Genuine multipartite entanglement measures based on multi-party teleportation capability

Quantifying entanglement is vital to understand entanglement as a resource in quantum information processing, and many entanglement measures have been suggested for this purpose. When mathematically defining an entanglement measure, we should consider the distinguishability between entangled and separable states, the invariance under local transformation, the monotonicity under local operations and classical communication, and the convexity. These are reasonable requirements but may be insufficient, in particular when taking into account the usefulness of quantum states in multi-party quantum information processing. Therefore, if we want to investigate multipartite entanglement as a resource, then it can be necessary to consider the usefulness of quantum states in multi-party quantum information processing when we define a multipartite entanglement measure. In this paper, we define new multipartite entanglement measures for three-qubit systems based on the three-party teleportation capability, and show that these entanglement measures satisfy the requirements for being genuine multipartite entanglement measures. We also generalize our entanglement measures for N-qubit systems, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 4$$\end{document}N≥4, and discuss that these quantities may be good candidates to measure genuine multipartite entanglement.


Introduction
Entanglement is a crucial resource in quantum computing and quantum information tasks that cannot be explained classically.Typical two-party quantum applications of entanglement as a resource include quantum teleportation 1 and quantum key distribution 2 , which are performed on bipartite entangled states.However, there also exist multi-party quantum applications, such as multi-party quantum teleportation 3,4 , conference key agreement 5 , and quantum secret sharing 6 , where multipartite entanglement is considered as a resource.In particular, genuine multipartite entanglement (GME) is a significant concept for multipartite systems, since it plays an essential role in quantum communication 3,7 and quantum cryptography 5,6,8 .GME is also a critical resource in measurement-based quantum computing 9 , quantum-enhanced measurements 10 , quantum phase transitions 11,12 , and quantum spin chains 13 .Therefore, in order to make use of GME as a resource, its quantification is necessary.
Entanglement measures are mathematical tools to quantify entanglement.For bipartite systems, concurrence is one of the well-known entanglement measures [14][15][16] .It can distinguish between entangled and separable states and does not increase under local operations and classical communication (LOCC), which are important requirements for quantifying entanglement.For multipartite systems, it is more complicated to investigate entanglement, in particular GME.Even for three-qubit pure states, we should consider three different bipartition scenarios, A|BC, B|AC, and C|AB.In addition, there are two inequivalent classes, the Greenberger-Horne-Zeilinger (GHZ) class and the W class, differentiated by stochastic LOCC 17 .A straightforward approach to define entanglement measures for quantifying GME is to deal with bipartite entanglement measures for all bipartitions.For instance, the minimum and the geometric mean of concurrences for all bipartitions 18,19 satisfy the conditions for being a GME measure 18,20 ; the distinguishability between genuinely multipartite entangled states and biseparable states, the invariance under local transformations, the monotonicity under LOCC, and the convexity.The concurrence fill 21 , which is the square root of the area of the three-qubit concurrence triangle, was also proposed as a GME measure, but it has recently been shown that this measure does not satisfy the monotonicity under LOCC 22 .
We now ask whether a GME measure can compare the usefulness of any pure states in some specific multipartite quantum information processing.This question is natural when we use GME as a resource.For example, a monotonic relationship exists between bipartite entanglement and teleportation fidelity for pure states 23,24 .Indeed, such a concept of the proper GME measure has been discussed 21 , which makes the GHZ state rank as more entangled than the W state.This concept stems from the fact that the GHZ state can be more useful than the W state in three-qubit teleportation 25 .However, teleportation capabilities for other arbitrary pure states have not been taken into account.In fact, the minimum and the geometric mean of concurrences for all bipartitions are proper GME measures, but it is not difficult to find quantum states for which these GME measures and the three-qubit teleportation capability 26 give different orders.
In order to appropriately utilize GME as a resource, we need a GME measure that can compare the usefulness of quantum states in a given quantum information processing.In this paper, we first take account of three-qubit teleportation, and propose novel GME measures for three-qubit systems based on three-qubit teleportation capability.To this end, we consider the maximal average teleportation fidelity of resulting states on the other parties obtained after a measurement by one of the parties, and prove that our measures based on the fidelity can be used to observe separability on three-qubit systems and does not increase on average under LOCC.By comparing our GME measures with other GME measures, we show that there are quantum states such that their usefulness in three-qubit teleportation cannot be explained by the other GME measures, while it can naturally be done from ours.We also show that our GME measures can be defined by using only two of the possible three fidelities, unlike the minimum and the geometric mean of concurrences should consider concurrences for all bipartitions.In other words, we can make GME measures that have a simpler form.
This paper is organized as follows.We first introduce the maximal average teleportation fidelity obtained after one of the parties measures his/her system, and look into its properties.After defining entanglement measures based on the three-qubit teleportation capability, we prove that they fulfill the conditions for the GME measures.We give examples to show that our newly defined GME measures are more appropriate than the other GME measures to compare the capability of threequbit teleportation.We finally generalize our entanglement measures to N-qubit systems, and discuss that these N-partite entanglement measures have the potential to be GME measures by showing that GME is related to N-qubit teleportation capability when N ≥ 4.

Three-qubit teleportation capability and its properties
Three-qubit teleportation we consider proceeds as follows.Suppose that three parties, Alice, Bob, and Charlie, share a threequbit state.After one performs an orthogonal measurement on his/her system, the rest carry out the standard teleportation 1 over the resulting state with the measurement outcome.For instance, if the initial state is then having one of them measures his/her system in the X basis {|0 x ⟩ , |1 x ⟩}, where |t x ⟩ = 1 √ 2 (|0⟩ + (−1) t |1⟩) for t = 0 or 1, makes it possible for them to perfectly accomplish three-qubit teleportation since it can be written as Let us first look at the maximal fidelity of two-qubit teleportation.For a given teleportation scheme Λ ρ AB over a two-qubit state ρ AB , the fidelity of two-qubit teleportation is defined as 23 It has been proven that when Λ ρ AB represents the standard teleportation scheme over ρ AB to attain the maximal fidelity, the following equation holds 27,28 : where f is the fully entangled fraction 14 , which is given by where the maximum is over all maximally entangled states |e⟩ of two qubits.The given state ρ AB is said to be useful for teleportation if and only if F(Λ ρ AB ) > 2/3 23,24,29 .We now consider three-qubit teleportation capability.Let F i j be the maximal average fidelity of teleportation over the resulting state in the systems i and j after measuring the system k, where i, j, and k are distinct systems in {A, B,C}.Since three-qubit teleportation consists of a one-qubit measurement and two-qubit teleportation, it is straightforwardly obtained that 26

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where Here, U k is a unitary operator, that is, {U k † |0⟩ ,U k † |1⟩} describes a one-qubit orthogonal measurement on the system k, and ρ U k ,t i j is the resulting state with the outcome t.We say that a given state ρ ABC is useful for three-qubit teleportation if and only if min{F AB (ρ ABC ) , F BC (ρ ABC ) , F CA (ρ ABC )} > 2/3.
Before showing the properties of the maximal average teleportation fidelity F i j , let us first consider the two-qubit maximal teleportation fidelity F. Hereafter, we denote |ψ⟩ S 1 S 2 ∈ SEP(S 1 : S 2 ) when |ψ⟩ S 1 S 2 is separable between the systems S 1 and S 2 for simplicity.By the Schmidt decomposition, any two-qubit pure state |φ ⟩ AB can be written in the form a(1 − a).Note that the concurrence 15,16 for a pure state |ψ⟩ S 1 S 2 is defined as where ρ S 1 is the reduced density operator of |ψ⟩ S 1 S 2 , so we have From this equation, we can see that |φ ⟩ AB ∈ SEP(A : B) if and only if F Λ |φ ⟩ AB = 2 3 , and for two-qubit pure states, the more entangled state with respect to the concurrence, the higher the maximal teleportation fidelity F.Moreover, since the concurrence satisfies the monotonicity under LOCC on pure states, so does the maximal teleportation fidelity F.
We now show that the maximal average teleportation fidelity F i j on three-qubit pure states has similar properties.For three-qubit pure states, it has been shown that the following equation holds 26 : where τ is the three-tangle 30 and C i j is the concurrence for the reduced density operator ρ i j of |φ ⟩ ABC .We note that the three-tangle τ satisfies for any distinct i, j, and k, where C i( jk) denotes the concurrence between i and the other system jk.For mixed states, the concurrence is defined by means of the convex roof extension.In particular, for two-qubit systems, the concurrence of mixed state can be computed by 16 where the λ l s are eigenvalues of the matrix ρ in decreasing order, σ y is the Pauli Y operator, and ρ * is the conjugate of ρ.Hence, it is not difficult to calculate F i j for a given three-qubit pure state.
By using Eq. ( 8), we obtain the following lemmas, which are important properties when we define our GME measures.
Lemma 1 Let |φ ⟩ ABC be a three-qubit pure state.Then for any distinct i, j, and k in {A, B,C}, F i j (|φ Lemma 2 For three-qubit pure states, the maximal average teleportation fidelities F AB , F BC , and F CA does not increase on average under LOCC. We remark that Lemma 1 and Lemma 2 are not directly derived from the properties of the three-tangle τ and the concurrence C i j although we use Eq. ( 8) to prove them.If |φ ⟩ ABC ∈ SEP(i : jk) for some i, j, and k, then τ(|φ ⟩ ABC ) = 0, but the converse is not true.For example, τ(|W ⟩ ABC ) = 0, where ∈ SEP(i : jk) for any distinct i, j and k.In addition, the concurrence C i j can increase under LOCC on three-qubit states.Indeed, C i j (|GHZ⟩) = 0, but after measuring the system k in the X basis, C i j of the resulting state becomes 1.

GME measures based on three-qubit teleportation capability
Here, k < N and { j 1 , ..., j k | j k+1 , ..., j N } is a bipartition of the whole system.An N-partite mixed state ρ is called biseparable if it can be written as a convex sum of biseparable pure states ρ = ∑ i p i |ψ i ⟩ ⟨ψ i |, where the biseparable pure states {|ψ i ⟩} can be biseparable with respect to different bipartitions.If an N-partite state is not biseparable, then it is a genuinely N-partite entangled state.
Note that minimal conditions for being a good entanglement measure have been suggested as follows 18,20 : (i) E(ρ) > 0 if and only if ρ is a nonbiseparable state.
(ii) E is invariant under local unitary transformations.
(iii) E is not increasing on average under LOCC.That is, if we have states {ρ k } with probabilities {p k } after applying a LOCC transformation to ρ, then (iv) E is convex.
If a multipartite entanglement measure satisfies these conditions, then we call it a GME measure.Our approach is to define a multipartite entanglement measure on pure states and generalize it to mixed states through the convex roof extension where the minimum is over all possible decompositions ρ = ∑ l p l |ψ l ⟩ ⟨ψ l |.This approach has the advantage that it suffices to define an entanglement measure on pure states that satisfies the conditions (i), (ii), and (iii) in order to construct a GME measure.
Let us now define entanglement measures based on the three-qubit teleportation fidelity.
Definition 3 For a three-qubit pure state |φ ⟩ ABC , let where F i j is the maximal average teleportation fidelity in Eq. ( 8).We define multipartite entanglement measures T min and T GM as respectively, on three-qubit pure states.For three-qubit mixed states, we generalize them via the convex roof extension.
The reason why we use T i j instead of F i j itself is to set the values of T min and T GM between 0 and 1.It directly follows from Lemma 1 that T min and T GM satisfy the condition (i).We know that they are invariant under local transformations, which is the condition (ii), from the definition of F i j .From Lemma 2, we can also prove that they fulfill the condition (iii).The condition (iv) is guaranteed by the convex roof extension.Therefore, we have the following theorem.
Theorem 4 Entanglement measures T min and T GM are GME measures.
In Lemma 1, we also showed that for a three-qubit pure state |φ ⟩ ABC , if 3 for any distinct i, j, and k in {A, B,C}.Therefore, only two quantities T i j and T ik are enough to define a GME measure.Definition 5 For any distinct i, j, and k in {A, B,C}, we define multipartite entanglement measures T on three-qubit pure states.For three-qubit mixed states, we generalize them through the convex roof extension.
We have the following theorem by applying the same proof method in Theorem 4.
Theorem 6 Entanglement measures T  GM for the state |ψ(r)⟩ of the form in Eq. ( 16).For any two of these measures, we can choose r for which these measures have different values.Note that for the state |φ (t)⟩ of the form in Eq ( 14), these measures have the same value 2t , and these values vary from 0 to 1. Hence, we can easily find states in which the values of the measures have different orders.For instance, let t ′ be a value such that C GM (φ (t ′ )) = T GM (φ (t ′ )) = 0.8.Then C GM (ψ(0.7))> C GM (φ (t ′ )), but T GM (ψ(0.7))< T GM (φ (t ′ )).This means that C GM and T GM rank the states ψ(0.7) and φ (t ′ ) differently.
We can interpret entanglement measures T (i) min and T (i) GM as the minimum and the average teleportation capability of the system i, respectively.Remark that if one defines an entanglement measure with concurrence in this way, then it cannot be a GME measure.For example, let us think of the biseparable state The following examples show that our GME measures are more suitable to capture the usefulness of a given state for threequbit teleportation.We note that GME measures C min and C GM in References 18,19 are given by , respectively, on three-qubit pure states.
GM , C min , and C GM for some states, and show that they give the opposite order for the states, which means that these GME measures are distinct from one another.For Then all GME measures return the same value 2t √ 1 − t 2 on the state |φ (t)⟩ ABC .Note that h(t) ≡ 2t √ 1 − t 2 is a continuous function of t, which has the minimum value 0 at t = 0, 1 and the maximum value GM ,C min ,C GM , then it follows from the intermediate value theorem that there exists t ′ with which means that these GME measures E and E ′ provide the opposite order for the quantum states |ψ⟩ ABC and |φ (t ′ )⟩ ABC .

Example 2
The GME measure T min is defined based on three-qubit teleportation capability.Thus, we can say that if |ψ⟩ is more entangled than |ξ ⟩ with respect to T min , then |ψ⟩ is more useful than |ξ ⟩ in three-qubit teleportation.Let where 0 ≤ r ≤ 1.In FIG. 2, we can see that for 0.7 < r < 0.9, where |ψ(r)⟩ ABC is the state in Eq. ( 16).In other words, although |ψ(r)⟩ ABC is more valuable for three-qubit teleportation than |ξ (r)⟩ ABC in this case, C min does not catch this fact.Similar examples can be readily found for other GME measures as well.

GME and N-qubit teleportation capability
We here discuss the relation between GME and N-qubit teleportation capability, where N ≥ 4. Note that f i j in Eq. ( 5) can be generalized in the following two ways.Let 6/14 where ρ where K l = K \ {k l } and f i j = f i j in Eq. ( 5).The difference between these two definitions is whether or not communication between assistants is allowed.Hence, we obtain two different maximal average teleportation fidelities as follows: Let us define new quantities based on these N-qubit teleportation capabilities with a similar way for three-qubit case as follows.
Definition 7 For a N-qubit pure state |φ ⟩ respectively, on N-qubit pure states, where m = N 2 .For N-qubit mixed states, we generalize them via the convex roof extension.In the same way, we define T (N) min and T (N) GM by using F(N) i j .
Now, we discuss that the quantities proposed in Definition 7 may be good candidates to be GME measures for N-qubit systems, where N ≥ 4. As we have already shown in Lemma 1, for a three-qubit pure state |φ ⟩ ABC , F i j (|φ ⟩ ABC ) = 2 3 if and only if |φ ⟩ ABC ∈ SEP(i : jk) or |φ ⟩ ABC ∈ SEP( j : ik) for any distinct i, j, and k in {A, B,C}.Here, we have a similar argument for an N-qubit pure state.
Let us now assume that for an N-qubit pure state |φ ⟩ A 1 •••A N , there exists a bipartition {G i |G j } of the whole system with i ∈ G i and j ∈ G j such that |φ ⟩ where k / ∈ {i, j}, then we clearly have that the resulting state after performing an orthogonal measurement in the system k is a pure state which belongs to SEP( Gi : G j ), where Gi = G i \ {k}.By continuing this process, we can see that the quantum state obtained after measuring all systems except i and j is a pure state and separable between systems i and j.Therefore, we have the following proposition.
We note that a genuinely N-partite entangled state.However, we do not need to check all possible fidelities to verify that a given state is a genuinely N-partite entangled state.Indeed, it suffices to check min j F for a bipartition {P|P ′ } of the whole system, then we can always choose a system k from the party that does not contain i and 3 by Proposition 8.For example, when N = 4, if , then the remaining fidelities are also greater than 2 3 and so, |φ ⟩ ABCD is a genuinely quadripartite entangled state.
Conversely, does 3 implies separability for a bipartition {G i |G j } with i ∈ G i and j ∈ G j ?If this holds for F (N) i j , then this also holds for F(N) For N = 4, by checking all the cases, we get the following proposition.
3 , then |φ ⟩ ABCD ∈ SEP(G i : G j ) for some bipartition {G i |G j } with i ∈ G i and j ∈ G j .

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To sum up, we can distinguish genuinely quadripartite entangled states and biseparable states by means of F GM , T (4) min , and T (4) GM satisfy the GME condition (i), (ii), and (iv).
Note that we expect to have the same argument with more complicated calculations when N ≥ 5. Therefore, T min , T GM , T (4) min , and T (4) GM have the potential to be GME measures.

Conclusion
In this paper, we have introduced GME measures for three-qubit states based on three-qubit teleportation capability.In order to do that, we have considered the maximal average teleportation fidelity of the resulting states on the other parties obtained after one of the parties measures his/her system.We have shown that the fidelity can be used to observe separability, and does not increase on average under LOCC for three-qubit pure states, and by using these properties, we have proven that our entanglement measures defined using the fidelity satisfy the conditions for the GME measure.
For three-qubit mixed states, we have defined our entanglement measures by means of the convex roof extension.This method is a good way to make our entanglement measures satisfy the conditions for being a good entanglement measure on the mixed states, but it is not easy to get an exact value because it is defined as a minimum for all possible ensembles.For a profound understanding of multipartite entanglement, it is necessary to find this value or at least find its lower bound.Furthermore, it would be important to see how this value relates to multipartite teleportation capability.
We have shown that the maximal average fidelity of four-qubit teleportation can be used to distinguish genuinely quadripartite entangled states and biseparable states.This could be generalized to N-qubit systems, where N ≥ 5. Hence, the quantities defined in Definition 7 have the potential to be GME measures.It is not easy to show that the entanglement measures satisfy the conditions for GME measures, in particular the monotonicity under LOCC, because no analytic form such as Eq. ( 8) has been known for N ≥ 4. Our future work is to rigorously prove that these quantities are GME measures.Moreover, it would also be intriguing to explore entanglement measures for N-qudit systems using a similar approach since we can define N-qudit teleportation capability analogously as we define N-qubit teleportation capability in this work.
We note that there are other quantum information tasks that use GME, such as conference key agreement or secret sharing.It has been shown that any multipartite private state, which is the general form of quantum state capable of conference key agreement, is a genuinely multipartite entangled state 8 .Hence, it could be interesting to see if we can define GME measures based on those quantum information tasks.Besides, there have been recently known quantum information tasks such as the quantum secure direct communication 31 or controlled quantum teleportation based on quantum walks 32 .It would be also a possible future work to see how these tasks can be related to GME measures and how we define entanglement measures based on them.

Proof of Lemma 2
It is sufficient to show that τ +C 2 AB , τ +C 2 BC , and τ +C 2 CA do not increase on average under LOCC, thanks to Eq. ( 8).One important observation is that it is always possible to decompose any local protocol into positive operator-valued measures (POVMs) such that only one party implements operations on his/her system.Moreover, we also remark that a generalized local POVM can be carried out by a sequence of POVMs with two outcomes 17,34 .Without loss of generality, let us assume that Alice performs POVM consisting of two elements, say A 0 and A 1 .By using the singular value decomposition, they can be written as A t = U t D t V , where U t and V are unitary matrices, and D t are diagonal matrices with entries (a, b) and Here, the same unitary operation V can be chosen for both A 0 and A 1 because they compose the POVM.
Let |φ ⟩ ABC be an initial state of the form in Eq. (23) By definition, with some tedious calculations, we can obtain where From Eq. ( 9) and Eq. ( 24), we have One can readily check that Hence, τ +C 2 AB and τ +C 2 CA do not increase on average under LOCC for three-qubit pure states.Now, it remains to show that τ +C 2 BC does not increase on average under LOCC for three-qubit pure states, or equivalently, Observe that it can be written as where with the normalization factor P. We can rewrite with some nonzero values γ 0 , γ with a normalization factor P. We can rewrite with some complex values γi and δ j .Reordering the systems, we can have which means |φ ⟩ ABCD ∈ SEP(AC : BD). 12/14 (i) min and T (i) GM are GME measures for any i ∈ {A, B,C}.

Figure 1 .
Figure 1.Graphs of the GME measures C min , T min , T (A) min , C GM , T GM , and T (A) GM ,C min ,C GM , it is easy to find r ′ such that E (ψ(r ′ )) > E ′ (ψ(r ′ )) as seen in FIG.1.Hence, they are all different GME measures.
GM satisfy the GME condition (ii) and (iv), and hence, we have the following corollary.