Abstract
Monogamy and polygamy relations of quantum entanglement characterize the sharing and distribution of entanglement in a multipartite system. Multiqubit entanglement can be characterized entirely with bipartite combinations by saturating the monogamy and polygamy inequalities. In this paper, we tighten monogamy and polygamy constraints for the squared convex-roof extended negativity and its dual measure by employing a genetic algorithm. This evolutionary algorithm optimizes inequality residual functions to improve the monogamy and polygamy relations of these entanglement measures.
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Introduction
Entanglement is a quantum mechanical phenomenon enabling spatially separated parties to share quantum correlations in a manner that is not possible in classical systems1,2,3. To characterize, utilize, and quantify this unique phenomenon, entanglement measures, properties, and applications have been reported in the literature4,5,6,7,8.
One distinct property of entanglement is its limited shareability. This property is eloquently captured by the monogamy relation of entanglement9. The entanglement monogamy states that if two parties A and B are maximally entangled with each other, then they cannot be entangled with any third party C. More generally, individual bipartite quantum correlations are highly restricted by the amount of quantum correlations between C and AB. This statement can be further generalized to a multipartite scenario and similar restrictions on the amount of individual correlations can be imposed10,11,12,13. These monogamy relations provide a way to characterize different types of entanglement sharing. The monogamy of entanglement is also an important element in the analysis of quantum information protocols, such as quantum cryptography14 and quantum channel discrimination15. Entanglement of assistance is a notion that is dual to entanglement measures16. It can be viewed as the maximum amount of entanglement that the party C can distribute between A and B by performing measurements on his own subsystem17. While the quantum entanglement is monogamous, the entanglement of assistance is known to be polygamous18.
The concepts of monogamy and polygamy of the multipartite entangled state are concretely represented in the form of mathematical inequalities. Saturation of the monogamy inequality implies the complete characterization of multipartite entanglement19. On the other hand, the saturation of the polygamy inequality provides a finer characterization of the entanglement distribution20. Therefore, there are recent attempts to tightening these relations by raising the entanglement measures to a power and then utilizing some elementary mathematical inequalities21,22,23,24,25.
In this paper, we use a genetic algorithm (GA) to tighten the monogamy and polygamy inequalities. The GA belongs to a broad class of algorithms known as evolutionary algorithms (EAs)26. The EAs mimic the process of evolution in species over multiple generations27. Each generation consists of individuals whose fitness for an objective function is calculated. Survivals demonstrating higher fitness for the objective function are passed on to the next generation (either directly, or after crossover and mutation with other individuals), whereas the weaker individuals are removed. This process is stochastic and generally spans several generations28. The main advantage of the GA is that it can solve any optimization problem even if it is not convex. One problem associated with the GA is its possibility to give local minima29. Since the GA is easy to implement without any constraints, we use this algorithm to solve the optimization problem for tightness.
Our main focus in this paper is to tighten the monogamy and polygamy inequalities based on the squared convex-roof extended negativity (SCREN) and the SCREN of assistance (SCRENoA)13 for multipartite qubit systems. We first fit a shape (in terms of a mathematical expression) of the residual (the difference between both sides) of each inequality. We then optimize key parameters of the residual expression to tighten the inequality using the GA. This framework provides monogamy and polygamy inequalities that are significantly tighter than the other known bounds.
Results
Measures of Entanglement
Concurrence and negativity are well-known bipartite entanglement measures4,5,6. The monogamy inequality of concurrence holds true only for qubit systems while being violated for higher-dimensional (qudit) quantum systems9,12,30. In contrast, entanglement negativity, which is based on the positive partial transposition (PPT) criterion, holds the monogamy relation for some qudit systems as well13. For any bipartite quantum state ρAB with its partial transpose \({\rho }_{AB}^{{T}_{B}}\), its negativity is defined as6
where \(\parallel \rho {\parallel }_{1}={\rm{tr}}\sqrt{\rho {\rho }^{\dagger }}\) denotes the trace norm. Although the negativity is a computable measure of mixed-state entanglement for arbitrary dimensional quantum systems, there exist entangled states, known as PPT-bound entangled states, whose negativity is zero. For the ability to distinguish the PPT-bound entanglement from separability, an extension of negativity has been presented by the convex-roof construction7,31.
For a bipartite mixed state \({\rho }_{AB}={\sum }_{i}\,{p}_{i}{|{\psi }_{i}\rangle }_{AB}\langle {\psi }_{i}|\) where 0 ≤ pi ≤ 1, \(\forall i\), and \({\sum }_{i}\,{p}_{i}=1\), the SCREN and SCRENoA are defined respectively as23
where the minimization and maximization are over all possible pure-state decompositions of ρAB. Since the SCREN and SCRENoA reduce to the squared concurrence and its dual quantity (concurrence of assistance) for any two-qubit state, these measures provide their generalization without any known examples violating their properties even in higher dimensional quantum systems13. Hence, the monogamy and polygamy inequalities of multiqubit entanglement are given in terms of the SCREN and SCRENoA respectively as
for any N-qubit pure state \({|\psi \rangle }_{A{B}_{1}\cdots {B}_{N-1}}\) and its two-qubit reduced density matrices \({\rho }_{A{B}_{n}}\) of subsytems ABn, n = 1, 2, …, N − 1.
Tightening Monogamy and Polygamy Relations
We use two inequalities in this section, which will be obtained using the GA in Sec. Methods. Two inequalities are given by
where
Theorem 1.
For any multipartite pure state \({|\psi \rangle }_{{A}_{1}{A}_{2}\cdots {A}_{N}}\), we can always have \({|\psi \rangle }_{A{B}_{1}{B}_{2}\cdots {B}_{N-1}}\) after ordering and reindexing of its subsystems, such that
for n = 1, 2, …, N − 2. For any α ≥ 1, we have the monogamy relation as
Proof.
Since the SCREN is nonnegative, the monogamy inequality (4) can be rewritten as
for any α ≥ 1. For a multiqubit pure state \({|\psi \rangle }_{A{B}_{1}{B}_{2}\cdots {B}_{N-1}}\) with its reduced density matrices \({\rho }_{A{B}_{n}}\) for n = 1, 2, …, N − 1, we have
where the inequality (13) follows from (6), (10), and the fact that
and the last two inequalities are obtained by induction. Hence, using (12) and (14), we complete the proof. \(\square \)
Remark 1.
Since f(x; α) ≥ 2α − 1 ≥ α ≥ 1 for 0 ≤ x ≤ 1, we have
where the bounds (16–18) are used in the monogamy relations13,23,24. Hence, Theorem 1 provides a tighter inequality than these known bounds.
Remark 2.
Theorem 1 also holds true for any N-qubit mixed state \({\rho }_{A{B}_{1}\cdots {B}_{N-1}}\) due to the inequality32
Using the same arguments in Theorem 1, we get the tight monogamy inequality for multipartite mixed states as follows:
Theorem 2.
For any multipartite pure state \({|\psi \rangle }_{{A}_{1}{A}_{2}\cdots {A}_{N}}\), we can always have \({|\psi \rangle }_{A{B}_{1}{B}_{2}\cdots {B}_{N-1}}\) after ordering and reindexing of its subsystems, such that
for n = 1, 2, …, N − 2. For any 0 ≤ β ≤ 1, we have the polygamy relation as
Proof.
Since the SCRENoA is nonnegative, the polygamy inequality (5) can be rewritten as
for 0 ≤ β ≤ 1. Using the same arguments in the proof of Theorem 1, we get
which complete the proof.\(\square \)
Remark 3.
Since g(x; β) ≤ 2β − 1 ≤ β ≤ 1 for 0 ≤ β ≤ 1, we have
where the bounds (25) and (26) are used in the polygamy relations13,23. Hence, Theorem 2 provides a tighter polygamy inequality than these known bounds.
Discussion
For a numerical example, we consider a generalized tripartite qubit system33
where μi ≥ 0, \(\forall i\), and \({\sum }_{i=0}^{4}\,{\mu }_{i}^{2}=1\). For this tripartite qubit system, the SCREN and SCRENoA are computed as
To demonstrate the tightness of monogamy and polygamy inequalities in Theorems 1 and 2, we distribute μi’s in (27) using the eigenvalues of the following exponential correlation matrix34
where \(\zeta \in [0,1]\) is a correlation coefficient. Let λ0 ≥ λ1 ≥ λ2 ≥ λ3 ≥ λ4 ≥ 0 be the eigenvalues of R in decreasing order and set \({\mu }_{i}^{2}={\lambda }_{i}\), \(i=0,1,2,3,4\). Since tr(R) = 1, we have \({\sum }_{i=0}^{4}\,{\mu }_{i}^{2}=1\). Using (28) and (29), we plot the monogamy and polygamy inequalities in Theorems 1 and 2 as a function of ζ for the tripartite qubit system (27) in Fig. 1 when (a) α = 5 and (b) β = 0.5, respectively. The known bounds (16–18), (25), and (26) for the monogamy and polygamy relations are also depicted for comparison.
We can see from (28) and (29) that a tripartite state with μ4 = 0, e.g., the W-class state (μ1 = μ4 = 0), saturates the monogamy and polygamy inequalities, while a tripartite state with μ2 = μ3 = 0, e.g., the GHZ-class state (μ1 = μ2 = μ3 = 0), yields the maximum residuals of monogamy and polygamy inequalities. Tightening the monogamy and polygamy inequalities enables us to precisely characterize the entanglement sharing and distribution in a multipartite scenario. Our framework can also be used in other entanglement measures such as the entanglement of formation, Tsallis entropy, Rényi entropy, and unified entropy for qubit systems.
Methods
In this section, we first tighten a known inequality, which is used for obtaining tight monogamy relations, by identifying a mathematical expression for the residual of this inequality and fine-tuning its parameters by the GA. Next, we derive an inequality, which is tighter than the known results for polygamy, and then further tighten this inequality by using again the GA.
Tightening the inequality by fitting a parametric form of its residual is a nonlinear optimization problem. We can employ the GA to perform this optimization due to its ability to handle discontinuous, nonlinear, and nondifferentiable objective functions26,27,28,29. Specifically, the GA randomly generates candidate solutions within given constraints and mimics the evolution process (the survival of the fittest) in searching the optimal solution. The promising candidates from one generation are identified and utilized to produce the next generation of candidate solutions. This process is iterated over several generations until a stopping condition is satisfied. A stochastic search of the GA sometimes leads to local optima that can steer the search in a wrong direction. To overcome this problem, we can work on the population size, mutation rate, crossover probability, and termination condition26,27,28,29. In our optimization problem, we increase the population size more than 10,000, set the termination condition to be strictly 10−30, and restrict the lower limit of parameter variables to be nonnegative by looking at the landscape of our fitness function.
We begin with the known inequality22
where 0 ≤ x ≤ 1 and α ≥ 1. The inequality residual (1 + x)α − 1 − (2α − 1)xα is plotted as a function of (x, α) in Fig. 2(a). We identify this curve to be of the form
where u = [u1, u2, u3, u4, u5] is a parameter vector to be optimized for tightening the inequality; u1 is a scaling parameter; and u2, u3, u4, and u5 are shape parameters. Now, we can formulate our optimization problem for 0 ≤ x ≤ 1 and α ≥ 1 as follows:
By solving this optimization problem with the GA, we find the best parameter vector
leading to a tigher inequality (6). The residual (1 + x)α − 1 − f(x; α)xα of the inequality (6) is plotted as a function of (x, α) in Fig. 2(b). It can be seen from Fig. 2 that this optimized inequality (6) is significantly tighter than the inequality (31).
For the polygamy inequality, we first derive an opposite-side inequality and then take the same steps to tighten it using the GA.
Lemma 1.
For 0 ≤ x ≤ 1 and 0 ≤ β ≤ 1, we have
Proof.
Let h(y; β) = (1 + y)β − yβ. Then, for y ≥ 1 and 0 ≤ β ≤ 1, we have
which implies that the function is decreasing in y ≥ 1. Hence,
for y ≥ 1. Plugging y = 1/x in (37), we complete the proof.\(\square \)
The residual 1 + (2β − 1) − xβ(1 + x)β is plotted as a function of (x, β) in Fig. 3(a) and this curve is fitted to the following expression
where v = [v1, v2, v3, v4] is a parameter vector to be optimized for tightening the inequality. To further tighten the inequality (35), we now formulate an optimization problem for 0 ≤ x ≤ 1 and 0 ≤ β ≤ 1 as follows:
Using the GA, we obtain the parameter vector
leading to a tigher inequality (7). The residual 1 + g(x; β)xβ − (1 + x)β of the inequality (7) is plotted as a function of (x, β) in Fig. 3(b) for comparison of tightness with the inequality (35).
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Acknowledgements
This work was supported, in part, by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1A2B2014462), and by the ICT R&D program of MSIP/IITP [R0190-15-2030, Reliable crypto-system standards and core technology development for secure quantum key distribution network].
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A.F. and J.S.K. contributed the idea. A.F., J.u.R., Y.J. and J.S.K. developed the theory. H.S. improved the manuscript and supervised the research. All the authors contributed in analyzing and discussing the results and improving the manuscript.
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Farooq, A., ur Rehman, J., Jeong, Y. et al. Tightening Monogamy and Polygamy Inequalities of Multiqubit Entanglement. Sci Rep 9, 3314 (2019). https://doi.org/10.1038/s41598-018-37731-z
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DOI: https://doi.org/10.1038/s41598-018-37731-z
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