. Based on (17), the expectation value of this witness can be used to bound the fidelity as F ≥ 0.6 − W (P 3) D(6,3) /2.5 =: F ′ . Here we will demonstrate how well the fidelity estimation works for our witness for noisy states. We consider first white noise, then non-white noise of the form with p D63 = 4/7, which is one of the relevant types of noise for the experiment of [32]. Note that the noise contains the original state |D Note that it is also possible to design a witness for the largest possible tolerance to the noise in (30). Due to the special form of the noise, the fidelity estimate turns out to be equal to the fidelity. This is remarkable: The fidelity can be obtained exactly with only three local measurements.

Measuring the projector-based witness
For measuring the projector-based witness (7) for N = 6, one has to decompose the projector in an efficient way. The straightforward decomposition into the weighted sum of products of Pauli spin matrices leads to a scheme that needs 183 settings, since for all local operators all the permutations have to be measured. The number of settings needed can dramatically be decreased if one is looking for a decomposition of the form (9). Observation 1 makes it possible to decompose the projector in this way such that only 25 settings are needed. We could further decrease the number of settings needed and found the following decomposition Here where σ 0 = 1 1. That is, it is the sum of terms with even number of σ a 's and σ b 's, with the sign of the terms depending on the number of σ a 's. The expectation value of the operators Mermin a,b can be measured based on the decomposition [23] Mermin , and x ± y ± z. The settings are also shown in figure 2(a) [50].

B. Witness independent from the projector witness
So far we constructed witnesses that detected fewer states than the projector-based witness, in return, they were easier to measure. When proving that they were witnesses, we used the simple relation (2). Following the example of [37], we now look for a two-setting witness of the form (26) for N = 6 that is independent from the projector witness. For determining c D6 , we need to compute the maximum of J 2 x + J 2 y for biseparable states for all the possible bipartitions. As we have discussed in section II C, instead of looking for the maximum for states that are separable with respect to a certain bipartition, we can also look for the maximum for PPT states. (See Appendix A.) We obtain c D6 := 11.0179.
W (I2) D (6,3) detects genuine multipartite entanglement if for white noise p noise < 0.1091. Simple numerical optimization leads to the same value for the maximum for biseparable states [51]. Hence we find that our witness is optimal. Finally, the list of witnesses presented in this section are shown in the top part of table 1.

VIA PROJECTIONS
By projective measurements of one or two of the qubits we can obtain several states that are inequivalent under stochastic local operations and classical communication (SLOCC). Surprisingly, these states still possess genuine multipartite entanglement [32,44]. Next, we discuss how to detect the entanglement of these states.
A. Witnesses for the superposition of five-qubit Dicke states: After measuring one of the qubits in some basis and post-selecting for one of the two outcomes, one can obtain states of the form where |c 1 | 2 + |c 2 | 2 = 1. For such states, the expectation value of J 2 x + J 2 y is maximal, thus a witness of the form (26) for N = 5 is used to detect their entanglement. Both semidefinite programing and simple numerical optimization leads to c D5 := 7.8723. Naturally, W is minimal not only for states of the form (35), but for any mixture of such states.

B. Witness for the four-qubit W-state
Now we will construct witnesses for a four-qubit W-state, which is obtained from |D if two qubits are measured in the σ z basis, and the measurement result is +1 in both cases. We consider a witness of the form (27) for N = 4 and m = 1. We try several values for q and determine c q for the witness W D(4,1) (q) as a function of q using semidefinite programming. For each witness we also compute the noise tolerance. The results of these computations can be seen in figure 3. It turns out, that the best witness is obtained for q = 1.47 and c q = 4.1234. It tolerates white noise if p noise < 0.1476. , namely if the measurement outcomes are +1 and −1 for two consecutive σ z measurements. For that case, we look for a three-setting witness, based on the projector witness. For white noise, the result is The witness tolerates white noise if p noise < 0.2759. It is easy to check that W is a witness: One has to notice that W We can also measure the projector witness W The 9 measurement settings are x, y, z, x ± y, x ± z, and y ± z, shown also in figure 2(b). The list of witnesses presented in this section are given in the bottom part of table 1.

V. CONCLUSIONS
In summary, we presented general methods for constructing entanglement witnesses for detecting genuine multipartite entanglement in experiments. In particular, we considered projector-based witnesses and found efficient decompositions for them. Then, we constructed two-and three-setting witnesses for symmetric Dicke states that were based on the projector witness, as well as independent from the projector witness. We applied our methods to design witnesses for the recent experiment observing a six-qubit symmetric Dicke state with three excitations [32]. Our methods can be generalized for future experiments. As a first step, in Appendix C we list some entanglement witnesses for systems with 5 − 10 qubits. Moreover, recent results on the symmetric tensor rank problem suggest that decompositions more efficient than the one in Observation 1 are possible, however, they involve complex algorithms [45]. Thus, it would be interesting to look for better upper bounds for the number of settings used for symmetric operators.
APPENDIX C: WITNESSES FOR SYSTEMS WITH 5 − 10 QUBITS A three-setting witness based on the projector witness for the state |D A three-setting witness independent of the projector witness for the N -qubit W -state is given by (27) for m = 1. For N = 5, we have c 5 = 5.6242, q 5 = 2.22, and the witness tolerates white noise if p noise < 0.0744. For N = 6, we have c 6 = 7.1095, q 6 = 3.13, and noise is tolerated if p noise < 0.0401.