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Non-Gaussian Entanglement and Wigner Function

By Mustapha Ziane and Morad El Baz

Submitted: February 1st 2019Reviewed: April 19th 2019Published: October 23rd 2019

DOI: 10.5772/intechopen.86426

Downloaded: 40

Abstract

A measure of non-Gaussian entanglement in continuous variable (CV) systems based on the volume of the negative part of the Wigner function is proposed. We analyze comparatively this quantity with a numerical evaluation of the negativity of the partial transpose (NPT) considering a system of Bell states formed in the coherent state basis (quasi-Bell states).

Keywords

  • Wigner function
  • negativity
  • non-Gaussian state
  • nonclassicality
  • non-Gaussianity
  • quasi-Bell states
  • coherent states

1. Introduction

Continuous variable (CV) quantum optical systems are well-established tools for both theoretical and experimental investigations of quantum information processing (QIP) [1, 2]. Entangled states represent key resources, both for quantum computers and for many communication schemes [1, 3], an can be realized with Gaussian two-mode states; these states are relatively easy to work with theoretically and are also commonly produced in a laboratory. It has been successfully applied to implement various important protocols, such as quantum teleportation [4, 5, 6], quantum dense coding [7, 8, 9], and entanglement swapping [10]. This advancement comes from the development of Gaussian optical operations, such as beam splitting, phase shifting, squeezing, displacement, and homodyne detection. Recently, it became evident that the understanding of entanglement behavior beyond Gaussian systems is a necessity [11, 12, 13]. Furthermore, recent theoretical investigations have shown some limits to the Gaussian operations. For example, the no-go theorem relating to the distillation of entanglement shard by distant parties using only Gaussian local operations and classical communications (LOCC) [14, 15]. Moreover, on the theoretical level, the study of entanglement in many-body systems has been limited to Gaussian states [16, 17, 18, 19] where the quantification of quantum correlations (QC) reduces to the study of the covariance matrix, but the non-Gaussian entanglement doesn’t have such a simplified approach.

The problem of quantifying entanglement in non-Gaussian systems, in a way that is independent of particular external parameters, hasn’t solved yet; it is our main objective in this paper. An entanglement measure Eof the state ρshould satisfy some criteria [20] to be an entanglement monotone. Many quantities have been proposed as a quantifier of entanglement in discrete variables (DV) and CV Gaussian states. Recently, however, two entanglement measures that are much more amenable to evaluation have been proposed, the negativity of the partial transpose (NPT) and its logarithmic extension [21].

In this chapter, we are interested in establishing a direct measure of entanglement in non-Gaussian systems. This measure is based on the Wigner representation in the phase space of the non-Gaussian states. That is, they are defined in terms of the quantification of the degree of the negativity of Wigner function (NWF) [22, 23]. The most distinctive feature of this entanglement measure is the ease of calculated with a numerical integration

2. Two-mode quasi-Bell state: an entangled non-Gaussian state

The simplest example of a non-Gaussian state is the single-photon state. There are also other examples that can be generated by excitations of Gaussian states [24, 25]. Here we are going to use quasi-classical state that has been extensively studied for its nonclassical proprieties and violation of Bell inequalities; it is the superposition of two-mode standard coherent states (SCS). Let us consider two modes of electromagnetic fields A and B with corresponding annihilation operators âand b̂. Two-mode coherent states are defined by α,β=DaαDbβ0,0, where 0,0is the two-mode vacuum state and Diαis the displacement operator of the mode ii=AB. The state α,βcan be expressed into the form

α,β=eα2+β2/2n,mαnβnn!m!n,m,E1

where n1,n2are the two-mode Fock states. The quasi-Bell coherent states (QBS) are defined by the following superpositions of two-mode coherent states:

ψ±=N±αβ±αβ,E2
ϕ±=N±αβ±αβ,E3

where N±=12exp2α22β2+2is the normalization factor.

The Wigner function WR̂αβof the state (1) is given by

WR̂αβ=1πexpfαq1p1+fβq2p2,E4

where R̂=q1p1q2p2Tis the quadrature operators vector and fxyz=2x2+2x+xy+i2xxzy2z2. For the quasi-Bell entangled coherent states Eq. (2), the Wigner function is given by [26, 27].

WQCS±R̂αβ=Nα,β,±2WR̂1ααWR̂2ββ±WR̂1ααWR̂2ββ±WR̂1ααWR̂2ββ+WR̂1ααWR̂2ββ,E5

where R̂1and R̂2are the quadrature operators vectors of the first and second modes and WR̂ixyis the Wigner function of one-mode coherent state with i=1,2; xy=±α±βsatisfies the normalization condition R̂WR̂αβdR̂=1. Hence the doubled volume of the integrated negative part of the Wigner function of the state (2) may be written as

δψ±=R̂WQCS±R̂αβdR̂1.E6

By definition, the quantity δis equal to 0 for coherent and squeezed vacuum states, for which Wis nonnegative. In this work we shall treat δas a parameter characterizing the properties of the state under consideration.

It is clear from expression (5) and the plot in Figure 1 that the Wigner function of the quasi-Bell state (2) is non-Gaussian. In order to characterize this non-Gaussianity, several measures of the degree of non-Gaussianity were proposed [28, 29]. According to [29], the degree of non-Gaussianity of state ρis defined by

Figure 1.

Wigner function of quasi-Bell states (2). (a) Wigner function of ψ+ for α=0. (b) Wigner function of ψ+ for α=2. (c) Wigner function of ψ+ for α=1. (d) Wigner function of ψ+ for α=2. (e) Wigner function of ψ− for α=0.5. (f) Wigner function of ψ− for α=1. (g) Wigner function of ψ− for α=1.5. (h) Wigner function of ψ− for α=2.

δNGρ=Sρτ.E7

where Sρ1ρ2is the quantum relative entropy between states ρ1and ρ2. Here τis the reference Gaussian state with the same first and second moments of ρ. This property of reference state τleads to Trρlnτ=Trτlnτ, so that

δNGρ=SτSρ,E8

where Sρis the Von Neumann entropy of the state ρ. Also Sτ=hd++hd, where

hx=x+12lnx+12x12lnx12E9

and d±2=12Δδ±Δδ24I4are the symplectic eigenvalues of the covariance matrix σof the reference Gaussian state τ. Here Δδ=I1+I2+2I3, where I1=detA, I2=detB, I3=detC, and I4=detσare the four local symplectic invariants of the covariance matrix:

σ=ACCB,E10

where

σij=12RiRjRiRj.E11

For the considered states (2), we suppose that the two fields have the same mode (α=β); we find

σψ+=u+0r+00v+0s+r+0u+00s+0v+,E12
σψ=u0r00v0sr0u00s0v,E13

where we have defined

u±=Nα,β,±24α2±Γ2+1,r±=4α2Nα,β,±2,E14
v±=Nα,β,±24α2Γ2  Γ2+1,s±=4Γ2α2Nα,β,±2,E15

with Γ=αα=Expα22. Figure 2 shows the behavior of non-Gaussianity of states (2) in terms of α. These figures show that non-Gaussianity increases with increasing α(this behavior will be discussed in the fourth section).

Figure 2.

Non-Gaussianity versus α. (a) for states ψ+ and (b) for state ψ−.

3. Numerical evaluation of negativities (NPT and NWF)

In this section, we briefly review the NPT as a computable entanglement measure that possesses the proprieties of an entanglement monotone given in [21]. The NPT, Nρ̂of a state ρ̂is defined as the absolute value of the sum of the negative eigenvalues of the partial transpose of ρ̂denoted ρ̂PT. We may write it as

Nρ̂=12Trρ̂PT2ρ̂PT=ρ̂PT12,E16

where .denotes the trace norm [21].

The quasi-Bell coherent state (2) is defined in a non-orthonormal basis, and it is typically not possible to obtain an analytical expression for the negativity. However, as shown in the following, one can compute it numerically. First, we expand the quasi-Bell state (2) in the Fock basis:

ρ̂±=n1,n2,m1,m2ρn1,n2,m1,m2±m1n1m2n2,E17

where

ρn1,n2,m1,m2±=N02e2ααn1+n2n1!n2!+αm1+m2m1!m2!.E18

The partial transpose of this state with respect to mode two is

ρ̂±=n1,n2,m1,m2ρn1,m2,m1,n2±m1n1m2n2.E19

The eigenvalues are obtained by numerical diagonalization of the partial transpose density matrix (19). With this result, we can obtain the NPT straightforwardly using Eq. (16), and Figure 3a and b shows the numerical values of this NPT.

Figure 3.

Negativity of the partial transpose versus α: for the quasi-Bell state ψ+ (a) and ψ− (b). Negativity of the Wigner function versus α: for the quasi-Bell state ψ+ (c) and ψ− (d).

4. Discussion

In this section, we will discuss the different behaviors of the non-Gaussian entanglement and the variation of the negativity of the WF for the bipartite system considered early in terms of the coherent state amplitude α.

Figure 2 shows the variation of the degree of non-Gaussianity for the states in Eq. (2) as a function of coherent state amplitude α. We see that the non-Gaussianity δNGmeasured by (8) equal to 0 for small values of αincreases with increasing values of the parameter αto larger values much higher than 1 and does not establish in a maximum value. On the other hand, the NPT plots are shown in Figure 3a and b for the state (2) equal to 0 for α=0and increase with increasing values of the parameter αto reach its maximum value that is, equal to 1 for α  1.3. Furthermore, it is seen that the entanglement for large values of αreaches its maximum value. It is worthwhile noting that, at the limit of large values of the parameter α, the coherent states αand αbecome orthogonal; thus the behavior of quasi-Bell state (2) is, as expected, exactly that of the Bell state.

The plot in Figure 3c and d shows the behavior of the NWF as a function of αfor the non-Gaussian system (2). These two plots show that the NWF δWFhas the same behavior as the NPT. This allows to show that they behave identically and they have the same inflection points. Which confirms that the NWF is a direct computable measure of non-Gaussian bipartite entanglement that posses the proprieties of an entanglement quantifier [21].

For our measure, 1δWF0, equal to zero when α became null and the state in Eq. 2 is now nothing but a two-vacuum product state, and it is maximal for large values of αwhere the state (2) is maximally entangled (Bell state).

5. Conclusion

In this work, we have evaluated the negativity of Wigner function and the negativity of the partial transpose in non-Gaussian states formed by two modes of field coherent states. We have shown that the negative parts of the Wigner function can be used as a detector of non-Gaussian entanglement. Interestingly, as used in this work, the degree of Wigner function negativity can be used as a direct quantifier of non-Gaussian bipartite entanglement.

This work allows us to describe the best characterization of the non-Gaussian Wigner function and the important use of its negativity in bipartite non-Gaussian systems, which gives more efficiency in CV quantum information theory, particularly in quantum computing [30], because the Wigner function can be measured experimentally [31, 32], including the measurements of its negative values [33]. The interest put on such experiments has triggered a search for operational definitions of the Wigner functions, based on the experimental setup [34, 35]. It does represent a major step forward in the detection and the quantification of non-Gaussian entanglement in bipartite systems.

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Mustapha Ziane and Morad El Baz (October 23rd 2019). Non-Gaussian Entanglement and Wigner Function [Online First], IntechOpen, DOI: 10.5772/intechopen.86426. Available from:

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