Lower and upper bounds for entanglement of Rényi-α entropy

Entanglement Rényi-α entropy is an entanglement measure. It reduces to the standard entanglement of formation when α tends to 1. We derive analytical lower and upper bounds for the entanglement Rényi-α entropy of arbitrary dimensional bipartite quantum systems. We also demonstrate the application our bound for some concrete examples. Moreover, we establish the relation between entanglement Rényi-α entropy and some other entanglement measures.


Introduction
Quantum entanglement is one the most remarkable features of quantum mechanics and is the key resource central to much of quantum information applications.For this reason, the characterization and quantification of entanglement has become an important problem in quantum-information science. 1 A number of entanglement measures have been proposed for bipartite states such as the entanglement of formation (EOF), 2 concurrence, 3 relative entropy, 5 geometric entanglement, 6 negativity 7 and squashed entanglement. 8,9 mong them EOF is one of the most famous measures of entanglement.For a pure bipartite state |ψ AB in the Hilbert space, the EOF is given by where S (ρ A ) := −Trρ A log ρ A is the von Neumann entropy of the reduced density operator of system A.Here "log" refers to the logarithm of base two.The situation for bipartite mixed states ρ AB is defined by the convex roof where the minimum is taken over all possible pure state decompositions of ρ AB = ∑ i p i |ψ i AB ψ i | with ∑ i p i = 1 and p i > 0.
The EOF provides an upper bound on the rate at which maximally entangled states can be distilled from ρ and a lower bound on the rate at which maximally entangled states needed to prepare copies of ρ. 10 For two-qubit systems, an elegant formula for EOF was derived by Wootters in. 3 However, for the general highly dimensional case, the evaluation of EOF remains a nontrivial task due to the the difficulties in minimization procedures. 4At present, there are only a few analytic formulas for EOF including the isotropic states, 11 Werner states 12 and Gaussian states with certain symmetries. 135][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] Especially, Chen et al 18 derived an analytic lower bound of EOF for an arbitrary bipartite mixed state, which established a bridge between EOF and two strong separability criteria.4][35][36] While the entanglement of formation is the most common measure of entanglement, it is not the unique measure.There are other measures such as entanglement Rényi-α entropy (ERαE) which is the generalization of the entanglement of formation.The ERαE has a continuous spectrum parametrized by the non-negative real parameter α.For a bipartite pure state |ψ AB , the ERαE is defined as 37 where S α (ρ A ) is the Rényi-α entropy.Let µ 1 , • • • , µ m be the eigenvalues of the reduced density matrix ρ A of |ψ AB .We have where µ is called the Schmidt vector (µ 1 , µ 2 , • • • , µ m ).The Rényi-α entropy is additive on independent states and has found important applications in characterizing quantum phases with differing computational power, 38 ground state properties in many-body systems, 39 and topologically ordered states. 40,41 imilar to the convex roof in (2), the ERαE of a bipartite mixed state ρ AB is defined as It is known that the Rényi-α entropy converges to the von Neumann entropy when α tends to 1.So the ERαE reduces to the EOF when α tends to 1.Further ERαE is not increased under local operations and classical communications (LOCC). 37So the ERαE is an entanglement monontone, and becomes zero if and only if ρ AB is a separable state.An explicit expression of ERαE has been derived for two-qubit mixed state with α ≥ ( √ 7 − 1)/2 ≃ 0.823. 37,42 ecently, Wang et al 42 further derived the analytical formula of ERαE for Werner states and isotropic states.However, the general analytical results of ERαE even for the two-qubit mixed state with arbitrary parameter α is still a challenging problem.
The aim of this paper is to provide computable lower and upper bounds for ERαE of arbitrary dimensional bipartite quantum systems, and these results might be utilized to investigate the monogamy relation [43][44][45][46] in high-dimensional states.The key step of our work is to relate the lower or upper bounds with the concurrence which is relatively easier to dealt with.We also demonstrate the application of these bounds for some examples.Furthermore, we derive the relation of ERαE with some other entanglement measures.

Lower and upper bounds for entanglement of R ényi-α entropy
For a bipartite pure state with Schmidt decomposition |ψ = ∑ m i=1 √ µ i |ii , the concurrence of |ψ is given by c(|ψ A is also known as the mixedness and linear entropy. 50,51 he concurrence of a bipartite mixed state ρ is defined by the convex roof c (ρ) = min ∑ i p i c (|ψ i ) for all possible pure state decompositions of ρ = ∑ i p i |ψ i ψ i |.A series of lower and upper bounds for concurrence have been obtained in Refs. 19,24,25 Fr example, Chen et al 19 provides a lower bound for the concurrence by making a connection with the known strong separability criteria, 47,48 i.e., for any m ⊗ n(m ≤ n) mixed quantum system.The • denotes the trace norm and T A denotes the partial transpose.Another important bound of squared concurrence used in our work is given by 24,25 Tr (ρ with − ⊗(P − ⊗I (2) ), K 2 = 4(I (1) ⊗ P (2) − ) and P (i) + ) is the projector on the antisymmetric (symmetric) subspace of the two copies of the ith system.These bounds can be directly measured and can also be written as Tr Below we shall derive the lower and upper bounds of ERαE based on these existing bounds of concurrence.Different states may have the same concurrence.Thus the value of H α ( µ) varies with different Schmidt coefficients µ i for fixed concurrence.We define two functions The derivation of them is equivalent to finding the maximal and minimal of H α ( µ).Notice that the definition of H α ( µ), it is equivalent to find the maximal and minimal of ∑ m i=1 µ α i under the constraint 2(1 − ∑ m i=1 µ 2 i ) ≡ c since the logarithmic function is a monotonic function .With the method of Lagrange multipliers we obtain the necessary condition for the maximum and minimum of ∑ m i=1 µ α i as follows where λ 1 , λ 2 denote the Lagrange multipliers.This equation has maximally two nonzero solutions γ and δ for each µ i .Let n 1 be the number of entries where µ i = γ and n 2 be the number of entries where µ i = δ .Thus the derivation is reduced to maximizes or minimizes the function under the constrains where From Eq. ( 16) we obtain two solutions of γ with max{ 2 we should only consider the case for γ + n 1 n 2 .When n 2 = 0, γ can be uniquely determined by the constrains thus we omit this case.When m = 3, the solution of Eq.( 15) is R with (iii) When α = 2, these lower and upper bounds give the same value.We use the denotation co(g) to be the convex hull of the function g, which is the largest convex function that is bounded above by g, and ca(g) to be the smallest concave function that is bounded below by g.Using the results presented in Methods, we can prove the main result of this paper.
Theorem.For any m ⊗ n(m ≤ n) mixed quantum state ρ, its ERαE satisfies where and Next we consider how to calculate the expressions of co (R L (c)) and ca (R U (c)).As an example, we only consider the case m = 3.In order to obtain co (R L (c)), we need to find the largest convex function which bounded above by R L (c).We first set the parameter α = 3, then we can derive We plot the function R 11 , R 12 and R 21 in Fig. 1  Combining the above results, we get Similarly, we can calculate that R ′′ 11 ≥ 0 and R ′′ 21 ≥ 0, thus ca (R U (c)) is the broken line connecting the following points: In Fig. 3 we have plotted the lower and upper bounds with dashed and dotted line respectively.Then we choose the parameter α = 0.6, and we get Since R ′′ 11 ≤ 0, R ′′ 21 ≤ 0, we have that co (R L (c)) is the broken line connecting the points: [0, 0], [1, log 2], [2/ √ 3, log 3].In order to obtain ca (R U (c)), we need to find the smallest concave function which bounded below by R U (c).We find R ′′ 11 ≤ 0, R ′′ 12 ≥ 0, therefore ca (R U (c)) is the curve consisting R 11 for 0 < c ≤ 1 and the line connecting points [1, R 12 (1)] and As shown in Fig. 3, the lower and upper bound both consists of two segments in this case.Generally, we can get the expression of co (R L (c)) and ca (R U (c)) for other parameters α and m using similar method.

5/12 examples
In the following, we give two examples as applications of the above results.
Example 1.We consider the d ⊗ d Werner states where −1 ≤ f ≤ 1 and F is the flip operator defined by F (φ ⊗ ψ) = ψ ⊗ φ.It is shown in Ref. 49 that the concurrence C ρ f = − f for f < 0 and C ρ f = 0 for f ≥ 0. According to the theorem we obtain that Example 2. The second example is the 3 ⊗ 3 isotropic state ρ = (x/9)I + (1 − x) |ψ ψ|, where |ψ = a, 0, 0, 0, 1/ √ 3, 0, 0, 0, 1/ √ 3 t / a 2 + 2/3 with 0 ≤ a ≤ 1.We choose x = 0.1, it is direct to calculate that When α = 0.6, we can calculate the lower and upper bounds and the results is shown in Fig. 4. The solid red line corresponds to the lower bound of E α by choosing the lower bound of concurrence is C 1 , and the dash-dotted and dashed line correspond to the cases when we choose the lower bound of concurrence is C 2 and C 3 , respectively.We can choose the maximum value of the three curves as the lower bound of E α .The blue solid line is the upper bound of E α .

relation with other entanglement measures
In this section we establish the relation between ERαE and other well-known entanglement measures, such as the entanglement of formation, the geometric measure of entanglement, 62 the logarithmic negativity and the G-concurrence.
The inequality follows from the concavity of logarithm function.The last equality follows from the fact ∑ j µ j = 1.Hence the ERαE is monotonically non-increasing with α ≥ 0. Since it becomes the von Neumann entropy when α tends to one, we have where 0 ≤ α ≤ 1 and β ≥ 1.Using the convex roof, one can show that (36) also holds for mixed bipartite states ρ.

geometric measure of entanglement
The geometric measure (GM) of entanglement measures the closest distance between a quantum state and the set of separable states. 62The GM has many operational interpretations, such as the usability of initial states for Grovers algorithm, the discrimination of quantum states under LOCC and the additivity and output purity of quantum channels, see the introduction of 51 for a recent review on GM.For pure state |ψ we define G l (ψ) = − log max | ϕ|ψ | 2 , where the maximum runs over all product states |ϕ .it is easy to see that max | ϕ|ψ | 2 is equal to the square of the maximum of Schmidt coefficients of |ψ .
For mixed states ρ we define where the minimum runs over all decompositions of ρ = ∑ i p i |ψ i ψ i |. 51 We construct the linear relation between the GM and ERαE as follows.
Lemma .If α > 1 then α If α = 1 and ρ is a pure state then If α < 1 then where d is the minimum dimension of H A and H B .The details for proving the lemma can be seen from Methods.

logarithmic negativity
In this subsection we consider the logarithmic negativity. 52It is the lower bound of the PPT entanglement cost, 52 and an entanglement monotone both under general LOCC and PPT operations. 53The logarithmic negativity is defined as Suppose ρ = ∑ i p i |ψ i ψ i | is the optimal decomposition of ERαE E α (ρ), and the pure state |ψ i has the standard Schmidt form |ψ i = ∑ j √ µ i, j |a i, j , b i, j .For 1/2 ≤ α ≤ (2n − 1)/2n and n > 1, we have where the first inequality is due to the property proved in, 53 the second inequality is due to the concavity of logarithm function, and in the last inequality we have used the inequality 2n ≥ 1/(1 − α) for 1/2 ≤ α ≤ (2n − 1)/2n, n ≥ 1.

G-concurrence
The G-concurrence is one of the generalizations of concurrence to higher dimensional case.It can be interpreted operationally as a kind of entanglement capacity. 54,55 t has been shown that the G-concurrence plays a crucial role in calculating the average entanglement of random bipartite pure states 56 and demonstration of an asymmetry of quantum correlations. 57Let |ψ be a pure bipartite state with the Schmidt decomposition |ψ = ∑ d i=1 √ µ i |ii .The G-concurrence is defined as the geometric mean of the Schmidt coefficients 54,55 G (|ψ For α > 1, we have For 0 < α < 1, we have

Discussion and conclusion
Entanglement Rényi-α entropy is an important generalization of the entanglement of formation, and it reduces to the standard entanglement of formation when α approaches to 1. Recently, it has been proved 58 that the squared ERαE obeys a general monogamy inequality in an arbitrary N-qubit mixed state.Correspondingly, we can construct the multipartite entanglement indicators in terms of ERαE which still work well even when the indicators based on the concurrence and EOF lose their efficacy.However, the difficulties in minimization procedures restrict the application of ERαE.In this work, we present the first lower and upper bounds for the ERαE of arbitrary dimensional bipartite quantum systems based on concurrence, and these results might provide an alternative method to investigate the monogamy relation in high-dimensional states.We also demonstrate the application our bound for some examples.Furthermore, we establish the relation between ERαE and some other entanglement measures.These lower and upper bounds can be further improved for other known bounds of concurrence. 59,60 2

Methods
Proof of the theorem.
Suppose ρ = ∑ j p j ψ j ψ j is the optimal decomposition of ERαE E α (ρ), and the concurrence of ψ j is denoted as c j .
Thus we have where the first inequality is due to the definition of co(g); in the second inequality we have used the monotonically increasing and convex properties of co (R L (c j )) as a function of concurrence c j ; and in the last inequality we have used the lower bound of concurrence.On the other hand, we have E α (ρ) = ∑ j p j E α ( ψ j ) = ∑ j p j H α ( µ) ≤ ∑ j p j ca(R U (c j )) ≤ ca[R U ( ∑ j p j c j )] where the first inequality is due to the definition of ca(g); the second inequality is due to the monotonically increasing and concave properties of ca(R U (c j )) as a function of concurrence c j ; and in the last inequality we have used the upper bound of concurrence.Thus we have completed the proof of the theorem.

Proof of the lemma.
Suppose the minimum in ( 37) is reached at ρ = ∑ i p i |ψ i ψ i |.Let the Schmidt decomposition of |ψ i be |ψ i = ∑ j √ µ i, j |a i, j , b i, j where µ i,1 is the maximum Schmidt coefficient.For α > 1, we have We have proved (38).For α = 1, let µ i be the Schmidt coefficients of ρ, we have

9/12
We have proved (39).For α < 1, we have The inequality holds because the pure state |ψ i is in the d × d space.So we have proved (40).

Figure 1 .
Figure 1.(color online).The plot of lower bound (dashed line) and upper bound (dotted line) for α = 3, m = 3.The upper bound consists of two segments and the lower bound consists of three segments.The solid line corresponds to R 11 , R 12 and R 21 .

Figure 3 .
Figure 3. (color online).The plot of lower bound (dashed line) and upper bound (dotted line) for α = 0.6, m = 3.The upper bound consists of two segments and the lower bound also consists of two segments.The solid line corresponds to R 11 , R 12 and R 21 .

Figure 4 .
Figure 4. (color online).Lower and upper bounds of E α (ρ) for α = 0.6 where we have set x = 0.1.Red solid line is obtained by C 1 , the dash-dotted and dashed line is obtained by C 2 and C 3 , respectively.The blue solid line is the upper bound of E α (ρ).
21(c) we only need to compare the value of them at the endpoint c = 1.For convenience we divide the problem into three cases.If 0 < α < 2, then R 12 (1) > R 21(1);If α = 2, then R 12 (1) = R 21 (1); If α > 2, then R 12 (1) < R 21(1).Thus we conclude that the maximal and minimal function of H α ( µ) is given by R 21 (c) and R 12 (c) respectively for α > 2. When α < 2, the maximal and minimal function of H α ( µ) is R 12 (c) and R 21 (c) respectively.When α = 2, we can check that the two functions R 21 (c) and R 12 (c) always have the same value for 1 < c ≤ 2/ √ 3.In the general case for m = d, numerical calculation shows the following results 12 (c) and R 21 (c) for 1 < c ≤ 2/ √ 3.After a direct calculation we find R 12 (c) and R 21 (c) are both monotonically function of the concurrence c, and R 12 (2/ √ 3) = R 21 (2/ √ 3).In order to compare the value of 3/12 R 12 (c) and R