J. Phys. A 37, 6807 (2004)
Quantum Entanglement in Secondquantized Condensed Matter Systems Yu Shi
arXiv:quantph/0204058v5 17 Jun 2004
Department of Physics, University of Illinois at UrbanaChampaign, Urbana, IL 61801, USA The entanglement between occupationnumbers of different single particle basis states depends on coupling between different single particle basis states in the secondquantized Hamiltonian. Thus in principle, interaction is not necessary for occupationnumber entanglement to appear. However, in order to characterize quantum correlation caused by interaction, we use the eigenstates of the singleparticle Hamiltonian as the single particle basis upon which the occupationnumber entanglement is defined. Using this socalled proper single particle basis, if there is no interaction, then the manyparticle secondquantized Hamiltonian is diagonalized and thus cannot generate entanglement, while its eigenstates can always be chosen to be nonentangled. If there is interaction, entanglement in the proper single particle basis arises in energy eigenstates and can be dynamically generated. Using the proper single particle basis, we discuss occupationnumber entanglement in important eigenstates, especially ground states, of systems of many identical particles, in exploring insights the notion of entanglement sheds on manyparticle physics. The discussions on Fermi systems start with Fermi gas, HatreeFock approximation, and the electronhole entanglement in excitations. In the ground state of a Fermi liquid, in terms of the Landau quasiparticles, entanglement becomes negligible. The entanglement in a quantum Hall state is quantified as −f ln f − (1 − f ) ln(1 − f ), where f is the proper fractional part of the filling factor. For BCS superconductivity, the entanglement is a function of the relative momentum wavefunction of the Cooper pair gk , and is thus directly related to the superconducting energy gap, and vanishes if and only if superconductivity vanishes. For a spinless Bose system, entanglement does not appear in the HatreeGrossPitaevskii approximation, but becomes important in the Bogoliubov theory, as a characterization of twoparticle correlation caused by the weak interaction. In these examples, the interactioninduced entanglement as calculated is directly related to the macroscopic physical properties. PACS numbers: 03.65.w, 05.30.d, 74.20.z, 73.43.f
1. Introduction
Quantum entanglement is the situation that a quantum state of a composite system is not a direct product of states of the subsystems [1]. It is an essential quantum feature without classical analogy [2, 3]. For many decades, the notion of entanglement has been mostly used in foundations of quantum mechanics. Recently it was found to be crucial in quantum information processing. For a bipartite pure state ψAB i, the entanglement can be quantified as the von Neumann entropy of the reduced density matrix of either party, S = −trA ρA ln ρA = −trB ρB ln ρB , where ρA = trB (ψAB ihψAB ), ρB = trA (ψAB ihψAB ) [4]. Thus 0 ≤ S ≤ ln D, where D is the smaller one of the dimensions of the Hilbert spaces of A and B. The larger S, the stronger the entanglement. Recall that the von Neumann entropy of a density matrix is a measure of the distribution of its eigenvalues; the more homogeneous this distribution, the larger the von Neumann entropy. Since quantum entanglement is an essential quantum correlation, it is natural and interesting to consider useful or even fundamental insights that the notion of entanglement may provide on quantum manybody physics and quantum field theory. Historically, similar consideration was made in Yang’s study of offdiagonal longrange order [5] and in Leggett’s study of disconnectivity [6].
The recent development of quantum information theory may be useful to some important issues in frontiers of physics [7]. Some investigations have been made on entanglement between spins at different sites in some spin lattice models [8, 9, 10, 11, 12, 13, 14]. Nevertheless, the bulk of quantum manybody physics concerns identical particles, with the localized spin models as special cases. Hence in this regard, it is inevitable to address the issue of entanglement in systems of identical particles. This topic, related to both quantum information and condensed matter physics, is pursued in various approaches [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. As the quantum correlation beyond the symmetrization or antisymmetrization of identical particles, entanglement between occupation numbers of different single particle basis states (modes) is an appropriate characterization. Lloyd and coworkers when recognized the occupation number basis as the suitable basis for quantum simulation of second quantized manyparticle systems, exemplified by using the Hubbard model [15]. Furthermore, occupation was also proposed as the degree of freedom to implement qubit [16, 17, 18, 19, 20]. Zanardi noted the isomorphism between the full Fock space and qubits space and investigated the entanglement in grand canonical ensembles [21]. Afterwards, from a physical standpoint and the relation between occupation num
2 ber state and the (anti)symmetrized particle state, the present author carefully justified the use of Fock space in investigating entanglement issue, even in the case of particle number conservation, thus helped to establish the applicability of this approach to manyparticle pure states [23, 24]. It is also noted that the magnetic spin entanglement is a special case of occupationnumber entanglement of identical particles [23]. Some related papers have appeared after the present work was actually done [24]. Vedral made some very interesting investigations, where the application of twomode squeezing to Bose condensates and related systems were analyzed, and twoparticle fermionic entanglement due to symmetrization were computed [27]. In this approach, clearly the entanglement in the manyparticle system depends on which single particle basis is chosen. This point might seem uncomfortable to some researchers, since they remember that entanglement should not be affected by local operations. Let us emphasize that the subsystems are defined by modes, not by particles. The choice of single particle basis defines how to partition the system into subsystems, and actually defines the single particles [31]. Once the single particle basis is chosen, i.e. the partition into subsystems is defined, the entanglement is invariant under unitary operations on individual subsystems, i.e. the modes, in fully consistent with the general wisdom about entanglement. Naturally a question arises: which single particle basis does one choose? The answer is that it depends on the relevance to the question one is concerned, or which single particle basis corresponds to the particles that are detected in the circumstance. Given that in this approach, the entanglement is between the occupationnumbers of different single particle basis states, whether it can be generated by, or whether it exists in an eigenstate of, a manyparticle secondquantized Hamiltonian H depends on whether there is coupling between different single particle basis states in H, in contrast with the case of distinguishable particles, for which the entanglement depends on interaction of particles. Hence generically speaking, interaction is not necessary for generation or existence of occupationnumber entanglement. For example, in a single particle basis in which H is not diagonal, the occupationnumber entanglement exists in eigenstates of H, and can be dynamically generated. Here we note, however, there is a special single particle basis in which, if there is no interaction, entanglement cannot be generated from a nonentangled state, while each energy eigenstate must be nonentangled (except the insignificant case of a superposition of two degenerate nonentangled eigenstates with different occupation numbers in at least two single particle basis states). In this single particle basis, entanglement in a nondegenerate energy eigenstate is only caused by interaction. Hence this single particle basis is very suitable in characterizing
the quantum correlation due to interaction, rather than the entanglement that appears merely as a consequence of Bogoliubov mode transformation. For convenience, let us call this special single particle basis the proper single particle basis. The socalled proper single particle basis is just the set of eigenstates of the single particle Hamiltonian. It is indexed by the (continuous or discrete) momentum in the case of free particles, the Bloch wavevector plus the band index in the case of particles in a periodic potential, the degree of the Hermit polynomial and the perpendicular momentum in the case of electrons in a magnetic field, etc. The inclusion of spin as an additional index is straightforward. It is instructive and amusing to consider our method of characterizing interactioninduced entanglement as an extension of the novel way of counting states of a system of identical particles invented by Bose, Einstein and Dirac [32]. They considered ideal gas, hence the underlying manyparticle states are just all the possible occupationnumber basis states in the momentum basis, which is the proper single particle basis in this question. Each of these occupationnumber basis states is a direct product of the occupation states of single particle basis states (modes). No superposition of these occupationnumber basis states. No entanglement between the proper single particle basis states. Hence the classical Boltzmann counting is applicable when one considers the occupations of the single particle states, rather than the particles themselves. The entanglement between these different single particle states emerges when there is interaction, as discussed in this paper. In this paper, using the proper single particle basis, we shall explore the interactioninduced entanglement in representative manyparticle states, which are of fundamental importance in condensed matter physics and the like. In particular, we emphasize the role of Hamiltonian and the relation between entanglement with macroscopic physical properties. Energy eigenstates, especially the ground states, are of utmost importance in manybody and statistical physics. Besides, adiabatically controlled ground states is also used in some quantum computing schemes [33, 34, 35]. Hence it is important to address the issue of entanglement in the energy eigenstates, especially the ground state. These aspects further motivate our work. The organization of this paper is the following. First an introduction and clarification is made on occupationnumber entanglement in a system of many identical particles, and especially to the socalled proper single particle basis. Then we discuss the ground state and excitations of normal Fermi systems, especially the electronhole entanglement in the HatreeFock approach. In the next two sections, we make detailed investigations on entanglement in quantum Hall effect and BardeenCooperSchrieffer (BCS) superconductivity, respectively. After
3 wards there is a section on Bosonic entanglement, in which entanglement in Bogoliubov theory is calculated. We summarize after making some additional remarks.
2. The proper single particle basis
In the standard formalism of second quantization, one can write a state of many identical particles in terms of an arbitrarily chosen single particle basis, as X f (n1 , · · · , n∞ )n1 , · · · , n∞ i, (1) ψi = n1 ,···,n∞
where ni is the occupation number of single particle state i in the chosen single particle basis, ni i ≡ √ ni (1/ ni !)a†i 0i n1 , · · · , n∞ i corresponds to a Slater determinant or permanent wavefunction in the configuration space. For a fixed number of particles, whether a manyparticle state is entangled means whether the wavefunction is a single Slater determinant or permanent. In principle, entanglement in a system of identical particles is a property dependent on which single particles and which single particle basis is chosen in representing the manyparticle system, and can be quantified as that among occupation numbers of different single particle states. Choosing a different single particle basis means partitioning the system into a different set of subsystems, based on which the entanglement is then defined. But once a single particle basis is chosen, the entanglement in invariant under any unitary operation on individual single particle basis states, i.e. when there is no coupling between different single particle basis states. In other words, in the present case, the meaning of “local operations” as previously used in quantum information theory is generalized to operations on the corresponding single particle basis states, as indexed by the subscript i above. Of course, it is constrained that some kinds of generalized “local” unitary operations do not exist physically. Once this generalization of the meaning of subsystems and local operations is made, the usual method of calculating the amount of entanglement, as developed in quantum information theory, can be applied. Quantitatively, one considers the Fockstate reduced density matrix of a set of single particle basis states 1, · · · , l, hn′1 , · · · , n′l ρl (1 · · · l)n1 , · · · , nl i ≡ ′ ′ nl+1 ,···,n∞ hn1 , · · · , nl , nl+1 , n∞ ρn1 , · · · , nl , nl+1 , n∞ i. (2) Its von Neumann entropy measures the entanglement of this set of single particle basis states and the rest of the system, relative to the empty state. If the total number of particles is conserved, then it is constrained that the only matrix elements which may be nonzero are those P
Pl Pl with i=1 n′i = i=1 ni . In particular, the reduced density matrix of one single particle basis state is always diagonal, indicating entanglement whenever there are more than one nonzero diagonal elements. In this approach, the statistics determines the dimensions of the Hilbert space of each mode. For fermions, ni = 0, 1, D = 2, hence the entanglement between one single particle basis state and the rest of the system is 0 ≤ S ≤ ln 2. For bosons, ni is arbitrary, hence D is infinity. This point does not pose real difficulties, but further investigation on it is interesting. One can also define the entanglement relative to the ground state, by considering only the effect of creation and annihilation operators acting on the ground state. Then ni in (2) is understood as the number of the excited particles, which are absent in the ground state Gi, i.e. √ n1 n∞ n1 , · · · , n∞ i ≡ (1/ n1 ! · · · n∞ !)a†1 · · · a†∞ Gi. The nonrelativistic field theoretic or second quantized Hamiltonian is R R ˆ + d3 rψˆ† (r)h′ (r)ψ(r) ˆ H = d3 rψˆ† (r)h(r)ψ(r) R R (3) 1 † ′ ′ ˆ ′ ˆ 3 3 ′ ˆ† ˆ + 2 d r d r ψ (r)ψ (r )V (r, r )ψ(r )ψ(r), where h(r) is the single particle Hamiltonian including the kinetic energy, V (r, r′ ) is the particleparticle interaction, h′ (r) is some external potential which is not included in h(r) for convenience. In the examples in this paper, h′ 6= 0 only in the issue of generating electronhole excitations by electronlight interaction; h′ = 0 in all discussions on entanglement in manyparticle enˆ ergy eigenstates. The field operator ψ(r) can be expanded P in an arbitrarily chosen single particle basis as ˆ ψ(r) = i φi (r)ai , where i is the collective index of the single particle state, which may include spin if needed, ai is the annihilation operator, φi (r) is the single particle wavefunction in position space. We use the same notations for fermions and bosons. Thus H can also be written as P hihjia†i aj H = ij P P + hih′ jia†i aj + 12 hijV lmia†i a†j am al , ij
ijlm
(4) The generalization to the existence of more species of identical particles is straightforward. Single particle basis transformation leads to a unitary transformation in the creation and annihilation operators. There may be more general transformations of the creation and annihilation operators, and some even involve combination of operators of different species. Such a transformation means describing the system in terms of a different set of single particles or quasiparticles. Even if V = 0 and h′ = 0, as far as hihji 6= 0, H can generate occupationnumber entanglement between single particle basis state i and j. An eigenstate of a second quantized interacting Hamiltonian is often entangled. In the chosen single particle
4 basis, if an eigenstate of (4) is nonentangled, then it must be of the form ψi = ⊗i ni i. Consequently for each i, Hˆ ni ψi = n ˆ i Hψi, where n ˆ i = a†i ai . It can be seen that this is often not satisfied by H in (4). However, when we use entanglement to characterize the quantum correlation caused by interaction, it is suitable to use the set of eigenstates of the single particle Hamiltonian h, which we call proper single particle basis. In this single particle basis, with hφµ = ǫµ φµ , R 3 † P ˆ d rψˆ (r)h(r)ψ(r) = µ ǫµ a†µ aµ , whose eigenstates are of the form ⊗µ nµ i, where µ is the collective index of the proper single particle basis. Therefore in the proper single particle basis, entanglement can be used to characterize the effect of interaction. In case h′ = 0, it characterizes the effect of the particleparticle interaction. Each nondegenerate energy eigenstate of the noninteracting system must be nonentangled. When there is degeneracy, an entangled energy eigenstate of a free system may be constructed as a superposition of degenerate nonentangled states that differ in the occupationnumbers of at least two single particle basis states (on the other hand, particle number conservation constrains that it is impossible to be different only in one single particle basis state). But one can always use a set of nonentangled eigenstates. If in the proper single particle basis, entangled energy eigenstates inevitably arise, then there must be interaction. Besides, the proper single particle basis directly corresponds to the energy spectrum of single particle excitations, and is more experimentally accessible. For the socalled strongly correlated systems, e.g. Luttinger liquid and fractional quantum Hall state discussed below, peculiar physical properties are caused by the strong (Coulomb) interaction, hence it is particularly interesting to consider occupationnumber entanglement in the proper single particle basis. By generalizing the method to relativistic field theory, it may be useful for quantum chromodynamics. On the other hand, when an improper single particle basis is used, even the onebody term in H is not diagonal, and the eigenstates are entangled even when there is no interaction, as seen by transforming a†µ in p nµ ⊗µ nµ i ≡ ⊗µ (1/ nµ !)a†µ 0i. Nevertheless, entanglement in an improper basis may be interesting in problems such as hopping, tunnelling, Mott transition, etc. For example, in a twostate problem, of which the double well potential problem is an example, the proper basis states are linear superpositions of the two states, but in many cases it is these two states that are observed. As occupationnumber entanglement in an improper single particle basis presents even when there is no interaction, it may be valuable for quantum information processing. When there are more than one index in the single particle basis, one of them can be used as the tag effectively distinguishing the particles, and the other indices deter
mine whether they are entangled in these degrees of freedom. With this effective distinguishability, the state in the configuration space of the remaining degrees of freedom can be directly obtained from the secondquantized state. For example, in √12 (a†k′ ↑ a†k↓ +a†k′ ↓ a†k↑ )0i, where k′ and k represent momenta, one can say that the particle in k′ i and the particle in ki are spinentangled. One can also say that the particle in  ↑i and the particle in  ↓i are momentumentangled. With the momentum as the distinguishing tag, the spin state is √12 ( ↑ik′  ↓ik +  ↑ ik′  ↓ik ). Alternatively, with the spin as the distinguishing tag, the momentum state is √12 (k′ i↑ ki↓ + ki↑ k′ i↓ ). The ideas about the occupationnumber entanglement can beP consistently applied even to a oneparticle state φi = i ci ii, where ii’s are a set of basis states. In terms of occupation numbers of different basis Q states, P c 1i the state can be written as φi = i i j6=i 0ij . i Thus the occupationnumber of basis state ii is entangled with other basis states, with the amount of entanglement −ci 2 ln ci 2 − (1 − ci 2 ) ln(1 − ci 2 ). When φi and ii ’s are eigenstates of the Hamiltonian, φi = Ii, thus cI = 1 while cj = 0 for j 6= I, consequently in φi, each basis state is nonentangled with other basis states. In the example of an electron in a superposition of a state  − k′ ie in a Fermi sea and a state kie out of the Fermi sea, written in terms of the occupationnumbers of these two electronic states, a0iek 0ie−k′ + b1iek 1ie−k′ , can also be written in terms of occupationnumbers of the electron state kie and the hole state k′ ih , as a0iek 0ihk′ + b1iek 1ihk′ . This becomes a superposition of absence and presence of an electronhole pair. But this kind of electronhole entanglement is different from the entanglement between an existing electron and an existing hole. Now we start our discussions on occupationnumber entanglement in important energy eigenstates in manyparticle physics, using the socalled proper single particle basis. These systems play fundamental roles in condensed matter physics.
3. Fermi systems
First let us consider a fermi gas, which plays a fundamental role in understanding condensed matter physics. The proper single particle basis here is the tensor product of singleparticle momentum and spin states. The ground state of a free Fermi gas is Gi = Qk
is clearly nonentangled. An excited state like a†ks b†k′ s′ Gi is still separable, where k > kF > k′ , b†ks = a−k−s is the hole operator. It is simple to check that for each of these nonentangled states, a Fockspace reduced density matrix, as in (2), always has only one nonzero element, hence the entanglement between the occupationnumbers
5 of any set of single particle basis states and the rest of the system indeed vanishes. There may be entanglement in an excited state of a Fermi gas, because of degeneracy due to spin degree of freedom. For example, there is maximal entanglement in the electronhole pair √12 (a†k↑ b†k′ ↓ + a†k↓ b†k′ ↑ )Gi. For a free gas, it is a superposition of the degenerate nonentangled states a†k↑ b†k′ ↓ Gi and a†k↓ b†k′ ↑ Gi. With respect to the empty state, it is an entanglement between the occupationnumber of the excited electron state and others. Since Gi is nonentangled, entanglement in an excited state relative to the empty state is equal to the entanglement relative to Gi. The entanglement in † † † † √1 (a b ′ + a b ′ )Gi is simply electronhole entank↓ k ↑ 2 k↑ k ↓ glement with respect to the ground state. Moreover, an electron and a hole, by definition, corresponds to different single particle states, and can be regarded as distinguishable particles, as tagged by that a creation operator of a hole corresponds to annihilation of an electron. In the absence of interaction, however, one can always use a set of nonentangled energy eigenstates as the orthonormal set. More realistic treatment, in the context of solid state physics, takes into account the Coulomb interaction between the electrons, as well as the crystal structure, which provides a single particle (periodic) potential. A basic method is the HatreeFock approach [36]. The ground state is still nonentangled, since the HatreeFock treatment only modifies the single particle states and ground state energy. But entanglement inevitably arises in excited states. To illustrate the idea, the simplest model of electronic excitations in solids is considered in the following. Consider one electron is excited from a valence band to a conduction Pband. An eigenstate of this excitation, an exciton, is k,k′ Ak,k′ a†k b†k′ Gi, in the spinless case. For brevity, the band indices are omitted, as one corresponds to the electron operator while the other corresponds to the hole operator. The occupation numbers of the basis states kie and k′ ih respectively occupied by the excited electron and by the hole are the same as those in relative to the ground state, since they are zero in the ground state. The Fockspace reduced density matrixPelements of k can be obtained as h1ρ1 (k)1i = αk ≡ k′ Ak,k′ 2 , h0ρ1 (k)0i = 1 − αk . Therefore the occupationnumber entanglement between the electron basis state kie and the rest of the system is −αk ln αk − (1 − αk ) ln(1 − αk ). The occupationnumber entanglement between the hole basis state k′ ih and the rest of the system P is −αk′ ln αk′ − (1 − αk′ ) ln(1 − αk′ ), where αk′ = k Ak,k′ 2 . The Fockspace reduced density matrix ρ1,1 of the electron basis state kie plus the hole basis state k′ ih as a subsystem is calculated by considering that the electron and the hole belong to different species of identical particles. h1, 1ρ1,1 (k, k′ )1, 1i =
P 2 Ak,q′ 2 , h1, 0ρ1,1 (k, k′ )1, 0i = γk ≡ q′ 6=k′ Ak,q′  , P 2 ′ Furh0, 1ρ1,1 (k, k )0, 1i = γk′ ≡ q6=k Aq,k′  . thermore, ρ1,1 (k, k′ ) must be diagonal. Hence their occupationnumber entanglement with the rest of the system is −Ak,k′ 2 ln Ak,k′ 2 − γk ln γk − γk′ ln γk′ − (1 − Ak,k′ 2 − γk − γk′ ) ln(1 − Ak,k′ 2 − γk − γk′ ). With the electron and the hole effectively distinguishable, P the state can be written, in the configuration space, as k,k′ Ak,k′ kie k′ ih . The entanglement between these two distinguishable particles is obtained by finding the eigenvalues of the reduced density matrix for either particle. PWith spin degeneracy, the excitonic states are ′ ′ ′ k,k′ Ak,k S, Sz ik,k , where S, Sz ik,k represents three triplet states as the ground states, 1, 1ik,k′ = a†k↑ b†k′ ↑ Gi, 1, 0ikk′ =
† † √1 (a b ′ 2 k↑ k ↓
− a†k↓ b†k′ ↑ )Gi and 1, −1ik,k′ =
a†k↓ b†k′ ↓ Gi, and one singlet state 0, 0ik,k′ =
† † √1 (a b ′ + 2 k↑ k ↓
a†k↓ b†k′ ↑ )Gi. The occupationnumber entanglement, with the full collective index including Bloch wavevector and spin, can be calculated in a way similar to the spinless case. The above discussions on the spinless P ′ 1, ±1ik,k′ . For A case applies similarly to k,k′ k,k P † † † † 1 √ k,k′ Ak,k′ 2 (ak↑ bk′ ↓ ± ak↓ bk′ ↑ )Gi, one can find, for example, that the occupationnumber entanglement between the electron basis state k, ↑ie plus the hole basis state k′ , ↓ih as a subsystem and the rest of the system is −(Ak,k′ 2 /2) ln(Ak,k′ 2 /2) − (γk /2) ln(γk /2) − (γk′ /2) ln(γk′ /2) − (1 − γk /2 − γk′ /2 − Ak,k′ 2 /2) ln(1 − γk /2 − γk′ /2 − Ak,k′ 2 /2), and that the occupationnumber entanglement between the electron basis state k, ↑ie plus the hole basis state k′ , ↑ih as a subsystem and the rest of the system is −(αk /2) ln(αk /2) − (αk′ /2) ln(αk′ /2)−(1−αk/2−αk′ /2) ln(1−αk /2−αk′ /2). The entanglement considered above is determined by the Coulomb interaction, as Ak,k′ is determined by the Schr¨odinger equation in momentum representation. When Coulomb interaction is negligible, Ak,k′ = 1 for a specific pair of values of k and k′ , and consequently various entanglements concerning the k and k′ as discussed above consistently vanish. The spin part of the eigenstates can be chosen to be nonentangled. Interaction causes spread of Ak,k′ and thus nonvanishing entanglement in the Bloch wavevectors, as well as the spinentanglement. Noteworthy is that the detail of the interaction only affects Ak,k′ , but does not affect the structure of the spin states. With the electron and the hole effectively distinguishable, Pthe state can be′ written, in the configuration space, as k,k′ Ak,k′ kie k ih S, Sz i. So the orbital and spin degrees of freedom are actually separated, as consistent with the fact that the spinorbit coupling has been neglected here. An excited state is often generated by electronlight in
6 teraction switched on for a period. The light is treated as classical. The electronlight interaction corresponds to h′ in Section 2. With coupling between different single electron and hole basis states, it can generate electronhole entanglement. This underlies a recent experimental result [37], on which a theoretical analysis, with spinorbit coupling taken into account, will be given elsewhere [38]. If an interacting fermi system can be described as a Fermi liquid, then there is a onetoone correspondence between the particles in the noninteracting system and the quasiparticles of the interacting system, obtained by adiabatically turning on the interaction [39]. Therefore in terms of the quasiparticles, the ground state of a Fermi liquid is nonentangled. One may say that the electron entanglement caused by the interaction can be renormalized away. In contrast, the ground state of Luttinger liquid is a global unitary transformation of a Fermi sea [40], and is entangled. New ground states emerge in phenomena like quantum Hall effect and superconductivity, in which entanglement is important, as shown in the next two sections.
4. Quantum Hall Effect
Quantum Hall states are obtained by filling the spin polarized electrons in the degenerate (single particle) Landau levels [41]. The single particle Hamiltonian, corresponding to the proper single particle basis, is the Hamiltonian of a twodimensional electron in a magnetic field. One knows that the degeneracy of each Landau level, i.e. the number of different states of each energy eigenstate, is the same. The key quantity in quantum Hall effect is the filling factor ν, which is the number of electrons divided by the degeneracy of each Landau level, and manifested in the quantized Hall resistivity. We show that the filling factor ν determines the entanglement. First, the entanglement vanishes in integer quantum Hall effect, which appears when ν = n is an integer. The n lowest Landau levels are completely filled while others are empty. Because of energy gap, the interaction is not important, and the ground state of the interacting systems can be smoothly connected to that of the noninteracting Q system. Thus the ground state is just a product state µ a†µ 0i, where µ runs over the filled states. Hence the occupation number of each single particle state belonging to a completely filled Landau levels is 1, while the occupation number of every other single particle state is 0. In the Fockspace reduced density matrix of a single particle basis state, for a state µ belonging to a completely filled Landau level, only h1ρ1 (µ)1i = 1 is nonzero, while for a state µ belonging to an empty Landau level, only h0ρ1 (µ)0i = 1 is nonzero. It is like the ground state of a free Fermi gas. All single particle basis states are separable from one another, and there is no entanglement.
In a fractional quantum Hall state of ν = n+f , where f is the proper fractional part, n ≥ 0 lowest Landau levels are completely filled, f of the next Landau level is filled, the higher Landau level are empty. Because of partial filling, the interaction cannot be treated perturbatively, and in the ground state, electrons are strongly correlated. Each single particle basis state belonging to one of the n completely filled levels or the empty levels is separated, i.e. the occupation number is either 0 or 1 and is just a factor in the manyparticle state in the particle number representation. For each single particle basis state in the partially filled landau level, its entanglement with the rest of the system is obtained as follows. Consider the identity h1ρ1 (µ)1i =
X n1 ···n∞
nµ hn1 · · · n∞ ρn1 · · · n∞ i = hˆ nµ i,
(5) where nµ = 0, 1 (µ = 1, · · · , ∞), hˆ nµ i ≡ T r(ρˆ nµ ) is the expectation value of the particle number at state µ. The first equality is valid only for fermions while the second is valid for both fermions and boson. On the other hand, in an isotropic uniform state, for each single particle basis state belonging to the partially filled Landau level, hˆ nµ i = f.
(6)
Therefore the entanglement between a single particle basis state belonging to the partially filled Landau level and the rest of the system is S = −f ln f − (1 − f ) ln(1 − f ).
(7)
This simple expression of entanglement, in terms of the proper fractional part of filling factor, gives a useful measure of the quantum correlation in a quantum Hall state. The entanglement increases from 0 at f = 0, corresponding to the integer quantum Hall effect, towards the maximum ln 2, after which it decreases towards 0 at f = 1, corresponding to the integer quantum Hall effect again. Note that the filling factor, hence the entanglement, is extremely precisely measured, with topological stability. Anyons have been proposed as a candidate to implement faulttolerant quantum computing [17, 44]. The present result confirms the intrinsic entanglement, which is needed for quantum computing. The amount of entanglement obtained above is consistent with the fact that for an integer quantum Hall effect, the manyparticle wavefunction is a single Slater determinant, indicating separability, while for fractional quantum effect, it is not [42], indicating the existence of entanglement. The Laughlin state is indeed isotropic uniform. The fractional quantum Hall effect can be understood in terms of the composite fermions or composite bosons [41], which are nonentangled. For example, the
7 state at ν = 1/(2p + 1) can be viewed as νef f = 1 integer quantum Hall state of composite fermions, or equivalently as Bose condensation of composite bosons, while ν = 1/2p state is a free Fermi gas with a Fermi surface. In each of these descriptions, the system of the composite particles is separable. This separability can also be inferred from the offdiagonal longrange order (ODLRO) exhibited by these composite particles [43], because disentanglement of the condensate mode from other modes underlies ODLRO [25]. ODLRO is an important notion in manyparticle physics, and is the hallmark of Bose condensation and superconductivity [5].
5. BCS Superconductivity
ρ1 (k, s) = diag(1 − xk , xk ).
k
where N0 = k (1+gk 2 )−1/2 is the normalization factor. For ψ0 i, in which the particle number is not conserved, The entanglement is only between each pair (k, s) and (−k, −s), with the amount Q
(9)
where zk = 1/(1 + gk 2 ). There is no entanglement between different pairs. However, for a system of N electrons, with N fixed, the exact state is the projection of ψ0 i onto the N particle space, which is [46][55]. X ψ(N )i = NN gk1 a†k1 ↑ a†−k1 ↓ · · · gkN/2 a†kN/2 ↑ a†−kN/2 ↓ 0i, (10) P P represents where NN = ( gk1 2 · · · gkN/2 2 )−1/2 , summations over k1 , · · · , kN/2 , with the constraint ki 6= kj . One can observe that this state is given by the superposition of all kinds of products of N/2 different gk a†k↑ a†−k↓ . This feature leads to entanglement between different Cooper paired modes. Let us investigate the entanglement in ψ(N )i. First we evaluate the elements of the Fockspace reduced density matrix of mode (k, s), denoted as hnk,s ρ1 (k, s)nk,s i. One can obtain P′ gk 2 gk′2 2 · · · gk′N/2 2 , h1k,s ρ1 (k, s)1k,s i = xk = P gk1 2 · · · gkN/2 2
(11)
One can also obtain that the element of the reduced density matrix for one pair of modes with the opposite k and s, denoted as hnk,s , n−k,−s ρ2 (k, s; −k, −s)nk,s , n−k,−s i, is xk when nk,s = n−k,−s = 1, is 1 − xk when nk,s = n−k,−s = 0, and is 0 otherwise. Hence in the basis (0k,s 0−k,−s i, 0k,s 1−k,−s i, 1k,s 0−k,−s i, 1k,s 1−k,−s i), ρ2 (k, s; −k, −s) = diag(1 − xk , 0, 0, xk ).
As another example of using the concept of entanglement to further our understanding of manyparticle physics, we now consider BCS superconductivP ity [45, 46]. The Hamiltonian is H = k,s ǫk nk,s + P † † ′ ′ The proper k,k′ hk , −k V k, −kiak′ ↑ a−k′ ↓ a−k↓ ak↑ . single particle basis is (k, s), in which the onebody term in H is diagonalized. The BCS superconducting ground state is, Y ψ0 i = N0 (1 + gk a†k↑ a†−k↓ )0i, (8)
S0 = −zk ln zk − (1 − zk ) ln(1 − zk ),
P′ where represents the summations over k′2 , · · · , k′N/2 , with the constraint k′i 6= k′j and k′i 6= k, where i, j = 2 · · · N/2. One can obtain h0k,s ρ1 (k, s)0k,s i = 1 − xk . Hence in the basis (0k,s i, 1k,s i),
(12)
Therefore the entanglement between the occupationnumber at mode (k, s) and others is S = −xk ln xk − (1 − xk ) ln(1 − xk ),
(13)
so is also the entanglement between the occupationnumbers of the pair (k, s) and (−k, −s) on one hand, and the rest of the system on the other. Note that in ψ(N )i, there is no entanglement between each pair (k, s) and (−k, −s), as can be simply confirmed by the fact that (12) is diagonal. If gk is 1 for k < kf and is 0 for k > kf , then xk is 1 for k < kf and is 0 for k > kf . Consequently for any k, each of those Fockspace reduced density matrices only has one nonvanishing element. Therefore the entanglement S reduces to zero, consistent with the fact that under this limit, the state (10) reduces to the ground state of a free Fermi gas [46]. In the superconducting state, gk differs from that of the free Fermi gas in the vicinity of the Fermi surface, consequently the amount of entanglement becomes nonzero. gk is just the relative momentum wavefunction of each Cooperpaired electron, and is directly related to the superconductingp energy gap ∆k as gk /(1 + gk2 ) = ∆k /2Ek , where Ek = k2 /2m + ∆2k . As the order parameter, superconducting energy gap is a key physical property of superconductivity. Therefore we have obtained a direct relation between entanglement and the superconducting energy gap and thus various physical properties of superconductivity. The entanglement vanishes if and only if the superconductivity vanishes. Although superconductivity may be loosely described as Bose condensation of Cooper pairs, it is understood that a Cooper pair is still different from a boson, the strong overlap and correlations between Cooper pairs gives rise to the gap which is absent in the case of a Bose gas [46]. The crossover between Bose condensation and BCS superconductivity has been an interesting topic for a long time. Here we have found that entanglement
8 in ψ(N )i provides a quantitative characterizations of the correlations between Cooper pairs and thus may be useful in studying of the crossover between Bose condensation and superconductivity. After this work was done, there appeared a preprint on entanglement in BCS state involving strong interaction [48]. 6. Bose Systems
Consider a system of spinless Bosons. The proper single particle basis is the momentum state. An eigenstate of a free spinless boson system is simply nq1 , nq2 , · · ·i = (a†q1 )nk1 (a†q2 )nq2 · · · 0i. In the ground state (a†0 )N 0i, all particles occupy the zero momentum state. This is BoseEinstein condensation. The system is obviously nonentangled, in the proper single particle basis, in all the eigenstates. Thus there is entanglement in position basis, in consistent with a related work [47]. For a weakly interacting spinless boson gas, entanglement between occupationnumbers of different momentum states is still absent under HatreeGrossPitaevskii approximation. In this approach, an energy eigenstate is approximated as a product of single particle states, with symmetrization, hence there is no occupationnumber entanglement. The ground state is a product of a same single particle state. The weak interaction only affects the single particle state. Nevertheless, there may be entanglement when there is spin degree of freedom or in other complex situations [49, 50]. These features are like those of the HatreeFock approach of a Fermi gas. The next level of treatment is Bogoliubov theory [51], nonzero entanglement appears, even in the ground state. It is first hinted by the Bogoliubov transformation in the original, particle nonconserving, formulation, which defines a new annihilation operator which is a superposition of a annihilation operator aq and the creation operator for the opposite momentum a†−q , namely, bq = uq aq + vq a†−q . This transformation diagonalizes the second quantized Hamiltonian, hence in terms of the newly defined quasiparticles, there is no entanglement, signalling that there exists entanglement in terms of the original particles. Similar to BCS superconductivity, in the particle nonconserving theory, entanglement only exists between the each pair of modes q and −q (q > 0). The ground state is [52, 53] P P nq1 (−vq2 /uq2 )nq2 · · ·] Ψ0 i ∝ nq2 · · · [(−vq1 /uq1 ) nq1 n0 ; nq1 , nq1 ; nq2 , nq2 ; · · ·i, (14) in which there are n0 particles with zero momentum while nq pairs of particles with q and −q. Therefore, the entanglement at q and −q is P P between occupation numbers S = − i xi ln xi , where xi = yi / i yi , n = 0, 1, · · · , ∞. where yi = vq /uq 2i . The condensate mode is indeed
disentangled from the rest of the system, in consistent with our result obtained from ODLRO. Vedral studied entanglement in a Bose condensate, using a state similar to Eq. (14), and calculated a different quantity defined there to measure the amount of entanglement [27]. We now focus on the particle number conserving version of the Bogoliubov theory, which gives the ground state as [54] Ψ(N )i ∝ (a†0 a†0 −
X q6=0
cq a†q a†−q )N/2 0i,
(15)
where cq , with cq  < 1 is determined by the Hamiltonian and is the effect of the weak interaction. It can be found that P Ψ(N )i ∝ p(N/2; n0 · · · n∞ )(−cq1 )n1 · · · n0 ,n1 ···n∞
×(−cq∞ )n∞ 2n0 i0 n1 iq1 n1 i−q1 · · · n∞ iq∞ n∞ i−q∞ (16) where n0 + n1 + · · · + n∞ = N/2, p(N/2; n0 · · · n∞ ) = (N/2)! n0 !n1 !···n∞ ! is the number of partitions of N/2 objects into different boxes, with n0 objects in the box labelled 0, n1 in the box labelled 1, and so on. Then one obtains the Fockspace reduced density matrices of different momentum states. The nonvanishing elements of ρ1 (0) are x2n0 (0)P≡ h2n0 ρ1 (0)2n0 i =A p2 (N/2 − n0 ; n1 · · · n∞ )cq1 2n1 · · · cq∞ 2n∞ , n1 ···n∞
(17) where n1 + · · · + n∞ = N/2 − nP the normalization factor A = 0, [ n0 ,···,n∞ p2 (N/2; n0 , · · · , n∞ )cq1 2n1 · · · cq∞ 2n∞ ]−1 , with n0 + n1 + · · · + n∞ = N/2. It can be seen that x2n0 (0) ≈

P
q6=0
N/2 P

cq N −2n0
P
n0 =0 q6=0
cq
,
(18)
N −2n0
P under the assumption that c∗q1 cq2 ≈ 0, where the summation is over all nonzero q1 6= q2 . The entanglement between the zero momentum state and the rest of the system is N/2
S(0) = −
X
x2n0 (0) ln x2n0 (0).
(19)
n0 =0
For a momentum q1 6= 0, the nonvanishing elements of ρ1 (q1 ) are xn1 (q1 ) ≡ hn1 ρP 1 (q1 )n1 i = Acq1 2n1 p2 (N/2; n0 · · · n∞ )cq1 2n2 · · · cq∞ 2n∞ , n0 ,n2 ···n∞
(20)
9 where the summation is subject to n0 + n2 + · · · + n∞ = N/2 − n1 . It can be seen that
xn1 (q1 ) ≈
N/2 P
n1 =0
cq1  =
N/2−n P 1
cq1 2n1
n0 =0
cq1 2n1
2n1
[cq1
n1 =0
q6=0,q1
n0 =0
cq1 
− ( P
2n1
P
N/2−n P 1
q6=0,q1
N/2 P

cq 
q6=0,q1
cq N −2n0 −2n1
P
7. Summary and remarks
q6=0,q1
2n1
cq1 
− ( P
 )
cq 

cq N −2n0 −2n1
P
q6=0,q1
)2n1 
cq 
P
q6=0,q1
N +1
(21) cq
N +1 ]
The entanglement between a nonzero momentum state q1 and the rest of the system is N/2
S(q1 ) = −
X
or twoparticle reduced density matrix (in the case of superconductivity), the condensate mode is disentangled with the rest of the system [25].
xn1 (q1 ) ln xn1 (q1 ).
(22)
n1 =0
Obviously, the entanglement between −q1 and the rest of the system, as well as the entanglement between the pair q1 plus −q1 and the rest of the system, are both also S(q1 ). It can also be seen that there is no entanglement between q1 and −q1 . On this aspect, there is a similarity with BCS superconductivity. P = Consider the identity nq nq hnq ρ1 (q)nq i P′ n hn · · · n ρn · · · n i = hˆ n i, for ∞ 0 ∞ q {ni } q 0 P′ nq = 0, 1, · · ·, and for different q’s, where {ni } represents summations over n0 , · · · , n∞ , except nq . For the Bogoliubov ground state Ψ(N )i, it is known that hˆ n0 i is close to N , while hˆ nq1 i ≪ N for q1 6= 0. It is thus constrained that only a small number (compared with N ) of the Fock space matrix elements hnq ρ1 (q)nq i is considerable for mode q. Thus the entanglement is small. But it is not zero, as in the Hatree approximation. Furthermore, (18) indicates that x2n0 (0) exponentially P decays with the n0 , with the rate 1/2 ln  q6=0 cq . Hence indeed very small number of matrix elements hn0 ρ1 (0)n0 i is considerable, and thus S(0) is small. On the other hand, in (21), the change of xn1 (q1 ) with n1 is slower since it involves the counteracting of two exponentially increasing terms. Consequently S(q1 ) > S(0). The small but nonzero entanglement is a characterization of the twoparticle correlation caused by the weak interaction, which is the essence of Bogoliubov theory [54]. This can be seen from Eqs. (18) and (21), which indicates that the entanglement is only dependent on the function cq , which is determined by the weak interaction. The result in last and this section is the entanglement in terms of the original particles. As consistent with the fact that entanglement depends on which single particle is used in representing the manyparticle system, it can be shown that in the set of eigenstates of the oneparticle reduced density matrix (in the case of Bose condensation)
It is known that quantum correlation in a system of identical particles can be characterized in terms of entanglement between occupationnumbers of different single particle basis states, and thus depends on which single particle basis is chosen. Consequently, in general, occupationnumber entanglement may be generated, or exists in the energy eigenstates, even in absence of interaction of particles. Indeed, it is caused by coupling between different single particle basis states (modemode coupling) in the manyparticle secondquantized Hamiltonian, which may exist even in the onebody term in the Hamiltonian. However, our purpose in this paper is to use entanglement as a characterization of effects of interaction. For this purpose, we choose the set of eigenstates of the single particle Hamiltonian as the single particle basis on which the entanglement is defined. For convenience, we call it proper single particle basis. In this single particle basis, if there is no interaction, the secondquantized Hamiltonian is diagonal in different single particle basis states, and thus the manyparticle eigenstates can always be chosen to be nonentangled. Using the socalled proper single particle basis state, we examined entanglement in eigenstates, especially the ground states, of some important manyparticle Hamiltonians. These examples demonstrate that entanglement in the proper single particle basis can indeed characterize the effect of interaction, vanishing as the interaction vanishes. Moreover, the amount of the entanglement calculated is directly related to the macroscopic physical properties. In other words, it is demonstrated that the microscopic entanglement is manifested in the macroscopic physical properties. It appears that entanglement in the proper single particle basis is useful especially for studying the strongly correlated systems, in which interactions are important. For an interacting Fermi gas, electronhole entanglement inevitably appears in some excited eigenstates, as described in the HatreeFock approach. Electronhole entanglement can be generated by electronlight interaction, which is not included in the single particle Hamiltonian which defines the proper single particle basis. When the ground state of a Fermi liquid is expressed in terms of the Landau quasiparticles, i.e. electrons dressed by the interaction, it becomes nonentangled. We found the nice result that the entanglement in a quantum Hall state is just the entropy of the probability distribution f and 1 − f , where f is the proper fractional part of the filling factor of a Landau level. Hence
10 entanglement here can be extremely precisely measured, with topological stability. This gives a support to the wellknown proposal of using anyons for faulttolerant quantum computing. We also made a detailed calculation of entanglement in BCS ground state. Both the particlenumber nonconserved and the particle number conserved states are considered. In each case, the amount of entanglement is a function of the relative momentum wavefunction gk of every two Cooperpaired particles, and thus directly related to the superconducting energy gap. The entanglement vanishes if and only if the superconductivity vanishes. Finally, we turned to Bose systems. For a spinless system, the entanglement is absent in the eigenstates in the HatreeGrossPitaevskii approximation. However, though small, it is nonvanishing in the Bogoliubov theory, using which we calculate the entanglement in the ground state, where there is a kind of pairing between opposite momenta. Entanglement in the proper single particle basis provides a characterization of the twoparticle correlation due to interaction, which is the essence of Bogoliubov theory. Manybody entangled states as those in condensed matter physics may be useful for quantum information processing. One may adiabatically control a timedependent manybody ground state which encodes the quantum information. If there is a finite energy gap between the ground state and the excited states, as existing in many condensed matter systems, such a quantum information processing should naturally possess some robustness against environmental perturbation. But more caution is needed in using entanglement in a condensed matter system to demonstrate Bell theorem and such. A reason is that in condensed matter physics, many Hamiltonians, usually instantaneous, are effective ones on a certain time scale, with many degrees of freedom renormalized. The formal entangled state and the instantaneous correlations may be meaningful only on a certain coarsegrained time scale. For identical particles, there is intrinsic builtin nonseparability because of the precondition that the spatial wavefunctions overlap, for example, spin magnetism based on exchange interaction originates in antisymmetrizing the spinorbit states of the electrons interacting with “instantaneous” Coulomb interaction which is always there. Deeper understanding is still needed on the occupationnumber entanglement.
Acknowledgments
I thank Professors Tony Leggett, Peter Littlewood, John Preskill, William Wootters, YongShi Wu and Chen Ning Yang for useful discussions.
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