Highly-efficient estimation of entanglement measures for large experimentally created graph states via simple measurements

Quantifying experimentally created entanglement could in principle be accomplished by measuring the entire density matrix and calculating an entanglement measure of choice thereafter. Due to the tensor-structure of the Hilbert space, this approach becomes infeasible even for medium-size systems. In this letter we present methods to quantify the entanglement of arbitrarily large two-colorable graph states from simple measurements. The measurement data considered here is merely given by stabilizer measurements, thus leading to an exponential reduction in the number of measurements required. We provide analytical results for the robustness of entanglement and the relative entropy of entanglement.

3 graph. Graph states satisfy the following eigenvalue equation: K i |G k 1 ,...,k n = (−1) k i |G k 1 ,...,k n . The n operators K i generate an Abelian group S, called the stabilizer. The elements of the stabilizer will be denoted by g j , j = 1, . . . , 2 n . An experimentally created graph state could in principle be verified by measuring the 2 n elements of the stabilizer. As mentioned in the introduction, full-state tomography is not an option for determining the properties of a quantum many-body system due to an exponentially fast growing measurement effort. We will see that merely the measurement results of the generators of the stabilizer suffice to attain highly useful bounds on entanglement measures.
Let us suppose that the goal of an experiment is the creation of a two-colorable graph state, and the generators of the stabilizer are measured with outcomes a i = tr(ρ K i ), i = 1, . . . , n. As the convention for the coloring we use |A| Amber and |B| Blue qubits, taking |A| |B|. A generator K i is said to be Amber (Blue) if i corresponds to an Amber (Blue) qubit.
Given this tomographically incomplete data, one is now interested in finding the minimal entanglement (according to a certain entanglement measure) compatible with the measurement data. Mathematically, this is formulated as the semidefinite program [15,16]: where E(ρ) is a convex entanglement quantifier of choice. We will consider the following entanglement measures. The relative entropy of entanglement is defined as [17] where SEP denotes the set of fully separable states. The global robustness of entanglement is given by the minimum amount of an unnormalized state σ that has to be mixed into the given state ρ to wash out all entanglement [18]: Let us return to the estimation of entanglement measures from stabilizer measurements. The crucial point is that the minimization (1) does not need to be carried out over all states ρ. Instead, it suffices to minimize the entanglement quantifier over stabilizer diagonal states only. This can be seen in the following way. Since the stabilizer operators are mutually commuting, the measurement outcomes tr(ρ K i ) are invariant under any rotation of the density matrix of the form ρ → g j ρg j , g j ∈ S. Due to convexity of the entanglement quantifier, it is legitimate to apply a local symmetrization procedure, colloquially referred to as 'twirling', to the state. This is performed by averaging over all stabilizer rotations ρ → 1 2 n 2 n j=1 g j ρg j . In so doing, the optimization is restricted to stabilizer diagonal states of the form where some coefficients are determined by the measurement outcomes, while the rest are variables. Since the stabilizer operators are mutually commuting, and their spectrum is given by {−1, +1}, it is straightforward to compute the eigenvalues of ρ as λ j = 1 a stabilizer diagonal state corresponds to a mixture of graph states generated by these stabilizer operators. These facts can easily be seen in the following way:

Estimating the robustness of entanglement
In order to estimate the global robustness of entanglement, we begin by bounding it from below in the following way. For any mixed state that can be written as a convex mixture of other mixed states as ρ = k λ k ρ k , and any index m, it holds that [19] R(ρ) λ m (1 + R(ρ m )) − 1.
To see this, consider the state ρ, a real and positive number t and another state κ, which fulfill ρ + tκ ∈ SEP (up to normalization). It follows that withκ = (tκ + k>0 λ k ρ k )/(t + k>0 λ k ). Using this result and the definition of the robustness, one finds As shown in section 2, we need to consider only twirled states of the form ρ = k λ k |G k G k | in the minimization of the robustness. At this point, note that ρ is a sum of graph states with equal entanglement, since those states are equivalent up to local unitaries. In [19], it is shown that the robustness of a pure two-colorable graph state is 2 |B| − 1. Hence, the estimation of the robustness reduces to a fidelity estimation of the desired graph state. In an experiment that aims at creating the graph state |G (k 1 ,...,k n ) , it is then possible to estimate the fidelity of the experimentally created state with the target state from stabilizer measurements. The minimization of the least fidelity compatible with stabilizer measurements reads F = min ρ [tr(ρ|G (k 1 ,...,k n ) G (k 1 ,...,k n ) |) : tr(ρ K i ) = a i , ρ 0]. As a convention we use K 0 = 1 and a 0 = 1. By Lagrange duality, one finds the dual problem: . As described in [16], a solution to the primal is given by ρ = 1 2 n i c i K i 1 1 . . .K i n n , where the coefficients are given by c i = n k=1 i k a k − n k=1 i k + 1. And one can show that a solution to the dual problem is provided by choosing µ 0 = 1 − N 2 and µ i = 1 2 for i 1. Since primal and dual objective functions coincide, they must be optimal, and we attain the following analytic solution for the fidelity estimation: Combining this with equation (7) and using the fact that the robustness of a pure two-colorable graph state is 2 |B| − 1 [19] provides us with the following lower bound on the global robustness of entanglement that can be achieved from stabilizer measurements:

Estimating the relative entropy
In a similar fashion, we now calculate the lower bound on the relative entropy on entanglement in the case of stabilizer measurements. First, note that once more the optimization may be restricted to stabilizer diagonal states, resp. mixtures of graph states. Then, the lower bound on the relative entropy is given by We can prove this in the following way. First, note that the two-coloring divides the system in the two partitions A and B. Now one uses the fact that the relative entropy is lower bounded by the difference between the entropy of system A, resp. B, and the entropy of the total system [22]: In our case, we consider only mixtures of two-colorable graph states, so that tracing out system A results in a maximally mixed state with entropy |B|. Hence, the minimization of the relative entropy involves an entropy maximization. This can be achieved as outlined in [20]: measuring the generators of the stabilizer group gives rise to probability distribution p (±) k = 1±a k 2 for the projections upon the stabilizer eigenspaces. Furthermore, we denote the probability distribution of the joint state of the system by λ i 1 ...i n . A crucial feature of the entropy is subadditivity: where H denotes the classical entropy function. A little thought shows that the above inequality holds with equality for the probability distribution given by , thus giving the exact maximal entropy To conclude, the lower bound on the relative entropy of entanglement that can be inferred from stabilizer measurements is computed as

Upper bounds
In the previous paragraphs we have derived lower bounds on the minimal entanglement that is consistent with the statistics obtained from measuring individual stabilizer operators. It is also of use to derive upper bounds. A simple approach to doing this is available using the results of [19]. The total Hilbert space can be divided up into subspaces, where each subspace is labeled 6 by a deterministic outcome for all the Amber stabilizers. For any pure two-colorable graph state the entanglement E R is given by |B|, and the robustness of entanglement is given by 2 |B| − 1. In [19], it was shown that for a mixed (twirled) state supported entirely in such a subspace, E R is given by E R (ρ) = |B| − S(ρ), whereas R(ρ) = 2 |B| max λ k − 1. Let us use the symbol a to denote a possible set of outcomes for the Amber measurements, and let b denote a possible set of outcomes for the Blue measurements. Hence any state that is diagonal in the graph state basis can be described as a probability distribution p(a, b) = p(a) p(b|a) corresponding to the probabilities for getting the various possible stabilizer outcomes. We can partition such a state into a mixture of states that are individually supported on each of the Amber subspaces, such that ρ = p(a)ρ a , where a is a bit string corresponding to the positive/negative stabilizer subspaces of the Amber qubits. By concavity of the entropy function and convexity of E R , we find that E R (ρ) |B| − a p(a)S(ρ a ). But S(ρ a ) is given by a classical entropy H ( p(b|a)) where p(b|a) is the conditional probability distribution for getting outcomes b upon finding a. Thus we obtain Similarly, since the A subspace entanglement is given by R(ρ a ) = 2 |B| (λ max (ρ a ) − 1), we find Hence to get upper bounds to the minimal entanglement consistent with the measurement outcomes, we need to pick the p(a, b) consistent with the marginal distributions that minimize these expressions. The relative entropy can now be estimated by noticing that the conditional entropy is upper bounded by H ( p(b)) and choosing a product distribution p(b) = p(b 1 ) . . . p(b |B| ), for which it is well known that it maximizes H ( p(b)). Thus, we obtain We need to minimize a p(a) max b p(b|a) (21) given fixed marginals p(a), p(b). The solution is as follows. Let b * be the string of b outcomes that has maximal probability p(b). For a given value of a, we always have that This is true whatever the joint distribution p (a, b) is, provided that it is consistent with the marginals p(a), p(b). But this lower bound can be attained by simply selecting p(a, b) = p(a) p(b). Hence we have that Since we do not have complete information about p(b * ), we need to minimize it subject to the individual stabilizer statistics p(b 1 ), . . . , p(b |B| ). This problem is equivalent to minimizing the l ∞ norm of a joint probability distribution p(b) = p(b 1 , . . . , b |B| ) constrained to fixed marginals p(b 1 ), . . . , p(b |B| ), which is equivalent to the fidelity minimization (only considering the Blue stabilizers). Hence we obtain R min (ρ) + 1 2 |B| max 0, 1 2 i∈B a i − |B| + 2 . (24)

Quality and scaling of the bounds
In order to check the quality of the entanglement estimates, we consider noisy two-colorable graph states. Assuming the experiments start from a perfect graph state, which is then subjected to local dephasing for a certain time, we can take the density matrix time evolution to be governed by the following master equation: where γ is the dephasing constant. The effect of such noise on graph states has been studied in detail in [21]. Due to the dephasing, the stabilizer coefficients suffer a decay exponential in the dephasing constant. For our test we consider a linear chain of qubits subject to this noise. It can be shown that the stabilizer coefficients obey the following time evolution in this noise model: c i 1 ...i n (t) = exp(−γ t k i k ). Estimates according to the described methods for the logarithm of the global robustness of entanglement and the relative entropy of entanglement are shown in figures 1 and 2, respectively. The robustness of entanglement can be estimated up to a certain number of qubits, for which a non-zero fidelity can be inferred with the target state. This effectively sets a threshold to the estimation. For the noise model considered here, the minimal fidelity F consistent with stabilizer measurements is given by F = In contrast to the robustness of entanglement there is no limit to the lower bound, but the difference between upper and lower bounds grows with system size.
For a fixed dephasing time γ t, this condition will be violated when the number n of vortices in the graph is increased. In contrast, the relative entropy may be estimated even for larger noisy systems without suffering from the threshold problem. However, the difference between lower and upper bounds apparently grows with system size.
In many cases, stabilizer measurements are carried out via local measurements. This local information on the quantum state could in principle be used to improve the bounds on minimization of entanglement measures from incomplete information on the density matrix, since they restrict the set of compatible states involved in the optimization. Note, however, that local measurement operators generally do not commute with the stabilizer operators. This implies that symmetries cannot be exploited. For this reason we do not consider local measurement data in our scheme.

Conclusion
Here we have shown how entanglement of arbitrarily large graph states can be estimated from simple measurements. High-quality bounds on the robustness of entanglement and the relative entropy of entanglement have been derived for stabilizer measurements. The stabilizers of two-colorable graph states can be measured in two measurement settings (assuming the measurements can be performed simultaneously); thus our scheme avoids the exponential overhead required by full-state tomography. In addition, the results presented here are of an analytical form that allows for extremely efficient post-processing. In contrast, quantum state tomography requires computationally hard post-processing of the measurement data to create an estimate of the real density matrix, and schemes for entanglement estimation from incomplete measurement data usually rely on numerical methods such as convex optimization which is limited to systems no larger than 20 qubits-even if symmetries can be exploited. Our scheme should therefore be invaluable for future graph-state experiments.
Our results may also be interesting to study the effects of various noise models on the entanglement dynamics of two-colorable graph states. One step in this direction has been made in [23], where the entanglement of graph states under the influence of Pauli maps is investigated.