University:
Purdue University
Course:
CHM 67300 | Computational Quantum Chemistry
Academic year:

2007
Views:

321

Pages:

43
Author:

Pilzstiel1j

ENTANGLEMENT, ELECTRON CORRELATION, AND DENSITY MATRICES CONTENTS I. Introduction A. Entanglement of Formation and Concurrence B. Entanglement Measure for Fermions C. Entanglement and Ranks of Density Matrices II. Entanglement for Spin Systems A. Entanglement for Two-Spin Systems B. Entanglement for One-Dimensional N-Spin Systems C. Numerical Solution of the One-Dimensional Spin-12 Systems D. Entanglement and Spin Reduced Density Matrices E. Some Numerical Results F. Thermal Entanglement and the Effect of Temperature G. Entanglement for Two-Dimensional Spin Systems III. Entanglement for Quantum Dot Systems A. Two-Electron Two-Site Hubbard Model 1. Exact Solution 2. Hartree–Fock Approximation 3. Correlation Entropy 4. Entanglement B. One-Dimensional Quantum Dots System C. Two-Dimensional Array of Quantum Dots IV Ab Initio Calculations and Entanglement V. Dynamics of Entanglement and Decoherence VI. Entanglement and Density Functional Theory VII. Future Directions Acknowledgments References I. INTRODUCTION In quantum chemistry calculations, the correlation energy is defined as the energy error of the Hartree–Fock wavefunction, that is, the difference between the Hartree–Fock limit energy and the exact solution of the nonrelativistic Schrödinger equation [1]. Different types of electron correlation are often distinguished in quantum chemistry such as dynamical and nondynamical [2], radial versus angular correlation for atoms, left–right, in–out and, radial correlation for diatomic molecules, and weak and strong correlation for solids. There also exists other measures of electron correlation in the literature such as the statistical correlation coefficients [3] and more recently the Shannon entropy as a measure of the correlation strength [4–8]. Correlation of a quantum many-body state makes the one-particle density matrix nonidempotent. Therefore the Shannon entropy of the natural occupation numbers measures the correlation strength on the one-particle level [7]. Electron correlations have a strong influence on many atomic, molecular [9], and solid properties [10]. The concept of electron correlation as defined in quantum chemistry calculations is useful but not directly observable; that is, there is no operator in quantum mechanics that its measurement gives the correlation energy. Moreover, there are cases where the kinetic energy dominates the Coulomb repulsion between electrons, so the electron correlation alone fails as a correlation measure [6]. Entanglement is a quantum mechanical property that describes a correlation between quantum mechanical systems and has no classical analogue [11–15]. Schrödinger was the first to introduce these states and gave them the name ‘‘Verschränkung’’ to a correlation of quantum nature [16]: ‘‘For an entangled state the best possible knowledge of the whole does not include the best possible knowledge of its parts.’’ Latter, Bell [17] defined entanglement as ‘‘a correlation that is stronger than any classical correlation.’’ Thus it might be useful as an alternative measure of electron–electron correlation in quantum chemistry calculations. Ever since the appearance of the famous EPR Gadanken experiment [18], the phenomenon of entanglement [19], which features the essential difference between classical and quantum physics, has received wide theoretical and experimental attention [17, 20–25]. Generally, if two particles are in an entangled state then, even if the particles are physically separated by a great distance, they behave in some respects as a single entity rather than as two separate entities. There is no doubt that the entanglement has been lying in the heart of the foundation of quantum mechanics. A desire to understand quantum entanglement is fueled by the development of quantum computation, which started in the 1980s with the pioneering work of Benioff [26], Bennett [27], Deutsch [28], Feynman [29] and Landauer [30] but gathered momentum and research interest only after Peter Shor’s revolutionary entanglement, electron correlation, and density matrices discovery [31] of a quantum computer algorithm in 1994 that would efficiently find the prime factors of composite integers. Since integer factorization is the basis for cryptosystems used for security nowadays, Shor’s finding will have a profound effect on cryptography. The astronomical power of quantum computations has researchers all over the world racing to be the first to create a practical quantum computer. Besides quantum computations, entanglement has also been at the core of other active research such as quantum teleportation [32, 33], dense coding [34, 35], quantum communication [36], and quantum cryptography [37]. It is believed that the conceptual puzzles posed by entanglement have now become a physical source of novel ideas that might result in applications. A big challenge faced by all of the above-mentioned applications is to prepare the entangled states, which is much more subtle than classically correlated states. To prepare an entangled state of good quality is a preliminary condition for any successful experiment. In fact, this is not only an experimental problem but also poses an obstacle to theories, since how to quantify entanglement is still unsettled; this is now becoming one of the central topics in quantum information theory. Any function that quantifies entanglement is called an entanglement measure. It should tell us how much entanglement there is in a given mutipartite state. Unfortunately, there is currently no consensus as to the best method to define an entanglement for all possible multipartite states. And the theory of entanglement is only partially developed [13, 38–40] and for the moment can only be applied in a limited number of scenarios, where there is an unambiguous way to construct suitable measures. Two important scenarios are (i) the case of a pure state of a bipartite system, that is, a system consisting of only two components and (ii) a mixed state of two spin1 2 particles. When a bipartite quantum system AB describe by HA HB is in a pure state, there is an essentially well-motivated and unique measure of the entanglement between the subsystems A and B given by the von Neumann entropy S. If we denote with rA the partial trace of r 2 HA HB with respect to subsystem B, rA ¼ TrB ðrÞ, the entropy of entanglement of the state r is defined as the von Neumann entropy of the reduced density operator rA, SðrÞ Tr½rA log2 rA . It is possible to prove that, for the pure state, the quantity S does not change if we exchange A and B. So we have SðrÞ Tr½rA log2 rA Tr½rB log2 rB : For any bipartite pure state, if an entanglement EðrÞ is said to be a good one, it is often required to have the following properties [14]: Separability: If r is separable, then EðrÞ ¼ 0. Normalization: The entanglement of a maximally entangled state of two d-dimensional systems is given by E ¼ logðdÞ. No Increase Under Local Operations: Applying local operations and classically communicating cannot increase the entanglement of r. Continuity: In the limit of vanishing distance between two density matrices, the difference between their entanglement should tend to zero. Additivity: A certain number N of identical copies of the state r should contain N times the entanglement of one copy. Subadditivity: The entanglement of the tensor product of two states should not be larger that the sum of the entanglement of each of the states. Convexity: The entanglement measure should be a convex function, that is, Eðlr þ ð1 lÞsÞ lEðrÞ þ ð1 lÞEðsÞ for 0 < l < 1: For a pure bipartite state, it is possible to show that the von Neumann entropy of its reduced density matrix, Sðrred Þ ¼ Trðrred log2 rred Þ, has all the above properties. Clearly, S is not the only mathematical object that meets the requirement, but in fact, it is now basically accepted as the correct and unique measure of entanglement. The strict definitions of the four most prominent entanglement measures can be summarized as follows [14]: Entanglement of Entanglement of Entanglement of Relative entropy distillation ED . cost EC . formation EF . of entanglement ER . The first two measures are also called operational measures, while the second two don’t admit a direct operational interpretation in terms of entanglement manipulations. Suppose E is a measure defined on mixed states that satisfy the conditions for a good measure mentioned above. Then we can prove that for all states r 2 ðH A H B Þ, ED ðrÞ EðrÞ EC ðrÞ, and both ED ðrÞ and EC ðrÞ coincide on pure states with the von Neumann reduced entropy as demonstrated earlier. A. Entanglement of Formation and Concurrence At the current time, there is no simple way to carry out the calculations with all these entanglement measures. Their properties, such as additivity, convexity, and continuity, and relationships are still under active investigation. Even for the best-understood entanglement of formation of the mixed states in bipartite systems AB, once the dimension or A or B is three or above, we don’t know how to express it simply, although we have the general definitions given previously. However, for the case where both subsystems A and B are spin-12 particles, there exists a simple formula from which the entanglement of formation can be calculated [42]. entanglement, electron correlation, and density matrices Given a density matrix r of a pair of quantum systems A and B and all possible pure-state decompositions of r r¼ X pi jci ihci j ð1Þ i where pi are the probabilities for ensembles of states jci i, the entanglement E is defined as the entropy of either of the subsystems A or B: EðcÞ ¼ TrðrA log2 rA Þ ¼ TrðrB log2 rB Þ ð2Þ The entanglement of formation of the mixed r is then defined as the average entanglement of the pure states of the decomposition [42], minimized over all decompositions of r: EðrÞ ¼ min X pi Eðci Þ ð3Þ i For a pair of qubits this equation can be written [42–44] EðrÞ ¼ eðCðrÞÞ ð4Þ where e is a function of the ‘‘concurrence’’ C: eðCÞ ¼ h 1þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 C2 2 ð5Þ where h in the binary entropy function [20] hðxÞ ¼ x log2 x ð1 xÞ log2 ð1 xÞ ð5Þ In this case the entanglement of formation is given in terms of another entanglement measure, the concurrence C [42–44]. The entanglement of formation varies monotonically with the concurrence. From the density matrix of the two-spin mixed states, the concurrence can be calculated as follows: CðrÞ ¼ max½0; l1 l2 l3 l4 ð6Þ where li are the eigenvalues in decreasing order of the Hermitian matrix qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ R r r r with r ¼ ðsy sy Þr ðsy sy Þ. Here sy is the Pauli matrix of the spin in the y direction. The concurrence varies from C ¼ 0 for a separable state to C ¼ 1 for a maximally entangled state. The concurrence as a measure of entanglement will be used in Section II to discuss tuning and manipulating the entanglement for spin systems. B. Entanglement Measure for Fermions As we discussed in the previous section, for distinguishable particles, the most suitable and famous measure of entanglement is Wootters’ measure [42], the entanglement of formation or concurrence. Recently, Schlieman and co-workers [45, 46] examined the influence of quantum statistics on the definition of entanglement. They discussed a two-fermion system with the Slater decomposition instead of Schmidt decomposition for the entanglement measure. If we take each of the indistinguishable fermions to be in the single-particle Hilbert space CN , with fm ; fmþ ðm ¼ 1; . . . ; NÞ denoting the fermionic annihilation and creation operators of single-particle states and ji representing the vacuum state, then a pure two-electron state can be written X omn fmþ fnþ ji; m;n where omn ¼ onm . Analogous to the Schmidt decomposition, it can be proved that every ji can be represented in an appropriately chosen basis in C N in a form of Slater decomposition [45], K X 1 zi faþ1 ðiÞ faþ2 ðiÞ ji ji ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PK 2 i¼1 i¼1 jzi j ð7Þ where faþ1 ðiÞ ji; faþ2 ðiÞ ji, i ¼ 1; :::; K; form an orthonormal basis in CN . The number of nonvanishing coefficients zi is called the Slater rank, which is then used for the entanglement measure. With similar technique, the case of a two-boson system is studied by Li et al. [47] and Paškauskas and you [48]. Gittings and Fisher [49] put forward three desirable properties of any entanglement measure: (i) invariance under local unitary transformations; (ii) noninvariance undernonlocal unitary transformations; and (iii) correct behavior as distinguishability of the subsystems is lost. These requirements make the relevant distinction between one-particle unitary transformation and one-site unitary transformations. A natural way to achieve this distinction [49] is to use a basis based on sites rather than on particles. Through the Gittings–Fisher investigation, it is shown that all of the above-discussed entanglement measures fail the tests of the three criteria. Only Zanardi’s entanglement, electron correlation, and density matrices measure [50] survives, which is given in Fock space as the von Neumann entropy, namely, Ej ¼ Trrj log2 rj ; rj ¼ Trj jcihcj ð8Þ where Trj denotes the trace over all but the jth site and c is the antisymmetric wavefunction of the studied system. Hence Ej actually describes the entanglement of the jth site with the remaining sites. A generalization of this one-site entanglement is to define an entanglement between one L-site block with the rest of the system [51], EL ¼ TrðrL log2 rL Þ C. ð9Þ Entanglement and Ranks of Density Matrices In this section we review the known theorems that relate entanglement to the ranks of density matrices [52]. The rank of a matrix r, denoted as rankðrÞ, is the maximal number of linearly independent row vectors (also column vectors) in the matrix r. Based on the ranks of reduced density matrices, one can derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement [53]. For convenience, let us introduce the following definitions [54–56]. A pure state r of N particles A1 ; A2 ; . . . ; AN is called entangled when it cannot be written r ¼ rA1 rA2 rAN ¼ N O rAi ð10Þ i¼1 where rAi is the single-particle reduced density matrix given by rAi TrfAj g ðrÞ for fAj jall Aj 6¼ Ai g. A mixed state r of N particles P A1 ; A2j; :::; AN , described by M probabilities pj and M pure states rj as r ¼ M j¼1 pj r , is called entangled when it cannot be written r¼ M X j¼1 pj N O rjAi ð11Þ i¼1 P where pj > 0 for j ¼ 1; 2; :::; M with M j¼1 pj ¼ 1. Now we are in a position to list the separability conditions without proof. (The reader who is interested in the formal proofs can consult the paper by Chong, Keiter, and Stolze [53]. Lemma 1 A state is pure if and only if the rank of its density matrix r is equal to 1, that is, rankðrÞ ¼ 1. Lemma 2 A pure state is entangled if and only if the rank of at least one of its reduced density matrices is greater than 1. Lemma 3 Given a pure state r, if its particles are separated into two parts U and V, then rankðrU Þ ¼ 1 holds if and only if these two parts are separable, that is, r ¼ rU rV . Now we can discuss the necessary conditions for separable states. For convenience, we will use the following notation. For a state r of N particles A1 ; A2 ; . . . ; AN , the reduced density matrix obtained by tracing r over particle Ai is written rRðiÞ ¼ TrAi ðrÞ, where RðiÞ denotes the set of the remaining ðN 1Þ particles other than particle Ai . In the same way, rRði;jÞ ¼ TrAj ðrRðiÞ Þ ¼ TrAj ðTrAi ðrÞÞ ¼ TrAi ðTrAj ðrÞÞ denotes the reduced density matrix obtained by tracing r over particles Ai and Aj , rRði;j;kÞ ¼ TrAi ðTrAj ðTrAk ðrÞÞÞ, and so on. In view of these relations, r can be called the 1-level-higher density matrix of rRðiÞ and 2-level-higher density matrix of rRði;jÞ ; rRðiÞ can be called the 1-level-higher density matrix of rRði;jÞ and 2-level-higher density matrix of rRði;j;kÞ ; and so on. Now let us define the separability condition theorem [53]. If a state r of N particles A1 ; A2 ; :::; AN is separable, then the rank of any reduced density matrix of r must be less than or equal to the ranks of all of its 1-level-higher density matrices; that is, rankðrRðiÞ Þ ð12Þ rankðrÞ holds for any Ai 2 fA1 ; A2 ; . . . ; AN g; and rankðrRði;jÞ Þ rankðrRðiÞ Þ; rankðrRði;jÞ Þ rankðrRðjÞ Þ ð13Þ holds for any pair of all particles. This will lead to the conditions for a mixed state to be entangled. Given a mixed state r, if the rank of at least one of the reduced density matrices of r is greater than the rank of one of its 1-level-higher density matrices, then the state r is entangled. II. ENTANGLEMENT FOR SPIN SYSTEMS A. Entanglement for Two-Spin Systems We consider a set of N localized spin-12 particles coupled through exchange interaction J and subject to an external magnetic field of strength B. In this section we entanglement, electron correlation, and density matrices will demonstrate that (i) entanglement can be controlled and tuned by varying the anisotropy parameter in the Hamiltonian and by introducing impurities into the systems; (ii) for certain parameters, the entanglement is zero up to a critical point lc , where a quantum phase transition occurs, and is different from zero above lc ; and (iii) entanglement shows scaling behavior in the vicinity of the transition point. For simplicity, let us illustrate the calculations of entanglement for two spin-12 particles. The general Hamiltonian, in atomic units, for such a system is given by [57] J J H ¼ ð1 þ gÞsx1 sx2 ð1 gÞsy1 sy2 Bsz1 I2 BI1 sz2 2 2 ð14Þ where sa are the Pauli matrices (a ¼ x; y; z) and g is the degree of anisotropy. For g ¼ 1 Eq. (14) reduces to the Ising model, whereas for g ¼ 0 it is the XY model. This model admits an exact solution; it is simply a ð4 4Þ matrix of the form 0 2B B 0 B H¼@ 0 Jg 0 0 J 0 1 0 Jg J 0 C C 0 0 A 0 2B ð15Þ with the following four eigenvalues, l1 ¼ J; l2 ¼ J; pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l3 ¼ 4B2 þ J 2 g2 ; l4 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4B2 þ J 2 g2 ð16Þ and the corresponding eigenvectors, 0 1 0 p ﬃﬃ ﬃ B 1= 2 C B C jf1 i ¼ B pﬃﬃﬃ C; @ 1= 2 A 1 0 p ﬃﬃ ﬃ B 1= 2 C B pﬃﬃﬃ C jf2 i ¼ B C @ 1= 2 A 0 0 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 a þ 2B B 2a C B C B C 0 B C jf3 i ¼ B C; B C 0 B rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C @ a 2B A 2a 0 0 0 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 a 2B B 2a C B C B C 0 B C jf4 i ¼ B C B C 0 B rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C @ a þ 2B A 2a ð17Þ ð18Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where a ¼ 4B2 þ J 2 g2 . In the basis set fj ""i; j "#i; j #"i; j ##ig, the eigenvectors can be written 1 jf1 i ¼ pﬃﬃﬃ ðj #"i þ j "#iÞ 2 1 jf2 i ¼ pﬃﬃﬃ ðj #"i j "#iÞ 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a 2B a þ 2B jf3 i ¼ j ##i þ j ""i 2a 2a rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a þ 2B a 2B j ##i j ""i jf4 i ¼ 2a 2a ð19Þ ð20Þ ð21Þ ð22Þ Now we confine our interest to the calculation of entanglement between the two spins. For simplicity, we take g ¼ 1; Eq. (14) reduces to the Ising model with the ground-state energy l3 and the corresponding eigenvector jf3 i. All the information needed for quantifying the entanglement in this case is contained in the reduced density matrix rði; jÞ [42–44]. For our model system in the ground state jf3 i, the density matrix in the basis set ð""; "#; #"; ##) is given by 0 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 a þ 2B a2 4B2 0 0 B C B 4a2 C 2a B C 0 0 0 0 C ð23Þ r¼B B C 0 0 ﬃ 0 0 B rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C @ a2 4B2 a þ 2B A 0 0 4a2 2a The eigenvalues of the Hermitian matrix R needed to calculate the concurrence [42], C, Eq. (6), can be calculated analytically. We obtained l2 ¼ l3 ¼ l4 ¼ 0 and therefore sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l2 ð24Þ CðrÞ ¼ l1 ¼ 4 þ l2 where l ¼ J=B. Entanglement is a monotonically increasing function of the concurrence and is given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ EðCÞ ¼ hðyÞ ¼ y log2 y ð1 yÞ log2 ð1 yÞ; y ¼ 12 þ 12 1 C 2 ð25Þ Substituting the value of the concurrence C, Eq. (24) gives pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 1 1 4 þ l2 2 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ log2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E ¼ log2 2 2 4 4þl 4 þ l2 4 þ l2 þ 2 ð26Þ entanglement, electron correlation, and density matrices This result for entanglement is equivalent to the von Neumann entropy of the reduced density matrix rA . For our model system of the form AB in the ground state jf3 i, the reduced density matrix rA ¼ TrB ðrAB Þ in the basis set ð"; #Þ is given by 0 1 a þ 2B 0 B C ð27Þ rA ¼ @ 2a a 2B A 0 2a As we mentioned before, when a biparticle quantum system AB is in a pure state, there is essentially a unique measure of the entanglement between the subsystems A and B given by the von Neumann entropy S Tr½rA log2 rA . This approach gives exactly the same formula as the one given in Eq. (26). This is not surprising since all entanglement measures should coincide on pure bipartite states and be equal to the von Neumann entropy of the reduced density matrix (uniqueness theorem). This simple model can be used to examine the entanglement for two-electron diatomic molecules. The value of J, the exchange coupling constant between the spins of the two electrons, can be calculated as half the energy difference between the lowest singlet and triplet states of the hydrogen molecule. Herring and Flicker [58] have shown that J for the H2 molecule can be approximated as a function of the interatomic distance R. In atomic units, the expression for large R is given by JðRÞ ¼ 0:821 R5=2 e2R þ OðR2 e2R Þ ð28Þ Figure 1 shows the calculated concurrence CðrÞ as a function of the distance between the two electronic spins R, using JðRÞ of Eq. (28), for different values of the magnetic field strength B. At the limit R ! 1 the exchange interaction J vanishes as a result of the two electronic spins being up and the wavefunction being factorizable; that is, the concurrence is zero. At the other limit, when R ¼ 0 the concurrence or the entanglement is zero for this model because J ¼ 0. As R increases, the exchange interaction increases, leading to increasing concurrence between the two electronic spins. However, this increase in the concurrence reaches a maximum limit as shown in the figure. For large distance, the exchange interaction decreases exponentially with R and thus the decrease of the concurrence. Figure 1 also shows that the concurrence increases with decreasing magnetic field strength. This can be attributed to effectively increasing the exchange interaction. This behavior of the concurrence as a function of the internuclear distance R is typical for two-electron diatomic molecules. We will show later in Section IV that by using accurate ab initio calculations we essentially obtain qualitatively the same curve for entanglement for the H2 molecule as a function of the internuclear distance R. 1 B = 0.01 0.9 Concurrence (C ) 0.8 0.7 B = 0.005 B = 0.05 0.6 0.5 0.4 B = 0.1 0.3 0.2 0.1 0 0 1 2 3 4 R(Å) 5 6 7 8 Figure 1. The concurrence (C) as a function of the distance R between the two spins for different values of the magnetic field strength B. B. Entanglement for One-Dimensional N-Spin Systems Now let us generalize it to a one-dimensional lattice with N sites in a transverse magnetic field and with impurities. The Hamiltonian for such a system is given by [59] H¼ N N N X 1þgX 1 gX Ji;iþ1 sxi sxiþ1 Ji;iþ1 syi syiþ1 Bi szi 2 i¼1 2 i¼1 i¼1 ð29Þ where Ji;iþ1 is the exchange interaction between sites i and i þ 1, Bi is the strength of the external magnetic field at site i, sa are the Pauli matrices (a ¼ x; y; z), g is the degree of anisotropy, and N is the number of sites. We assume cyclic boundary conditions, so that sxNþ1 ¼ sx1 ; syNþ1 ¼ sy1 ; szNþ1 ¼ sz1 ð30Þ For g ¼ 1 the Hamiltonian reduces to the Ising model and for g ¼ 0 to the XY model. For the pure homogeneous case, Ji;iþ1 ¼ J and Bi ¼ B, the system exhibits a quantum phase transition at a dimensionless coupling constant entanglement, electron correlation, and density matrices l ¼ J=2B ¼ 1. The magnetization hsx i is different from zero for l > 1 and it vanishes at the transition point. The magnetization along the z direction hsz i is different from zero for any value of l. At the phase transition point, the correlation length x diverges as x jl lc jn with n ¼ 1 [60]. C. Numerical Solution of the One-Dimensional Spin-12 Systems The standard procedure used to solve Eq. (29) is to transform the spin operators into fermionic operators [61]. Let us define the raising and lowering operators aþ i , ai ,: y 1 x aþ i ¼ 2 ðsi þ isi Þ; y 1 x a i ¼ 2 ðsi isi Þ Then we introduce the Fermi operators ci ,cþ i defined by a i ¼ exp pi i1 X ! cþ j cj ci ; aþ i ¼ cþ i exp pi i1 X j¼1 ! cþ j cj j¼1 . So the Hamiltonian assumes the following quadratic form: H¼ N X þ þ Ji;iþ1 ½ðcþ i ciþ1 þ gci ciþ1 Þ þ h:c: 2 i¼1 N X i¼1 1 Bi ðcþ i ci Þ 2 ð31Þ l ¼ J=2B We can write the parameters Ji;iþ1 ¼ Jð1 þ ai;iþ1 Þ, where a introduces the impurity in the exchange interactions, and the external magnetic field takes the form Bi ¼ Bð1 þ bi Þ, where b measures the impurity in the magnetic field. When a ¼ b ¼ 0 we recover the pure XY case. Introducing the matrices A, B, where A is symmetrical and B is antisymmetrical, we can rewrite the Hamiltonian: N X 1 þ þ þ ci Ai;j cj þ ðci Bi; j cj þ h:cÞ H ¼ 2 i; j¼1 0 Introducing linear transformation, we have Zk ¼ N X i¼1 gki ci þ hki cþ i ; Zþ k ¼ N X i¼1 gki cþ i þ hki ci with the gki and hki real and which will give the Hamiltonian form H¼ N X k Zþ k Zk þ constant k From these conditions, we can get a set of equations for the gki and hki : k gki ¼ N X ðgkj Aji hkj Bji Þ ð32Þ ðgkj Bji hkj Aji Þ ð33Þ j¼1 k hki ¼ N X j¼1 By introducing the linear combinations fki ¼ gki þ hki ; cki ¼ gki hki we can get the coupled equation fk ðA BÞ ¼ Kk ck and ck ðA þ BÞ ¼ Kk fk Then we can get both fk and ck vectors from these two equations by the numerical method [62]. The ground state of the system corresponds to the state of ‘‘noparticles’’ and is denoted as j0 i, and Zk j0 i ¼ 0; for all k D. Entanglement and Spin Reduced Density Matrices The matrix elements of the reduced density matrix needed to calculate the entanglement can be written in terms of the spin–spin correlation functions and the average magnetization per spin. The spin–spin correlation functions for the ground state are defined as [62] Sxlm ¼ 14 h0 jsxl sxm j0 i Sylm ¼ 14 h0 jsyl sym j0 i Szlm ¼ 14 h0 jszl szm j0 i and the average magnetization per spin is Miz ¼ 12 h0 jszi j0 i ð34Þ entanglement, electron correlation, and density matrices These correlation functions can be obtained using the set ck and fk from the previous section. The structure of the reduced density matrix follows from the symmetry properties of the Hamiltonian. However, for this case the concurrence Cði; jÞ depends on i; j and the location of the impurity and not only on the difference ji jj as for the pure case. Using the operator expansion for the density matrix and the symmetries of the Hamiltonian leads to the general form 0 r1;1 B 0 r¼B @ 0 r1;4 0 r2;2 r2;3 0 0 r2;3 r3;3 0 1 r1;4 0 C C 0 A r4;4 ð35Þ with pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r1;1 r4;4 þ jr1;4 j; lb ¼ r2;2 r3;3 þ jr2;3 j pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ lc ¼ j r1;1 r4;4 jr1;4 jj; ld ¼ j r2;2 r3;3 jr2;3 jj la ¼ ð36Þ ð37Þ Using the definition < A > ¼ TrðrAÞ, we can express all the matrix elements in the density matrix in terms of different spin–spin correlation functions [62]: r1;1 ¼ 12 Mlz þ 12 Mmz þ Szlm þ 12 ð38Þ r2;2 ¼ 12 Mlz 12 Mmz Szlm þ 14 ð39Þ r3;3 ¼ 12 Mmz 1 z 2 Ml Szlm þ 14 ð40Þ r4;4 ¼ 12 Mlz 12 Mmz þ Szlm þ 14 ð41Þ r2;3 ¼ Sxlm þ Sylm ð42Þ r1;4 ¼ Sxlm Sylm ð43Þ E. Some Numerical Results Let us show how the entanglement can be tuned by changing the anisotropy parameter g by going from the Ising model (g ¼ 1) to the XY model (g ¼ 0). For the XY model the entanglement is zero up to the critical point lc and is different from zero above lc . Moreover, by introducing impurities, the entanglement can be tuned down as the strength of the impurity a increases [59]. First, we examine the change of the entanglement for the Ising model ðg ¼ 1Þ for different values of the impurity strength a as the parameter l, which induces the quantum phase transitions, varies. Figure 2 shows the change of the nearest-neighbor concurrence Cð1; 2Þ with the impurity located at im ¼ 3 as a 0.3 α=0 γ = 0.7 0.5 1 0.2 C(1,2) γ=1 0.1 1.5 0 0.3 γ = 0.3 γ=0 0.2 0.1 0 0 0.5 1 1.5 0 λ 0.5 1 1.5 2 Figure 2. The nearest-neighbor concurrence C(1,2) for different values of the anisotropy parameter g ¼ 1, 0.7, 0.3, 0 with an impurity located at im ¼ 3 as a function of the reduced coupling constant l ¼ J=2h, where J is the exchange interaction constant and h is the strength of the external magnetic field. The curves correspond to different values of the impurity strength a ¼ 0,0.5,1,1.5 with system size N ¼ 201. function of l for different values of a. One can see clearly in Figure 2 that the entanglement can be tuned down by increasing the value of the parameter a. For a ¼ 1:5, the concurrence approaches zero above the critical lc ¼ 1. The system size was taken as N ¼ 201 based on finite size scaling analysis. Analysis of all the results for the pure case (a ¼ 0) for different system sizes ranging from N ¼ 41 up to N ¼ 401 collapse into a single curve. Thus all key ingredients of the finite size scaling are present in the concurrence. This holds true for the impurity problem as long as we consider the behavior of the value of l for which the derivative of the concurrence attains its minimum value versus the system size. As expected, there is no divergence of the derivative dCð1; 2Þ=dl for finite N, but there are clear anomalies. By examining lnðlc lm Þ versus ln N for a ¼ 0:1, one obtains that the minimum lm scales as lm lc þ N 0:93 and dCð1; 2Þ=dl diverges logarithmically with increasing system size. For a system with the impurity located at larger distance im ¼ 10 and the same a ¼ 0:1, lm lc þ N 0:85 , showing that the scaling behavior depends on the distance between the impurity and the pair of sites under consideration. Figure 2 also shows the variation of nearest-neighbor concurrence as the anisotropy parameter g decreases. For the XY model ðg ¼ 0Þ, the concurrence for a ¼ 0 is zero up to the critical point lc ¼ 1 and different from zero above lc ¼ 1. However, as a increases the concurrence develops steps and the entanglement, electron correlation, and density matrices results strongly depend on the system size. For small system size, such as N ¼ 101, the steps and oscillations are large but become smaller as the system size increases as shown in Fig. 2 for N ¼ 201. But they disappear in the limit N ! 1. To examine the different behavior of the concurrence for the Ising model and the XY model, we took the system size to be infinite, N ! 1, where the two models have exact solutions. However, the behavior is the same for a finite system with N ¼ 201. For larger values of im the concurrence gets larger and approaches its maximum value, the pure case with a ¼ 0, at large values im >> 1. It is worth mentioning that, for the Ising model, the range of entanglement [63], which is the maximum distance between spins at which the concurrence is different from zero, vanishes unless the two sites are at most nextnearest neighbors. For g 6¼ 1, the range of entanglement is not universal and tends to infinity as g tends to zero. So far we have examined the change of entanglement as the degree of the anisotropy g varies between zero and one and by introducing impurities at fixed sites. Rather than locating the impurity at one site in the chain, we can also introduce a Gaussian distribution of the disorder near a particular location [62]. This can be done by modifying a, the exchange interaction, where a introduces the impurity in a Gaussian form centered at ðN þ 1Þ=2 with strength or height : 2 ai;iþ1 ¼ eEðiðNþ1Þ=2Þ ð44Þ The external magnetic field can also be modified to take the form hi ¼ hð1 þ bi Þ, where b has the following Gaussian distribution [62]: 2 bi ¼ xeEðiðNþ1Þ=2Þ ð45Þ where E is a parameter to be fixed. Numerical calculations show that the entanglement can be tuned in this case by varying the strengths of the magnetic field and the impurity distribution in the system. The concurrence is maximum close to lc and can be tuned to zero above the critical point. F. Thermal Entanglement and the Effect of Temperature Recently, the concept of thermal entanglement was introduced and studied within one-dimensional spin systems [64–66]. The state of the system described by the Hamiltonian H at thermal equilibrium is rðTÞ ¼ expðH=kTÞ=Z; where Z ¼ Tr½expðH=kTÞ is the partition function and k is Boltzmann’s constant. As rðTÞ represents a thermal state, the entanglement in the state is called the thermal entanglement [64]. For a two-qubit isotropic Heisenberg model, there exists thermal entanglement for the antiferromagnetic case and no thermal entanglement for the ferromagnetic case [64]; while for the XY model the thermal entanglement appears for both the antiferromagnetic and ferromagnetic cases [67, 68]. It is known that the isotropic Heisenberg model and the XY model are special cases of the anisotropic Heisenberg model. Now that the entanglement of the XY Hamiltonian with impurities has been calculated at T ¼ 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble r ¼ ebH =Z, where b ¼ 1=kB T, and Z ¼ Tr ðebH Þ is the partition function. The thermal density matrix is diagonal when expressed in terms of the Jordan–Wigner fermionic operators. Our interest lies in calculating the quantum correlations present in the system as a function of the parameters b, g, l, and a. For the pure Ising model with a ¼ 0, the constructed two-site density matrices [66] are valid for all temperatures. By using these matrices, it is possible to study the purely two-party entanglement present at thermal equilibrium because the concurrence measure of entanglement can be applied to arbitrary mixed states. For this model the influence that the critical point has on the entanglement structure at nonzero temperatures is particularly clear. The entanglement between nearest-neighbor in the Ising model at nonzero temperature is shown in Fig. 3. The entanglement is nonzero only in a certain region in the kB T–l plane. It is in this region that quantum effects are likely to dominate the behavior of the system. The entanglement is largest in the vicinity of the critical point l ¼ 1, kB T ¼ 0. 0.4 C 0.2 0 1 2.5 0.5 2 1.5 kBT 1 0 0.5 0 λ Figure 3. Nearest-neighbor concurrence C at nonzero temperature for the transverse Ising model. entanglement, electron correlation, and density matrices Figure 3 shows that, for certain values of l, the two-site entanglement can increase as the temperature is increased. Moreover, it shows the existence of appreciable entanglement in the system for temperatures kB T above the ground-state energy gap . It has been argued that quantum systems behave classically when the temperature exceeds all relevant frequencies. For the transverse Ising model, the only relevant frequency is given by the ground-state energy gap ho. The presence of entanglement in the system for temperatures above the energy gap indicates that quantum effects may persist past the point where they are usually expected to disappear. The zero-temperature calculations of the previous section Section, the XY model with impurities, represent a highly idealized situation; however, it is unclear whether they have any relevance to the system at nonzero temperature. Since the properties of a quantum system for low temperatures are strongly influenced by nearby quantum critical points, it is tempting to attribute the effect of nearby critical points to persistent mixed-state entanglement in the thermal state. G. Entanglement for Two-Dimensional Spin Systems Quantum spin systems in two-dimensional lattices have been the subject of intense research, mainly motivated by their possible relevance in the study of high-temperature superconductors [69]. On the other hand, high magnetic field experiments on materials with a two-dimensional structure, which can be described by the Heisenberg antiferromagnetic model in frustrated lattices, have revealed novel phases as plateaus and jumps in the magnetization curves [70] and might be useful for quantum computations. Among the many different techniques that have been used to study such systems, the generalization of the celebrated Jordan–Wigner transformation [71] to two spatial dimensions [72] has some appealing features. It allows one to write the spin Hamiltonian completely in terms of spinless fermions in such a way that the S ¼ 12 single-particle constraint is automatically satisfied due to the Pauli principle, while the magnetic field enters as the chemical potential for the Jordan–Wigner fermions. This method has been applied to study the XXZ Heisenberg antiferromagnet [73–75]. For this case one can use the Jordan–Wigner transformation since it is a generalization of the well-known transformation in one dimension that we have used in previous sections. The Jordan–Wigner transformation is exact but the resulting Hamiltonian is highly nonlocal and some kind of approximation is necessary to proceed. One can use numerical methods such as Monte Carlo and variational approach to deal with the transformed Hamiltonian. This will allow us to explore the ground state of two-dimensional lattice spin 12 systems, in a way that could be applied to arbitrary lattice topologies. The method can also be used in the presence of an external magnetic field, at finite temperature, and can even be applied to disordered systems. Once this is solved and we have the density matrix, we can follow the previous procedure to examine the entanglement as the parameters of external magnetic field, temperature, lattice topologies, and impurities vary. III. ENTANGLEMENT FOR QUANTUM DOT SYSTEMS A. Two-Electron Two-Site Hubbard Model Many electron systems such as molecules and quantum dots show the complex phenomena of electron correlation caused by Coulomb interactions. These phenomena can be described to some extent by the Hubbard model [76]. This is a simple model that captures the main physics of the problem and admits an exact solution in some special cases [77]. To calculate the entanglement for electrons described by this model, we will use Zanardi’s measure, which is given in Fock space as the von Neumann entropy [78]. 1. Exact Solution The Hamiltonian of the two-electron two-site Hubbard model can be written [77] H¼ X tX y ^ni" ^ni# cis cis þ 2U 2 i;s i ð46Þ where cyis and cis are the Fermi creation and annihilation operators at site i and with spin s ¼"; # and ^ nis¼cy cis is the spin-dependent occupancy operator at site is i. For a two-site system i ¼ 1 and 2, i ¼ 3 i, t=2 is the hopping term of different sites, and 2U is the on-site interaction (U > 0 for repulsion in our case). The factors t=2 and 2U are chosen to make the following expressions for eigenvalues and eigenvectors as simple as possible. This Hamiltonian can be solved exactly in the basis set j1 "; 1 #; 2 "; 2 #i; it is simply a ð4 4Þ matrix of the form 0 1 2U t=2 t=2 0 B t=2 0 0 t=2 C C H¼B @ t=2 0 0 t=2 A 0 t=2 t=2 2U ð47Þ with the following four eigenvalues and eigenvectors, l1 ¼ U pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t2 þ U 2 ; l2 ¼ 0; l3 ¼ 2U; l4 ¼ U þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t2 þ U 2 ð48Þ entanglement, electron correlation, and density matrices and the corresponding eigenvectors, 0 1 1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p B x þ 1 þ x2 C B pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C jf1 i ¼ B C; @ x þ 1 þ x2 A 0 1 0 0 1 B 1 C B C jf2 i ¼ B C; @ 1 A 1 1 0 0 1 1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p B x 1 þ x2 C B pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C jf4 i ¼ B C @ x 1 þ x2 A B 0 C B C jf3 i ¼ B C; @ 0 A 1 ð49Þ ð50Þ 1 with x ¼ U=t. The eigenvalue and eigenvector for the ground state are E¼U pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ t2 þ U 2 ð51Þ and jGSi ¼ j1; x þ 2. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ x2 ; x þ 1 þ x2 ; 1i ð52Þ Hartree–Fock Approximation In quantum chemistry, the correlation energy Ecorr is defined as Ecorr ¼ Eexact EHF . In order to calculate the correlation energy of our system, we show how to calculate the ground state using the Hartree–Fock approximation. The main idea is to expand the exact wavefunction in the form of a configuration interaction picture. The first term of this expansion corresponds to the Hartree–Fock wavefunction. As a first step we calculate the spin-traced oneparticle density matrix [5] (1PDM) g: gij ¼ hGSj X cyis cjs jGSi ð53Þ s We obtain g¼ 1 2a 2ab 1 ð54Þ where 1 a ¼ pﬃﬃﬃ 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x 1 1 þ x2 and 1 b ¼ pﬃﬃﬃ 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x 1þ 1 þ x2 Diagonalizing this 1PDM, we can get the binding (þ) and unbinding () molecular natural orbitals (NOs), 1 ji ¼ pﬃﬃﬃ ðj1i j2iÞ 2 ð55Þ and the corresponding eigenvalues 1 n ¼ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ x2 ð56Þ where j1i and j2i are the spatial orbitals of sites 1 and 2, respectively. The NOs for different spins are defined as 1 j si ¼ pﬃﬃﬃ ðcy1s cy2s Þj0i cys j0i 2 ð57Þ where j0i is the vacuum state. After we define the geminals j i ¼ cy" cy# j0i, we can express jGSi in terms of NOs as rﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃ nþ n j þ þi sgn U j i jGSi ¼ 2 2 ð58Þ In the Hartree–Fock approximation, the GS is given by jHFi ¼ j þ þi and EHF ¼ t þ U. Let us examine the Hartree–Fock results by defining the ionic and nonionic geminals, respectively: 1 jAi ¼ pﬃﬃﬃ ðcy1" cy1# þ cy2" cy2# Þj0i 2 1 y y jBi ¼ pﬃﬃﬃ ðc1" c2# þ cy2" cy1# Þj0i 2 ð59Þ If x ! 0, the system is equally mixed between ionic and nonionic germinal, jHFi ¼ jAi þ jBi. When x ! þ1, jGSi ! jBi, which indicates that as x becomes large, our system goes to the nonionic state. Similarly, jGSi ! jCi, as x ! 1, where 1 jCi ¼ pﬃﬃﬃ ðcy1" cy1# cy2" cy2# Þj0i 2 Thus the HF results are a good approximation only when x ! 0. The unreasonable diverging behavior results from not suppressing the ionic state jAi in jHFi entanglement, electron correlation, and density matrices when jxj ! 1. In order to correct this shortcoming of the Hartree–Fock method, we can combine different wavefunctions in different ranges to obtain a better wavefunction for our system. This can be done as follows: Range GS Energy U>t t U U < t Correlation Energy Wavefunction pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jBi U U 2 þ t2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 2 pﬃﬃﬃ ðjAi þ jBiÞ t U þt pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 jCi U U þ t 0 t t þ U 2U 3. nþ n 1 1 2 1 0 1 Correlation Entropy The correlation entropy is a good measure of electron correlation in molecular systems [5, 7]. It is defined using the eigenvalues nk of the one-particle density matrix 1PDM, X X nk ð ln nk Þ; nk ¼ N ð60Þ S¼ k k This correlation entropy is based on the nonidempotency of the NONs nk and proves to be an appropriate measure of the correlation strength if the reference state defining correlation is a single Slater determinant. In addition to the eigenvalues nk of the ‘‘full’’ (spin-dependent) 1PDM, it seems reasonable to consider also the eigenvalues nk of the spin-traced 1PDM. Among all the nk there are a certain number N0 of NONs nk0 with values between 1 and 2 and all the other N1 NONs nk1 also have values between 0 and 1. So one possible measure of the correlation strength of spin-traced 1PDM is S1 ¼ X ðnk0 1Þ lnðnk0 1Þ k0 X nk1 ln nk1 ð61Þ k1 Since all the nk =2 have values between 0 and 1, there is another possible measurement of the correlation strength: S2 ¼ X nk k 4. 2 ln nk 2 ð62Þ Entanglement The entanglement measure is given by the von Neumann entropy [78] Ej ¼ Trðrj log2 rj Þ; rj ¼ Trj ðjihjÞ ð63Þ where Trj denotes the trace over all but the jth site, ji is the antisymmetric wavefunction of the fermions system, and rj is the reduced density matrix. Hence Ej actually describes the entanglement of the jth site with the remaining sites [79]. In the Hubbard model, the electron occupation of each site has four possibilities; there are four possible local states at each site, jnij ¼ j0ij , j "ij , j #ij , j "#ij . The reduced density matrix of the jth site with the other sites is given by [80, 81] rj ¼ zj0ih0j þ uþ j "ih" j þ u j #ih# j þ wj "#ih"# j ð64Þ with w ¼ hnj" nj# i ¼ Trðnj" nj# rj Þ ð65Þ þ ð66Þ u ¼ hnj" i w; þ u ¼ hnj# i w z ¼ 1 u u w ¼ 1 hnj" i hnj# i þ w ð67Þ And the entanglement between the jth site and other sites is given by Ej ¼ z Log2 z uþ Log2 uþ u Log2 u w Log2 w ð68Þ For the one-dimensional Hubbard model with half-filling electrons, we have hn" i ¼ hn# i ¼ 12, uþ ¼ u ¼ 12 w, and the entanglement is given by ð54Þ Ej ¼ 2w log2 w 2 12 wÞlog2 12 wÞ For our case of a two-site two-electron system w¼ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ 2½x þ 1 þ x2 2 Thus the entanglement is readily calculated from Eq. (69). In Fig. 4, we show the entanglement between the two sites (top curve) and the correlation entropy S1 and S2 as a function of x ¼ U=t. The entanglement measure is given by the von Neumann entropy in which the density matrix of the system is traced over the other site to get the reduced density matrix. The reduced density matrix describes the four possible occupations on the site: j0i, j "i, j #i, j "#i. The minimum of the entanglement is 1 as x ! 1. It can be understood that when U ! þ1, all the sites are singly occupied; the only difference is the spin of the electrons at each site, which can be referred to as spin entanglement. As U ! 1, all the sites are either doubly occupied or empty, which is referred to as space entanglement. The maximum of the entanglement is 2 at U ¼ 0; all four entanglement, electron correlation, and density matrices 2 1.8 1.6 1.4 1.2 1 S2 Combined Wavefunction 0.8 0.6 0.4 S1 0.2 0 −10 S2 −5 0 x = U/t 5 10 Figure 4. Two-site Hubbard model. Upper curve is the entanglement calculated by the von Newmann entropy. The curves S1 and S2 are the correlation entropies of the exact wavefunction as defined in the text. The dashed line is the S2 for the combined wavefunction based on the range of x values. S1 for the combined wavefunction is zero. occupations are evenly weighted, which is the sum of the spin and space entanglements of the system. The correlation entropy S1 vanishes for x ! 0 and x ! 1 and has a maximum near jxj ¼ 1; the correlation entropy S2 vanishes for x ! 0 and increases monotonically and approaches ln 2 for x ! 1. For x ! þ1 it can be viewed as t ! 0 for fixed U > 0 or as U ! þ1 for fixed t. B. One-Dimensional Quantum Dots System We consider an array of quantum dots modeled by the one-dimensional Hubbard Hamiltonian of the form [82] H¼ X hiji;s tij cþ is cjs þ U X ni" ni# ð70Þ i where tij stands for the hopping between the nearest-neighbor sites for the electrons with the same spin, i and j are the neighboring site numbers, s is the electron spin, cþ is and cjs are the creation and annihilation operators, and U is the Coulomb repulsion for the electrons on the same site. The periodic boundary condition is applied. The entanglement measure is given by the von Neumann entropy [78]. In the Hubbard model, the electron occupation of each site has four possibilities; there are four possible local states at each site, jnij ¼ j0ij ; j "ij ; j #ij ; j "#ij . The dimensions ofQ the Hilbert space of an L-site system is 4L and jn1 n2 nL i ¼ Lj¼1 jnj ij can be used as basis vectors for the system. The entanglement of the jth site with the other sites is given in the previous section by Eq. (65). In the ideal case, we can expect an array of the quantum dots to have the same size and to be distributed evenly, so that the parameters t and U are the same everywhere. We call this the pure case. In fact, the size of the dots may not be the same and they may not be evenly distributed, which we call the impurity case. Here, we consider two types of impurities. The first one is to introduce a symmetric hopping impurity t0 between two neighboring dots; the second one is to introduce an asymmetric electron hopping t0 between two neighboring dots, the right hopping is different from the left hopping, while the rest of the sites have hopping parameter t. Consider the particle–hole symmetry of the one-dimensional Hubbard model. One can obtain wðUÞ ¼ 12 wðUÞ, so the entanglement is an even function of U, Ej ðUÞ ¼ Ej ðUÞ. The minimum of the entanglement is 1 as U ! 1. As U ! þ1, all the sites are singly occupied; the only difference is the spin of the electrons on each site, which can be referred to as spin entanglement. As U ! 1, all the sites are either doubly occupied or empty, which is referred to as space entanglement. The maximum of the entanglement is 2 at U ¼ 0, which is the sum of the spin and space entanglements of the system. The ground state of the one-dimensional Hubbard model at half-filling is metallic for U < 0, and insulating for U > 0; U ¼ 0 is the critical point for the metal–insulator transition, where the local entanglement reaches its maximum. In Fig. 5 we show the 2 1.8 1.6 Ev 1.4 1.2 1 –40 –20 0 U/t 20 40 Figure 5. Local entanglement given by the von Neumann entropy, Ev , versus U=t in the pure case. entanglement, electron correlation, and density matrices entanglement as a function U=t for six sites and six electrons. Our results are in complete agreement with the exact one obtained by Bethe ansatz [80]. C. Two-Dimensional Array of Quantum Dots Using the Hubbard model, we can study the entanglement scaling behavior in a two-dimensional itinerant system. Our results indicate that, on the two sides of the critical point denoting an inherent quantum phase transition (QPT), the entanglement follows different scalings with the size just as an order parameter does. This fact reveals the subtle role played by the entanglement in QPT and points to its potential application in quantum information processing as a fungible physical resource. Recently, it has been speculated that the most entangled systems could be found at the critical point [83] when the system undergoes a quantum phase transition; that is, a qualitative change of some physical properties takes place as an order parameter in the Hamiltonian is tuned [84]. QPT results from quantum fluctuations at the absolute zero of temperature and is a pure quantum effect featured by long-range correlations. So far, there have already been some efforts in exploring the above speculations, such as the analysis of the XY model about the single-spin entropies and two-spin quantum correlations [59, 85], the entanglement between a block of L contiguous sites and the rest of the chain [51], and also the scaling of entanglement near QPT [60]. But because there is still no analytical proof, the role played by the entanglement in quantum critical phenomena remains elusive. Generally, at least two difficulties exist in resolving this issue. First, until now, only two-particle entanglement is well explored. How to quantify the multiparticle entanglement is not clear. Second, QPT closely relates to the notorious many-body problems, which is almost intractable analytically. Until now, the only effective and accurate way to deal with QPT in critical region is the density-matrix renormalization group method [86]. Unfortunately, it is only efficient for one-dimensional cases because of the much more complicated boundary conditions for the two-dimensional situation [87]. In this chapter, we will focus on the entanglement behavior in QPT for the two-dimensional array of quantum dots, which provide a suitable arena for implementation of quantum computation [88, 89, 103]. For this purpose, the real-space renormalization group technique [91] will be utilized and developed for the finite-size analysis of entanglement. The model that we will be using is the Hubbard model [83], X X 1 1 ½cþ ð71Þ H ¼ t is cjs þ h:c: þ U 2 ni" 2 ni# hi; ji;s i where t is the nearest-neighbor hopping term and U is the local repulsive interaction. cþ is ðcis Þ creates(annihilates) an electron with spin s in a Wannier orbital located at site i; the corresponding number operator is nis ¼ cþ is cis and h i denotes the nearest-neighbor pairs; h.c. denotes the Hermitian conjugate. For a half-filled triangular quantum lattice, there exists a metal–insulator phase transition with the tuning parameter U=t at the critical point 12.5 [92–94]. The corresponding order parameter for metal–insulator transition is the charge gap defined by 4g ¼ EðNe 1Þ þ EðNe þ 1Þ 2EðNe Þ, where EðNe Þ denotes the lowest energy for an Ne -electron system. In our case, Ne is equal to the site number Ns of the lattice. Unlike the charge gap calculated from the energy levels, the Zanardi measure of the entanglement is defined based on the wavefunction corresponding to EðNe Þ instead. Using the conventional renormalization group method for the finite-size scaling analysis [92–94], we can discuss three schemes of entanglement scaling: single-site entanglement scaling with the total system size, Esingle ; single-block entanglement scaling with the block size, Eblock ; and block–block entanglement scaling with the block size, Eblockblock . Our initial results of the single-site entanglement scaling indicate that Esingle is not a universal quantity. This conclusion is consistent with the argument given by Osborne and Nielsen [85], who claim that the single-site entanglement is not scalable because it does not have the proper extensivity and does not distinguish between the local and the distributed entanglement. This implies that only a limited region of sites around the central site contributed significantly to the single-site entanglement. Using the one-parameter scaling theory, near the phase transition point, we can assume the existence of scaling function f for Eblockblock such that Eblockblock ¼ qyE f ðL=xÞ, where q ¼ ðU=tÞ ðU=tÞc measures the deviation distance of the system away from the critical state with ðU=tÞc ¼ 12:5, which is exactly equal to the critical value for metal–insulator transition when the same order parameter U=t is used [92–94]. x ¼ qn is the correlation length of the system with the critical exponent n and N ¼ L2 for the two-dimensional systems. In Fig. 6, we show the results of Eblockblock as a function of ðU=tÞ for different system sizes. With proper scaling, all the curves collapse into one curve, which can be expressed as Eblock--block ¼ f ðqN 1=2 Þ. Thus the critical exponents are yE ¼ 0; n ¼ 1. It is interesting to note that we obtained the same n as in the study of the metal–insulator transition. This shows the consistency of the initial results since the critical exponent n is only dependent on the inherent symmetry and dimension of the investigated system. Another significant result lies in the finding that the metal state is highly entangled while the insulating state is only partly entangled. It should be mentioned that the calculated entanglement here has a corresponding critical exponent yE ¼ 0. This means that the entanglement is constant at the critical point over all sizes of the system. But it is not a constant over all values of U=t. There is an abrupt jump across the critical point as L ! 1. If we divide the regime of the order parameter into noncritical regime and critical entanglement, electron correlation, and density matrices 2 7 3 1 6 4 5 (a) 2.0 7 72 Eblock-block 73 74 1.5 75 76 77 78 79 1.0 –50000 50000 0 (U/t – (U/t)c)N1/2 (b) 2.0 7 Eblock 72 73 74 1.5 75 76 1.0 –8000 0 (U/t – (U/t)c)N1/2 8000 (c) Figure 6. (a) Schematic diagram displays the lattice configuration with central block and the surrounding ones. (b) Scaling of block–block for various system size and (c) scaling of block entanglements with the block size. regime, the results can be summarized as follows. In the noncritical regime, that is, U=t is away from ðU=tÞc , as L increases, the entanglement will saturate onto two different values depending on the sign of U=t ðU=tÞc ; at the critical point, the entanglement is actually a constant independent of the size L. These properties are qualitatively different from the single-site entanglement discussed by Osborne and Nielsen [85], where the entanglement with Zanardi’s measure increases from zero to the maximum at the critical point and then decreases again to zero as the order parameter g for the XY mode is tuned. These peculiar properties of the entanglement found here can be of potential interest to make an effective ideal ‘‘entanglement switch.’’ For example, with seven blocks of quantum dots on a triangular lattice, the entanglement among the blocks can be regulated as ‘‘0’’ or ‘‘1’’ almost immediately once the tuning parameter U=t crosses the critical point. The switch errors will depend on the size of the blocks. Since it is already a well-developed technique to change U=t for the quantum dot lattice [95, 103], the above scheme should be workable. To remove the special confinement we have made upon the calculated entanglement, namely, only the entanglement of blocks 1 and 7 with the rest ones are considered, in the following, we will prove that the average pairwise entanglement also has the properties shown in Fig. 6. As we change the size of the central block, its entanglement with all the rest of the sites follows the same scaling properties as Eblockblock . It is understandable if we consider the fact that only a limited region round the block contributes mostly to Eblock . This result greatly facilitates the fabrication of realistic entanglement control devices, such as quantum gates for a quantum computer, since we don’t need to consider the number of component blocks in fear that the next neighboring or the next-next neighboring quantum dots will influence the switching effect. IV. AB INITIO CALCULATIONS AND ENTANGLEMENT For a two-electron system in 2m-dimensional spin-space orbital, with ca and cya denoting the fermionic annihilation and creation operators of single-particle states and j0i representing the vacuum state, a pure two-electron state j i can be written [57] X oa;b cya cyb j0i ð72Þ j i¼ a;b2f1;2;3;4;...;2mg where a; b run over the orthonormal single-particle states, and Pauli exclusion requires that the 2m 2m expansion coefficient matrix o is antisymmetric: oa;b ¼ ob;a , and oi;i ¼ 0. In the occupation number representation ðn1 "; n1 #; n2 "; n2 #; . . . ; nm "; nm #Þ, where " and # mean a and b electrons, respectively, the subscripts entanglement, electron correlation, and density matrices denote the spatial orbital index and m is the total spatial orbital number. By tracing out all other spatial orbitals except n1 , we can obtain a ð4 4Þ reduced density matrix for the spatial orbital n1 rn1 ¼ Trn1 j ih j 0 Pm1 4 i;j¼1 jo2iþ1;2jþ2 j2 B B 0 ¼B @ 0 0 0 Pm1 4 i¼1 jo2;2iþ1 j2 0 0 0 0 P 2 4 m i¼2 jo1;2i j 0 1 0 C 0 C C 0 A 4jo1;2 j2 ð73Þ The matrix elements of o can be calculated from the expansion coefficient of the ab initio configuration interaction method. The CI wavefunction with single and double excitations can be written j i ¼ c0 j0 i þ X ar cra jra i þ X r;s cr;s a;b ja;b i ð74Þ a 3 Å the correlation between the two electrons is almost zero but the entanglement is maximal until around R 4 Å; the entanglement vanishes for R > 4 Å. V. DYNAMICS OF ENTANGLEMENT AND DECOHERENCE In this section, we investigate the dynamics of entanglement in one-dimensional spin systems with a time-dependent magnetic field. The Hamiltonian for such a system is given by [98] N N N X X X J J sxi sxiþ1 ð1 gÞ syi syiþ1 hðtÞszi H ¼ ð1 þ gÞ 2 2 i¼1 i¼1 i¼1 ð76Þ where J is the coupling constant, hðtÞ is the time-dependent external magnetic field, sa are the Pauli matrices (a ¼ x; y; z), g is the degree of anisotropy, and N is the number of sites. We can set J ¼ 1 for convenience and use periodic boundary conditions. Next, we transform the spin operators into fermionic operators. So the Hamiltonian assumes the following form: H¼ N=2 X p¼1 ap ðtÞ½cþ p cp þ cþ p cp þ þ idp ½cþ p cp þ cp cp þ 2hðtÞ ¼ N=2 X p¼1 ~p H ð77Þ entanglement, electron correlation, and density matrices where ap ðtÞ ¼ 2 cos fp 2hðtÞ, dp ¼ 2g sin fp , and fp ¼ 2pp=N . It is easy to ~ q ¼ 0, which means the space of H ~ decomposes into noninteracting ~ p; H show ½H subspace, each of four dimensions. No matter what hðtÞ is, there will be no transitions among those subspaces. Using the following basis for the pth þ þ þ subspace, ðj0i; cþ p cp j0i; cp j0i; cp j0iÞ, we can explicitly get 0 2hðtÞ idp B idp 4 cos fp 2hðtÞ ~ p ðtÞ ¼ B H @ 0 0 0 0 0 0 2 cos fp 0 1 0 C 0 C A 0 2 cos fp ð78Þ We only consider the systems that, at time t ¼ 0, are in thermal equilibrium at temperature T. Let rp ðtÞ be the density matrix of the pth subspace; we have ~ rp ð0Þ ¼ ebHp ð0Þ , where b ¼ 1=kT and k is Boltzmann’s constant. Therefore, using Eq. (78), we have rp ð0Þ. Let Up ðtÞ be the time-evolution matrix in the ~ p ðtÞ, with the boundary pth subspace, namely, ( h ¼ 1): idUp ðtÞ=dt ¼ Up ðtÞH condition Up ð0Þ ¼ I. Now the Liouville equation of this system is i drðtÞ ¼ ½HðtÞ; rðtÞ dt ð79Þ which can be decomposed into uncorrelated subspaces and solved exactly. Thus, in the pth subspace, the solution of the Liouville equation is rp ðtÞ ¼ Up ðtÞrp ð0ÞUp ðtÞy . As a first step to investigate the dynamics of the entanglement, we can take the magnetic field to be a step function then generalize it to other relevant functional forms such as an oscillating one [98]. Figure 8 shows the results for nearest-neighbor concurrence Cði; i þ 1Þ at temperature T ¼ 0 and g ¼ 1 as a function of the initial magnetic field a for the step function case with final field b. For the a < 1 region, the concurrence increases very fast near b ¼ 1 and reaches a limit Cði; i þ 1Þ 0:125 when b ! 1. It is surprising that the concurrence will not disappear when b increases with a < 1. This indicates that the concurrence will not disappear as the final external magnetic field increases at infinite time. It shows that this model is not in agreement with obvious physical intuition, since we expect that increasing the external magnetic field will destroy the spin–spin correlation functions and make the concurrence vanish. The concurrence approaches a maximum Cði; i þ 1Þ 0:258 at ða ¼ 1:37; b ¼ 1:37Þ and decreases rapidly as a 6¼ b. This indicates that the fluctuation of the external magnetic field near the equilibrium state will rapidly destroy the entanglement. However, in the region where a > 2:0, the concurrence is close to zero when b < 1:0 and maximum close to 1. Moreover, it disappear in the limit of b ! 1. Figure 8. Nearest-neighbor concurrence C at zero temperature as a function of the initial magnetic field a for the step function case with final field b. Now let us examine the system size effect on the entanglement with three different external magnetic fields changing with time t [99]: a; hI ðtÞ ¼ b þ ða bÞeKt ; a; hII ðtÞ ¼ a a sinðKtÞ; 0; hIII ðtÞ ¼ a a cosðKtÞ; t 0 t>0 t 0 t>0 t 0 t>0 ð80Þ ð81Þ ð82Þ where a, b, and K are varying parameters. We have found that the entanglement fluctuates shortly after a disturbance by an external magnetic field when the system size is small. For larger system size, the entanglement reaches a stable state for a long time before it fluctuates. However, this fluctuation of entanglement disappears when the system size goes to infinity. We also show that in a periodic external magnetic field, the nearest-neighbor entanglement displays a periodic structure with a period related to that of the magnetic field. For the exponential external magnetic field, by varying the constant K, we have found that as time evolves, Cði; i þ 1Þ oscillates but it does not reach its equilibrium value at t ! 1. entanglement, electron correlation, and density matrices This confirms the fact that the nonergodic behavior of the concurrence is a general behavior for slowly changing magnetic field. For the periodic magnetic field hII ¼ að1 sin ðKtÞÞ, the nearest-neighbor concurrence is a maximum at t ¼ 0 for values of a close to one, since the system exhibits a quantum phase transition at lc ¼ J=h ¼ 1, where in our calculations we fixed J ¼ 1. Moreover, for the two periodic sin ðKtÞ and cos ðKtÞ fields the nearest-neighbor concurrence displays a periodic structure according to the periods of their respective magnetic fields [99]. For the periodic external magnetic field hIII ðtÞ, we show in Fig. 9 that the nearest-neighbor concurrence Cði; i þ 1Þ is zero at t ¼ 0 since the external magnetic field hIII ðt ¼ 0Þ ¼ 0 and the spins align along the x-direction: the total wavefunction is factorizable. By increasing the external magnetic field, we see the appearance of nearest-neighbor concurrence but very small. This indicates that the concurrence cannot be produced without a background external magnetic field in the Ising system. However, as time evolves one can see the periodic structure of the nearest-neighbor concurrence according to the periodic structure of the external magnetic field hIII ðtÞ [99]. 0.10 C(i,i+1) a = 1.1 a = 2.0 a = 1.5 a = 5.0 0.05 0.00 8.0 a = 5.0 a = 1.5 h 6.0 a = 2.0 4.0 2.0 a = 1.1 0.0 0 100 200 300 400 500 t Figure 9. The nearest-neighbor concurrence Cði; i þ 1Þ (upper panel) and the periodic external magnetic field hIII ðtÞ ¼ að1 cos½Kt Þ; see Eq. (14) in the text (lower panel) for K ¼ 0:05 with different values of a as a function of time t. Recently, interest in solid state systems has increased because they facilitate the fabrication of large integrated networks that would be able to implement realistic quantum computing algorithms on a large scale. On the other hand, the strong coupling between a solid state system and its complex environment makes it a significantly challenging mission to achieve the high coherence control required to manipulate the system. Decoherence is considered as one of the main obstacles toward realizing an effective quantum computing system [100–103]. The main effect of decoherence is to randomize the relative phases of the possible states of the isolated system as a result of coupling to the environment. By randomizing the relative phases, the system loses all quantum interference effects and may end up behaving classically. In order to study the decoherence effect, we examined the time evolution of a single spin coupled by exchange interaction to an environment of interacting spin bath modeled by the XY-Hamiltonian. The Hamiltonian for such a system is given by [104] H¼ N N N X 1þgX 1 gX Ji;iþ1 sxi sxiþ1 Ji;iþ1 syi syiþ1 hi szi 2 i¼1 2 i¼1 i¼1 ð83Þ where Ji;iþ1 is the exchange interaction between sites i and i þ 1, hi is the strength of the external magnetic field at site i, sa are the Pauli matrices (a ¼ x; y; z), g is the degree of anisotropy, and N is the number of sites. We consider the centered spin on the lth site as the single-spin quantum system and the rest of the chain as its environment, where in this case l ¼ ðN þ 1Þ=2. The single spin directly interacts with its nearest-neighbor spins through exchange interaction Jl1;l ¼ Jl;lþ1 ¼ J 0 . We assume exchange interactions between spins in the environment are uniform and simply set it as J ¼ 1. The centered spin is considered as inhomogeneously coupled to all the spins in the environment by being directly coupled to its nearest neighbors and indirectly to all other spins in the chain through its nearest neighbors. By evaluating the spin correlator CðtÞ of the single spin at the jth site [104], Cj ðtÞ ¼ rzj ðt; bÞ rzj ð0; bÞ ð84Þ we observed that the decay rate of the spin oscillations strongly depends on the relative magnitude of the exchange coupling between the single spin and its nearest neighbor J 0 and coupling among the spins in the environment J. The decoherence time varies significantly based on the relative coupling magnitudes of J and J 0 . The decay rate law has a Gaussian profile when the two exchange couplings are of the same order, J 0 J, but converts to exponential and then a entanglement, electron correlation, and density matrices power law as we move to the regimes of J 0 > J and J 0 < J. We also show that the spin oscillations propagate from the single spin to the environment spins with a certain speed. Moreover, the amount of saturated decoherence induced into the spin state depends on this relative magnitude and approaches a maximum value for a relative magnitude of unity. Our results suggest that setting the interaction within the environment in such a way that its magnitude is much higher or lower than the interaction with the single spin may reduce the decay rate of the spin state. The reason behind this phenomenon could be that the variation in the coupling strength along the chain at one point (where the single spin exits) blocks the propagation of decoherence along the chain by reducing the entanglement among the spins within the environment, which reduces its decoherence effect on the single spin in return [104]. This result might be applicable in general to similar cases of a centered quantum system coupled inhomogeneously to an interacting environment with large degrees of freedom. VI. ENTANGLEMENT AND DENSITY FUNCTIONAL THEORY Density functional theory is originally based on the Hohenberg–Kohn theorem [105, 106]. In the case of a many-electron system, the Hohenberg–Kohn theorem establishes that the ground-state electronic density rðrÞ, instead of the potential vðrÞ, can be used as the fundamental variable to describe the physical properties of the system. In the case of a Hamiltonian given by H ¼ H0 þ Hext ¼ H0 þ X bl ll A ð85Þ l where ll is the control parameter associated with a set of mutually commuting b l for the ground state jci bl g, the expectation values of A Hermitian operators fA b are denoted by the set fal g fhcjAl jcig. For such a Hamiltonian Wu et al. [107] linked entanglement in interacting many-body quantum systems to density functional theory. They used the Hohenberg–Kohn theorem on the ground state to show that the ground-state expectation value of any observable can be interchangeably viewed as a unique function of either the control parameter fll g or the associated operator representing the observable fal g. The Hohenberg–Kohn theorem can be used to redefine entanglement measures in terms of new physical quantities: expectation values of observables, fal g, instead of external control parameters, fll g. Consider an arbitrary entanglement measure M for the ground state of Hamiltonian (85). For a bipartite entanglement, one can prove a central lemma, which very generally connects M and energy derivatives. Lemma Any entanglement measure M can be expressed as a unique functional of the set of first derivatives of the ground-state energy [107] : M ¼ Mðfal gÞ ¼ M qE qll ð86Þ The proof follows from the fact that, according to the generalized Hohenberg– Kohn theorem, the ground-state wavefunction ji is a unique functional of fal g, and since ji provides a complete description of the state of the system, everything else is a unique functional of fal g as well, including M. Wu et al. [107] use density functional theory concepts to express entanglement measures in terms of the first or second derivative of the ground-state energy. As a further application they discuss entanglement and quantum phase transitions in the case of mean field approximations for realistic models of many-body systems [107]. This interesting connection between density functional theory and entanglement was further generalized for arbitrary mixed states by Rajagopal and Rendell [108] using the maximum entropy principle. In this way they established the duality in the sense of Legendre transform between the set of mean values of the observables based on the density matrix and the corresponding set of conjugate control parameters associated with the observables. VII. FUTURE DIRECTIONS We have examined and reviewed the relation between electron–electron correlation, the correlation entropy, and the entanglement for two exactly solvable models: the Ising model and the Hubbard model for two sites. The ab initio calculation of the entanglement for the H2 system is also discussed. Our results show that there is a qualitatively similar behavior between the entanglement and absolute standard correlation of electrons for the Ising model. Thus entanglement might be used as an alternative measure of electron correlation in quantum chemistry calculations. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics. Dimensional scaling theory [109] provides a natural means to examine electron–electron correlation, quantum phase transitions [110], and entanglement. The primary effect of electron correlation in the D ! 1 limit is to open up the dihedral angles from their Hartree–Fock values [109] of exactly 90! . Angles in the correlated solution are determined by the balance between centrifugal effects, which always favor 90! , and interelectron repulsions, which always favor 180! . Since the electrons are localized at the D ! 1 limit, one might need to add the first harmonic correction in the 1=D expansion to obtain entanglement, electron correlation, and density matrices a useful density matrix for the whole system, thus the von Neumann entropy. The relation between entanglement and electron–electron correlation at the largedimensional limit for the dimensional scaling model of the H2 molecule [111] will be examined in future studies. A new promising approach is emerging for the realization of quantum chemistry calculations without wavefunctions through first-order semidefinite programming [112]. Mazziotti has developed a first-order, nonlinear algorithm for the semidefinite programing of the two-electron reduced density matrix method that reduces memory and floating-point requirements by orders of magnitude [113, 114]. The electronic energies and properties of atoms and molecules are computable simply from an effective two-electron reduced density matrix rðABÞ [115, 116]. Thus the electron–electron correlation can be calculated directly as effectively the entanglement between the two electrons, which is readily calculated as the von Neumann entropy S ¼ TrrA log2 rA , where rA ¼ TrB rðABÞ . With this combined approach, one calculates the electronic energies and properties of atoms and molecules including correlation without wavefunctions or Hartree–Fock reference systems. This approach provides a natural way to extend the calculations of entanglement to larger molecules. Quantum phase transitions are a qualitative change in the ground state of a quantum many-body system as some parameter is varied [84, 117]. Unlike classical phase transitions, which occur at a nonzero temperature, the fluctuations in quantum phase transitions are fully quantum and driven by the Heisenberg uncertainty relation. Both classical and quantum critical points are governed by a diverging correlation length, although quantum systems possess additional correlations that do not have a classical counterpart: this is the entanglement phenomenon. Recently, a new line of exciting research points to the connection between the entanglement of a many-particle system and the appearance of quantum phase transitions [60, 66, 118, 119]. For a class of one-dimensional magnetic systems, the entanglement shows scaling behavior in the vicinity of the transition point [60]. 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