Abstract
Along with the vast progress in experimental quantum technologies there is an increasing demand for the quantification of entanglement between three or more quantum systems. Theory still does not provide adequate tools for this purpose. The objective is, besides the quest for exact results, to develop operational methods that allow for efficient entanglement quantification. Here we put forward an analytical approach that serves both these goals. We provide a simple procedure to quantify GreenbergerHorneZeilinger–type multipartite entanglement in arbitrary threequbit states. For two qubits this method is equivalent to Wootters' seminal result for the concurrence. It establishes a close link between entanglement quantification and entanglement detection by witnesses, and can be generalised both to higher dimensions and to more than three parties.
Introduction
It is a fundamental strength of physics as a science that most of its basic concepts have quantifiability built into their definition. Just think of, e.g., length, time, or electrical current. Their quantifiability allows to measure and compare them in different contexts, and to build mathematical theories with them^{1}. There is no doubt that entanglement is a key concept in quantum theory, but it seems to resist in a wondrous way that universal principle of quantification. The reason for this is, in the first place, that entanglement comes in many different disguises related to its resource character, i.e., what one would like to do with it. In principle, there are numerous taskspecific entanglement measures^{2,3}. However, most of them cannot be calculated easily (nor measured or estimated) for generic mixed quantum states, and therefore it is difficult to use them.
There are notable exceptions, the concurrence^{4} and the negativity for bipartite systems^{5}. These measures have already provided deep insight into the nature of entanglement, but they also have their shortcomings. The concurrence is strictly applicable only to twoqubit systems while for the negativities it is not known how to distinguish entanglement classes. The generalisations of the concurrence (such as the residual tangle^{6}) do quantify taskspecific entanglement even for multipartite systems but again it is not known how to estimate them for general mixed quantum states.
There is another difficulty. An Nqubit density matrix ρ is characterised mathematically by 2^{2N} − 1 real parameters. Reducing it to its socalled normal form^{7}—which contains the essential entanglement information—removes 6N parameters. The entanglement measure is determined by the remaining exponentially many parameters which need to be processed to calculate the precise value. Even an operational method similar to that of WoottersUhlmann^{8,9} would quickly reach its limits with increasing N. Therefore it is desirable to develop methods which provide useful approximate answers even for larger systems. If one asks for mere entanglement detection, witnesses^{10} are such a tool because here the number of required parameters (both for measurement and processing) can be reduced substantially. There are also estimates of entanglement measures using witness operators^{11,12} which, however, have not yet produced practical methods for entanglement quantification.
Here we develop an easytohandle quantitative witness for GreenbergerHorneZeilinger (GHZ) entanglement^{13} in arbitrary threequbit states. It yields the exact threetangle for the family of GHZsymmetric states^{14}, and those states which are locally equivalent to them. For all other states, the method gives an optimised lower bound to the threetangle. Due to this feature we call the approach a witness.
We start by defining the GHZ symmetry^{14} and stating our central result. Then we prove the validity of the statement for two qubits. We obtain a method equivalent to that of WoottersUhlmann, i.e., it gives the exact concurrence for arbitrary density matrices. Subsequently we explain the extension of the approach to arbitrary threequbit states.
Results
The procedure
The Nqubit GHZ state in the computational basis is defined as . It is invariant under: (i) Qubit permutation. (ii) Simultaneous spin flips i.e., application of . (iii) Correlated local z rotations: where σ_{x}, σ_{y}, σ_{z} are Pauli matrices. An Nqubit state is called GHZ symmetric and denoted by ρ^{S} if it remains invariant under the operations (i)–(iii). An arbitrary Nqubit state ρ can be symmetrized by the operation where the integral denotes averaging over the GHZ symmetry group including permutations and spin flips. Notably, the GHZsymmetric Nqubit states form a convex subset of the space of all Nqubit states.
Observation:
If an appropriate entanglement measure µ is known exactly for GHZsymmetric Nqubit states ρ^{S}, it can be employed to quantify GHZtype entanglement in arbitrary Nqubit states ρ. Here, µ(ψ) is a positive invariant function of homogeneous degree 2 in the coefficients of a pure quantum state ψ, and µ(ρ) is its convexroof extension^{15}. The estimate for µ(ρ) is found in the following sequence of steps:
(1) Given a state ρ, derive a normal form ρ^{NF}(ρ), i.e., apply local filtering operations so that all local density matrices are proportional to the identity^{7} (see Section Methods). If ρ^{NF}(ρ) = 0 the procedure terminates here, and µ(ρ) = 0.
(2) Renormalise ρ^{NF}/tr ρ^{NF} and transform it using local unitaries to obtain the state according to appropriate criteria (see below) so that the entanglement of ρ^{S}(ρ^{NF}/tr ρ^{NF}) is enhanced.
(3) Project the state onto the GHZsymmetric states . The estimate for µ(ρ) is obtained after renormalisation
Two qubits
For two qubits the entanglement measure under consideration is the concurrence C(ρ) (Refs. 4,8). From the symmetrization ρ^{S}(ρ) of an arbitrary twoqubit state ρ we find (for details see Supplementary Information): In the symmetrization entanglement may be lost, as illustrated by the state for which inequality (3) gives the poor estimate C(Ψ^{−}) ≥ 0. Therefore, the optimisation steps (1) and (2) are necessary to avoid unwanted entanglement loss in the symmetrization (3). The goal is to augment the righthand side of inequality (3) up to the point that equality is reached. We will show now that for two qubits this can indeed be achieved.
It is fundamental that the maximum of an invariant function µ(ρ) under general local operations can be reached by applying the optimal transformation where A = A_{1} A_{N} and is an invertible local operation^{7}. Consider first the normal form ρ^{NF}(ρ) which is obtained from ρ by iterating determinantone local operations^{7} (see also Methods). Such operations (represented by matrices) describe stochastic local operations and classical communication (SLOCC). Consequently, the normal form is locally equivalent to the original state ρ, that is, it lies in the entanglement class of ρ. Note that the iteration leading to the normal form minimises the trace of the state. Subsequent renormalisation increases the absolute values of all matrix elements in equation (3). Here, the correct rescaling of the mixedstate entanglement measure is crucial. This is why homogeneity degree 2 of µ(ψ) is required^{16,17}.
Hence, transforming ρ to its normal form increases the moduli of ρ_{00,00}, ρ_{00,11}, ρ_{11,00}, ρ_{11,11} (and also the concurrence) as much as possible for a state that is SLOCC equivalent with ρ. The sum of the offdiagonal matrix elements in equation (3) reaches its maximum if ρ_{00,11} is real and positive. As this can be achieved by a z rotation on one qubit we may consider it part of finding the normal form and drop the absolute value bars in equation (3). Then, the sum of matrix elements equals, up to a factor 1/2, the fidelity of ρ^{NF}/tr ρ^{NF} with the Bell state . The question is how large this fidelity may become.
To find the answer we transform ρ^{NF}/tr ρ^{NF} to a Belldiagonal form using local unitaries (this is always possible^{7,18,19}). If then we apply another operation to maximise (see Supplementary Information). The result is a Belldiagonal with maximum real offdiagonal element (please note that denotes a normalised state, whereas ρ^{NF} is not normalised). However, Belldiagonal twoqubit density matrices with this property can be made GHZ symmetric without losing entanglement^{4} (see also Supplementary Information).
Hence, our optimised symmetrization procedure (1)–(3) leads to the exact concurrence for arbitrary twoqubit states ρ. In passing, we have demonstrated that the concurrence is related via C(ρ) = max(0, 2f − 1) · tr ρ^{NF} to the maximum fidelity that can be achieved by applying invertible local operations to ρ.
Three qubits
For three qubits, the GHZsymmetric states are described by two parameters^{14} and therefore form a twodimensional submanifold in the space of all threequbit density matrices. It turns out that it has the shape of a flat isosceles triangle, see Fig. 1. A convenient parametrisation is as it makes the HilbertSchmidt metric in the space of density matrices conincide with the Euclidean metric. This way geometrical intuition can be applied to understand the properties of this set of states. All entanglementrelated properties of GHZsymmetric states are symmetric under sign change x −x as this is achieved by applying σ_{z} to one of the qubits.
The GHZclass entanglement of threequbit states is quantified by the threetangle τ_{3} (Refs. 6,17, see also Methods). For GHZsymmetric threequbit states ρ^{S}(x_{0}, y_{0}) the exact solution for the threetangle^{20} (see also Methods) is where x_{0} ≥ 0 and (, ) are the coordinates of the intersection of the GHZ/W line with the direction that contains both GHZ_{+} and ρ^{S}(x_{0}, y_{0}) (cf. Fig. 1). The grey surfaces in Fig. 2 illustrate this solution.
Now we turn to constructing a quantitative witness for the threetangle of arbitrary threequbit states by using the solution in equation (6). As before, the main idea is that an arbitrary state can be symmetrized according to equation (2) and thus is projected into the GHZsymmetric states. Again, we assume ρ_{000,111} real and nonnegative, so that x(ρ) ≥ 0. From Figs. 1 and 2 it appears evident that the entanglement of the symmetrization image ρ^{S}(ρ) can be improved by moving its point (x(ρ), y(ρ)) closer to GHZ_{+}. More precisely, the entanglement measure is enhanced upon increasing one of the coordinates without decreasing the other (cf. equations (3) and (6)).
In this spirit, finding the normal form in step (1) is appropriate as it yields the largest possible threetangle for a state ρ^{NF}/tr ρ^{NF} locally equivalent to the original ρ (cf. Ref. 7). As the normal form is unique only up to local unitaries it does not automatically give the state with minimum entanglement loss in the symmetrization. Therefore, the unitary optimisation step (2) is required to generate the best coordinates.
In the symmetrization the information contained in various matrix elements is lost. For two qubits, however, the concurrence of the optimised Belldiagonal states depends only on and the loss of in the symmetrization does not harm. In contrast, the threequbit normal form depends on 45 parameters. We may not expect that τ_{3}(ρ) depends only on two of them and, hence, entanglement loss in the symmetrization (3) is inevitable (cf. Supplementary Information). Consequently, steps (1)–(3) lead to a lower bound for the threetangle that coincides with the exact τ_{3}(ρ) at least for those states which are locally equivalent to a GHZsymmetric state. The most straightforward optimisation criterion in step (2) is to maximise . Alternative criteria which generally do not give the best τ_{3}(ρ) but can be handled more easily (possibly analytically) are maximum fidelity , minimum HilbertSchmidt distance of from GHZ_{+}, or maximum Re .
Discussion
Evidently this approach can be generalised. Therefore we conclude with a discussion of some of its universal features. The essential ingredients are an exact solution of the entanglement measure for a sufficiently general family of states with suitable symmetry, and the entanglement optimisation for a given arbitrary state ρ via general local operations. The former determines the border where the entanglement vanishes. The latter ensures an appropriate fidelity of the image ρ^{S}(ρ) with the maximally entangled state. This reveals a remarkable relation between entanglement quantification through SL(2, ) invariants and the standard entanglement witnesses which we briefly explain in the following.
A wellknown witness for twoqubit entanglement is . It detects the entanglement of an arbitrary normalised twoqubit state ρ^{2qb} if On the other hand, from our concurrence result we see, by dropping the optimisation over SLOCC A = A_{1} A_{2}, that is a (nonoptimised) quantitative witness for twoqubit entanglement. In other words, yields one of the many possible lower bounds to the exact result. Analogously it is straightforward to establish the relation between the standard GHZ witness and the nonoptimal quantitative witness . The latter represents a linear lower bound to the threetangle obtained via the optimisation steps (1)–(3) (see Supplementary Information).
Finally we mention that our approach can be used without optimisation, i.e., either without step (1), or (2), or both. This renders the witness less reliable but more efficient. At best it requires only four matrix elements (for any N). We note that, if we apply the witness to a tomography outcome the measurement effort can be reduced by using the prior knowledge of the state and choosing the local measurement directions such that the fidelity with the expected GHZ state is measured directly. This implements optimisation step (2) right in the measurement.
Methods
Normal form of an Nqubit state
The normal form of a multipartite quantum state is a fundamental concept that was introduced by Verstraete et al.^{7}. It applies to arbitrary (finitedimensional) multiqudit states. Here we focus on Nqubit states only.
In the normal form of an Nqubit state ρ, all local density matrices are proportional to the identity. Therefore the normal form is unique up to local unitaries. Remarkably, the normal form can be obtained by applying an appropriate local filtering operation where . Therefore ρ^{NF} is locally equivalent to the original state ρ. The normal form ρ^{NF} is peculiar since it has the minimal norm of all states in the orbit of ρ generated by local filtering operations. Practically, the normal form can be found by a simple iteration procedure described in Ref. 7. It is worth noticing that GHZsymmetric states – which play a central role in our discussion – are naturally given in their normal form.
Threetangle of threequbit GHZsymmetric states
The purestate entanglement monotone that needs to be considered for threequbit states is the threetangle τ_{3}(ψ), i.e., the square root of the residual tangle introduced by Coffman et al.^{6}: Here ψ_{jkl} with are the components of a pure threequbit state in the computational basis. The threetangle becomes an entanglement measure also for mixed states via the convexroof extension^{15} i.e., the minimum average threetangle taken over all possible purestate decompositions {p_{j}, ψ_{j}}. In general it is difficult to carry out the minimisation procedure in equation (9), but there exist various approaches for special families of states^{16,17,20,21,22,23}. For GHZsymmetric threequbit states, the convex roof of the threetangle can be calculated exactly (see equation (6)). This solution is shown in Fig. 2 and can be understood as follows. The border between the W and the GHZ states is the GHZ/W line which has the parametrised form^{14} with −1 ≤ v ≤ 1. The solution for the convex roof is obtained by connecting each point of the GHZ/W line (x^{W}, y^{W}, τ_{3} = 0) with the closest of the points (, , τ_{3} = 1). That is, the threetangle is nothing but a linear interpolation between the points of the border between GHZ and W states, and the maximally entangled states GHZ_{±}.
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Acknowledgements
This work was funded by the German Research Foundation within SPP 1386 (C.E.), and by Basque Government grant IT47210 (J.S.). The authors thank R. Fazio, P. Hyllus, K.F. Renk, and A. Uhlmann for comments, and J. Fabian and K. Richter for their support.
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Affiliations
Institut für Theoretische Physik, Universität Regensburg, D93040 Regensburg, Germany
 Christopher Eltschka
Departamento de Química Física, Universidad del País Vasco UPV/EHU, E48080 Bilbao, Spain
 Jens Siewert
IKERBASQUE, Basque Foundation for Science, E48011 Bilbao, Spain
 Jens Siewert
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The authors contributed equally to this work.
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Correspondence to Christopher Eltschka.
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