Quantum Engineering
Volume 2025, Issue 1 7480764
Research Article
Open Access

Entangled State Representation: From Original EPR Wave Function to Beam Splitter

Rui He, 

Corresponding Author

Rui He

School of Electrical and Photoelectronic Engineering , West Anhui University , Lu’an , Anhui, 237012 , China , wxc.edu.cn

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First published: 16 July 2025
https://doi.org/10.1155/que2/7480764
Academic Editor: Yu-Bo Sheng

Abstract

We show that the state vector of the wave function of Einstein, Podolsky, and Rosen, 1935, (EPR) is a special entangled state representation. Then, taking the entangled state representation as an intermediary, we reveal the intrinsic connection between the HG mode and the LG mode in paraxial optics. Finally, we study how to generate the entangled state representation using an optical beam splitter.

1. Introduction

In 1935, Einstein, Podolsky, and Rosen (EPR) published a paper titled “Can quantum-mechanical description of physical reality be considered complete?” [1]. In this article of epoch-making significance, the concept of quantum entanglement stepped onto the forefront of physics for the first time. It not only promoted the philosophical thinking of generations of physicists on the essence of quantum mechanics but also gave birth to the brand-new discipline of quantum information science.

Leaving aside the philosophical significance of the EPR paradox, what we focus on is the ideal experiment designed by Einstein in the EPR paper. In this experiment, a particle splits into two particles that move backwards, and after a sufficient time, they are already far apart. The relative coordinate operator Q1 − Q2 and its total momentum operator P1 + P2 of two particles are commutative and can be accurately measured simultaneously. To be more precise, Einstein also conceived the state of a 1 − 2 particle combination system for this ideal experiment, represented by the following wave function:
()
where q0 is a constant. Obviously, this is the wave function of the state vector |ψ⟩ in a two-dimensional coordinate representation. As for what the |ψ⟩ is, it is not mentioned in the EPR article.

It should be noted that it is not enough to just give the wave function of the two-body system composed of Particle 1 and Particle 2 mentioned above, because the wave function is just a representation of the two-body quantum state in the coordinate representation. To deeply understand the quantum entanglement of two-body continuous variables, we must give the specific form of the quantum state |ψ⟩. This is the entangled state representation first derived by He et al. [2] using the technique of integration within an ordered product (IWOP) of operators [2], which deeply reveals the mystery of Dirac’s symbolic method. Based on the above discussion, our work starts with the original wave function of the EPR entangled state.

2. Derive the EPR-Entangled State Representation From the Original EPR Wave Function

For the convenience of writing, we can let ℏ = 1 and add 1/(2π) to the right side of equation (1). Using the Dirac function, we can obtain that
()
Then, we have the EPR entangled state
()
where |q1, q2⟩ is a two-mode coordinate eigenstate. It is well-known that
()
where
()
are two-mode creation and annihilation operators, which satisfy
()
Substituting equations (2) and (4) into equation (3), we have
()

Equation (7) is the explicit expression of the EPR state |ψ⟩ [2]. It can be seen that it is a state vector of the two-mode Fock space and it belongs precisely to the entangled state representation to be discussed in the next section.

3. Entangled State Representation |η⟩ and |ξ⟩

Inspired by the quantum entanglement ideas of EPR, and combined with the IWOP technique [2], Fan Hong-yi first established the entangled state representation [2] and widely applied it to the theoretical research in multiple fields such as Fourier optics, quantum optics, and condensed-matter physics. This section will start from the known completeness relations of representations, expand and construct new completeness relations, and thus naturally derive the two-body entangled state representation.

We can directly construct the following integral:
()
where :: denotes the normal product, η = η1 + iη2 is a complex number, and d2η = dη1dη2. Decomposing the integrand operator function in equation (8), we can obtain
()
noticing that . It can be seen from the form of equation (9) that it is equivalent to a pure state operator |η⟩⟨η|. Then, equation (8) can be written as
()
In the two-mode Fock space, this new state |η⟩ is
()
and we called it as entangled state representation. There are the following relations:
()

By comparing equation (7) with equation (11), it can be known that the original EPR state |ψ⟩ is actually a special case of the entangled state representation.

By using the relationship and , j = 1, 2, we can easily from equation (12) obtain that |η⟩ satifies with the following eigen equations:
()
and the orthogonal relation
()
Equation (13) shows that |η⟩ is the common eigenstate of the two-particle relative-coordinate Q1 − Q2 and total-momentum P1 + P2 operators due to [Q1 − Q2, P1 + P2] = 0, which means that |η⟩ is precisely the generalized EPR state. In fact, using Schmidt decomposition, it can also be proved that |η⟩ is a continuous-variable entangled state [2], that is,
()
or
()

Equations (15) and (16) are the Schmidt decomposition forms of |η⟩ in the coordinate representation and the momentum representation, respectively, where |q〉j and |p〉j are the eigenstates of the coordinate and momentum, respectively.

Similar to equation (8), another new completeness relation can be constructed as follows:
()
Decomposing the integrand operator function in equation (17), we can obtain another EPR entangled state representation |ξ⟩ and
()
()
It can be easily proved that |ξ⟩ is the common eigenstate of the two-particle masscenter-coordinate Q1 + Q2 and relative-momentum P1 − P2 operators due to [Q1 + Q2, P1 − P2] = 0, that is,
()
whose orthogonal relation is
()
The entangled nature of |ξ⟩ can be also analyzed by the Schmidt decomposition as follows:
()
or
()

4. Two-Dimensional Isotropic Harmonic Oscillator and Hermite–Gaussian (HG) Mode or Laguerre–Gaussian (LG) Mode of Paraxial Light

Known research work shows that the LG light beams of paraxial light have definite angular momentum [3], and they have been widely used in many research fields such as the preparation of high-dimensional quantum entangled states, quantum information processing, and optical tweezer technology [4]. In this section, we are committed to exploring the correlation between the energy eigenstates of a two-dimensional harmonic oscillator and the HG mode and LG mode of paraxial light.

In the case of two-dimensional isotropy, the Hamiltonian of the harmonic oscillator is
()
where for the convenience of describing the problem, the x-axis is denoted by the Subscript 1, and the y-axis is denoted by the Subscript 2. The corresponding particle-number operator is
()
The total number operator is
()
thus,
()
Since N1 and N2 form a complete set of commuting observables, a basis composed of the tensor products of a set of one-dimensional eigenstates can be chosen, and the elements of this basis are given by
()
which are just HG modes. In the coordinate representation, they can be expressed as
()
where
()

The axes x and y are not the only ways to describe this problem. Since the energy is invariant under rotation in the x − y plane, we can also choose any other rotating reference frame. To make better use of this symmetry, we consider angular momentum. The following analysis shows that we can construct another basis.

The angular momentum operator can be defined as
()
We will prove that {N, L} is another complete set of commuting observables. Supposing
()
and
()
we get a set of the commutation relationship
()
We also consider and ar as creation and annihilation operators, whose corresponding particle-number operator is
()
Thus, the question is how to find the common eigenstates of N+ and N−. We find that N+ and N− also form a complete set of commuting observables, that is, for every (n+, n−), they all have a common eigenvector denoted by |n+, n−⟩. In fact,
()
which form a complete set of orthogonal eigenbasis and
()
()
We find that
()
and
()

So, N and L also form a complete set of commuting observables. Actually, |n+, n−⟩ are just LG modes.

Now, we try to change |q1, q2⟩ into another form. Let
()
then,
()
Due to equations (4), (32), and (33), we have
()

Comparing equations (19) with equation (43), we find that |q1, q2⟩ transforms into an entangled state representation. This means that a nonentangled two-dimensional coordinate eigenstate in a rectangular coordinate frame transforms into a two-mode entangled state in a polar coordinate frame, which is a thought-provoking issue.

Nienhuis and Allen use operator-algebra method to describe the LG light beams and find that the LG mode is an analog of the laser mode of the angular momentum eigenstate of a two-dimensional isotropic harmonic oscillator [5]. In reference [2], by inserting the completeness relation of coherent state representation, we also showed that 〈q1, q2|n+, n−〉 is actually a LG mode of paraxial light, that is,
()
in which, n = n+ + n−, l = n+ − n−.
On the other hand, according to equations (32) and (33), we get
()
and
()
Due to equations (45) and (46), we suppose that |η⟩ = |p1, p2〉, where |p1, p2⟩ is a two-dimensional momentum eigenstate. Now, we set and equation (47) shows that the entangled state representation |η⟩ is just the form of the two-dimensional momentum eigenstate |p1, p2⟩ in polar coordinates.
()

5. Examine the Correlation Between HG Modes and LG Modes From the Perspective of Unitary Transformation

From the content of the above section, we can see that the HG mode is actually the wavefunction 〈q1, q2|n1, n2〉 of the two-mode number state |n1, n2⟩ in the two-dimensional coordinate representation, while the LG mode is the wavefunction 〈q1, q2|n+, n−〉 of the state |n+, n−⟩ in the two-dimensional coordinate representation.

We can calculate the expression of the LG mode from another perspective, but we need to introduce the unitary transformation . It is already known that , then
()
Now, we use the creation and annihilation operators of the two-dimensional harmonic oscillator in Schwinger’s formalism to represent the angular momentum operator, that is,
()
By using the operator identity
()
we get
()
So, we have
()
noting that . Thus,
()
By the inverse transformations of equations (43) and (51), it can be obtained that
()
Furthermore,
()
it can be seen that equation (55) is actually an entangled state representation.
Now, we let ξ ≡ q1 − iq2 = ρe−iϕ, then
()
From the Schmidt decomposition equation (22) of the entangled state representation, we have
()
then
()
Thus, due to equation (29), we can obtain
()
Now, we introduce the integral formula [6].
()
where , and
()
where . Comparing with equation (30), we can see that Hn(x) = hn(2x, −1). Thus, equation (59) can be changed as
()
where . So, , then
()
where we have set η = q1 + iq2. Let q1 = ρ cos θ, q2 = ρ sin θ again, and from
()
we can obtain
()

Equation (65) is just the expression of the LG mode.

In Sections 4 and 5, we derive the analytical expressions of LG modes using two methods, respectively. We can see that although the method provided in Section 4 has a certain theoretical value, it is not operable in practical applications. The method given in Section 5, however, is experimentally feasible. We can prepare a quantum state by applying two realizable continuous unitary operations and to the two-mode particle state |n1, n2⟩ and then measure this state in the two-mode position representation. In principle, the spatial image of the LG mode can be obtained in this way. Figure 1 shows the intensity and phase distributions of the LG beams output by the mode converter when different HG beams are incident. Our mechanism should be able to reasonably explain the black-box mechanism of the mode converter.

Details are in the caption following the image
Figure 1
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When different Hermite–Gaussian beams are incident, the intensity and phase distributions of the Laguerre–Gaussian beams output by the mode converter.

6. Entangled State Representation Generated by the Beam Splitter

The beam splitter is one of the basic linear devices in quantum optics, and it plays an important role in the preparation and measurement of quantum entanglement. According to [7], we know that when the beam splitter operator B(θ) acts on the two-mode coordinate eigenstate , we have
()
where . Using the completeness relation of the coordinate states, it can be obtained immediately that
()
which is the expression of the beam splitter operator in the coordinate representation. According to [7], by integrating equation (67) using the IWOP technique, the compact representation form of the beam splitter operator can be derived as
()
Let θ = π/4, we have
()
where is the Schwinger representation of the angular momentum y-component operator in the above section. Obviously, from equation (67), we can get
()
and we know that |q1, p2⟩ is the common eigenvector of Q1 and P2, that is,
()
However, the expression of |q1, p2⟩ in Fock space is
()
Now, we let the beam splitter operator B(π/4) act on |q1, p2⟩. According to equation (70), we have
()
Let ξ1 = q1, ξ2 = −p2, ξ = ξ1 + iξ2, then
()
which is a typical entangled state representation. In this way, after the π/4 beam splitter acts on the direct-product state of the coordinate eigenstate of Mode 1 and the momentum eigenstate of Mode 2, the entanglement of Mode 1 and Mode 2 is generated.
Next, we will consider the decomposition of the beam splitter operator. Note that when θ ≠ π/2, that is, in the case of noncomplete transmission, the matrix N can be decomposed into the form as follows:
()
Using the completeness relation of the coordinate eigenstates, we have
()
Then, the beam splitter operator can be decomposed into the form as follows:
()
in which,
()
By using the completeness relation of the coordinate eigenstates and the properties of the coordinate state translation operator
()
we can express as
()
()
Noting that the natural representation of the single-mode squeezed operator in the coordinate representation
()
we know that is the direct product of two single-mode squeezed operators (with a squeezing phase difference of π), that is,
()
It can be seen that we can conveniently decompose the beam splitter operator into the following form:
()
If θ = π/4 is taken, then
()
Let the beam splitter operator in the form of equation (85) act on the direct-product state |p1, q2⟩ of the momentum eigenstate of Mode 1 and the coordinate eigenstate of Mode 2, and we have
()

It can be seen from equations (15) that (86) is actually in the form of the Schmidt decomposition of another entangled state representation |η = q2 − ip1⟩.

In this section, a significant work is presented: by using two methods to apply a π/4 beam splitter to the direct product state of two-mode momentum and coordinate, the entangled state representation can be directly obtained. It is found that Appendix A of reference [8] also explicitly gives similar results, but there are certain differences between their operation procedures and our mechanism. From reference [8], we further realize the practical value of our work, which can be fully applied to the theoretical frameworks of continuous-variable quantum teleportation and bosonic mode error-correcting codes. This may also become the focus of our further research.

7. Conclusion

The transformation theory is the essence of theoretical physics. Starting from the original form of the EPR wave function, this paper derives the EPR-entangled state representation. Then, by the transformation of the boson modes of the two-dimensional isotropic harmonic oscillator in the Cartesian coordinate frame and the polar coordinate frame, the two-dimensional coordinate eigenstates are related to the EPR-entangled state representation. Based on this, we conclude in a certain sense that quantum entanglement is just a relative effect, which provides a valuable idea for the essential characteristics of quantum entanglement. In addition, we also investigate the transformation between HG modes and LG modes from the perspective of unitary transformation and interestingly discuss the entangled state representation generated by the beam splitter, all of which are of certain heuristic significance. We believe that the concepts proposed in this paper will possibly be further extended to other fields of quantum optics and quantum mechanics.

Conflicts of Interest

The author declares no conflicts of interest.

Funding

No funding was received for this manuscript.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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