Multipartite entanglement in spin chains

We investigate the presence of multipartite entanglement in macroscopic spin chains. We discuss the Heisenberg and the XY model and derive bounds on the internal energy for systems without multipartite entanglement. Based on this we show that in thermal equilibrium the above mentioned spin systems contain genuine multipartite entanglement, even at finite modest temperatures.


I. INTRODUCTION
The study of entanglement in condensed matter systems has become a major line of research in quantum information science. In these studies, different aspects of entanglement have been investigated. First, by investigating the entanglement in the reduced state of two qubits in a spin chain, it has been shown that entanglement theory can help to understand the physical properties of systems undergoing quantum phase transitions [1]. A slightly different approach studies the entanglement between two qubits which can be generated by local operations on the remaining qubits, the so-called localizable entanglement [2]. A third field of study investigates the entanglement between a block and the remaining qubits in the ground state. The entanglement can be quantified by the entropy of the reduced state and then the question arises, whether the entanglement scales like the surface of the block [3]. Furthermore, the insights from entanglement theory made it possible to understand the limits of known simulation techniques (like the density matrix renormalization group) in condensed matter physics as well as the design of new techniques which are superior to the ones known before [4]. Finally, it has been shown that macroscopic properties of solids can be related to entanglement properties of microscopic degrees of freedom, allowing for a detection of entanglement in macroscopic objects by observing macroscopic observables [5,6,7].
In all of the research lines specified above, entanglement was studied as a bipartite phenomenon; the investigation of multipartite entanglement in spin chains has only a limited literature [8,9]. In entanglement theory, however, multipartite entanglement has been intensively studied and it is known that quantum correlations in the multipartite setting have a much richer structure than in the bipartite setting [10,11]. The main results known are for three or four particles, results for large ensembles of particles are still rare. Nevertheless, studying multipartite entanglement in large systems can already benefit from the results for small systems in two respects. First, on a more local scale, one can pick up sub-ensembles of three or four particles and investigate the multipartite entanglement of the corresponding reduced state. Second, on a more global scale, one can ask what types of multipartite entanglement are necessary to form a given state of the total system. In this paper we investigate multipartite entanglement in macroscopic spin systems. The tool we use is similar to the one introduced in Ref. [5]: We derive bounds for the internal energy U = H which have to hold for states without multipartite entanglement, violation of these bounds implies the presence of multipartite entanglement in the system. We investigate in detail two one-dimensional spin models: the Heisenberg model and the isotropic XY model. Our results lead to the insight that for these models even at modest temperatures the correlations cannot be explained without assuming the presence of genuine multipartite entanglement. However, we would like to stress that in our approach we do not assume that the spin system is in the ground state or in a thermal state, our theorems can also be applied to states out of equilibrium.
Our paper is organized as follows: In Section II we introduce the basic definitions of multipartite entanglement we want to apply to spin systems. We first recall the notion n-separability and genuine multipartite entanglement. Then, we also introduce the notion of k-producibility, which is well suited for the investigation of multipartite entanglement in macroscopic systems. In Section III we apply these terms to the antiferromagnetic Heisenberg chain. We calculate energy bounds below which either reduced states are genuine multipartite entangled or the total state requires multipartite entanglement for its creation. This shows that already at temperatures of kT ≈ J the effect of multipartite entanglement cannot be neglected. In Section IV we demonstrate that our proofs also work for other models by calculation similar thresholds for the XY model. Finally, we summarize our results and name some open problems.

II. NOTIONS OF MULTIPARTITE ENTANGLEMENT
Let us start by clarifying the terms we will use to classify multipartite entanglement in spin chains. For a pure state |ψ of a quantum systems of N parties we may ask for a given n ≤ N , whether it is possible to cluster the parties into n groups, such that |ψ is a product state with respect to this partition, If this is the case, we call the state |ψ n-separable. A state which is N -separable is a product state with respect to all subsystems, these we call fully separable. If a state is not 2-separable, we call it genuine N -partite entangled. For a mixed state described by a density matrix ̺ these terms can be extended through convex combination. If we can write ̺ = i p i |ψ i ψ i | with p i ≥ 0, i p i = 1 and n-separable |ψ i we call ̺ nseparable. Physically, this means that a production of ̺ requires only n-separable pure states and mixing [12].
Besides asking for n-separability, we may also ask questions like "Do two party entangled states suffice to create the mixed state ̺?" This leads to the following definition. We call a state |ψ producible by k-party entanglement (or k-producible, for short) if we can write where the |φ i are states of maximally k parties. So, in this definition m ≥ N/k has to hold. This definition means that it suffices to generate specific k-party entanglement to arrive at the state |ψ . Conversely, we call a state containing genuine k-party entanglement if it is not producible by (k − 1)-party entanglement. This definition can be extended to mixed states as before via convex combinations. Again, a mixed state which is k-producible requires only the generation of k-party entangled states and mixing for its production. Consequently, a mixed state ̺ contains k-party entanglement, iff the correlations cannot be explained by assuming the presence of (k − 1)-party entanglement only. Obviously, there are some relations between the notions of k-producibility and n-separability. For instance, the states containing N -party entanglement are just the genuine multipartite entangled states and the 1producible states are the fully separable states. Furthermore, a state of which some reduced state of m parties is genuine m-partite entangled, contains m-partite entanglement, while the converse is in general not true [13].
In the thermodynamic limit N ≫ 1, however, the notion of n-separability for the total system becomes problematic. This is mainly due to two reasons. First, it does not take into account how many particles are entangled. A biseparable state can be a product state for two partitions of equal size or just one qubit might be separated from a large genuine multipartite entangled state; these cases are not distinguished. Second, statements about n-separability require the exact knowledge of N, -a precondition which is usually not fulfilled in realistic situations. In stark contrast, the notion of k-producibility is designed to be sensitive to the question "How many qubits are entangled?" Also, deciding whether a state is k-producible does not require the exact knowledge of N. If one controls only k ≪ N subsystems, one may still conclude that the total state contains k-party entanglement, e.g. if the reduced state is genuine k-party entangled.

III. THE HEISENBERG MODEL
Let us start the discussion of spin models with the anti-ferromagnetic isotropic Heisenberg model. Here we assume a one-dimensional chain of N spin 1/2 particles with periodic boundary conditions. The Hamiltonian of this model is given by where σ (i) x denotes the Pauli matrix σ x , acting on the ith qubit and J > 0 the coupling between the spins. For simplicity, we assume that N is even. For completeness, we always mention also known results for fully separable states [5,8]. We have the following theorems: Theorem 1. Let ̺ be an N qubit state of a system described by the Heisenberg Hamiltonian in Eq. (3). If there exists two neighboring qubits i, i + 1 in the chain such that the reduced state ̺ i,i+1 is a two-qubit entangled state. Furthermore, if then there exist three neighboring qubits i, i + 1, i + 2 such that the reduced state ̺ i,i+1,i+2 of these qubits is genuine tripartite entangled.
Proof. It suffices to prove these bounds for pure states. To prove first Eq.
The idea is to view the W ijk as entanglement witnesses, detecting genuine tripartite entanglement on the qubits i, j, k. Let us denote etc. and consider a biseparable state |ψ = |φ i |φ jk . We have | W ijk | = |x i ·x j +y i · y j +z i ·z j +x j x k +y j y k +z j z k | ≤ x 2 j + y 2 j + z 2 j +|x j x k + y j y k +z j z k |. For the two-qubit state |φ jk the first term is the purity of the reduced state and invariant under local unitaries, while the second is maximal, when the 3 × 3 correlation matrix λ µν = µ j ν k for µ, ν = x, y, z is diagonal. Expressing a general state in this form (see Eq. (14) in Ref. [14]) leads to | W ijk | ≤ 1 + √ 5. Since there are N/2 of the W ijk , Eq. (5) implies that one of the reduced three-qubit states cannot be biseparable. Finally, Eq. (4) can be proved similarly, using |x i · x j + y i · y j + z i · z j | ≤ 1 for separable two-qubit states [5,8]. Theorem 2. Let ̺ be an N qubit state as in Theorem 1. If ̺ is 1-producible, then holds, while for 2-producible states holds. Thus, if H H < C C3 the state contains genuine tripartite entanglement.
Proof. The bound Eq. (6) follows from Eq. (4) and has already been derived in [5,8]. Let us first consider a two-producible pure state, where neighboring spins are allowed to be entangled. I.e., we have a state of the type |ψ = |φ 12 |φ 34 ...|φ N −1,N , where the state |φ 12 is a state of the qubits 1 and 2, etc. Let us define for the state |ψ two vectors with 6N real components each, via Since the v j have 6N entries, in v 1 (and similarly for v 2 ) some coefficients (like x 1 ) may appear twice, namely iff N ∈ 4 . It is straightforward to see that for the given state 2 · H H = J v 1 · v 2 holds. Now, we need χ = x 2 i +y 2 i +z 2 i +(x i x i+1 ) 2 +(y i y i+1 ) 2 +(z i z i+1 ) 2 + x 2 i+1 + y 2 i+1 + z 2 i+1 ≤ χ max = 3, which holds for any twoqubit state ̺, due to (1 + χ)/4 ≤ T r(̺ 2 ) ≤ 1. Based on that, we have v 1 2 ≤ (N/2)(χ max + 3) = 3N. The same bound holds for v 2 2 . So, due to the Cauchy-Schwarz inequality we have | H H | ≤ 1/2J v 1 · v 2 ≤ 3/2JN which proves the claim. Now we have to consider arbitrary two-producible states. A general two-producible pure state is always a tensor product of two-qubit states |φ ij of the qubits i and j and single qubit states |φ k . If for one of the twoqubit states |φ ij the qubits i and j are not neighboring, we can replace it by two one-qubit reduced states ̺ i ⊗ ̺ j , since in this case, the Hamiltonian is only sensitive to the reduced states. Furthermore, if two neighboring single qubit states |φ i ⊗ |φ j , appear, we replace them by |φ ij , since |φ ij is allowed to be separable.
Thus it suffices to prove the bound for a state where entanglement is only present between neighboring qubits and the single qubit states are isolated in the sense that if |φ k appears, then |φ k−1 and |φ k+1 do not appear, e.g. |ψ = |φ 12 |φ 3 |φ 45 |φ 67 |φ 8 . Let M be the (even) number of isolated qubits. We can associate to any isolated qubit one neighboring two-qubit state (on the left or right) to form a three-party group. There is an ambiguity in doing that, and we can choose the three-qubit groups in such a way that the number of two-qubit groups between the three-qubit groups is always even. Then, the state can be viewed as a sequence g 1 , g 2 , g 3 , g 4 ... of M three-party groups and (N −3M )/2 two party groups, all in all there are (N −M )/2 groups. We can double this sequence by setting g (N −M)/2+k = g k etc. In the example this grouping can be (12), (345), (678), (12), (345), (678). Now we define the vectors v 1 and v 2 as follows: v 1 collects terms from g 1 , g 3 , g 5 ... while v 2 consists of terms from g 2 , g 4 , ... In detail, we have where [g 1 ] denotes the three terms x i x j , y i y j , z i z j if g 1 is a two-qubit group and corresponds to the four terms x i x j , y i y j , z i z j , (x j x k + y j y k + z j z k ) when g 1 is a threequbit group with the isolated qubit k. [g 1 |g 2 ] etc. denotes the coupling terms between the groupings g 1 and g 2 , in the Hamiltonian, e.g. x i , y i , z i for v 1 and x i+1 , y i+1 , z i+1 for v 2 or vice versa. The symbol [½] denotes a sequence of three or four times "1", depending on whether in the other vector there is a two-or a three-qubit group.
Again, we have 2 · H = J v 1 · v 2 and it remains to bound v i 2 . First, note that by construction v 1 and v 2 contain the same number of two and three-qubit groups, namely (N −3M )/2 two-qubit groups and M three-qubit groups each. Also, both contain 3(N − 3M )/2 + 4M times the "1". Then, note that for states of the type |ψ = |φ ij |φ k the bound x 2 i +y 2 i +z 2 i +(x i x j ) 2 +(y i y j ) 2 + (z i z j ) 2 +(x j ·x k +y j ·y k +z j ·z k ) 2 +x 2 k +y 2 k +z 2 k ≤ 5 is valid. So we have v i 2 ≤ (4 + 5)M + (3 + 3)(N − 3M )/2 = 3N, which proves the claim.
To start the discussion, first note that the bounds in Eqs. (6,7) are sharp. Equation (7) is saturated for the singlet chain |ψ = |ψ − |ψ − ...|ψ − where |ψ − = (|01 − |10 )/ √ 2. In general, any energy bound H H ≥ C X corresponds to a certain temperature T X . Below this temperature the state has with certainty some degree of entanglement. Corresponding to the Eqs. (4,5,6,7) there are thus the temperatures T R2 , T R3 , T C2 and T C3 below which either reduced states of two or three parties are entangled or the total state contains two-or three-party entanglement. Obviously, T R2 = T C2 > T C3 > T R3 has to hold here.
Let us estimate these temperatures. For small N one can solve the Heisenberg model by diagonalizing H H numerically [15]. Then, the threshold temperatures can directly be computed. Results are shown in Fig. 1. As expected, the values for T C2 = T R2 coincide with the ones of Ref. [8]. The given values for T C3 and T R3 show that in the Heisenberg chain of ten spins at kT ≈ J multipartite entanglement plays a role, namely at least one reduced state is genuine tripartite entangled and the total state contains tripartite entanglement.
In the thermodynamic limit N ≫ 1 the ground state energy of the Heisenberg model is known to be E 0 /N = −J(4 ln 2 − 1) ≈ −1.773J [16]. Thus three qubits can be found such that their reduced state is genuine tripartite entangled. One can infer from numerical calculations [15,17] that the threshold temperatures are determined by kT C3 ≈ 1.61J and kT R3 ≈ 1.23J in agreement with the values of the ten-qubit Heisenberg chain. Finally, note that our results also shed light on the characterization of the ground state itself. For instance, they show that the ground state cannot be a GHZ state since for that state the reduced three-qubit states are separable. Finally, note that when a state is translationally invariant, Eq. (5) guarantees that all reduced three-qubit states are genuine tripartite entangled.

IV. THE XY MODEL
Let us now investigate the isotropic XY model. The Hamiltonian of this model is given by Again, we assume periodic boundary conditions and an even number of spins. For this model we have: Theorem 3. Let ̺ be an N qubit state whose dynamics is governed by the Hamiltonian in Eq. (8). If ̺ is one-producible, then holds. If H XY < −JN this implies that there are two neighboring qubits such that their reduced state is entangled. For two-producible states holds. If H XY < −9/8JN the state contains thus tripartite entanglement and if then there are three neighboring qubits such that their reduced state is genuine tripartite entangled.
for all odd i. This corresponds to a chain of non-maximally entangled two-qubit states. This also motivates the definition of α and β i,j in the proof of Eq. (10). α and β i,j were chosen such that for the state |ψ we have v 1 = v 2 , thus the Cauchy-Schwarz inequality is sharp.
In the thermodynamic limit the XY model is exactly solvable [18]. The ground state energy is E 0 /N = −4J/π ≈ −1.273J. Thus the ground state contains tripartite entanglement and fulfills the condition of Eq. (11). Furthermore, the results of Ref. [18] imply that if kT < 0.977J the mixed state contains tripartite entanglement and if kT < 0.668J some reduced state is genuine tripartite entangled.

V. CONCLUSION
In conclusion, we showed for two important spin models that the internal energy can be used as a signature for the presence of multipartite entanglement in these models. Based on this, we computed threshold temperatures below which any realistic description of the system cannot neglect the effects of multipartite entanglement. Our results may stimulate the research on entanglement in phase transitions, since they suggests the use of multipartite entanglement measures as a tool for the investigation of phase transitions in these regimes [19].
A natural continuation of the present work is the extension of the Theorems presented here to other spin systems, e.g. higher dimensional or frustrated systems. Furthermore, it is very tempting to analyze the results of performed experiments whether they can be interpreted as giving evidence for multipartite entanglement, similarly as it has been done for bipartite entanglement in Ref. [20]. Here, we leave this as an open problem.