Non-Gaussian entangled states (NGESs) as a new physical carrier can be very useful in both theoretical and experimental aspects of quantum information mainly because such states can overcome the shortage of the local Gaussian unitary operations and their induced Gaussian states.^[1–3] Especially, the NGESs effectively improve some certain quantum information protocols including quantum teleportation,^[4] quantum key distribution,^[5] and quantum metrology.^[6–8]

Among all the NGESs, pair coherent states (PCSs) have some prominent nonclassical properties including squeezing and sub-Poissionian distribution,^[9] entanglement quantified via quantum Fisher information or Neumann entropy, and nonlocality.^[10] Hence, it is a valuable quantum information resource which can be widely used to realize or improve quantum interferometric lithography,^[11] quantum teleportation,^[12] Heisenberg-limited interferometry,^[13] and measurement-device-independent quantum key distribution.^[14,15] Experimentally, the PCSs can be generated in an idealistic scenario via weak cross-Kerr media, the competition of four-wave mixing and amplified spontaneous emission,^[16] the motion of an ion in a two-dimensional trap,^[17] the nondegenerate parametric oscillator,^[18] or state-projective scheme in traveling-wave optical fields.^[19] In addition, the PCSs can be used to prepare some useful quantum states, such as Schröinger cat states,^[20] superpositions of PCSs,^[21] generalized PCSs^[22,23] and quantum mixture coming from phase randomization of a PCS.^[10]

As an induced NGES of the PCS, in this paper we introduce the finite-dimensional pair coherent state (FDPCS) in the nonlinear Bose operator realization and investigate its entanglement and nonclassicality via the photon number distribution (PND), entangled entropy, and Wigner function (WF), and detailedly discuss how the controlled parameters q (i.e., two-mode photon number sum) and τ influence them. The work is arranged as follows. In Section 2 the FDPCS is theoretically generated via the nonlinear Bose operator realization. In Sections 3 and 4 the PND of the FDPCS is numerically examined in the two-mode Fock space and the entanglement degree of the FDPCS is quantified via its Neumann entropy. Section 5 is devoted to using two conjugated entangled state representations to analytically derive the WF for the FDPCS and its marginal distribution functions, and investigate the nonclassicality of the FDPCS in terms of the partial negativity of the WF. Some main results are summarized in Section 6.

Notice that the nonlinear Bose operator realization of SU(2) generators gives

As a vital quantum feature of the state |ξ,q⟩, its PND (|⟨n_a,n_b. |ξ,q⟩|²) means the probabilities of finding (n_a,n_b) photons in this optical field.^[25,26] Hence, we now calculate the PND of the state |ξ,q⟩ as

In Fig. 1, we present the PND P(n_a,n_b) of the state |ξ,q⟩ in the Fock space (n_a,n_b) for different parameters q and τ. From Fig. 1, it is clearly seen that the PND only emerges in several (n_a,n_b) numbers of photons and is equal to zero for other photon numbers since it is restricted from the condition n_a + n_b = q, as the above analytical result in Eq. (6). For a smaller value of q, the maximum probability of finding photons appears where n_b takes a maximum value (or n_a takes a minimum value). As the parameter q increases, more photons appear in number states with smaller n_b, thus the maximum probability of finding (n_a,n_b) photons gradually decreases and moves only along the diagonal direction toward the larger n_a numbers of photons [see Figs. 1(a)–1(c)]. However, the changes of the PND with the increase of the parameter τ are just contrary [see Figs. 1(d)–1(f)]. Overall, the values of the parameters q and τ directly affect the positions and values of the peaks of the PND.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Photon number distribution P(na,nb) of the FDPCS in the two-mode Fock space (na,nb) for some different values of (q, τ), where (5,4), (8,4), (10,4), (8,1), (8,3), and (8,5), respectively, refer to (a), (b), (c), (d), (e), and (f).

In the present section, we quantify the degree of quantum entanglement of the FDPCS via computing its Neumann entropy. For a pure bipartite entangled state |φ⟩_ab with the Schmidt decomposition, i.e., |φ⟩_ab = ∑_{n = 1}d_n|α_n⟩_a |β_n⟩_b, where the states |α_n⟩_a,|β_n⟩_b are mutually orthonormal and d_n is positive and real. Hence, its entanglement degree can be characterized via the partial Neumann entropy of the reduced density operator, i.e., , where ρ_a = tr_b(|φ⟩_ab ⟨φ|).^[27–30] Using the Schmidt expansion of the state |ξ,q⟩ in Eq. (4), and thus its entanglement entropy can be easily calculated by

In Fig. 2, we present the entanglement entropy of the FDPCS as a function of the parameter τ for different values of q. Obviously, for any q, the entanglement of the FDPCS always increases very quickly at first stage of the parameter τ and then decreases slowly for the remaining stages as τ is increased, and is always close to zero when τ is large enough. However, for a large value of q, the entanglement tends to zero more slowly. When a threshold of the parameter τ is reached, the maximum value of the entanglement is shown. Also, as q increases, the threshold of τ and the maximum value of the entanglement gradually increases.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Entanglement entropy of the FDPCS as a function of the parameter τ for q = 1 (black solid line), q = 2 (red dashed line), q = 4 (green dotted line), q = 6 (blue dash-dotted line), q = 9 (cyan dash-dot-dotted line), q = 12 (magenta short-dashed line).

In quantum statistical theory, the WF of quantum state in phase space contains some crucial information of this state, especially its quantum interference character and partial negativity can frequently be used to accurately describe the nonclassicality of the state. Hence, in this section we analytically and numerically study the Wigner phase-space distributions for the FDPCS.

5.1. Two conjugated entangled states

To derive the WF for the state |ξ,q⟩, we next review two conjugated two-mode entangled states |ρ⟩ and |θ⟩ and their main properties. In Refs. [31] and [32] the authors pointed out that the entangled state |ρ⟩ simultaneously obeys the eigen-equations

because of the communative relation [a+b^†,b+a^†] = 0, or the states |ρ⟩ are the common eigenstates of the relative coordinate operator (Q_a - Q_b) and the total momentum operator (P_a + P_b), i.e., where , , (i = a,b), thus |ρ⟩ represents an entangled state defined as in two-mode Fock space^[32–34] where ρ is a complex parameter, ρ = ρ₁ + iρ₂. In term of Eq. (10), and using the normal ordering of two-mode vacuum projection operator^[35,36] and the integration technique within an ordered operator product (IWOP),^[37,38] we can prove the completeness relation of the state |ρ⟩, i.e., , and its orthonormalized property, i.e., ⟨ρ|. ρ^′⟩ = πδ⁽²⁾(ρ − ρ^′). Further, using the generating function of two-variable Hermite polynomials H_m,n(ε,ε^*), we have We can obtain the expansion of the entangled state |ρ⟩ under the basis of two-mode number state |m,n⟩, that is, Similarly, another two-mode entangled state |ϑ⟩ conjugated to |ρ⟩ is obtained as which indeed represents the common eigenstates of the mass-center coordinate operator (Q_a + Q_b) and the relative momentum operator (P_a − P_b), i.e., Its expansion in two-mode Fock space is expressed as and there exists the relation between |ϑ⟩ and |ρ⟩, i.e., |ϑ⟩ = (−1)^a†a|ρ = − ϑ⟩.

5.2. Partial negativity of Wigner function

In the two-mode entangled state |ρ⟩ representation, the two-mode Wigner operator is neatly expressed as^[39–41]

where ς and σ are complex parameters. Indeed, using Eqs. (10), (16) and the IWOP method, we can find the identity Δ(σ,σ) = Δ(α) ⊗ Δ (β) via letting σ = α +iβ^*, σ = α −iβ^* in Eq. (16), where Δ(α) and Δ(β) are two independent single-mode Wigner operators.^[42,43] Hence, the WF for the FDPCS reads Combining Eqs. (4) and (12), we calculate the scalar product ⟨ρ|ξ,q⟩ as thus the function W(ς,σ) is obtained as Further, using the power-series expansion of H_m,n(ε,ε^*),^[44,45] and the mathematics integration formula^[46,42,48] which holds for the constrained condition Reζ < 0, thus we have where ϰ = ς −σ and υ = ς + σ. It shows that the function W(ς_,σ) is related to two-variable Hermite polynomials with the maximum order q. In terms of the standard definition of H_m,n(·,·) and the exponential term (−1)^{m + n}, we decide that the function W(ς_,σ) can show some negative regions in the whole phase space for some certain values of the parameters q and τ, as an evidence of the nonclassicality of the FDPCS.

In Fig. 3, we present the WF W(ς_,σ) for the FDPCS as a function of Reς (taking Reσ = 0) for different values of q and τ. Obviously, Figure 3(a) shows that the WF always has a upward main peak for even q but a downward main peak for odd q at the position of Reσ = 0, and for any even (or odd) q, the main peak value is unchanged, whereas the numbers and positions of the secondary peaks are completely different. However, the total number of the downward peaks is always equal to two-mode photon number sum q of the FDPCS. Intuitively, the partial negative volume of the WF for odd q is always larger than that for even q as the presence of a downward main peak for odd q, which shows that the nonclassicality of the FDPCS for odd q is stronger than that for even q. However, the nonclassicality of the FDPCS cannot monotonously change with the odd (even) q.

Fig. 3.

Figure Option ViewDownloadNew Window

Fig. 3. Wigner function for the FDPCS of Reς (letting Reσ = 0) for two different cases: (a) τ = 0.5, and q = 1 (black solid line), q = 2 (red dashed line), q = 4 (green dotted line), q = 5 (blue dash-dotted line), q = 6 (cyan dash-dot-dotted line), q = 9 (magenta short-dashed line); (b) q = 4, and τ = 0.1 (black solid line), τ = 0.5 (red dashed line), τ = 1.2 (green dotted line), τ = 2 (blue dash-dotted line), τ = 4 (cyan dash-dot-dotted line), τ = 7 (magenta short-dashed line), τ = 20 (yellow short-dotted line), τ = 100 (dark-yellow short-dash-dotted line).

From Fig. 3(b), we clearly see that, for a fixed value of even q (e.g., q = 4), the upward main peak remains unchanged as τ increases, but the numbers and the peak values of the secondary peaks gradually decreases, even the downward secondary peaks (the partial negative volume) are completely gone and the WF becomes Gaussian when τ reaches a certain threshold value. Continuing to increase the value of τ, the downward secondary peaks representing the negative region will appear once again. This result means that the nonclassicality of the FDPCS weakens first and then enhances with the increase of τ. Especially, when the parameter τ is large or small enough, the WF has almost similar distributions, that is, the FDPCS has similar nonclassicality for the two cases. For odd q case, a similar result occurs for the FDPCS, so we ignore it here.

5.3. Marginal distribution functions

In this subsection, we want to calculate two marginal distribution functions for the function W(ς,σ) in the ς–σ phase space. Surprisingly, performing two integrations of the Wigner operator Δ(ς,σ) over the variables σ and ς in the whole phase space, we find that the relationship between the operator Δ(ς,σ) and two conjugate entangled states |ρ⟩ and |θ⟩ yields^[49]

Thus, using Eqs. (4), (12) and (17), we can arrive at a marginal distribution function of W(ς_,σ) in the ς variable, i.e., Similarly, using Eqs. (4), (17) and the expansion of the entangled state |θ⟩ in Eq. (15), we obtain another marginal distribution function of W(ς_,σ) in the σ variable, that is, Obviously, equation (24) [or (25)] represents the probabilities for finding the two particles with the relative coordinate and the total momentum [or the mass-center coordinate and the relative momentum ] in the FDPCS, which is just related to the module-square of two-variable Hermite polynomials. Hence, we get a conclusion, that is, for a bipartite entangled state, the physical significance of its function W(σ,σ) is that its marginal distribution functions show the probabilities of finding two entangled particles in the ς–σ phase space.

In Fig. 4, we plot the marginal distribution functions P(ς) and P(σ) for different values of q and τ. Figures 4 shows that the marginal distribution P(ς) [or P(σ)] is symmetrical by the axis of symmetry Reς = 0 (or Reσ = 0) and cannot monotonously change with the increase of Reς (or Reσ). For a fixed value of τ and any q, the peak values in the distribution P(ς) and the valley values of P(ς) and P(σ) increase first and then decreases as q increases. However, the peak values of P(ς) always decreases. Also, the maximum marginal distribution probability moves toward the larger Reς| (or |Reσ|) with the increase of q.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. Marginal distribution functions P(σ) and P(σ) for the function W(ς, σ) in the ς–σ phase space for different values of q and τ, where (a) τ = 0.5, and q = 1 (black solid line), q = 2 (red dashed line), q = 4 (green dotted line), q = 5 (blue dash-dotted line), q = 6 (cyan dash-dot-dotted line), q = 9 (magenta short-dashed line); (b) q = 4, and τ = 0.1 (black solid line), τ = 0.5 (red dashed line), τ = 1.2 (green dotted line), τ = 2 (blue dash-dotted line), τ = 4 (cyan dash-dot-dotted line), τ = 7 (magenta short-dashed line), τ = 20 (yellow short-dotted line), τ = 100 (dark-yellow short-dash-dotted line).

On the other hand, for a given q and any τ, the distributions P(ς) (or P(σ)) are symmetrical but cannot monotonously change with Reς (or Reσ). Specifically, in the distribution P(ς), the peak values decrease first and then increase as τ increases, but the change of the valley value with τ is just contrary. However, the peak values of the distribution P(σ) increase first and then decrease as τ increases and there exist a series of small peaks near the valley, which are different from the distribution P(ς). Moreover, for a large τ enough, the distributions P(ς) and P(σ) are similar to those for a small τ.

In summary, we have introduced the FDPCS by using the nonlinear Bose operator realization of SU(2) generators and investigated its PND and entanglement entropy. It is found that the PND is restricted from the condition n_a + n_b = q, i.e., the two-mode photon number sum q of the FDPCS, and the parameter values of q and τ have a very different impact on the positions and values of the peak of the PND, the entanglement of the FDPCS increases in the first stage of the parameter τ and then decreases gradually for the remaining stages. Furthermore, using the entangled state |ρ⟩ representation of two-mode Wigner operator, we have analytically obtained the WF for the FDPCS and its marginal distributions, and numerically discussed its nonclassical properties based on the negativity of the WF. It is shown that the FDPCS for odd q has more stronger nonclassicality than that for even q for a fixed τ, however the nonclassicality of the FDPCS decreases first and then increases with increasing τ for any q. Moreover, a fixed q and any τ, the marginal distributions P(ς) (or P(σ)) are symmetrical and cannot monotonously change with the parameter Reς (or Reσ).