Quantum decoherence in the entanglement entropy of a composite particle and its relationship to coarse graining in the Husimi function

I investigate quantum decoherence in a one-body density matrix of a composite particle consisting of two correlated particles. Because of a two-body correlation in the composite particle, quantum decoherence occurs in the one-body density matrix that has been reduced from the two-body density matrix. As the delocalization of the distribution of the composite particle grows, the entanglement entropy increases, and the system can be well-described by a semi-classical approximation, wherein the center position of the composite particle can be regarded as a classical coordinate. I connect the quantum decoherence in the one-body density matrix of a composite particle to the coarse graining in a phase space distribution function of a single particle and associate it with the Husimi function.

In recent decades, quantum entanglement has attracted a great deal of interest in various fields. To estimate correlations in quantum systems, entanglement measures such as the entanglement entropy (EE) have been intensively studied [1][2][3][4][5][6][7][8][9]. Many entanglement measures are defined by reduced density matrices which describe the structure of the Schmidt decomposition and contain information about entanglements in quantum systems. In entangled states, the EE is produced by quantum decoherence caused by a reduction in the number of degrees of freedom (DOF). In my previous papers, I calculated entanglement measures of the one-body density matrix in nuclear systems [10,11] and showed that the EE is enhanced by the delocalization of the distribution of clusters, which are composite particles of spatially correlated nucleons. My first aim in the present paper is to understand how quantum decoherence occurs and how the EE is produced in the one-body density matrix of correlating particles.
Quantum decoherence-that is, the quantum entropy-has been also investigated with the coarse graining of distribution functions in a phase space. The Husimi function [12][13][14] is known to have finite the Wehrl and Rényi-Wehrl entropies [15,16] because of the coarse graining by a Gaussian smearing of the Wigner function. It has been shown that the Wehrl and Rényi-Wehrl entropies are increased by delocalization of the distributions in quantum systems [17][18][19][20]. Campos et al. have discussed a correlation between the EE and the Rényi-Wehrl entropy in entangled states [20]. One of the fundamental questions in quantum physics is how the quantum decoherence in the reduced density matrix of entangled states can be connected to the coarse graining of distribution functions. My second aim in this paper is to understand the correspondence between the quantum decoherence in the one-body density matrix of correlating particles and the coarse graining in the Husimi function of a single-particle state.
In this paper, I investigate the EE of the one-body density matrix of a two-body system in which two particles are strongly correlated to form a composite particle, and I discuss how quantum decohence occurs in the reduction of the DOF. To describe two-body wave functions, I adopt a cluster wave function in the generator coordinate method in nuclear physics [21,22]. Let us consider a system where two particles (c1 and c2) with masses m and um form a bound state with an attractive interparticle force. I assume that the bound state is described by the lowest state of a harmonic oscillator (ho) potential and can be approximately treated as an inert composite particle, where intrinsic excitations cost a relatively high amount of energy compared with the center of mass (cm) motion of the composite particle. In this approximation, a total two-body wave function is given as where b 2 = b/ √ u, and Ψ (2) |Ψ (2) = 1. r 1 (r 2 ) is the coordinate of c1(c2). Here, I describe the one-dimensional case, but the present model can also be extended to the three-dimensional case. |s; b 1 |s; b 2 2 indicates the composite particle localized around the mean position s, and Ψ (2) is given by the superposition of different s states with the weight factor F (s). |Ψ (2) can be expressed by the cm motion and the intrinsic wave functions as with the cm coordinate R = u 1 r 1 + u 2 r 2 (u 1 = 1/(u + 1) and u 2 = u/(u + 1)), the relative coordinate r = r 1 − r 2 , is the lowest intrinsic state for the ho potential, U ho (µ, b r ; r) = − 2 r 2 /2µb 4 r , with µ = u 2 m. Thus, general low momentum states of the inert composite particle can be expressed by the form (1), in which the cm motion Φ G (R) is expressed by the shifted Gaussian expansion as given in Eq. (4).
The one-body density matrixρ (1) Ψ (2) for c1 is defined by the matrix reduced from the many-body density matrix Ψ (2) ] and is given aŝ where W (s ′ , s) ≡ F * (s ′ )F (s) and Trρ (1) Ψ (2) for u = 3 equals to the one-body density matrix of an α cluster composed of four nucleons with an equal mass investigated in previous papers [10,11]. The Wigner transformation (Wigner function) ofρ The Rényi EE of order 2 (Rényi-2 EE) and von Neumann EE for Ψ (2) with the one-body density matrixρ If ρ W (ρ (1) Ψ (2) ; q 1 , p 1 ) ≥ 0 is satisfied in the entire phase space, I can consider the phase-space Shannon entropy which I call the "Wigner-Shannon EE".
In the one-body density matrixρ Ψ (2) and its Wigner transformation, quantum decoherence occurs and produces the EEs because of the factor , which originates in the reduction of the DOF of c2. Indeed, in the case of u = 0, without this factor,ρ (1) = {ρ (1) } 2 and the Rényi-2 and von Neumann EEs are zero that corresponds to a pure single-particle state.
Let us consider a semi-classical approximation ofρ Ψ (2) . The factor exp − u 4b 2 (s − s ′ ) 2 , which is the source of the quantum decoherence, has a sharp peak around s ′ ≈ s with a width 2b/ √ u. I assume that the function F (s) is a slowly varying function compared with exp − u 4b 2 (s − s ′ ) 2 , and it can be approximated as F (s ′ ) ≈ F (s). Then I obtain a semi-classical approximation, where f (s) ∝ F (s) whose normalization is determined by where the parameter s and |f (s)| 2 are regarded as a classical coordinate and a classical distribution of the composite particle, respectively. In the large u limit-that is, the large c2 mass limit-ρ(q 1 , q ′ 1 ) → ρ cl (q 1 , q ′ 1 ) and b r → b. Note that, |s and |s ′ are not orthogonal to each other because of the quantum fluctuations of the c1 position in the composite particle.
In the semi-classical approximation given by Eq. (12), the Wigner function is approximated as which is nonnegative definite. Usingρ For simple examples, I first consider the zeromomentum state of the composite particle in a finite volume V described by a constant F (s). I assume that V ≫ b and the contribution of the box boundary can be ignored and obtain ρ (1) In this case, ρ The Rényi-2, von Neumann, and Wigner-Shannon EEs are where V eff = V /b r denotes the effective Volume size for the cm motion. These results are not valid for a small V eff because of the box boundary. The one-body density matrix is diagonalized in the momentum space with a Gaussian distribution, exp[− b 2 r 2 p 2 1 ]. This indicates that the one-body density matrix of a free composite particle is equivalent to a thermal state of a single particle at finite temperature kT = 2 /2mb 2 r . The temperature is of the same order as the mean kinetic energy, 2 /2mb 2 , of constituent particles confined in the composite particle. Strictly speaking EEs are not thermodynamic entropies, however, by associating the onebody density matrix of the free composite particle with a quantum mixed state of a single particle, I can propose an interpretation of the entropy production and thermalization as follows: when the DOF of c2 are reduced, the quantum decoherence occurs, producing the entropy, and simultaneously, the intrinsic kinetic energy of the composite particle is converted into heat. Next, I consider a composite particle moving in an external ho potential, where the lowest state of the composite particle is given by the Gaussian distribution, F (s) = e − s 2 2B 2 /(B 2 π) 1/4 . This gives the exact solution to the two-body wave function, r 1 , r 2 |Ψ (2) = R, r|Ψ (2) , for ho potentials U ho (M, β; R) + U ho (µ, b r ; r), with M = (u + 1)m and β = √ B 2 + u 1 b 2 . In the B = 0 limit, Ψ (2) describes a localized composite particle that corresponds to a non-entangled (uncorrelated) state of two constituent particles and has zero Renyi-2 and von Neumann EEs. As B enlarges and the delocalization of the cm of the composite particle grows, the EEs increase. The Wigner function and EEs forρ where γ = (1 + (u + 1)v 2 eff )/(1 + uv 2 eff ), and v eff = B/b denotes the effective volume size. The EEs increase as v eff enlarges and approaches lnv eff in the large v eff limit. In the semi-classical approximation, the Wigner function and EEs are S W-Sh cl ≈ S W-Sh,cl = S R2,cl + 1 − ln2, (23) where v c,eff = B/b r . I show the EEs for u = 1 and u = 8 in Fig. 1. S vN is calculated numerically, as was done in the previous paper [10]. S R2,cl for the semi-classical approximation agrees well with S R2 in the v eff ≥ 2 case to within 10% error for u = 1, and the agreement is better for the larger mass ratio, u = 8. S W-Sh has a constant shift 1 − ln2 (a constant scaling e/2 in the e S plot in Fig. 1) from S R2 , and it is finite even at v eff = 0. S vN starts from zero at v eff = 0 and approaches S W-Sh as v eff increases. As the mass ratio u increases, the EEs converge on values in the large u limit. Finally, I connect the quantum decoherence in the onebody density matrix of the composite particle to coarse graining in the phase space distribution function of a single particle and associate it with the Husimi function. Let us start from the Wigner transformation of the full two-body density matrixρ It is rewritten by a separable form in the phase space for the cm and relative coordinates as ρ W (ρ   (2) ; q 1 , p 1 ) = dq 2 dp 2 2π ρ W (ρ Ψ (2) ; q 1 , p 1 , q 2 , p 2 ) Here I use relations q = (q 1 − Q)/u 2 , p = p 1 − u 1 P , and the transformation dq 2 dp 2 = |J|dQdP with the determinant of Jacobian |J| = 1/u 2 . This means that ρ W (ρ Ψ (2) ; q 1 , p 1 ) is regarded as a coarse grained distribution function of ρ W (ρ (G) ΦG ; Q, P ) with a Gaussian smearing. In other words, the quantum decoherence caused