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Quantifying the entanglement of quantum states under the geometry method

Quantifying entanglement is an important issue in quantum information theory. A straightforward method to quantify entanglement is to measure the distance between the entangled state and the separable sets under the fidelity, distance norm and so on. However, there are few results on the entanglement measure in terms of trace norm. Here we consider the entanglement measures through the trace norm in terms of three methods, the standard measure, the modified measure and the way proposed in [Phys. Rev. A 102. 062401 (2020)]. Specifically, we present some properties of the entanglement measures under the three methods. We also present the analytical expressions for a class of mixed states on two-qubit systems for the standard measure, the pure states for the modified measure and the mixed states on two-qubit systems for the method generated under the third way. And we also generalize the bipartite entanglement measure to a measure for the tripartite states.


I. INTRODUCTION
Quantum entanglement is an essential feature of quantum mechanics. It plays an important role in quantum information and quantum computation theory [1], such as superdense coding [2], teleportation [3] and the speedup of quantum algorithms [4].
To quantify the amount of entanglement for a multipartite system is key in quantum information theory. As it is linked with many areas of the entanglement theory, such as entanglement distillability [5], the transformation of quantum states [6][7][8], monogamy of entanglement [9][10][11][12][13], quantum speed limit [14] and so on. Due to the importance of this issue, it has been investigated almost since the end of the last century [5,6,15]. In 1996, Bennett et al. proposed the distillable entanglement and entanglement cost, as well as their operational interpretations [5]. In 1997, Vedral presented three necessary conditions that an entanglement measure should satisfy [6]. And Vidal presented a general method to quantify the entanglement for the bipartite entangled states [16]. This method is built on the function of pure states in the bipartite system and generalizes to the mixed states under the convex roof extended methods on the bipartite system. Other than the generic conditions and the building methods of entanglement measures, some meaningful thoughts to bulid the entanglement measures are presented. The robustness of an entangled state tells us how many degrees of the separable states needed to make the state separable [17]. The squashed entanglement was motived by the intrinsic information [18,19], it is monogamous for arbitrary dimensional systems [19] and faithful [20]. Moreover, the way to measure the entanglement can also be defined based on the geometrical ways, such as, the geomeric measure of entanglement [21,22], the quantum relative entropy [6] and the fidelity [23]. However, there are few results of the measure based on the distance between an entangled state and the set of separable states under the trace norm.
This article is organized as follows. In section II, first we present some preliminary knowledge needed. We also present some properties of the modified entanglement measure. In section III, we consider the properties of the entanglement mesurement in terms of the trace norm under three methods. Specifically, in section III A, we present the standand measurement under the trace norm is continuity and subadditivity. We also present an analytical expression for a class of two-qubit mixed states, which tells us that this measure is not equal to the concurrence. Next we generalize it to a measure for tripartite mixed states D f sep (·), which tells the distance between a tripartite state and the set of full separable states. We also get D f sep (W ) and D f sep (GHZ). In section III B, we present the solutions of the modified measure for pure states. We also present that the modified measure satisfies the similar properties with the standard one. In section III C, we consider the measure under the trace norm under the method proposed in [37,38]. In section IV, we end with a conclusion.

II. PRELIMINARY KNOWLEDGE
In the following, we denote D AB as the set of states on H A with finite dimensions and D • A as the set of semidefinite positive operators with trace less than or equal to 1 on H A with finite dimensions. Next for a state ρ AB ∈ D AB , if it can be written as ρ AB = i p i |ψ i A ψ i |⊗|φ i B φ i |, then it is separable, otherwise, it is an entangled state. We denote SEP AB as the set of separable states on H AB with finite dimensions. And we may leave out the systems in the absence of ambiguity in the following.
Assume |ψ AB is a bipartite pure state in H AB , |ψ AB = i √ λ i |ii , then its concurrence is defined as here ρ A is the reduced density matrix of ρ AB . When ρ AB is a mixed state, then where the minimum takes over all the decompositions Next we recall the definition of a distance measure in terms of trace norm for a bipartite mixed state. Assume ρ ∈ D AB , its distance measure in terms of trace norm is defined as, here β takes over all the separable states in the system here β takes over all the separable states in the system Then we recall E : D(H AB ) → R + is an entanglement measure [6] if it satifies: (II) E does not increase under the local operation and classical communication (LOCC), here Ψ is LOCC.
In [16], the author presented that when E satisfies the following two conditions, E is an entanglement monotone, k is any unilocal quantum operation performed by any party A or B, and i are on behave of the party A or B.
The condition (IV) can also be regarded as the convexity of an entanglement measure. Next we recall the simplified conditions when the function satisifies the LOCC monotone [39]: For a convex function f does not increase under LOCC if and only if 1 . f satisfies local unitary invariance (LUI) for X = A ′ , B ′ , where |i are local orthogonal flags.
The properties (I), (II) and (IV) is satisfied by D sep (·) [6,40], however, the property (III) for the entanglement measure D sep (·) is not valid [41]. Then we recall the modified version D ′ sep (·) of the entanglement measure D sep (·), Due to the properties of · 1 and the definition of D ′ sep (·), if we could prove here {E k } is a positive-operator-valued measurement (POVM) on the party A of ρ, D ′ sep (·) satisfies the property (III). First we recall the Naimark theorem [42]: A method is via the direct sum, requiring here ρ is an arbitrary operator in H A , 0 is the zero matrix with dimension d As the local operations E, addition of an uncorrelated ancilla system and the dismissal of a local part of the whole system satisfy the above inequality is due to the definition of D ′ sep (·) and the monotonicity of the trace norm under E. Combing the Naimark theorem, we only need to prove (8) is valid for the PVM, that is, we need to prove that Here in the first equality, σ is the optimal state in terms of D ′ sep (·) for ρ. In the first inequality, {Π k } is a PVM on the A-party of ρ AB , k Π k = I, then due to the contractive under the trace-preserving quantum operations, the first inequality is valid. In the second equality, p k = Tr Π k ρΠ k , σ k = Π k σΠ k p k . As we could always assume that after the operations of the {Π k }, σ is transformed into a separable state, and due to the definition of D sep (·), the last inequality is valid. We finish the proof of the inequality (12).
At last, we prove the convexity of D the first equality is due to the definition of (7), the first inequality is due to the convexity of the · 1 . In the second inequality, we denote that σ be the optimal in terms of D ′ sep (·) for ρ, and the second inequality is due to the definition of D ′ sep (·). Then we prove the property (IV) is satisfied by the entanglement measure D sep (·).

III. MAIN RESULTS
In this section, we will consider the entanglement measures based on trace norm in terms of three methods. First we will present some properties of the standard entanglement measure based on the trace norm and evaluate a class of entangled states on two-qubit systems, We generalize this bipartite measure to quantify the genuinely entangled states on a tripartite systems. Then we present the modified entanglement measure based on the trace norm of a class of entangled states, we also present some properties of the modified measure. At last, we consider the distance measure on the trace norm under the method presented in [37,38], we present this measure is monogamous for n-qubit systems .

A. The Standard Distance Measure
In this subsection, we first present the standard distance measure satisfies the subadditivity and continuity.
Assume ρ 1 A1B1 and σ 1 A2B2 are the optimal separable states in terms of D sep (·) for ρ A1B1 and σ A2B2 , respectively. Then we have Here the first inequality is due to the definition of D sep (ρ), the second inequality is due to the triangle inequality of the norm, the second equality is due to that The last equality is due to the assumption.
⊓ ⊔ Theorem 2 Assume ρ AB and σ AB are two mixed states with ρ AB − σ AB ≤ ǫ, then we have Proof.
Here we can always assume Here α AB is the optimal for σ AB . The first inequality is due to the definition of D sep (·), as α AB may not be the optimal for ρ AB , the second inequality is due to the triangle inequality Next we recall the result in [43] on D sep (·) for pure bipartite states.
j , and p l = s 2 l − 1 − lm l for l ∈ {1, · · · , d}. There exists a maximum integer k ∈ {1, · · · , d} satisfying Furthermore, According to the above lemma, we could present an example showing D sep (ρ AB ) does not satisfy the additivity property.
, then from the result in [43], we have Assume ρ AB is a bipartite mixed state,D(ρ AB ) under the convex roof extended method is defined as where the minimum takes over all the decompositions {p i , |ψ i } of ρ AB . Next we present an example to show that there exists a state χ AB such that D sep (χ AB ) = D(χ AB ).
Example 2 Assume χ AB is an arbitrary diagonal symmetric mixed state, Here we present the proof in Sec. VI.
In [43], the authors presented when |ψ is a two-qubit pure state, D sep (|ψ AB ) = C(|ψ AB ). Then due to [44], That is, we present a counterexample showing that D sep (ρ AB ) = D sep (ρ AB ) when ρ AB is a mixed state in two-qubit system. Recently, the triangle inequalities were showed valid for a rank-2 mixed state in terms of the l 1 norm and concurrence [45], however, the triangle inequality of D sep (·) is not invalid.

Example 3 Assume
As ρ TB ≥ 0, ρ is separable. Then we have At last, we formulate a semidefinite program to calculate the D sep (σ) when σ is a state on 2 ⊗ 2 or 2 ⊗ 3 systems. Here we mainly apply the following fact: assume ρ is a Hermitian, then it can be written as ρ = ρ + − ρ − , here ρ + and ρ − are two semidifinite positive matrix, and ρ + ⊥ ρ − . Next when σ is a state on 2⊗2 or 2⊗3 systems, σ is separable if and only if σ TB ≥ 0. Thus D sep (σ) is the result of the following semidefinite program: However, for a bipartite system d 1 ⊗d 2 , when d 1 d 2 > 6, SEP ⊂ P P T [46], and the problem to judge whether a state σ ∈ B(H d1 ⊗ H d2 ) is separable is hard. Hence, the above method to compute the value of entanglement measure for a mixed state in terms of D sep (·) is infeasible.
At the end of this subsection, we generalize the measure D sep (ρ) to a quantity D f sep (·) for a tripartite system which tells the distance from a tripartite state to the set of the fully separable states. First we recall the full separable states. Assume |ψ ABC is a pure state in H A ⊗ H B ⊗ H C that can be written as |ψ ABC = |φ 1 A |φ 2 B |φ 3 C , then it is fully separable. If a mixed state ρ ABC defined on the system H A ⊗ H B ⊗ H C can be represented as the following form, then ρ ABC is a fully separable state. Next we present the definition of D f sep (·) : assume ρ ABC is a tripartite state in B(H A ⊗ H B ⊗ H C ), here the minimum takes over all the states that are full separable.
In the following, we will compute the values D f sep (|W ) and D f sep (|GHZ ).
We place the proof of Example 4 in Sec. VI. From the above example, we have that D f sep (W ) > 1 = D f sep (GHZ), that is, when considering the full separable states as the free states for a three-qubit system, |W is more resourceful than |GHZ . However, when considering the biseparable states as the free states, as |W and |GHZ are symmetric states, |W and |GHZ can be written as the following respectively, Here |φ 0 = |00 , |φ 1 = 1 √ 2 (|10 + |01 ), |φ 3 = |00 , |φ 4 = |11 . Due to the Lemma 1, we have that |W is less rescourceful than |GHZ in terms of this scenario.

B.
The modified distance measure In the above section, we presented the modified distance measure that is defined as here σ takes over all the elements in the set {σ|σ = In [47], the authors presented the method to quantify the coherence in this way. In [48], they showed that the modified distance of almost all pure states is a constant, and they numerically showed that a similar result occurs for states with a fixed rank. Here we showed that the same result occurs for the entangled pure states.
Theorem 3 Assume |ψ = i λ i |ii is an arbitrary pure state, here λ i ≥ λ i+1 ,then the modifies distance measure of |ψ is We present the proof of this theorem in Sec. VI Remark 1 In [49], the authors considered the same problem for the pure states in terms of D ′ sep (·) and presented a solution. The method there is due to the convex analysis, ours is similar to the method of the Example 2 here. Moreover, comparing with the result in [49], ours is more apparent. And we can also generalize our methods to a class of mixed states.
Corollary 1 Assume ρ = a 0 |00 00| + a 1 |00 11| + a 1 |11 00| + a 2 |11 11| is a two-qubit mixed state, then D ′ sep (ρ) = 2|a 1 |. The main proof of the above corollary is the same as the proof for the pure states above, and by combing the theorem 1 in [48], we can get the corollary. Next by the similar proof of Theorem 2 of the standard distance measure, the modified distance measure satisfy the similar results.
Theorem 4 Assume ρ AB and σ AB are two mixed states with ρ AB − σ AB ≤ ǫ, then we have At last, combing the Theorem 3 and the convexity of D ′ sep , we can prove that D ′ sep also satisfies the subadditivity with the similar proof of Theorem 1.

C. A third way to build the distance measure
Here we consider another entanglement measure based on the trace norm under the method presented in [37,38]. There the authors presented a way to generate the entanglement measure E for the mixed states from the entanglement measure E on pure states. Assume here the infimum takes over all the pure states in the set T is a class of quantum channels. }, There we presented conditions when the entanglement measure is entanglement monotone and convex when T is LOCC. Here we specifically consider the properties of the entanglement measures based on the trace norm under the method there. That is, Due to the results in [37], we know that the above quantity is an entanglement measure, that is, D sep (ρ AB ) ≥ D sep (Λ(ρ AB )), Λ ∈ LOCC. Next as when |ψ AB ∈ H 2 ⊗ H 2 , D sep (|ψ AB ) = C(|ψ AB ) [43], by the results in [38], we have the following corollary, Corollary 2 When ρ AB is a two-qubit mixed state, D sep (ρ AB ) is entanglement monotone. Furthermore, where the minimum takes over all the decompositions {p i , |ψ i AB } of ρ AB . Furthermore, assume ρ A|B1B2···Bn−1 is an n-qubit mixed state, then At last, we summarize the main properties of three entanglement measures generated by the trace norm presented here in the table I.

IV. CONCLUSION
In this paper, we have considered the entanglement measures in terms of the trace norm under three methods. We have presented the measures proposed under the first two methods satisfy convexity and subadditivity for arbitrary dimensional systems. And we also have presented the analytical expressions of the measures considered here for some classes of states. However, we have only shown it is entanglement monotone for the 2-qubit systems and monogamous for the n-qubit systems in terms of the measure generated under the third method. We think the above results may be not valid for larger systems, although we havenot found a counterexample. At last, we believe that our results are helpful in the study of monogamy relations for multipartite entanglement systems. And we hope our work could shed some light on related studies.   Here we summarize the properties of the entanglement measures generated by the trace norm under three methods. The measure I is the measure generated by the standard method considered in section III A, The measure II is the measure considered in section III B, The measure III is the measure considered in section III C. And "Y" indicates the measure satisfies the property, "N" indicates the measure does not satisfy the property. "Y for the 2 ⊗ 2 system" indicates that the measure III satisfies the property for the 2 ⊗ 2 system, however, it may be invalid for larger systems.
Y for the 2 ⊗ 2 system Y for the 2 ⊗ 2 system Y Y Here we present the proof of Example 2, Example 4 and Theorem 3.
From (40), we have Combing the Theorem 1 in [50], the above can be written as the following, from computation, we have D f sep (|W ) = 19−9 Next we present the result on |GHZ . In [51], the authors presented that |GHZ satisfies the following properties: i. it is a symmetric state, As |GHZ is a symmetric state, that is, here σ, σ ′ are arbitrary permutations σ, σ ′ ∈ Ω N , Ω N isthe group of all permutations of an N -element set, and V σ |ψ 1 |ψ 2 · · · |ψ N = |ψ σ(1) |ψ σ(2) · · · |ψ σ(N ) . Assume θ is the state attaining the minimum, then here σ, σ ′ are arbitrary permutations. As s and s ′ are arbitrary, θ is a symmetric state. By the similar method, θ should satisfy the same properties (i),(ii) and (iii). That is, the matrix of θ should be the following: T heorem 3: Assume |ψ = i λ i |ii is an arbitrary pure state, here λ i ≥ λ i+1 ,then the modifies distance measure of |ψ is Proof. Assume |ψ is a pure state, |ψ = i √ λ i |ii , then we have (U ⊗ U )|ψ ψ|(U ⊗ U ) † = |ψ ψ|, here U is a diagonal unitary matrix. Next assume σ is an arbitrary positive operator which can be written as σ = i ω A i ⊗ θ B i , here ω A i and θ B i are positive operators on the systems A and B, respectively.
Then let σ ′ be the optimal in terms of the modified distance measure D as σ ′ is the optimal, and U∈D(U) (U ⊗ U)σ ′ (U ⊗ U ) † is an element in the set of λSEP , then we have the inequality in (44) is an equality. Let here D(U ) is the set consisting of all diagnoal unitary matrices, ̺ ijkl denotes the coefficient of the state ̺ in the basis |ij kl|. Hence we only need to compute the minimum of the following equality, here the minimum takes over all the positive operators ̺ that can be written as ̺ = i ω A i ⊗ θ B i , ω A i and θ B i are positive operators on the systems A and B, respectively.
In the first equality, as |ψ = i √ λ i |ii , the coefficients of |ψ ψ| in the basis |ij ij| are 0. The first inequality is due to that the positive operator ̺ satisfies the PPT condition, then |̺ iijj | ≤ |̺ ijij ̺ jiji | ≤ |̺ijij |+|̺jiji | 2 . The last inequality is due to the triangle inequality of the trace norm. In the last equality, |φ = i √ λ i |i . Next in [48], the authors showed that the optimal coherence states for a pure state in terms of the modified trace norm of coherence is a diagonal matrix, and in the last equality of (45), this problem is turned into the modified trace norm of coherence for a pure state, then we finish the theorem. ⊓ ⊔