A Supplemental material

General monogamy relation for the entanglement of formation in multiqubit systems

Abstract

We prove exactly that the squared entanglement of formation, which quantifies the bipartite entanglement, obeys a general monogamy inequality in an arbitrary multiqubit mixed state. Based on this kind of exotic monogamy relation, we are able to construct two sets of useful entanglement indicator: the first one can detect all genuine multiqubit entangled states even in the case of the two-qubit concurrence and -tangles being zero, while the second one can be calculated via quantum discord and applied to multipartite entanglement dynamics. Moreover, we give a computable and nontrivial lower bound for multiqubit entanglement of formation.

pacs:
03.65.Ud, 03.65.Yz, 03.67.Mn

For multipartite quantum systems, one of the most important properties is that entanglement is monogamous (1), which implies that a quantum system entangled with another system limits its entanglement with the remaining others (2). For entanglement quantified by the squared concurrence (3), Coffman, Kundu, and Wootters (CKW) proved the first quantitative relation (4) for three-qubit states, and Osborne and Verstraete proved the corresponding relation for -qubit systems, which reads (5)

(1)

Similar inequalities were also generalized to Gaussian systems (6); (7) and squashed entanglement (8); (9). As is known, the monogamy property can be used for characterizing the entanglement structure in many-body systems (4); (10). A genuine three-qubit entanglement measure named “three-tangle” was obtained via the monogamy relation of squared concurrence in three-qubit pure states (4). However, for three-qubit mixed states, there exists a special kind of entangled state that has neither two-qubit concurrence nor three-tangle  (11). There also exists a similar case for -qubit mixed states (12). To reveal this critical entanglement structure other exotic monogamy relations beyond the squared concurrence may be needed.

On the other hand, from a practical viewpoint, to calculate the entanglement measures appeared in the monogamy relation is basic. Unfortunately, except for the two-qubit case (3), this task is extremely hard (or almost impossible) for mixed states due to the convex roof extension of pure state entanglement (13). Quantum correlation beyond entanglement (e.g., the quantum discord (14); (15)) has recently attracted considerable attention, and various efforts have been made to connect quantum discord to quantum entanglement (16). It is natural to ask whether or not the calculation method for quantum discord can be utilized to characterize the entanglement structure and entanglement distribution in multipartite systems.

In this Letter, by analyzing the entanglement distribution in multiqubit systems, we prove exactly that the squared entanglement of formation (3) is monogamous in an arbitrary multiqubit mixed state. Furthermore, based on the exotic monogamy relation, we construct two sets of useful indicators overcoming the flaws of concurrence, where the first one can detect all genuine multiqubit entangled states and be utilized in the case when the concurrence and -tangles are zero, while the second one can be calculated via quantum discord and applied to a practical dynamical procedure. Finally, we give a computable and nontrivial lower bound for multiqubit entanglement of formation.

General monogamy inequality for squared entanglement of formation. – The entanglement of formation in a bipartite mixed state is defined as (13); (17),

(2)

where the minimum runs over all the pure state decompositions and is the von Neumann entropy of subsystem . For a two-qubit mixed state , Wootters derived an analytical formula (3)

(3)

where is the binary entropy and is the concurrence with the decreasing nonnegative s being the eigenvalues of the matrix .

A key result of this work is to show exactly that the bipartite entanglement quantified by the squared entanglement of formation obeys a general monogamy inequality in an arbitrary -qubit mixed state, i.e.,

(4)

where quantifies the entanglement in the partition (hereafter for qubit cases), and quantifies the one in the two-qubit system . Under two assumptions, a qualitative analysis on three-qubit pure states was given in Ref. (18). Before showing the general inequality, we first give the two propositions, whose analytical proofs are presented in the Supplemental Material (19).

Proposition I: The squared entanglement of formation in two-qubit mixed states varies monotonically as a function of the squared concurrence .

Proposition II: The squared entanglement of formation is convex as a function of the squared concurrence .

We now analyze the monogamy property of in an -qubit pure state . According to the Schmidt decomposition (20), the subsystem is equal to a logic qubit . Thus the entanglement can be evaluated using Eq. (3), leading to

(5)

where we have used the two propositions, with the details presented in (19).

At this stage, most importantly, we prove that the squared entanglement of formation is monogamous in an arbitrary -qubit mixed state . In this case, the analytical Wootters formula in Eq. (3) cannot be applied to , since the subsystem is not a logic qubit in general. But, we can still use the convex roof extension of pure state entanglement as shown in Eq. (2). Therefore, we have

(6)

where the minimum runs over all the pure state decompositions . We assume that the optimal decomposition for Eq. (6) takes the form

(7)

Under this decomposition, we have

(8)

where the is the average entanglement of formation under the specific decomposition in Eq. (7) and the parameter . Then we can derive the following monogamy inequality

(9)

where, in the second equation, the first term is non-negative because the is monogamous in pure state components, and the second term is also non-negative from a rigorous analysis shown in the Supplemental Material (19), justifying the monogamous relation. On the other hand, for the two-qubit entanglement of formation, the following relation is satisfied

(10)

since the is a specific average entanglement under the decomposition in Eq. (7), which is greater than in general. Combining Eqs. (9) and (10), we can derive the monogamy inequality of Eq. (4), such that we have completed the whole proof showing that the squared entanglement is monogamous in -qubit mixed states.

Two kinds of multipartite entanglement indicator. – Lohmayer et al (11) studied a kind of mixed three-qubit states composed of a state and a state

(11)

where , , and the parameter ranges in . They found that, when the parameter with and , the mixed state is entangled but without two-qubit concurrence and three-tangle. The three-tangle quantifies the genuine tripartite entanglement and is defined as (4) . It is still an unsolved problem on how to characterize the entanglement structure in this kind of states, although an explanation via the enlarged purification system was given (12).

Based on the monogamy inequality of in pure states, we can introduce a kind of indicator for multipartite entanglement in an -qubit mixed state as

(12)

where the minimum runs over all the pure state decompositions . This indicator can detect the genuine three-qubit entanglement in the mixed state specified in Eq. (11). After some analysis, we can get the optimal pure state decomposition for the three-qubit mixed state

(13)

where the pure state component and the parameter with . Then the indicator is

(14)

where and . In Fig.1, we plot the entanglement indicators , and in comparison to the indicators , and calculated originally in Ref. (11). As seen from Fig.1, although the three-tangle is zero when , the nonzero indicates the existence of the genuine three-qubit entanglement. This point may also be understood as a fact that the three-tangle indicates merely the -type entanglement while the newly introduced indicator can detect all genuine three-qubit entangled states.

Figure 1: (Color online) Entanglement indicators , and in comparison with the indicators , and in Ref. (11), where the nonzero detects the genuine three-qubit entanglement in the region.

For three-qubit mixed states, a state is called genuine tripartite-entangled if any decomposition into pure states contains at least one genuine tripartite-entangled component with and corresponding to the states of a single qubit or a couple of qubits (1). For the tripartite entanglement indicator , we have the following lemma and the proof can be found in the Supplemental Material (19).

Lemma 1: For three-qubit mixed states, the multipartite entanglement indicator is zero if and only if the quantum state is biseparable, i.e., .

When the three-qubit mixed state is genuine tripartite entangled, its optimal pure state decomposition contains at least one three-qubit entangled component. According to the lemma, we obtain that is surely nonzero.

For -qubit mixed states, when the indicator in Eq. (12) is zero, we can prove that there exists at most two-qubit entanglement in the partition (see lemmas b and c in (19)) and we further have the following lemma:

Lemma 2: In -qubit mixed states, the multipartite entanglement indicator

(15)

is zero if and only if the quantum state is -separable in the form , which has at most two-qubit entanglement with the superscript being all permutations of the qubits.

According to lemma 2, whenever an -qubit state contains genuine multiqubit entanglement, the indicator is surely nonzero. Thus this quantity can serve as a genuine multiqubit entanglement indicator in -qubit mixed states. The analytical proof of this lemma and its application to an -qubit mixed state (without two-qubit concurrence and -tangles) are presented in the Supplemental Material (19).

In general, the calculation of the indicators defined in Eqs. (12) and (15) is very difficult due to the convex roof extension. Here, based on the monogamy property of in mixed states, we can also introduce an alternative multipartite entanglement indicator as

(16)

which detects the multipartite entanglement (under the given partition) not stored in pairs of qubits (although this quantity is not monotone under local operations and classical communication  (19)). From the Koashi-Winter formula (8), the multiqubit entanglement of formation can be calculated by a purified state with ,

(17)

where is the quantum conditional entropy with being the von Neumann entropy, and the quantum discord is defined as (14); (15)

(18)

with the minimum running over all the POVMs and the measurement being performed on subsystem . Recent studies on quantum correlation provide some effective methods (21); (22); (23); (24); (25); (26); (27); (28); (29) for calculating the quantum discord, which can be used to quantify the indicator in Eq. (16). For all partitions, we may introduce a partition-independent indicator .

We now apply the indicator to a practical dynamical procedure of a composite system which is composed of two entangled cavity photons being affected by the dissipation of two individual -mode reservoirs. The interaction of a single cavity-reservoir system is described by the Hamiltonian (30) . When the initial state is with the dissipative reservoirs being in the vacuum state, the output state of the cavity-reservoir system has the form (30)

(19)

where with the amplitudes being and . As quantified by the concurrence, the entanglement dynamical property was addressed in Refs. (30); (31), but the multipartite entanglement analysis is mainly based on some specific bipartite partitions in which each party can be regarded as a logic qubit. When one of the parties is not equivalent to a logic qubit, the characterization for multipartite entanglement structure is still an open problem. For example, in the dynamical procedure, although the monogamy relation is satisfied, the entanglement is unavailable so far because subsystem is a four-level system and the convex roof extension is needed. Fortunately, in this case, we can utilize the presented indicator to indicate the genuine tripartite entanglement, where can be obtained via the quantum discord  (19). This indicator detects the genuine tripartite entanglement which does not come from two-qubit pairs. In Fig.2, the indicator and its entanglement components are plotted as functions of the time evolution and the initial amplitude , where the nonzero actually detects the tripartite entanglement area and the bipartite components of characterize the entanglement distribution in the dynamical procedure. By analyzing the multipartite entanglement structure, we can know that how the initial cavity photon entanglement transfers in the multipartite cavity-reservoir system, which provides the necessary information to design an effective method for suppressing the decay of cavity photon entanglement.

Figure 2: (Color online) The indicator and its entanglement components as functions of the time evolution and the initial amplitude , which detects the tripartite entanglement area and illustrates the entanglement distribution in the dynamical procedure.

Discussion and conclusion. – The entanglement of formation is a well-defined measure for bipartite entanglement and has the operational meaning in entanglement preparation and data storage (2). Unfortunately, it does not satisfy the usual monogamy relation. As an example, its monogamy score for the three-qubit state is . In this Letter, we show exactly that the squared entanglement is monogamous, which mends the gap of the entanglement of formation. Furthermore, in comparison to the monogamy of concurrence, the newly introduced indicators can really detect all genuine multiqubit entangled states and extend the territory of entanglement dynamics in many-body systems. In addition, via the established monogamy relation in Eq. (4), we can obtain

(20)

which provides a nontrivial and computable lower bound for the entanglement of formation.

In summary, we have not only proven exactly that the squared entanglement of formation is monogamous in -qubit mixed states but also provided a set of useful tool for characterizing the entanglement in multiqubit systems, overcoming some flaws of the concurrence. Two kinds of indicator have been introduced: the first one can detect all genuine multiqubit entangled states and solve the critical outstanding problem in the case of the two-qubit concurrence and -tangles being zero, while the second one can be calculated via quantum discord and applied to a practical dynamical procedure of cavity-reservoir systems when the monogamy of concurrence loses its efficacy. Moreover, the computable lower bound can be utilized to estimate the multiqubit entanglement of formation.

Acknowledgments. –This work was supported by the RGC of Hong Kong under Grant Nos. HKU7058/11P and HKU7045/13P. Y.-K.B. and Y.-F.X. were also supported by NSF-China (Grant No. 10905016), Hebei NSF (Grant No. A2012205062), and the fund of Hebei Normal University.

Note added. – Recently, by using the same assumptions as those made in (18), a similar idea on the monogamy of squared entanglement of formation was presented in (32), but the claimed monogamy for mixed states was not proven in that paper (33), in contrast to what we have done in the present work.

Appendix A Supplemental material

a.1 I. Proof of proposition I

Proposition I: The squared entanglement of formation in two-qubit mixed states varies monotonically as a function of the squared concurrence .

Proof: This proposition holds if the first-order derivative with . According to the formula in Eq. (3) of the main text, we have

(1)

The details for illustrating the positivity of Eq. (1) are as follows.

The inverse hyperbolic tangent function has the form

(2)

and the last two terms in Eq. (1) can be simplified as

(3)

Thus the first-order derivative is

(4)

in which

(5)

respectively. Due to , it is obvious that the term . For the term , we have and , which results in

(6)

Therefore, the second term in Eq. (4) is positive. For the third term, we have

(7)

where, in the first inequality, we use the property , and the last inequality is satisfied due to . Since , and , the first-order derivative in Eq. (1) is positive. Combining the fact with that corresponds to the minimum and corresponds to the maximum , we get that is a monotonically increasing function of , which completes the proof of the proposition.

a.2 II. Proof of proposition II

Proposition II: The squared entanglement of formation is convex as a function of the squared concurrence .

Proof: This proposition holds if the second-order derivative . After some deduction, we have

(8)

where is a non-negative factor.

Figure 1: (Color online)The factor is plotted as a function of , which is monotonically increasing in the region and decreasing in the region with the critical value .

The detailed derivation for the above result is as follows. In Eq. (8), when the parameter , the factor is positive. In this case, the positivity of is equivalent to

(9)

where . In order to analyze the sign of , we study its monotonic property. The first-order derivative of has the form

(10)

where the parameters are

(11)

respectively. Since , we have the factor . Therefore, according to Eq. (10), the function increases monotonically when the factor , and it decreases monotonically for the case . This property is shown in Fig.1, and the maximal value of corresponds to the critical point .

Now, we analyze two endpoints of for and . When , we can deduce

(12)

where in the second equation we have used the property . For the other endpoint , we can get . Combining it with the monotonic properties of , we can find , which is equivalent to in the region .

Furthermore, we analyze the value of second-order derivative at the endpoints. When , we can get

(13)

On the other hand, when , we have

(14)

Thus, we have shown the second-order derivative in the whole region , and then complete the proof of proposition II. In Fig.2, the derivative is plotted as a function of , which illustrates our result.

Figure 2: (Color online) The second-order derivative is plotted as a function of , which is positive and the two endpoints value are and , respectively.

a.3 III. Proof for the inequalities in Eq. (5)

According to the proposition in the Letter, we know that the squared entanglement of formation is monotonically increasing as a function of the squared concurrence . Combining this property with the monogamy relation of concurrence in Eq. (1) of the main text, we can derive the first inequality in Eq. (5) of Letter

(15)

Here, it should be emphasized that the composite system is in a pure state . The reason is that, in a generic mixed state , the relation between the entanglement of formation and the squared concurrence can not be characterized by Eq. (3) of Letter since the subsystem is not equivalent to a logic qubit.

Figure 3: (Color online) A schematic diagram for the monogamy inequality in Eq. (16). The blue line is the squared entanglement of formation and the green line is the squared concurrence , where the gradients and with , , and , respectively.

Furthermore, according to the proposition that the squared entanglement of formation is convex as a function of , we can derive

(16)

A schematic diagram for this inequality is shown in Fig.3. Due to the monotonic and convex property of ), we have the relations of gradients and , which give rise to

(17)

where , , and , respectively. By iterating the property of gradients and , we finally have the inequality

(18)

Combining Eqs. (18) and (15), we can obtain the inequalities in Eq. (5) of Letter.

a.4 IV. The second term in Eq. (9) being non-negative

In the Letter, the second term in Eq. (9) has the form

(19)

and here we prove that it is non-negative. For any pure state component in Eq. (7) of the main text, the monogamy property of is satisfied, and we have

(20)

For the two arbitrary pure state components and , we can get

(21)