Quantum Entanglement with Generalized Uncertainty Principle

We explore how the quantum entanglement is modified in the generalized uncertainty principle (GUP)-corrected quantum mechanics by introducing the coupled harmonic oscillator system. Constructing the ground state $\rho_0$ and its reduced substate $\rho_A = \mbox{Tr}_B \rho_0$, we compute two entanglement measures of $\rho_0$, i.e. ${\cal E}_{EoF} (\rho_0) = S_{von} (\rho_A)$ and ${\cal E}_{\gamma} (\rho_0) = S_{\gamma} (\rho_A)$, where $S_{von}$ and $S_{\gamma}$ are the von Neumann and R\'{e}nyi entropies, up to the first order of the GUP parameter $\alpha$. It is shown that ${\cal E}_{\gamma} (\rho_0)$ increases with increasing $\alpha$ when $\gamma = 2, 3, \cdots$. The remarkable fact is that ${\cal E}_{EoF} (\rho_0)$ does not have first-order of $\alpha$. Based on there results we conjecture that ${\cal E}_{\gamma} (\rho_0)$ increases or decreases with increasing $\alpha$ when $\gamma>1$ or $\gamma<1$ respectively for nonnegative real $\gamma$.


I. INTRODUCTION
As IC (integrated circuit) becomes smaller and smaller in modern classical technology, the effect of quantum mechanics becomes prominent more and more. As a result, quantum technology (technology based on quantum mechanics and quantum information theories [1]) becomes important more and more recently. The representative constructed by quantum technology is a quantum computer [2], which was realized recently by making use of superconducting qubits. In quantum information processing quantum entanglement [1,3,4] plays an important role as a physical resource. It is used in various quantum information processing, such as quantum teleportation [5,6], superdense coding [7], quantum cloning [8], quantum cryptography [9,10], quantum metrology [11], and quantum computer [2,12,13]. Furthermore, with many researchers trying to realize such quantum information processing in the laboratory for the last few decades, quantum cryptography and quantum computer seem to approaching the commercial level [14,15].
Then, it is natural to ask how the quantum entanglement is modified at the Planck scale.
This question might be important to unveil the role of the quantum information at the Planck scale or early universe. In order to explore this issue we choose the GUP as the simplest form proposed in Ref. [36]: where α is a GUP parameter, which has a dimension (momentum) −2 . Using ∆A∆B ≥ The existence of the ML can be seen in Eq. (1.1). If P = 0 for simplicity, the equality of Eq. (1.1) yields which arises when ∆P j = 0 (j = i). If α is small, Eq. (1.2) can be solved as where p i and q i obey the usual Heisenberg algebra [q i , p j ] = i δ ij . Thus, the ordering ambiguity occurs at O(α 2 ). We will use Eq. (1.4) in the following to compute the quantum entanglement within the first order of α.
As commented before the purpose of this paper is to examine how the quantum entanglement is modified in the GUP-corrected quantum mechanics. In order to explore the issue we consider the two harmonic oscillator systems, which are coupled with each other via the quadratic term. The Hamiltonian of the system is presented in section II. In section III we derive the vacuum state ρ 0 and its reduced substate ρ A . In this paper we adopt the entanglement measure for ρ 0 as von Neumann and Rényi entropies of the substate; where S von and S γ denote the von Neumann and Rényi entropies. It is easy to show that these entanglement measures are invariant in the choice of the substate due to the Schmidt decomposition [1]. The first entanglement measure is the most popular one called "entanglement of formation (EoF)" [37]. The second measure was used in Ref. [38,39] to explore the entanglement of the anisotropic XY spin chain with a transverse magnetic field in the various phases. In order to compute the entanglement measures in our system we compute Trρ n A up to O(α) in section IV. In section V we compute the entanglement of formation E EoF (ρ 0 ) and the second entanglement measure E γ (ρ 0 ) within the first order of α when γ is positive integer. In this section it is shown that E γ (ρ 0 ) increases with increasing α when γ = 2, 3, · · · . However, it is also shown that the first-order term of α in E EoF (ρ 0 ) is exactly zero. In section VI a brief conclusion is given.

II. HAMILTONIAN
Let us consider the two coupled harmonic oscillator system, whose Hamiltonian is where X i , P i obeys the GUP (1.2). If we set where ( x j , p j ) obeys the HUP, H 2 becomes Now, we introduce the new coordinates Then, the Hamiltonian H 2 reduces to In eq. (2.7) π 1 and π 2 are the canonical momenta of y 1 and y 2 , and the frequencies are In next section we will derive the ground state for H 2 up to the order of α by treating ∆ H as a small perturbation.

III. GROUND AND ITS REDUCED STATES FOR H 2
Before we solve the Schrödinger equation for H 2 , let us consider the one oscillator problem, whose Hamiltonian is where n = 0, 1, 2, · · · and Now, let us consider the Schrödinger equation for H 0 : Since H 0 is diagonalized, the eigenvalue and the corresponding eigenfunction are If we treat ∆ H as a small perturbation, the ground state Φ 0,0 and its eigenvalue E 0,0 for Thus, the density matrix for the ground state is In order to examine the entanglement of ρ 0 we should derive its reduced substate. After long and tedious calculation one can derive the reduced state ρ A = Tr B ρ 0 in a form g 1 = −m 2 (ω 8 1 + 5ω 7 1 ω 2 + 94ω 6 1 ω 2 2 + 459ω 5 1 ω 3 2 + 930ω 4 1 ω 4 2 + 459ω 3 1 ω 5 2 + 94ω 2 1 ω 6 2 + 5ω 1 ω 7 2 + ω 8 2 ) It is useful to note Also, one can show Using Eq. (3.12) one can explicitly show Trρ A = 1 within the leading order of α, which guarantees that ρ A is a mixed quantum state. shows that the reduced state ρ A becomes more mixed with increasing the GUP parameter α.
In order to quantify how much ρ A is mixed we compute the purity function, whose expression is J = 0, ρ A is a pure state regardless of α. With increasing J, ρ A becomes more and more mixed. The remarkable fact is that at fixed J the GUP parameter α makes ρ A to be more mixed. This is due to the minus sign in the bracket of Eq. (3.13).

IV. CALCULATION OF Trρ n A
The most typical way for computing the Rényi and von Neumann entropies of ρ A is to solve the eigenvalue equation If ρ A is a Gaussian state, the eigenvalue equation (4.1) can be solved straightforwardly [41].
Then, the Rényi entropy of order γ and von Neumann entropy can be computed by making use of the eigenvalue λ n as following: where γ is arbitrary nonnegative real. The problem is that ρ A is not Gaussian state if α = 0 as Eq. (3.8) shows. Thus, it seems to be extremely difficult to solve Eq. (4.1) directly.
Although we cannot solve the eigenvalue equation (4.1) explicitly, we can compute E γ=n (ρ 0 ) and E EoF (ρ 0 ) at least up to the O(α) by computing Trρ n A [42]. In this case E γ=n (ρ 0 ) can be computed by Then, E EoF (ρ 0 ) also can be computed from Eq. (4.3) by taking n → 1 limit. In this reason we will compute Trρ n A in this section within O(α).

From Eq. (3.8) one can show
where X is a n-dimensional row vector defined by X = (x 1 , x 2 , · · · , x n ) and G n is a n × n matrix given by In Eq. (4.5) the matrix components in the empty space are all zero. As shown in Ref. [42], Then, it is possible to show dx 1 · · · dx n exp −XG n X † = π n/2 √ det G n ≡ h n (4.8) Also, one can show [42] dx 1 · · · dx n (x 4 1 + · · · + x 4 n ) exp −XG n X † (4.9) = h n 3n 4 where H n is a n × n tridiagonal matrix given by It is straightforward to show Eq. (4.11) is valid for any nonnegative integer n.
(3.13). When n = 3, Eq. (4.12) yields (4.14) It is not difficult to show that as expected, Eq. (4.14) exactly coincides with . In next section we will discuss on the entanglement of ρ 0 by making use of Eq. (4.12). that it increases with increasing the GUP parameter α.

V. ENTANGLEMENT FOR ρ 0
The second entanglement measure E γ=n (ρ 0 ) can be derived by inserting Eq. (4.12) into Eq. (4.3), which is Now, let us compute the EoF of ρ 0 . This is achieved by taking n → 1 limit to Eq. (5.1).
One can show that J n (ω 1 , ω 2 ) in Eq. (4.13) satisfies Eq. (5.2) implies that the EoF of ρ 0 does not involve the first order of α. Thus, it is expressed as In Fig. 2 we plot the J-dependence of E γ=2 (ρ 0 ) when α is 0 (black solid line), 0. In this paper we examine how the quantum entanglement is modified in the GUPcorrected quantum mechanics. In order to explore this issue we consider the coupled harmonic oscillator system. Constructing the vacuum state ρ 0 and its substate ρ A , we compute the entanglement by choosing the EoF E EoF (ρ 0 ) = S von (ρ A ) and the Rényi entropy E γ (ρ 0 ) = S γ (ρ A ) of the substate as entanglement measures. It is shown that the second entanglement measure increases with increasing α when γ = 2, 3, · · · . Remarkable fact is that the EoF is invariant within the first-order of α in quantum mechanics with HUP and GUP. Since E EoF (ρ 0 ) = lim γ→1 E γ (ρ 0 ), we conjecture that E γ (ρ 0 ) decreases with increasing α when γ < 1.
In order to compute E γ (ρ 0 ) for nonnegative real γ we should derive the eigenvalue λ n in Eq. (4.1). However, it seems to be highly difficult (or might be impossible) to derive the eigenvalue due to non-Gaussian nature of ρ A . Thus, we cannot confirm our conjecture directly. If E γ (ρ 0 ) is equal to the right-hand side of Eq. (5.1) with changing only n → γ for all nonnegative real γ, it is possible to show that our conjecture is right. For example, we plot the J-dependence and α-dependence of E γ=0.7 (ρ 0 ) in Fig. 3(a) and Fig. 3(b) respectively.