Local entanglement generation in the adiabatic regime
We study entanglement generation in a pair of qubits interacting with an initially correlated system. Using time independent perturbation theory and the adiabatic theorem, we show conditions under which the qubits become entangled as the joint system evolves into the ground state of the interacting theory. We then apply these results to the case of qubits interacting with a scalar quantum field. We study three different variations of this setup; a quantum field subject to Dirichlet boundary conditions, a quantum field interacting with a classical potential and a quantum field that starts in a thermal state.
Quantum entanglement is widely believed to be the distinguishing resource of quantum computers. For this reason it is crucial that we fully understand how entanglement can be generated and how it evolves. Various studies already show that the time evolution of entanglement in open systems is often highly non trivial, see e.g. veitia (). This complicated time evolution often prevents us from studying general features of entanglement dynamics and forces us to focus on particular examples. In this work, we focus our attention on one particular time evolution scenario, namely, the adiabatic evolution of the ground state. This allows us to show that adiabatic evolution can naturally generate entanglement in a pair of qubits interacting with a correlated system.
One of the main applications of this setup is the extraction of entanglement from the vacuum. Indeed, it was shown in cliche-kempf () that qubits interacting adiabatically with a relativistic quantum field in the vacuum state can get entangled in a renewable fashion. This result is one of the many new features that arise as a consequence of special relativity considerations in quantum information theory, see e.g. terno (); Rezn2 (); hu0 (); hu (); Meni (). We follow-up on this work by studying modifications of this setup and analyzing whether they enhance or degrade the entanglement generated in the qubits.
The paper is organized as follows. In Sec. II we present the general framework and calculate explicitly the entanglement contained in a pair of qubits in the ground state of a weakly interacting theory. We also discuss how this entanglement can be generated with an adiabatic switch on of the interaction Hamiltonian. In Sec. III we use the tools previously developed to show that the entanglement available in a quantum field theory with Dirichlet boundary conditions is degraded. In Sec. IV we consider a quantum field weakly interacting with a classical field and show that depending on the type of interaction it can either enhance or degrade the entanglement generated in the pair of qubits. In Sec. V we study the case of a quantum field that starts in a thermal state instead of the vacuum and show how the entanglement decreases as the temperature of the system increases.
We work in the natural units . Wherever necessary to avoid ambiguity we will denote operators or states , corresponding to the Hilbert space , by a superscript , for example, and . Orders in perturbation theory will be denoted by a subscript (j), so for example we could have . We conveniently work in the Schrödinger picture of quantum mechanics.
Ii General framework
The system we study is illustrated in Fig. (1) which describes two localized qubits, and , unitarily and locally interacting with a correlated system, . Let the Hamiltonian of the free theory be of the form . More precisely, let
The assumption that both two-level systems are identical allows setting . In addition, for system B, we set and denote the corresponding eigenstate by . Thus, the ground state of the free theory is (where ) and we have It is convenient to choose the basis (for ) coinciding with the eigenstates of , that is
Then we have where and .
Let the local interaction between the qubits and and system B be described by a Hamiltonian of the form
For simplicity we assume that . Furthermore, we assume that the interaction described by is weak, so we can resort to perturbation theory to find the ground state of the interacting theory , that is, . Thus, up to second order, we may write where is a small parameter that sets the scale of . We have and using time-independent perturbation theory Coh () one can easily find explicit expressions for and . The density matrix describing the joint system AB after the interaction reads where , and . Using these expressions, we readily determine the reduced density matrix describing system when the joint system AB is in the ground state of the interacting theory. Thus, up to second order, we have with ,
Here the sums run over all the values of except those for which any denominator vanishes. For simplicity let us now focus on an important class of local interactions, namely . The nonvanishing matrix elements are then given by
We note that the above matrix elements may be written as , , and . The operators and for are defined as and . From these expressions one easily proves that , thus guaranteeing the positivity of ( up to second order in ). In order to quantify the degree of entanglement in system we make use of the negativity , defined as twice the absolute value of the negative eigenvalue of Werner (). In our particular case it reads
To generate this entanglement in the pair of qubits, we need to prepare the state of system in the ground state of the interacting theory . To do this, we assume that the state of system can easily be prepared in the ground state of the free theory . Moreover, we assume that the interaction Hamiltonian can be switched on with a switching function such that where and . If the interaction between the qubits and B is switched on adiabatically, then the evolution of the joint system is given by where is the ground state of . According to the validity condition for adiabatic behavior Saran (); Schiff (), we need at first order
Therefore, if this condition holds for some choice of , then the ground state of can easily be reached in a time scale of .
ii.1 Example: Qubits interacting with a scalar quantum field
As an application of this formalism, we consider qubits interacting locally with a smeared portion of a quantum scalar field of mass . This effectively models an atom interacting with a quantum field like the quantum electromagnetic field. This example was first explicitly considered in cliche-kempf (). The operators are then
and for simplicity we choose and such that the distance between the two qubits is . Note that the smearing functions describe the effective size of the qubits. In the limit the introduction of the smearing functions is equivalent to the introduction of a cut-off in momentum space. Therefore, we shall always replace these smearing functions with a momentum cut-off and set . From Eq. (6a), (6) and (8) we recover the results of cliche-kempf (),
where . Using these equations in the limits and one can easily show that if we have and similarly if we have . Moreover, using Eq. (9) with Eq. (10) it was show in cliche-kempf () that adiabatic evolution is possible in principle. In fact, in order to have a very small error in the ground state negativity at the end of the time evolution we roughly need . Thus, if we follow that prescription we can adiabatically switch on the interaction and keep all contributions in Eq. (13) intact.
Iii Quantum Field with Boundary Conditions
In this section we follow-up on the previous example by considering the case where the scalar quantum field is subject to Dirichlet boundary conditions. Such scenarios naturally arise when describing electromagnetic waves interacting with perfect conductors and have been extensively studied in the context of the Casimir effect casimir2 (). Here, our goal is to investigate whether the presence of boundary conditions augments or reduces the amount of entanglement generated in system For simplicity we only consider a massless field. In this case, the field operator is expanded in terms of creation and annihilation operators as
where the are solutions of Helmholtz equation satisfying the boundary conditions. The matrix elements , and are easily expressed in terms of the mode functions . From Eq. (6a), (6b) and (6) we obtain:
Let us consider the scenario in which the field satisfies the Dirichlet boundary conditions . In addition, we temporarily impose periodic boundary conditions on the y-z plane, i.e. . Under these assumptions, the mode functions read
Making use of Poisson summation formula one can rewrite the above sum as
where and . From the above equations one can determine the entanglement in for arbitrary positions of the qubits. We will however limit our discussion to two symmetric configurations. Let us first consider a symmetric configuration such that the qubits are located at with . Assuming and making use of Eq. (16a), (16c) and (LABEL:sum) we arrive at the following expressions:
Note that the free space situation (i.e. in the absence of boundary conditions) may be recovered by taking the limit . Indeed, in the regime , Eq. (20) and (21) reduce to Eq. (11) and (12)(with ). It is convenient to express Eq. (20) and (21) in terms of the dimensionless quantities , , and . After simple manipulations we obtain
Finally, by means of the formula Gradshteyn ()
where . Note that in the limit the above expressions vanish in accordance with the boundary conditions. Consequently, the entanglement in the qubits should vanish as . Numerical results for the entanglement generated in system versus are presented in Fig. (2).
Clearly, in this case the boundary conditions do not imply that matrix elements and should vanish as . Numerical results for this configuration are presented in Fig. (3). Thus, both graphs indicate that the entanglement generated in the pair of qubits reduces monotonically as the separation decreases. Note that in the regime , the orientation of the qubits relative to the planes becomes irrelevant and as a consequence the negativity values coincide for the two cases considered.
Iv Quantum field interacting with a classical potential
In this section we study entanglement generation in the qubits when system is either self-interacting or interacting with an external classical system. To do so, we consider the Hamiltonian
where and as usual . Here is a potential acting solely on system B. Throughout this section, we assume that
Clearly, the presence of the potential modifies the density matrix . Let us denote the new reduced density matrix by where contains all the contributions coming from the potential . Following similar steps to those in Sec. II, we apply perturbation theory to find the second order term (containing terms of the form )
We note that for potentials built out of even powers of the field , the above expression vanishes. In order to include this class of potentials into our framework, we need to include third order corrections. Making use of the condition Eq. (30), we obtain after some algebraic manipulations the third order correction to the reduced density matrix. It reads:
Here, we note that the matrix keeps its original form (as in (7)). In other words, the corrections do not generate new non-vanishing entries in the matrix . The relevant modifications of the matrix elements are given by
Naturally, may be obtained from upon replacing by and by in Eq. (33) . Note that when then and , obtained from the above expressions, coincide with the first order term appearing in the Taylor expansion of Eq. (6a) and (6). That is, we have plus an analogous relation for .
Let us now follow up on the proposal by Achim Kempf Kempf () to study the case of qubits interacting with a quantum scalar field which is interacting with a classical potential. We model this by choosing
The normal ordering Pesk () automatically guarantees that condition (30) is satisfied and it renders the matrix elements of Eq. (33) and (34) finite. This model can be seen as an analog of QED where the electromagnetic field is in a coherent state and therefore can be treated classically. For this reason the model is very similar to potential problems in non-relativistic quantum mechanics. We now proceed to compute the corrections and given by Eq. (33) and (34). Making use of Wick’s theorem Pesk () we obtain
where . Here note that if we set then we are simply dealing with a Klein-Gordon Hamiltonian with mass . The reader may check that in fact the above expressions reproduce the correct Taylor series expansions of (6a) and (6). That is, . We will make use of this simple observation in the next subsection.
iv.1 Example: Spherically symmetric Gaussian potential.
We now apply the above results to the situation where two identical detectors , located at , are interacting with the massive scalar field coupled to the spherically symmetric Gaussian potential
Its Fourier transform is easily found to be . The axial symmetry of the problem may be exploited by making use of plane wave expansion into spherical harmonics. Thus, after some algebraic manipulations, we obtain the useful identity