Lower and upper bounds for entanglement of Rényi-α entropy

Abstract

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 science¹. A number of entanglement measures have been proposed for bipartite states such as the entanglement of formation (EOF)², concurrence³, relative entropy⁴, geometric entanglement⁵, negativity⁶ and squashed entanglement^7,8. Among them EOF is one of the most famous measures of entanglement. For a pure bipartite state ${|\psi \rangle }_{AB}$ in the Hilbert space, the EOF is given by

$${E}_{F}({|\psi \rangle }_{AB})=S({\rho }_{A}),$$

(1)

where $S({\rho }_{A})\,:\,=-{\rm{Tr}}{\rho }_{A}\,\mathrm{log}\,{\rho }_{A}$ 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

$${E}_{F}({\rho }_{AB})=\,{\rm{\min }}\sum _{i}{p}_{i}{E}_{F}({|{\psi }_{i}\rangle }_{AB}),$$

(2)

where the minimum is taken over all possible pure state decompositions of ${\rho }_{AB}=\sum _{i}{p}_{i}{|{\psi }_{i}\rangle }_{AB}\langle {\psi }_{i}|$ 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 ⁹. For two-qubit systems, an elegant formula for EOF was derived by Wootters in ref. 3. However, for the general highly dimensional case, the evaluation of EOF remains a nontrivial task due to the the difficulties in minimization procedures¹⁰. At present, there are only a few analytic formulas for EOF including the isotropic states¹¹, Werner states¹² and Gaussian states with certain symmetries¹³. In order to evaluate the entanglement measures, many efforts have also been devoted to the study of lower and upper bounds of different entanglement measures^{14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32}. 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. Based on this idea, there are several improved lower and upper bounds for EOF presented in refs 33,34,35,36. 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 ${|\psi \rangle }_{AB}$, the ERαE is defined as³⁷

$${E}_{\alpha }({|\psi \rangle }_{AB})\,:={S}_{\alpha }({\rho }_{A})\,:=\frac{1}{1-\alpha }\,\mathrm{log}({\rm{tr}}{\rho }_{A}^{\alpha }),$$

(3)

where S _α (ρ _A ) is the Rényi-α entropy. Let ${\mu }_{1},\cdots ,{\mu }_{m}$ be the eigenvalues of the reduced density matrix ρ _A of ${|\psi \rangle }_{AB}$. We have

$${S}_{\alpha }({\rho }_{A})=\frac{1}{1-\alpha }\,\mathrm{log}(\sum _{i}{\mu }_{i}^{\alpha })\,:={H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow}),$$

(4)

where $\mathop{\mu }\limits^{\longrightarrow}$ is called the Schmidt vector $({\mu }_{1},{\mu }_{2},\cdots ,{\mu }_{m})$. The Rényi-α entropy is additive on independent states and has found important applications in characterizing quantum phases with differing computational power³⁸, ground state properties in many-body systems³⁹, and topologically ordered states^40,41. Similar to the convex roof in (2), the ERαE of a bipartite mixed state ρ _AB is defined as

$${E}_{\alpha }({\rho }_{AB})=\,{\rm{\min }}\sum _{i}{p}_{i}{E}_{\alpha }({|{\psi }_{i}\rangle }_{AB}\mathrm{).}$$

(5)

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)³⁷. So 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 $\alpha \ge (\sqrt{7}-\mathrm{1)/2}\simeq 0.823$ ^37,42. Recently, 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 Rényi-α entropy

For a bipartite pure state with Schmidt decomposition $|\psi \rangle ={\sum }_{i=1}^{m}\sqrt{{\mu }_{i}}|ii\rangle$, the concurrence of $|\psi \rangle$ is given by $c(|\psi \rangle )\,:=\sqrt{\mathrm{2(1}-{\rm{Tr}}{\rho }_{A}^{2})}=\sqrt{2(1-{\sum }_{i=1}^{m}{\mu }_{i}^{2})}$. The expression $1-{\rm{Tr}}{\rho }_{A}^{2}$ is also known as the mixedness and linear entropy^47,48. The concurrence of a bipartite mixed state ρ is defined by the convex roof $c(\rho )=\,{\rm{\min }}\sum _{i}{p}_{i}c(|{\psi }_{i}\rangle )$ for all possible pure state decompositions of $\rho =\sum _{i}{p}_{i}|{\psi }_{i}\rangle \langle {\psi }_{i}|$. A series of lower and upper bounds for concurrence have been obtained in refs 19,24,25. For example, Chen et al. ¹⁹ provides a lower bound for the concurrence by making a connection with the known strong separability criteria^49,50, i.e.,

$$c(\rho )\ge \sqrt{\frac{2}{m(m-2)}}(\max (\Vert {\rho }^{{T}_{A}}\Vert ,\Vert {\mathcal R} (\rho )\Vert )-1),$$

(6)

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 refs 24,25.

$${\rm{Tr}}(\rho \otimes \rho {V}_{i})\leq {[C(\rho )]}^{2}\leq {\rm{Tr}}(\rho \otimes \rho {K}_{i}),$$

(7)

with ${V}_{1}=\mathrm{4(}{P}_{-}^{(1)}-{P}_{+}^{(1)})\otimes {P}_{-}^{(2)}$, ${V}_{2}=4{P}_{-}^{(1)}\otimes ({P}_{-}^{(2)}-{P}_{+}^{(2)})$, ${K}_{1}=\mathrm{4(}{P}_{-}^{(1)}\otimes {I}^{(2)})$, ${K}_{2}=\mathrm{4(}{I}^{(1)}\otimes {P}_{-}^{(2)})$ and ${P}_{-}^{(i)}({P}_{+}^{(i)})$ 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

$${\rm{Tr}}(\rho \otimes \rho {V}_{1})=2({\rm{Tr}}{\rho }^{2}-{\rm{Tr}}{\rho }_{A}^{2}),$$

(8)

$${\rm{Tr}}(\rho \otimes \rho {V}_{2})=2({\rm{Tr}}{\rho }^{2}-{\rm{Tr}}{\rho }_{B}^{2}),$$

(9)

$${\rm{Tr}}(\rho \otimes \rho {K}_{1})=2(1-{\rm{Tr}}{\rho }_{A}^{2}),$$

(10)

$${\rm{Tr}}(\rho \otimes \rho {K}_{2})=2(1-{\rm{Tr}}{\rho }_{B}^{2})\mathrm{.}$$

(11)

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 ${H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow})$ varies with different Schmidt coefficients μ _i for fixed concurrence. We define two functions

$${R}_{U}(c)=\,{\rm{\max }}\{{H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow})|\sqrt{2(1-{\sum }_{i=1}^{m}{\mu }_{i}^{2})}\equiv c\},$$

(12)

$${R}_{L}(c)=\,{\rm{\min }}\{{H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow})|\sqrt{2(1-{\sum }_{i=1}^{m}{\mu }_{i}^{2})}\equiv c\}\mathrm{.}$$

(13)

The derivation of them is equivalent to finding the maximal and minimal of ${H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow})$. Notice that the definition of ${H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow})$, it is equivalent to find the maximal and minimal of ${\sum }_{i=1}^{m}{\mu }_{i}^{\alpha }$ under the constraint $\sqrt{\mathrm{2(1}-{\sum }_{i\mathrm{=1}}^{m}{\mu }_{i}^{2})}\equiv c$ 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 ${\sum }_{i=1}^{m}{\mu }_{i}^{\alpha }$ as follows

$$\alpha {\mu }_{i}^{\alpha -1}=2{\lambda }_{1}{\mu }_{i}-{\lambda }_{2},$$

(14)

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

$${R}_{{n}_{1}{n}_{2}}(c)=\frac{1}{1-\alpha }\,\mathrm{log}({n}_{1}{\gamma }^{\alpha }+{n}_{2}{\delta }^{\alpha }),$$

(15)

under the constrains

$${n}_{1}\gamma +{n}_{2}\delta =1,2(1-{n}_{1}{\gamma }^{2}-{n}_{2}{\delta }^{2})={c}^{2},$$

(16)

where n ₁ + n ₂ = d ≤ m. From Eq. (16) we obtain two solutions of γ

$${\gamma }_{{n}_{1}{n}_{2}}^{\pm }=\frac{{n}_{1}\pm \sqrt{{n}_{1}^{2}-{n}_{1}({n}_{1}+{n}_{2}\mathrm{)[1}-{n}_{2}\mathrm{(1}-{c}^{2}\mathrm{/2)]}}}{{n}_{1}({n}_{1}+{n}_{2})},$$

(17)

$${\delta }_{{n}_{1}{n}_{2}}^{\pm }=\frac{1-{n}_{1}{\gamma }_{{n}_{1}{n}_{2}}^{\pm }}{{n}_{2}},$$

(18)

with $\max \,\{\sqrt{2({n}_{1}-1)/{n}_{1}},\sqrt{2({n}_{2}-1)/{n}_{2}}\}\leq c\leq \sqrt{2(d-1)/d}$. Because ${\gamma }_{{n}_{2}{n}_{1}}^{-}={\delta }_{{n}_{1}{n}_{2}}^{+},{\delta }_{{n}_{2}{n}_{1}}^{-}={\gamma }_{{n}_{1}{n}_{2}}^{+}$, we should only consider the case for ${\gamma }_{{n}_{1}{n}_{2}}^{+}$. 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 ₁₂(c) and R ₂₁(c) for $1< c\leq \mathrm{2/}\sqrt{3}$. After a direct calculation we find R ₁₂(c) and R ₂₁(c) are both monotonically function of the concurrence c, and ${R}_{12}\mathrm{(2/}\sqrt{3})={R}_{21}\mathrm{(2/}\sqrt{3})$. In order to compare the value of R ₁₂(c) and R 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 ${H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow})$ is given by R ₂₁(c) and R ₁₂(c) respectively for α > 2. When α < 2, the maximal and minimal function of ${H}_{\alpha }(\mathop{\mu }\limits^{\longrightarrow})$ 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 $1\,< \,c\leq \mathrm{2/}\sqrt{3}$. In the general case for m = d, numerical calculation shows the following results

• (i) When α > 2,

  $${R}_{L}(c)=\frac{\mathrm{log}\,[({\gamma }_{\mathrm{1,}d-1}^{+}{)}^{\alpha }+{(d-\mathrm{1)}}^{1-\alpha }{\mathrm{(1}-{\gamma }_{\mathrm{1,}d-1}^{+})}^{\alpha }]}{1-\alpha },$$

  (19)
  $${R}_{U}(c)=\frac{\mathrm{log}\,[({\gamma }_{\mathrm{1,}d-1}^{-}{)}^{\alpha }+{(d-\mathrm{1)}}^{1-\alpha }{\mathrm{(1}-{\gamma }_{\mathrm{1,}d-1}^{-})}^{\alpha }]}{1-\alpha },$$

  (20)

  with $\sqrt{\mathrm{2(}d-\mathrm{2)/(}d-\mathrm{1)}}< c\leq \sqrt{\mathrm{2(}d-\mathrm{1)/}d}$, 1 ≤ d ≤ m − 1 and ${\gamma }_{\mathrm{1,}d-1}^{\pm }=\mathrm{(2}\pm \sqrt{\mathrm{2(}d-\mathrm{1)[}d\mathrm{(2}-{c}^{2})-\mathrm{2]}}\mathrm{)/2}d$.

• (ii) When α < 2,

  $${R}_{L}(c)=\frac{\mathrm{log}\,[({\gamma }_{\mathrm{1,}d-1}^{-}{)}^{\alpha }+{(d-\mathrm{1)}}^{1-\alpha }{\mathrm{(1}-{\gamma }_{\mathrm{1,}d-1}^{-})}^{\alpha }]}{1-\alpha },$$

  (21)
  $${R}_{U}(c)=\frac{\mathrm{log}\,[({\gamma }_{\mathrm{1,}d-1}^{+}{)}^{\alpha }+{(d-\mathrm{1)}}^{1-\alpha }{\mathrm{(1}-{\gamma }_{\mathrm{1,}d-1}^{+})}^{\alpha }]}{1-\alpha }\mathrm{.}$$

  (22)

• (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

$$co[{R}_{L}(\underline{C})]\leq {E}_{\alpha }(\rho )\leq ca[{R}_{U}(\overline{C})],$$

(23)

where

$$\overline{C}=\,{\rm{\min }}\,\{\sqrt{\mathrm{2(1}-{\rm{Tr}}{\rho }_{A}^{2})},\sqrt{\mathrm{2(1}-{\rm{Tr}}{\rho }_{B}^{2})}\},$$

(24)

and

$$\begin{array}{rcl}{\underline{C}}^{2} & = & {\rm{\max }}\{0,\mathrm{2/}m(m-\mathrm{1)}{(\Vert {\rho }^{{T}_{A}}\Vert -1)}^{2},\,\mathrm{2/}m(m-\mathrm{1)(}\Vert {\mathcal R} (\rho )\Vert -{\mathrm{1)}}^{2},\\ & & 2({\rm{Tr}}{\rho }^{2}-{\rm{Tr}}{\rho }_{A}^{2}),\,\,2({\rm{Tr}}{\rho }^{2}-{\rm{Tr}}{\rho }_{B}^{2})\}\mathrm{.}\end{array}$$

(25)

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

$$\begin{array}{rcl}{R}_{L}(c) & = & \{\begin{array}{l}{R}_{11},\,0\,< \,c\leq 1\\ {R}_{12},\,1\,< \,c\leq \mathrm{2/}\sqrt{3},\end{array}\\ {R}_{U}(c) & = & \{\begin{array}{l}{R}_{11},\,0\,< \,c\leq 1\\ {R}_{21},\,1\,< \,c\leq \mathrm{2/}\sqrt{3}\mathrm{.}\end{array}\end{array}$$

(26)

We plot the function R ₁₁, R ₁₂ and R ₂₁ in Fig. 1 which illustrates our result. It is direct to check that ${R}_{11}^{^{\prime\prime} }\ge 0$, therefore co(R ₁₁) = R ₁₁ for 0 < c ≤ 1. The second derivative of R ₁₂ is not convex near $c=\mathrm{2/}\sqrt{3}$ as shown in Fig. 2. In order to calculate co(R ₁₂), we suppose ${l}_{1}(c)={k}_{1}(c-\mathrm{2/}\sqrt{3})+\,\mathrm{log}\,3$ to be the line crossing through the point $[\mathrm{2/}\sqrt{3},{R}_{12}\mathrm{(2/}\sqrt{3})]$. Then we solve the equations l ₁(c) = R ₁₂(c) and dl ₁(c)/dc = dR ₁₂(c)/dc = k ₁ and the solution is k ₁ = 5.2401, c = 1.1533. Combining the above results, we get

$$co({R}_{L}(c))=\{\begin{array}{l}{R}_{11}\mathrm{(0}\,< \,c\leq \mathrm{1)}\\ {R}_{12}\mathrm{(1}\,< \,c\leq \mathrm{1.1533)}\\ 5.2401(c-\mathrm{2/}\sqrt{3})+\,\mathrm{log}\,3\\ \mathrm{(1.1533< }c\leq \mathrm{2/}\sqrt{3}\mathrm{).}\end{array}$$

(27)

Figure 1

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 2

The plot of the second derivative of R ₁₂ for ${\bf{1}}{\boldsymbol{< }}{\boldsymbol{c}}{\boldsymbol{\leq }}{\bf{2}}{\boldsymbol{/}}\sqrt{{\bf{3}}}$.

Similarly, we can calculate that ${R}_{11}^{^{\prime\prime} }\ge 0$ and ${R}_{21}^{^{\prime\prime} }\ge 0$, thus co(R _U (c)) is the broken line connecting the following points: $[0,0],[1,\,\mathrm{log}\,2],[\mathrm{2/}\sqrt{3},\,\mathrm{log}\,3]$. In Fig. 3 we have plotted the lower and upper bounds with dashed and dotted line respectively.

Figure 3

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.

Then we choose the parameter α = 0.6, and we get

$${R}_{L}(c)=\{\begin{array}{l}{R}_{11}\mathrm{(0}\,< \,c\leq \mathrm{1)}\\ {R}_{21}\mathrm{(1}\,< \,c\leq \mathrm{2/}\sqrt{3}),\end{array}$$

(28)

$${R}_{U}(c)=\{\begin{array}{l}{R}_{11}\mathrm{(0}\,< \,c\leq \mathrm{1)}\\ {R}_{12}\mathrm{(1}\,< \,c\leq \mathrm{2/}\sqrt{3}\mathrm{).}\end{array}$$

(29)

Since ${R}_{11}^{^{\prime\prime} }\leq 0$, ${R}_{21}^{^{\prime\prime} }\leq 0$, we have that co(R _L (c)) is the broken line connecting the points: $[0,0],[1,\,\mathrm{log}\,2],[\mathrm{2/}\sqrt{3},\,\mathrm{log}\,\mathrm{3]}$. 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 ${R}_{11}^{^{\prime\prime} }\leq 0$, ${R}_{12}^{^{\prime\prime} }\ge 0$, 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 $\mathrm{[2/}\sqrt{3},{R}_{12}\mathrm{(2/}\sqrt{3})]$ for $1\,< \,c\leq \mathrm{2/}\sqrt{3}$. 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.

Examples

In the following, we give two examples as applications of the above results.

Example 1. We consider the d ⊗ d Werner states

$${\rho }_{f}=\frac{1}{{d}^{3}-d}[(d-f)I+(df-1) {\mathcal F} ],$$

(30)

where −1 ≤ f ≤ 1 and ${\mathcal F}$ is the flip operator defined by ${\mathcal F} (\phi \otimes \psi )=\psi \otimes \phi$. It is shown in ref. 51 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 $1/(1-\alpha )\mathrm{log}[{(\mathrm{(1}+\sqrt{1-{f}^{2}}\mathrm{)/2})}^{\alpha }+{(\mathrm{(1}-\sqrt{1-{f}^{2}}\mathrm{)/2})}^{\alpha }]\leq {E}_{a}({r}_{f})\leq -f$ for −1 ≤ f ≤ 0 when m = 3.

Example 2. The second example is the 3 ⊗ 3 isotropic state $\rho =(x\mathrm{/9)}I+(1-x)|\psi \rangle \langle \psi |$, where $|\psi \rangle ={(a,\mathrm{0,}\mathrm{0,}\mathrm{0,}\mathrm{1/}\sqrt{3},\mathrm{0,}\mathrm{0,}\mathrm{0,}\mathrm{1/}\sqrt{3})}^{t}/\sqrt{{a}^{2}+\mathrm{2/3}}$ with 0 ≤ a ≤ 1. We choose x = 0.1, it is direct to calculate that

$${C}_{1}=\sqrt{2(Tr{\rho }^{2}-Tr{\rho }_{A}^{2})}=\sqrt{2(Tr{\rho }^{2}-Tr{\rho }_{B}^{2})}=\frac{2\sqrt{6.53+41.46{a}^{2}-1.71{a}^{4}}}{3(2+3{a}^{2})},$$

(31)

$${C}_{2}=\frac{1}{\sqrt{3}}(\Vert {\rho }^{{T}_{A}}\Vert \,-\,1)=\frac{2(5+6.9{a}^{2}-0.9{a}^{4}+9.353a\mathrm{(2}+3{a}^{2}))}{\mathrm{3(2}+3{a}^{2}{)}^{2}},$$

(32)

$${C}_{3}=\frac{1}{\sqrt{3}}(\Vert R(\rho )\Vert \,-\,1)=\frac{{\rm{0.346}}+{\rm{1}}{\rm{.2a}}}{{\rm{0.667}}+{a}^{2}},$$

(33)

$$\overline{C}=\sqrt{2(1-Tr{\rho }_{A}^{2})}=\sqrt{2(1-Tr{\rho }_{B}^{2})}=\frac{\sqrt{\mathrm{6(6.38}+33.72{a}^{2}+3.42{a}^{4})}}{3(2+3{a}^{2})}\mathrm{.}$$

(34)

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 .

Figure 4

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).

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 entanglement⁵², the logarithmic negativity and the G-concurrence.

Entanglement of formation

Let ρ be a bipartite pure state with Schmidt coefficients (μ ₁, μ ₂, …). We investigate the derivative of ERαE w.r.t. α as follows.

$$\begin{array}{rcl}\frac{d{E}_{\alpha }(\rho )}{d\alpha } & = & \frac{1}{{\mathrm{(1}-\alpha )}^{2}}(\sum _{j}\frac{{\mu }_{j}^{\alpha }}{{\sum }_{k}{\mu }_{k}^{\alpha }}\,\mathrm{log}\,{\mu }_{j}^{1-\alpha }+\,\mathrm{log}\sum _{k}{\mu }_{k}^{\alpha })\\ & \leq & \frac{1}{{\mathrm{(1}-\alpha )}^{2}}(\mathrm{log}\,\frac{{\sum }_{j}{\mu }_{j}}{{\sum }_{k}{\mu }_{k}^{\alpha }}+\,\mathrm{log}\sum _{k}{\mu }_{k}^{\alpha })\\ & = & 0.\end{array}$$

(35)

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

$${E}_{\alpha }(\rho )\ge {E}_{F}(\rho )\ge {E}_{\beta }(\rho )$$

(36)

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 states⁵². The 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 ref. 48 for a recent review on GM. For pure state |ψ〉 we define ${{\rm{G}}}_{{\rm{l}}}(\psi )=-\mathrm{log}\,{\rm{\max }}\,{|\langle \varphi |\psi \rangle |}^{2}$, where the maximum runs over all product states |φ〉. it is easy to see that ${\rm{\max }}\,{|\langle \varphi |\psi \rangle |}^{2}$ is equal to the square of the maximum of Schmidt coefficients of |ψ〉. For mixed states ρ we define

$${{\rm{G}}}_{{\rm{l}}}^{{\rm{c}}}(\rho )\,:=\,{\rm{\min }}\sum _{i}{p}_{i}{{\rm{G}}}_{{\rm{l}}}(|{\psi }_{i}\rangle )\,,$$

(37)

where the minimum runs over all decompositions of ρ = ∑ _i p _i |ψ _i 〉 〈ψ _i |⁴⁸. We construct the linear relation between the GM and ERαE as follows.

Lemma. If α > 1 then

$$\frac{\alpha }{\mathrm{2(}\alpha -\mathrm{1)}}{{\rm{G}}}_{{\rm{l}}}^{{\rm{c}}}(\rho )\ge {E}_{\alpha }(\rho \mathrm{).}$$

(38)

If α = 1 and ρ is a pure state then

$${{\rm{G}}}_{{\rm{l}}}^{{\rm{c}}}(\rho )\leq {E}_{\alpha }(\rho \mathrm{).}$$

(39)

If α < 1 then

$${E}_{\alpha }(\rho )+\frac{\alpha }{\mathrm{2(1}-\alpha )}{{\rm{G}}}_{{\rm{l}}}^{{\rm{c}}}(\rho )\leq \frac{1}{1-\alpha }\,\mathrm{log}\,d,$$

(40)

where d is the minimum dimension of ${ {\mathcal H} }_{A}$ and ${ {\mathcal H} }_{B}$. The details for proving the lemma can be seen from Methods.

logarithmic negativity

In this subsection we consider the logarithmic negativity⁵³. It is the lower bound of the PPT entanglement cost⁵³, and an entanglement monotone both under general LOCC and PPT operations⁵⁴. The logarithmic negativity is defined as

$$LN(\rho )=\,\mathrm{log}\Vert {\rho }^{{T}_{A}}\Vert \mathrm{.}$$

(41)

Suppose $\rho ={\sum }_{i}{p}_{i}|{\psi }_{i}\rangle \langle {\psi }_{i}|$ is the optimal decomposition of ERαE E _α (ρ), and the pure state |ψ _i 〉 has the standard Schmidt form $|{\psi }_{i}\rangle ={\sum }_{j}\sqrt{{\mu }_{i,j}}|{a}_{i,j},{b}_{i,j}\rangle$. For 1/2 ≤ α ≤ (2n − 1)/2n and n > 1, we have

$$\begin{array}{rcl}n\times LN(\rho ) & = & n\,\mathrm{log}\Vert {\rho }^{{T}_{A}}\Vert \ge n\,\mathrm{log}\sum _{i}{p}_{i}\Vert {(|{\psi }_{i}\rangle \langle {\psi }_{i}|)}^{{T}_{A}}\Vert \ge n\sum _{i}{p}_{i}\,\mathrm{log}\Vert {(|{\psi }_{i}\rangle \langle {\psi }_{i}|)}^{{T}_{A}}\Vert \\ & = & 2n\sum _{i}{p}_{i}\,\mathrm{log}(\sum _{j}\sqrt{{\mu }_{i,j}})\ge 2n\sum _{i}{p}_{i}\,\mathrm{log}\sum _{j}{\mu }_{i,j}^{\alpha }\ge \frac{1}{1-\alpha }\sum _{i}{p}_{i}\,\mathrm{log}\sum _{j}{\mu }_{i,j}^{\alpha }\\ & = & \sum _{i}{p}_{i}{E}_{\alpha }(|{\psi }_{i}\rangle )={E}_{\alpha }(\rho )\end{array}$$

(42)

where the first inequality is due to the property proved in ref. 54, 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^55,56. It has been shown that the G-concurrence plays a crucial role in calculating the average entanglement of random bipartite pure states⁵⁷ and demonstration of an asymmetry of quantum correlations⁵⁸. Let |ψ〉 be a pure bipartite state with the Schmidt decomposition $|\psi \rangle ={\sum }_{i=1}^{d}\sqrt{{\mu }_{i}}|ii\rangle$. The G-concurrence is defined as the geometric mean of the Schmidt coefficients^55,56

$$G(|\psi \rangle )\,:=d{({\mu }_{1}{\mu }_{2}\cdots {\mu }_{d})}^{\mathrm{1/}d}\mathrm{.}$$

(43)

For α > 1, we have

$$\begin{array}{rcl}{E}_{\alpha }(|\psi \rangle ) & = & \frac{1}{1-\alpha }\,\mathrm{log}\sum _{i}{\mu }_{i}^{\alpha }\\ & \leq & \frac{1}{1-\alpha }\,\mathrm{log}(d{({\mu }_{1}^{\alpha }\cdots {\mu }_{d}^{\alpha })}^{\frac{1}{d}})\\ & = & \frac{1}{(1-\alpha )}[\alpha \,\mathrm{log}\,d+\,\mathrm{log}\,{({\mu }_{1}\cdots {\mu }_{d})}^{\frac{\alpha }{d}}\\ & - & (\alpha -1)\mathrm{log}\,d]\\ & = & \frac{\alpha }{1-\alpha }\,\mathrm{log}\,G(|\psi \rangle )+\,\mathrm{log}\,d\mathrm{.}\end{array}$$

(44)

For 0 < α < 1, we have

$${E}_{\alpha }(|\psi \rangle )\ge \frac{\alpha }{1-\alpha }\,\mathrm{log}\,G(|\psi \rangle )+\,\mathrm{log}\,d\mathrm{.}$$

(45)

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 proved⁵⁹ 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^60,61. After completing this manuscript, we became aware of a recently related paper by Leditzky et al. in which they also obtained another lower bound of ERαE in terms of Rényi conditional entropy⁶².

Methods

Proof of the theorem

Suppose $\rho ={\sum }_{j}{p}_{j}|{\psi }_{j}\rangle \langle {\psi }_{j}|$ is the optimal decomposition of ERαE E _α (ρ), and the concurrence of $|{\psi }_{j}\rangle$ is denoted as c _j . Thus we have

$$\begin{array}{rcl}{E}_{\alpha }(\rho ) & = & \sum _{j}{p}_{j}{E}_{\alpha }(|{\psi }_{j}\rangle )=\sum _{j}{p}_{j}{H}_{\alpha }(\overrightarrow{\mu })\\ & \ge & \sum _{j}{p}_{j}co({R}_{L}({c}_{j}))\ge co[{R}_{L}(\sum _{j}{p}_{j}{c}_{j})]\\ & \ge & co[{R}_{L}(\underline{C})],\end{array}$$

(46)

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

$$\begin{array}{rcl}{E}_{\alpha }(\rho ) & = & \sum _{j}{p}_{j}{E}_{\alpha }(|{\psi }_{j}\rangle )=\sum _{j}{p}_{j}{H}_{\alpha }(\overrightarrow{\mu })\\ & \leq & \sum _{j}{p}_{j}ca({R}_{U}({c}_{j}))\leq ca[{R}_{U}(\sum _{j}{p}_{j}{c}_{j})]\\ & \leq & ca[{R}_{U}(\overline{C})],\end{array}$$

(47)

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 $|{\psi }_{i}\rangle ={\sum }_{j}\sqrt{{\mu }_{i,j}}|{a}_{i,j},{b}_{i,j}\rangle$ where μ _i,1 is the maximum Schmidt coefficient. For α > 1, we have

$$\begin{array}{rcl}\frac{\alpha }{\mathrm{2(}\alpha -\mathrm{1)}}{{\rm{G}}}_{{\rm{l}}}^{{\rm{c}}}(\rho ) & = & \frac{-\,\alpha }{\mathrm{2(}\alpha -\mathrm{1)}}\sum _{i}{p}_{i}\,\mathrm{log}\,{\mu }_{i\mathrm{,1}}^{2}\\ & = & \frac{-\,1}{\alpha -1}\sum _{i}{p}_{i}\,\mathrm{log}\,{\mu }_{i\mathrm{,1}}^{\alpha }\\ & \ge & \frac{-1}{\alpha -1}\sum _{i}{p}_{i}\,\mathrm{log}(\sum _{j}{\mu }_{i,j}^{\alpha })\\ & = & \sum _{i}{p}_{i}{E}_{\alpha }(|{\psi }_{i}\rangle )\\ & \ge & {E}_{\alpha }(\rho \mathrm{).}\end{array}$$

(48)

We have proved (38). For α = 1, let μ _i be the Schmidt coefficients of ρ, we have

$$\begin{array}{rcl}{E}_{\alpha }(\rho ) & = & S(\rho )=-\,\sum _{i}{\mu }_{i}\,\mathrm{log}\,{\mu }_{i}\\ & \ge & -\sum _{i}{\mu }_{i}\,\mathrm{log}\,\mathop{{\rm{\max }}}\limits_{j}\{{\mu }_{j}\}=-\,\mathrm{log}\,\mathop{{\rm{\max }}}\limits_{j}\{{\mu }_{j}\}\\ & = & {{\rm{G}}}_{{\rm{l}}}^{{\rm{c}}}(\rho ).\end{array}$$

(49)

We have proved (39). For α < 1, we have

$$\begin{array}{rcl}{E}_{\alpha }(\rho )+\frac{\alpha }{2(1-\alpha )}{{\rm{G}}}_{{\rm{l}}}^{{\rm{c}}}(\rho ) & = & {E}_{\alpha }(\rho )-\frac{\alpha }{\mathrm{2(1}-\alpha )}\sum _{i}{p}_{i}\,\mathrm{log}\,{\mu }_{i\mathrm{,1}}^{2}\\ & = & {E}_{\alpha }(\rho )-\frac{1}{1-\alpha }\sum _{i}{p}_{i}\,\mathrm{log}\,{\mu }_{i\mathrm{,1}}^{\alpha }\\ & = & {E}_{\alpha }(\rho )-\frac{1}{1-\alpha }(\sum _{i}{p}_{i}\,\mathrm{log}(d{\mu }_{i\mathrm{,1}}^{\alpha })-\,\mathrm{log}\,d)\\ & \leq & {E}_{\alpha }(\rho )-\frac{1}{1-\alpha }\sum _{i}{p}_{i}\,\mathrm{log}(\sum _{j}{\mu }_{i,j}^{\alpha })+11-\alpha \,\mathrm{log}\,d\\ & \leq & \frac{1}{1-\alpha }\,\mathrm{log}\,d\mathrm{.}\end{array}$$

(50)

The inequality holds because the pure state |ψ _i 〉 is in the d × d space. So we have proved (40).