Entanglement in pure and thermal cluster states

We present a closest separable state to cluster states. We start by considering linear cluster chains and extend our method to cluster states that can be used as a universal resource in quantum computation. We reproduce known results for pure cluster states and show how our method can be used in quantifying entanglement in noisy cluster states. Operational meaning is given to our method that clearly demonstrates how these closest separable states can be constructed from two-qubit clusters in the case of pure states. We also discuss the issue of finding the critical temperature at which the cluster state becomes only classically correlated and the importance of this temperature to our method.


Introduction
Entanglement plays an important role in modern physics and is the subject of intense research both for its implications in our fundamental understanding of nature [1] as well as its practical usefulness in quantum information processing [2] and [3]. Different models of quantum computing have been devised to harness this usefulness. The most prominent ones include the circuit model [4], topological computing [5] and measurement-based computing [6].
Measurement-based quantum computation (or one-way quantum computation) differs from the circuit model by using an initially prepared highly entangled resource state. The algorithm proceeds as a set of local measurements on this state. The basis and order of these measurements characterize the algorithm and outcomes of previous measurements are fed forward to make the computation deterministic.
The general resource for this computation model are graph states [7]. In this paper we will concentrate on particular graph states, namely cluster states [8]. Unlike graph states, cluster states have regular square structure. One of the advantages of one-way quantum computation is that all the entanglement used in the algorithm is present right at the beginning. The subsequent measurements only consume it. This makes it easier to identify the role of entanglement in the information processing compared to the traditional circuit model. Recently it has also been shown both theoretically and experimentally in [9] that the quantum computational power of cluster states can be extended by replacing some projective measurements with generalized quantum measurements. This makes cluster state quantum computation a very attractive and promising prospect when it comes to constructing a quantum computer.
Due to its clear role as a resource in one-way quantum computation it is highly desirable to know the entanglement properties of cluster states. Perhaps one of the most important problems in studying the properties of quantum states, and resource states for quantum computation in particular, is the quantification of entanglement itself. The entanglement scaling of pure d-dimensional graph states has been shown by Markham et al to be N/2 [10]. However any realistic quantum computer will operate at finite temperatures which makes it necessary to be able to characterize entanglement of mixed states. Quantifying entanglement is a notoriously difficult task [11] and the exact scaling of entanglement in thermal cluster states is still unknown though some advances have been made by investigating the localization of entanglement in noisy cluster states [12].
There are many entanglement measures that quantify the scaling of entanglement in quantum states. In this work we use relative entropy of entanglement (see [17] and [18]) to quantify the scaling. In general there are not many multipartite quantum states for which the relative entropy of entnaglement can be calculated analytically. D-dimensional symmetric states are one of the few examples [13], [14]. It is unknown how to find the closest separable state to a general entangled state and the problem of closed form of relative entropy still remains an open question. Though some progress has been made in this direction ( [15] and [16]) it is impossible to do this even for a general two-qubit state.
We present a method of constructing the closest separable state to a thermal cluster and show how it can reproduce the results of [10] for pure cluster states. We also show how bound entanglement complicates the procedure of finding the closest separable state for noisy cluster states.

Closest separable state for pure cluster states
The entanglement of cluster states is known to scale linearly with the size of the system as N 2 . This result was first obtained by Markham et al in [10]. The authors used a technique where they derived upper and lower bounds on the entanglement and then they showed that these two bounds are equal.
In this section we verify this result using geometrical ideas. To do this we employ the relative entropy of entanglement ( [17], [18]) E(σ) defined as where σ is the entangled state. The minimization is taken over the set D of all separable states ρ and S(σ||ρ) = Tr[σ log σ − σ log ρ] is the relative entropy. For pure states this simplifies to S(σ||ρ) = −Tr[σ log ρ]. Since we are considering only qubits we take the logarithm to base 2. State that minimizes the relative entropy in (1) will be designated by ρ * and will be called the closest separable state. In order to prove that this state ρ * achieves the minimum of relative entropy we employ the same approach as [19]. Consider a small region around the closest separable state (1 − x)ρ * + xτ for a small parameter x.
All we have to show now is that for a general separable state τ the gradient of relative entropy is non-negative in this region. Due to convexity of the set of separable states this will also mean that the minimum of relative entropy is global. A cluster state can be thought of as a state whose qubits are first prepared in the +1 eigenstate of σ x and then control-phase gates CZ are applied between nearest neighbours. For example the two-qubit cluster state is |ψ 2 = CZ|+ |+ = 1 2 [|00 + |01 + |10 − |11 ]. It can be easily seen that the closest separable state ρ * 2 is by calculating that E(σ 2 ||ρ * 2 ) = 1 where σ 2 = |ψ 2 ψ 2 |. So far all the states have been expressed in the computational basis {|0 , |1 }. It is however more useful to write the states in what we call the "mixed" basis {|0 , |1 , |+ , |− }. So the state vector and the closest separable state for the case of 2 qubits now become Similarly for 4 qubits the expressions take the following form From the form of ρ * 2 and ρ * 4 we can see that the closest separable state is obtainable from the state vector by writing the density matrix in the "mixed" basis and then keeping only the diagonal terms. This is why it is more instructive to work in the "mixed" basis, one can immediately see the form of the closest separable state from the state vector. Of course in the computational basis not all off-diagonal terms are zero.
Important thing to notice is the number of terms in the state vector of a cluster state is different in the computational basis and in the "mixed" basis. In the computational basis all the coefficients are non-zero, they are either 1 or −1. So for N-qubit cluster state there are 2 N basis coefficients. On the other hand in the "mixed" basis the number of coefficients drops to 2 N/2 . So when it comes to normalization coefficients for the density matrices we have From the above it is clear to see that the closest separable matrix ρ * N is of rank 2 N/2 . Another useful fact that will be used later is that when the matrices are expressed in the "mixed" basis it is easy to see that they commute with the cluster state [ρ * N , σ N ] = 0. Before we start the proof for the closest separable state a small note about our notation is in place. The proof contains places where in stead of just writing σ N or ρ * N it is more instructive to write out the state explicitly, for instance for 2 qubits . This is of course not possible in the case of general N. Therefore we use the following notation.

Pure vector
..) From the normalization factors it is clear that the states are in the "mixed" basis. This also implies that each round bracket contains 2 N/2 terms.
As mentioned above, to prove that states of the form ρ * 2 and ρ * 4 are really the closest separable states we need to show that the gradient of relative entropy in a small region around ρ * N is non-negative Substituting f (x, τ ) = S(σ||(1 − x)ρ * N + xτ ) and realizing that for a positive operator A we have the expression log A = ∞ 0 Starting with the first integral in Eq.
Evaluating the integral in Eq. (9): Substituting this back into Eq. (9): Now turning our attention to the second integral in Eq. (8). Here we will use the fact that σ and ρ * N commute with each other.
So finally the gradient of the relative entropy in Eq.(8) can be written as In order for inequality (13) to be satisfied the trace needs to scale at most as 2 −N/2 . This is in fact the case as can be proved by induction. First we will consider showing that the maximum of the trace is 1 and therefore the gradient of relative entropy is non-negative. Here a and b are parameters from τ representing the different amplitudes for the two subsystems. So τ = |αβ αβ| where |α = a|0 + 1 − |a| 2 |1 and similarly |β = b|0 + 1 − |b| 2 |1 . τ = |αβ αβ|. The inequality has been verified numerically for 2, 4 and 6 qubits. In the case of 2 qubits graphical solution has been obtained as well in Fig.(1).
Assuming that the above holds true for the case of N = k, namely Tr[σ k τ k ] ≤ 2 −k/2 . Now we need to prove using the above assumption that the case N = k + 2 holds true Starting with the trace where we have used the fact that |ψ k+2 = CZ|ψ k |ψ 2 and τ k+2 = τ k ⊗ τ 2 . The controlphase gate acts on qubits k and k + 1. Also the separable state τ does not commute with the controlled phase gate so their commutator is non-zero [τ k ⊗ τ 2 , CZ] = A so we can substitute τ k ⊗ τ 2 CZ = A + CZτ k ⊗ τ 2 into Eq. (16).
Taking a closer look at the product CZ † A and using the above commutation relation we get CZ † A = CZ † τ k ⊗ τ 2 CZ − τ k ⊗ τ 2 . Taking the trace of both sides Using the assumption for N = k this implies Therefore (∂f /∂x)(0, |αβ . . . αβ . . . |) ≥ 0. Since any separable state can be written in Therefore ρ * N is the closest separable state to σ N . Now we can compute the relative entropy between the cluster state σ N and ρ * N .
This concludes the proof that our closest separable state gives linear scaling of entanglement and reproduces the results of [10]. We can look at the form of the closest separable state from a more operational point of view. Imagine we are asked to prepare the closest separable state to an Nqubit one-dimensional cluster state. The only ingredients needed to construct this state are the 2-qubit closest separable state ρ * 2 and control-phase gates CZ. The state ρ * N can be written where the unitary applied has the form All the unitary (24) implements are controlled phase gates CZ between qubits 2-3 and 4-5 and so on. It connects the states ρ * 2 together in a chain as shown in Fig.(2). It is important to notice that although control-phase gates are entangling operations, they leave the state ρ * N in a separable form. This can easily be seen if we consider two separable 2-qubit states (2) joined by a control-phase operation. The total 4-qubit state can be written as |1011 1011|+|1100 1100|] where we have applied Hadamard operators on every second qubit I ⊗ H ⊗ I ⊗ H. The transformed state ρ * 4 is clearly separable. Since identical analysis can be applied to ρ * N we conclude that control-phase operations in the unitary (24) do not introduce any entanglement and ρ * N remains separable. It is important to note that the form of the closest separable state is the same for cluster states in higher dimensions as well. The majority of the proof for closest separable states is independent of the dimension of our cluster state. The dimensionality becomes important only in Eq. (16) where we assume one-dimensional cluster state by using |ψ k+2 = CZ|ψ k |ψ 2 where the control-phase gate is applied between qubits k and k + 1. For higher dimensions the number of control-phase gates in Eq.(16) increases due to larger number of neighbouring qubits. However the logic of the proof remains unchanged. Therefore it is possible to find the closest separable state to a pure cluster state of any dimensionality. This is crucial because unlike linear cluster chains, two and higher dimensional clusters are universal resources for quantum computation [6].

Thermal entanglement
Study of thermal entanglement is of great significance since any realistic scheme of implementing a quantum computer will operate at finite temperatures. In this section we look at entanglement in thermal cluster states. Specifically we are interested to see at what temperature the cluster state becomes separable. These considerations will lead us to a more general form for the closest separable state.
Consider the cluster Hamiltonian where J is the coupling constant and N is the neighbourhood of site j. The terms in the sum above are just a particular case of stabilizer operators [6]. Hence Hamiltonian (25) is also referred to as the stabilizer Hamiltonian. The ground state of this Hamiltonian is the cluster state. Excited states are achieved by local σ z j flips. The ground state has energy −NJ and is non-degenerate. The k th excited state has energy J(−N + 2k) with degeneracy N ! k!(N −k)! [20]. All the states in the spectrum of the Hamiltonian are equally entangled since they are all some σ z away from the ground state. Therefore our method of finding the closest separable state can also be used on these excited states.
The ground state of this Hamiltonian is highly entangled and the entanglement scales linearly [10] with the size of the cluster state as N 2 . An interesting question to ask is whether this entanglement persists at finite temperatures and where is the critical point beyond which the thermal entangled state becomes separable. Thermal cluster state has the following form [12] where K j is the stabilizer operator at site j. For simplicity we will now consider a two-qubit thermal cluster state of the form where ω = tanh(βJ). So in the low temperature limit as T → 0 we recover the pure state because ω → 1. As the temperature increases ω → 0 and for T → ∞ we obtain a maximally mixed state. To calculate the critical temperature at which the state becomes separable we use the Peres-Horodecki criterion [21] and [22]. To see when the state fails to be a positive definite we have to solve the following equation ω 2 c + 2ω c − 1 = 0. This equation has two solutions ω c = −1 ± √ 2 but we will disregard the negative solution because it is not physical. Substituting this back into ω = tanh(βJ) finally gives us the critical temperature The entanglement for T < T c happens to be of useful distillable nature [23]. Therefore by using local operations and classical communication on multiple copies of the thermal cluster pure entanglement can be distilled. This critical temperature for distillable entanglement remains unchanged for higher dimensions and any system size N as proved in [20]. For temperatures above the critical temperature, T ≥ T c , the N-qubit cluster state, where N > 2, is not fully separable but is bound entangled [24]. The critical temperature at which all entanglement vanishes is non-trivial to find due to the fact that properties of bound entanglement are still not fully understood. We have already calculated the closest separable state for a 2-qubit cluster. Now we will show that this state is not unique. In fact there are infinitely many states that satisfy condition (7). Since there is no bound entanglement for the case of two qubits we know that the thermal cluster state at the critical temperature T c is fully separable. It is natural to ask the question what the distance is between this state and a pure cluster state in terms of the relative entropy. So we want to calculate S(σ 2 ||σ 2 (ω c )) where σ 2 (ω c ) = 1 4 (I + ω c σ x ⊗ σ z )(I + ω c σ z ⊗ σ x ) and ω c = √ 2 − 1. It can be quickly checked that S(σ||σ 2 (ω c )) = 1 and the gradient of relative entropy in a small region around σ 2 (ω c ) is ∂f ∂x (0, τ ) = 1 − 2Tr[στ ] > 0 as can be seen from Eq. (14). Therefore the 2-qubit thermal cluster at critical temperature is another good closest separable state to the pure cluster state.
It is instructive to look at the thermal cluster state σ 2 (ω) in the "mixed" basis and compare it to the pure cluster closest separable state ρ * 2 . Transforming σ 2 (ω) we have From the form of the matrix it can be seen that the only way that σ 2 (ω) → ρ * 2 is when all the terms apart from 1 + ω vanish. So we arrive at the following four conditions that have to be satisfied simultaneously; ω(1 + ω) = 1 − ω = ω(ω − 1) = 0 and 1 + ω = 2. It is straightforward to see that the only solution is ω = 1 which corresponds to the pure cluster state case. Therefore as the temperature increases the thermal cluster state does not approach the pure closest separable state ρ * 2 . This last result can be seen also from the fact that the relative entropy between these two states is non-zero, ie. S(ρ * 2 ||σ 2 (ω c )) = 0. The relative entropy between two states S(A||B) vanishes if and only if A = B [26].

Thermal cluster states
Now we are in a position to calculate the entanglement of thermal cluster states. We will use a similar approach as above to show that the form of the closest separable state is the same as in Eq.(30). However this time the parameter λ that determines the mixture of our two closest separable states from Eq.(30) will not be completely free and its lower bound will depend on the temperature. We will show that as the temperature increases towards the critical temperature T → T c the parameter λ increases towards unity. We already have everything we need to compute the scaling of entanglement with temperature of a thermal 2-qubit cluster state given by Eq. (27). The only thing that remains to determine is a suitable candidate for the closest separable state. The most obvious choice would be σ 2 (ω c ) = 1 In fact this gives the correct behavior of entanglement E(σ 2 (ω)) = Tr[σ 2 (ω) log σ 2 (ω) − σ 2 (ω) log σ 2 (ω c )] as can be seen in Fig.(3). The gradient of the relative entropy in the neighbourhood of the state σ 2 (ω c ) is non-negative ∂f ∂x (0, τ ) ≥ 0 as expected. Now that we know that for 2 qubits the cluster state at critical temperature is a closest separable state we can ask whether there exists a more general form of the state just like in Sec. (3). It turns out that this is in fact possible, albeit with some needed modifications. We have seen that as the temperature increases the thermal cluster state does not approach the state ρ * N from Sec. (2). This can be also seen from the nonvanishing distance between the separable states σ N (ω c ) and ρ * N , ie S(σ N (ω c )||ρ * N ) = 0. As the temperature increases the distance between the cluster state σ N (ω) and σ N (ω c ) approaches zero as expected. However this is not true for the distance between σ N (ω) and ρ * N due to the fact that S(σ N (ω c )||ρ * N ) = 0. This problem can be overcome by not allowing the parameter λ from Eq.(30) to take any value from the interval [0, 1]. Rather we require that the lower bound of the possible values of λ increases with temperature. We call this bound λ * and require λ * = λ * (ω).
As before we will first demonstrate the general principle at work on a two-qubit cluster state σ 2 (ω). First we need to find the new parameter λ * by minimising the distance between the thermal cluster state σ 2 (ω) and the state (30) ρ 2 for a constant temperature. Solving ∂ ∂λ S(σ 2 (ω)||ρ 2 ) = 0 for λ gives the new λ * We can straight away confirm that the separable state given by this new parameter is also a closest separable state by calculating the gradient of relative entropy at this state ∂f ∂x (0, τ ) ≥ 0. Therefore we can construct a new general closest separable state where this time λ ∈ [λ * 2 , 1]. Relative positions of all the above states in Hilbert space are illustrated in Fig.(4). One thing that the figure does not capture is the fact that the distances S(σ 2 ||ρ * 2 ) and S(σ 2 ||σ 2 (ω c )) are the same. This is because not all properties of Hilbert space can be pictured on a two-dimensional drawing.
This method of finding closest separable states can be extended to general N-qubit cluster states. The crucial part is to find the critical temperature at which the cluster becomes fully separable. Due to presence of bound entanglement for N ≥ 3 this is a non-trivial task and at present it is unclear how it can be achieved. [24] calculates upper and lower bounds on the value of the critical temperature for cluster states of different dimensions. However all that this changes is that the separable state σ(ω c ) would be shifted to the left of the state σ 2 (ω c ) on the E-D boundary in Fig.(4) since the critical temperature for N ≥ 3 is higher for than the one calculated here.

Conclusion
We have demonstrated a systematic way of constructing closest separable state to Nqubit cluster states. Our method reproduces known results for pure states and also allows us to quantify entanglement for mixed cluster states.
For pure states the method relies on writing the density matrix of the state in the "mixed" basis and setting any off-diagonal elements to zero. Operationally we can construct this state from N/2 copies of 2-qubit closest separable state and joining them together by applying control-phase gates as demonstrated in Sec. (2). Our method applies to cluster states of all dimensions. This is particularly useful since cluster states of dimension d ≥ 2 are universal resource state for quantum computation.
It also turns out that state ρ * N is not the only closest separable state. Thermal cluster state at critical temperature σ N (ω c ) is also another closest separable state to pure cluster state. Therefore any convex mixture of these two states also minimizes the relative entropy. However at the moment it is difficult to determine the precise temperature at which the cluster becomes fully separable for cases of more than 2 qubits. This is due to the presence of bound entanglement.
Mixed states require an even more careful approach. When quantifying entanglement in thermal cluster states a convex mixture of ρ * N and σ N (ω c ) does not minimize the relative entropy anymore. Parameter λ that determines the mixture of these two states does not take any value from interval [0, 1]. Instead as the temperature increases the lower bound of this interval increases as well. So the range of values the parameter λ can take is constricted to λ ∈ [λ * , 1]. The new lower bound λ * is a function of temperature and as temperature approaches the critical value λ * → 1. In other words both the thermal cluster σ ω and the closest separable state ρ approach the same state.