Efficient detection of useful long-range entanglement in imperfect cluster states

Photonic cluster states are a crucial resource for optical quantum computing. Recently a quantum dot single photon source has been demonstrated to produce strings of photons in a linear cluster state, but high photon loss rates make it impossible to characterize the entanglement generated by conventional methods. We present a benchmarking method for such sources that can be used to demonstrate useful long-range entanglement with currently available collection/detection efficiencies below 1%. Measurement of the polarization state of single photons in different bases can provide an estimate for the three-qubit correlation function. This value constrains correlations spanning more than three qubits, which in turn provide a lower bound for the localizable entanglement between any two qubits in the large state produced by the source. Finite localizable entanglement can be established by demonstrating>2/3. This result enables photonic experiments demonstrating computationally useful entanglement with currently available technology.


I. INTRODUCTION
Measurement-based quantum computation (MBQC) [1] has become a promising candidate for the most resourceefficient way to build a universal quantum computer. The greatest challenge of MBQC is the generation of a sufficiently large entangled resource state. This step is critical because only certain types of multi-qubit entanglement are known to enable universal quantum computation [2], most prominently cluster state entanglement [3,4].
Photonic systems have been proposed to generate cluster states [5,6]. Recently a complete architecture for a linear optical quantum computer has been developed that relies on the probabilistic fusion of many small entangled states into one large cluster state [7]. In this proposal, the generation of the entangled resource state requires only the generation of many maximally entangled three-photon states, at the cost of a significant overhead of single-photon detection measurements and classical information processing. This overhead could be dramatically reduced by developing deterministic "machine gun" sources of photonic cluster states [8,9].
As experimental work [10] makes progress towards realizing the cluster state source proposed in [8], the challenge of benchmarking such systems comes to the fore. The state produced will certainly be affected by errors and one needs to quantify to what extent the functionality of the state as a resource for MBQC is affected. Since any useful resource state spans many qubits, any benchmarking method with exponential scaling, such as standard quantum state tomography, is out of the question. In fact a benchmarking method for cluster state sources has been presented in [11] which requires a number of single-photon measurements that scales only linearly in the size of the state being benchmarked. We briefly review this method before stating our result.
Localizable entanglement (LE i,j (ρ)) can be taken as the figure of merit of a cluster state approximation. It is defined with respect to two qubits i, j in a state ρ of n qubits as the * nutzat@gmail.com maximum entanglement between i and j that can be obtained by means of a sequence of single-qubit measurements on the n − 2 other qubits in ρ [12]. The maximum is taken over all measurement bases and we take concurrence to be the entanglement measure. The task of benchmarking a cluster state source is accomplished by lower bounding the localizable entanglement of the state that it produces. Particular triplets of expectation values B i , i = 1, 2, 3 provide a lower bound to the localizable entanglement, as shown in Appendix A. We refer to this bound as the direct bound in the following. The operators B i are tensor products of Pauli operators and their expectation values can be measured as correlations between outcomes of single-qubit measurements in the corresponding Pauli bases. [11] proposes a simple experimental setup to measure these expectation values for photonic systems such as the one proposed in [8]. This benchmarking method, however, is limited by the fact that the time required to measure B i increases exponentially with the number of photons in the state to be benchmarked and the inverse of the collection/detection efficiency. Hence only small states can be benchmarked using the direct bound in currently feasible experiments.
The method presented here overcomes this limitation. We find that if the expectation values Z i−1 X i Z i+1 on any three neighboring qubits in a linear state ρ are no smaller than some value ZXZ , then Hence measurement of ZXZ alone can suffice to establish localizable entanglement across many qubits, as illustrated in Fig. 1.
Furthermore, the expectation values Z i−1 X i Z i+1 give information on the usefulness of a state for practical quantum information processing tasks. For instance we find that a linear state ρ with expectation values Z i−1 X i Z i+1 ≥ ZXZ enables a quantum teleportation channel across n − 2 qubits of fidelity In the following we presents proofs of the two results (1) and (2) and describe their applicability.

II. THE CLUSTER BACKBONE AND TRANSLATIONAL INVARIANCE
Cluster states can be elegantly defined in the stabilizer formalism [13]. The linear cluster state on n-qubits |Ψ CS , for instance, is the unique eigenstate with eigenvalue +1 of the n − 2 Pauli operators K i = Z i−1 X i Z i+1 , where the subscripts label any three adjacent qubits, as well as the two operators K 1 = X 1 Z 2 and K n = Z n−1 X n at the ends. This collection of n Pauli operators is called the stabilizer generator, and the group of all 2 n products of any number of operators in the stabilizer generator is the stabilizer group of the linear cluster state. Clearly the cluster state is the +1 eigenstate of all operators in the stabilizer group, which we call stabilizers. In an imperfect cluster state the expectation values of stabilizers are generally smaller than unity and there may be nonzero expectation values of non-stabilizer operators. We call the collection of expectation values of stabilizers the cluster state backbone of a state, because these values determine how similar a state is to the ideal cluster state.
We can furthermore classify stabilizers K according to the number of generators K i that need to be multiplied to obtain K. We denote a stabilizer operator that is the product of m generators as K m . Clearly there are n m stabilizers K m where 1 ≤ m ≤ n.
In the following we make the assumption of weak translational invariance (TI) of the state to be characterized. In its strongest form TI requires the reduced density matrix of any number of adjacent qubits to be the same, i.e. independent of where in the n qubit state the segment is found. We only require the weaker assumption that K i = K j ∀ i, j and denote K i = ZXZ in the following. The assumption of translational invariance is reasonable because the large environments in semiconductor quantum light sources can be assumed to induce mainly Markovian errors, which give rise to TI states. Even the non-Markovian errors modelled in [14] satisfy TI.

III. HOW DOES ZXZ CONSTRAIN THE CLUSTER BACKBONE?
While generator expectation values ZXZ = 1 constrain the entire cluster backbone to K m = 1, the values K m are restricted to finite intervals when ZXZ < 1. To determine a lower bound on these intervals we consider two commuting multi-qubit Pauli operators P 1 and P 2 . The quantity gives the joint probability of measuring both P 1 and P 2 with outcome −1 in state ρ and is therefore non-negative. An alternative argument for its non-negativity is that the product of two commuting positive-semidefinite matrices is positivesemidefinite itself. Setting the r.h.s. of (3) non-negative yields a bound that is fundamental to this work: For the case of the cluster backbone for which only the expectation values of the generators K i = ZXZ are known, it follows that

IV. THE WC STATE
Imagine a device that is capable of emitting a long string of single photons in a TI state ρ with long-range entanglement potentially similar to that of the cluster state. All that can be measured on ρ is the value of ZXZ . What is the worstcase scenario in terms of long-range entanglement of the state produced, i.e. what is the minimal LE j,j+k (ρ) consistent with the known ZXZ ?
Note that when we consider the segment of the state ρ spanning qubits j to j+k, we mean the state on qubits j to j+k that remains after qubits j −1 and j +k +1 have been measured in the Z basis with outcomes +1 (see Fig. 2). Thus the two expectation values Z j−1 X j Z j+1 and Z j+k−1 X j+k Z j+k+1 become X j Z j+1 and Z j+k−1 X j+k , respectively, which constitute the boundary stabilizers of a cluster state. This construction is necessary because a segment of a perfect cluster state only becomes a cluster state itself after the boundary qubits are "clipped off" by Z measurements. For ease of notation, we denote the state of such a n-qubit segment by ρ n in the following and label the qubits from 1 to n.
Eq. (5) yields a lower bound to any unknown stabilizer expectation value K m , and certain triplets B i , i = 1, 2, 3, of these expectation values provide a direct bound to LE 1,n (ρ n ). One might now wonder what highest direct bound can be inferred in this way and whether a physical state ρ n can saturate it. Both these questions are answered by considering a state which we call the worst-case state (WC state) on n qubits and define as where |C n is the linear cluster state on n qubits. The parameter λ is chosen such that ρ W C n is consistent with a given value of ZXZ , i.e.
The defining property of the WC state is that it saturates all inequalities given by (5): The second line is obtained by noting that K m anticommutes with m of the Z i operators, yielding a −1 for m terms in the sum on the r.h.s. of the first line. The third line follows by substituting λ from (7).
Since ρ W C n is obviously a physical state and saturates all the inequalities of (5), these constraints are in fact a tight bound on the operators in the cluster backbone. Furthermore, ρ W C n is the state with the lowest possible direct bounds on LE consistent with the given ZXZ , because any other state cannot have expectation values in the cluster backbone smaller than Tr[K m ρ W C n ]. Certain triplets of these expectation values, however, guarantee a certain value of LE, as shown in Appendix A. Hence no other state can have LE lower than the direct bound that would be measured in ρ W C n , the value of which is derived in the next section.

V. THE ZXZ BOUND
A sequence of X or Y measurements on qubits 2, 3, ..., n − 1 performed on a linear cluster state results in a maximally entangled state on qubits 1 and n. This ideal resultant twoqubit state can be written as Figure 2. Example of a stabilizer description of a measurement sequence that localizes entanglement in a cluster state. The first and last photons (grey dots) are measured out first, such that the state segment has boundary stabilizers XZ and XZ. The state ρn describes the n = 5 qubits represented by black dots. Out of the entire stabilizer group only the three operators at the bottom commute with all three measurement projectors and therefore remain in the stabilizer group of the resultant state. It is easy to verify that the surviving stabilizers are a product of three, four, and three generators, respectively (top to bottom), such that mi take the values 3, 4, 3.
For example, a sequence of Y measurements on an ideal linear cluster state with +1 outcomes leads to σ 1 i = {Y, Z, X}, σ n i = {Z, Y, X}, and t = (1, 1, 1) [11]. We can can always take t i ≥ 0 by absorbing a minus sign in σ 1 i or σ 1 i . For states of this form the concurrence is given by [15] C(τ ) = max{0, Given a multi-qubit state ρ, the coefficients t i can be found as the expectation values of the three cluster state stabilizers that commute with every single-qubit measurement (see Fig.  2 for an example). In a faulty cluster state t i will take absolute values smaller than one, and measurement of these three expectation values yields the direct bound (see Appendix A) [11]. However, as mentioned in the introduction, the operators with expectation values t i have large support for long strings of photons produced and are therefore hard to measure.
In ρ W C n every expectation value takes the minimum value allowed by a given ZXZ , and therefore C(τ ) is minimized for every sequence of measurements and outcomes on qubits 2, 3, · · · , n − 1. In the following we derive the expression for this bound on LE of ρ W C n . Each t i is given by an expectation value K mi in ρ W C n and by (5) can be written as where m i gives the number of stabilizer generators contained in the operator K mi . Using (10) we obtain The number m 1 + m 2 + m 3 depends only on the number of single-qubit measurements in the X or Y basis and not on the sequence itself. For instance, a single X measurement leads to operators containing one, two, and three ZXZs, while a Y measurement will give three operators each containing two generators. Every further X or Y measurement appends another generator ZXZ to two of the three stabilizers that commute with the measurement sequence, so that where m is the number of X or Y measurements and n the number of qubits in ρ W C n . Hence the LE across n − 2 qubits in a state ρ W C n ( ZXZ ) is lower bounded by We refer to this value of concurrence as the ZXZ bound in the following. (1) follows by considering a segment ρ n of ρ as defined above and relabeling qubit 1, n as qubit j, j + k, respectively.

Localizable teleportation channel fidelity
While localizable entanglement certainly is necessary for MBQC, a state of a given LE does not necessarily enable any quantum computation. This discrepancy between entanglement and actual usefulness of a state for quantum information processing tasks has led to the introduction of the fully entangled fraction F as practical measure of quantum information processing significance [16]. The fully entangled fraction (FEF) is defined as the maximum fidelity of a given state ρ with a maximally entangled state |φ + that can be achieved by a local unitary: A two-qubit state ρ T shared between Alice and Bob enables Alice to teleport a quantum state to Bob with a fidelity of as shown in [17]. Remarkably, entanglement as measured by concurrence is a necessary but not sufficient condition for a nonzero FEF. There is, for instance, a state of both concurrence and FEF equal to 0.5. Many copies of this state can be used to distill a number of Bell pairs, yet a single copy offers no advantage over any separable pure state for quantum teleportation (F |ψ sep = 0.5) [17]. The triplet sum T := t 1 + t 2 + t 3 not only lower bounds the concurrence of the state ρ ti with the expectation values t i , but also its FEF: Hence the state ρ W C n with expectation values ZXZ enables a quantum teleportation channel between the first and nth qubit of fidelity Clearly there is no state that is consistent with K i = ZXZ of lower triplet sum T than ρ W C n . Hence (18) is a tight lower bound on the teleportation fidelity. This statement is stronger than the ZXZ bound, because it guarantees that the localizable entanglement present can be used for a single-qubit channel.

Is the ZXZ bound tight?
The ZXZ bound rules out the existence of TI states consistent with ZXZ of lower LE than the value given in (14). The question remains whether this bound is tight, i.e. whether there exists a state which has LE as low as its ZXZ bound. We believe that the ZXX bound is tight because we conjecture that Appendix B provides strong evidence in support of this conjecture.

A. The direct bound
Measurement of ZXZ can be accomplished with a simpler experimental setup and fewer photon counts that the direct bound. However the value of LE demonstrated by measuring the direct bound on a source is higher than the LE guaranteed by the ZXZ bound for this source, except for the unrealistic scenario where the source produces a WC state. Hence there is a trade-off between saving measurement resources and demonstrating a high value of LE.
An interesting scenario to investigate this trade-off is a cluster state source as proposed in [8] subject to uncorrelated Pauli Y -errors on the emitter spin between single-photon emissions. These errors are likely to be the dominant error mechanism in quantum dot implementations of the proposal. Both the direct and the ZXZ bound can provide two different figures of merit for such a source. Firstly the number of qubits across which the LE is non-zero can be considered; secondly the value of the LE across a fixed number of qubits constitutes a figure of interest. As shown in Fig. 3, the ZXZ bound is considerably below the direct bound for both quantities. The ZXZ bound approaches the direct bound as ZXZ → 1 (see inset in Fig. 3).
The ZXZ bound is therefore particularly useful for sources with low emission/detection efficiencies and little decoherence of the emitter spin. In that case no direct bound is available but ZXZ is close to unity and hence high values of LE across a fixed number of qubits as well as non-zero LE across many qubits can be demonstrated. Figure 3. The ZXZ bound is generally lower than the direct bound. In the main plot the maximum number of qubits across which the direct bound (blue) and the ZXZ bound (red) guarantee nonzero LE is shown. The x-axis gives the probability of a Pauli Y -error on the emitter spin (fundamental error) before each single-photon emission. The inset shows the value of LE that each bound establishes across three qubits, again plotted against the probability of a fundamental Y -error. The highlighted data points in the main plot correspond to the points in the inset plot where each curve hits zero.

B. Efficient tomography
The idea of characterizing a cluster state approximation by its stabilizer generator expectation values K i is not new. In fact a bound on the fidelity of a n-qubit state ρ n where Tr[K i ρ n ] = K i with the n-qubit cluster state |C n has been derived in [18]: For the case of ρ n satisfying weak TI this bound becomes This fidelity bound, however, does not directly give information about the long-range entanglement in a state with ZXZ approaching one. Clearly two states with the same fidelity with a third, entangled state can themselves have very different entanglement.
Interestingly, the worst-case state ρ W C n saturates the fidelity bound (21) and thereby sheds new light onto this result, which was obtained by a seemingly unrelated method. The fidelity of a state ρ n with |C n is given by where B i are the operators in the cluster backbone, k i = 1 2 (1 + Tr[B i |C n C n |]), and B 0 = 1. Using (8) for Tr[K m ρ W C n ] we find that ρ W C n saturates the fidelity bound of (21): where we have used the binomial theorem.
The WC state is therefore not only the worst-case scenario for LE, but also the worst-case scenario for fidelity given a value of ZXZ . Moreover since all expectation values in ρ W C n take on the minimal absolute values allowed by ZXZ , every state that saturates (21) must have the same cluster backbone as ρ W C n .

VII. CONCLUSION
Our result for the worst-case localizable entanglement in a cluster state approximation substantiates the idea of [18] that an expectation value ZXZ provides a meaningful benchmark for an experimentally produced cluster state. We have shown that as ZXZ approaches unity, useful entanglement increases at a rate lower-bounded by (14).
Measurement of an expectation value ZXZ can be accomplished with the simple setup presented in [11] and emission/detection efficiencies that seem within reach for the experimental implementations of [8], such as the originally proposed quantum dot system or charged NV-centres [18]. The threshold for establishing a cluster state source capable of producing long-range entanglement is therefore lowered to demonstrating ZXZ = 2 3 , which would demonstrate nonzero localizable entanglement across a single qubit.
We gratefully acknowledge fruitful discussions with David Jennings and Ilai Schwarz and the support of the United States Army Research Office and the EPSRC.
In [11] a bound on the LE was derived given three expectation values and assuming zero entanglement fluctuations [12] ("outcome-independent entanglement"). In the following we show that this bound holds even in the presence of entanglement fluctuations. We only deal with qubits and projective measurements here.
Theorem. The expectation values of three multi-qubit Pauli operators B 1 , B 2 , B 3 lower bound the LE as measured by a convex entanglement measure E between two target qubits t 1 , t 2 in any state ρ of n qubits as LE t1,t2 (ρ) ≥ E(ρ B ). (A1) Here ρ B = 1 4 (1⊗1+ B 1 Z ⊗Y + B 2 Y ⊗Z + B 3 X ⊗X) and {B i } need to satisfy the following: • There exists a collection of n − 2 single-qubit Pauli operators {P k }, each acting on one of the n qubits except the target qubits t 1 , t 2 , such that [B l , P k ] = 0 ∀ l, k. The label k runs from 0 to n − 2, while the target qubits are labelled t 1 , t 2 .
• All three operators B i have nontrivial support on both qubits t 1 and t 2 , i.e the component of of B i acting on the t 1 , t 2 subspace is some two-qubit Pauli operator B t i . Proof. The LE between qubits t 1 and t 2 in a n-qubit state ρ is defined as [12] Here {m} denotes all possible sequences of local measurements on the n − 2 qubits other than t 1 , t 2 , binary string s gives the outcomes of such a measurement sequence, and p s the probability of this outcome. ρ t1,t2 m,s is the state of qubits t 1 , t 2 after a particular measurement sequence m with outcomes s.
The state of qubits t 1 , t 2 obtained from ρ by measurement sequence {P k } with outcomes s is written ρ t1,t2 {P k },s . The average entanglement obtained by {P k } is and clearly lower bounds LE t1,t2 (ρ). Furthermore where ρ s : (additional terms cannot decrease entanglement [11]). The coefficients t i are the expectation values of the corresponding two-qubit Pauli operators B i in the state ρ t1,t2 {P k },s . We now associate strings of n − 2 bits with operators Q q on the n − 2 qubits other than t 1 , t 2 where a 1 (0) at position k in q means that the component of Q q acting on qubit k is P k (1 k ). For instance we have Q 01101 = 1 1 ⊗ P 2 ⊗ P 3 ⊗ 1 4 ⊗ P 5 .
With this notation the coefficients t i in ρ s are related to expectation values in the n qubit state ρ (write Ô = Tr[Ôρ]) as The sum runs over all 2 n−2 bit strings q of length n − 2. s · q denotes the modular sum of all those measurement outcomes s i where q i = 1. One of the terms in the sum of (A5) corresponds to the known expectation value B i = B t i ⊗ Q q B i . We can now exploit the convexity of our entanglement measure E and invariance of entanglement under local unitary operations U s to write The local unitary U s can be chosen as a tensor product of two Pauli operators such that it anticommutes with two of the three We then associate with every choice of U s a triplet of binary numbers . Note that f 1 (s) and f 2 (s) can be chosen freely, while f 3 (s) = f 1 (s) ⊕ f 2 (s). The particular choice of {U s } and associated f i (s) made here is where q Bi is the binary string giving the sequence of 1 and P k operators in B i .
Finally the expectation value of B t i for the state ρ mix := s p s U s ρ s U † s can be written using (A5) as Hence the functions f (s, 1) and f (s, 2) can be chosen such that ρ mix = 1 Without loss of generality any triplet of two-qubit Pauli operators satisfying the properties required in the theorem can be chosen as  [19]. By choosing O 1 and O 4 that achieve a signed singular value decomposition of T , we perform local unitary operations on ρ 1,4 that correspond to the transformation where S = sin θ 2 sin θ 3 and N ± = 1 ± √ 1 − S 2 . The above density matrix is manifestly in the form of an Xstate [20]. The concurrence of such a state is a simple function of the density matrix elements, which for us gives C(ρ 1,4 ) = max{0, 1 2 (3λ − 1)S + 1 2 (λ − 1)}.
The entanglement that is localized by measurements in the X-Z plane on ρ W 4 is therefore clearly maximized by the equatorial measurement sequence, which gives sin θ 2 = sin θ 3 = 1 and hence S = 1.

7
We have not found an analytic form for LE 1,n (ρ W C n ) when n > 4. To further investigate whether our conjecture LE 1,n (ρ W C n ) = 2λ − 1 holds we therefore perform a numerical optimization.
Using a Nelder-Mead simplex algorithm the following optimization was carried out: where N gives the normalization factor and | θ, ϕ = ⊗ 6 i=2 sin The results for six different values of λ are shown in Fig.  4. We find that the optimal measurement angles are θ i = π 2 and that ϕ i is arbitrary. This yields a value of concurrence C(ρ 1,7 ) = 2λ−1, again providing evidence for the conjecture LE 1,n (ρ W C n ) = 2λ − 1.