Optical and Quantum Electronics (2024) 56:12
https://doi.org/10.1007/s11082-023-05590-2
Quantum hyper‑entangled system with multiple qubits
based on spontaneous parametric down‑conversion
and birefringence effect
Yiqian Yang1 · Liangcai Cao1
Received: 18 July 2023 / Accepted: 13 October 2023 / Published online: 23 November 2023
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023
Abstract
Quantum hyper-entangled system with multiple qubits have garnered significant interest
and are considered essential resources for various quantum applications. The generation
of entangled photon pairs with multiple qubits has long been a sought-after goal in modern quantum technologies. However, the current quantum entangled system is limited to
small number of qubits due to the constrained down-conversion efficiency. In this work,
a multiple qubits hyper-entangled system in polarization and path is proposed by combining quantum conversion and classical conversion techniques. The spontaneous parametric down-conversion (SPDC) is achieved using a high-power pump laser and a BBO film
to generate two qubits of polarization-entangled photons, representing the quantum conversion. Subsequently, a BBO cube is employed to realize the birefringence effect (BE),
generating multiple qubits of path-entangled photons as part of the classical conversion.
To achieve highly efficient quantum hyper-entangled photon pair generation, BBO cubes
are cascaded, ensuring that the expansion of qubits does not compromise the conversion
efficiency of the entire system. The state function and the hyper-entangled correlation are
analysed to demonstrate the capabilities of the proposed system. The proposed SPDC-BE
method significantly extends the number of qubits involved in quantum entanglement. In
addition to the thought-provoking structure, this method offers new insights into the study
of quantum mechanics.
Keywords Hyper-entangled system · Spontaneous parametric down-conversion · BBO
film · Birefringence effect · BBO cube
* Liangcai Cao
clc@tsinghua.edu.cn
Yiqian Yang
yang-yq22@mails.tsinghua.edu.cn
1
Department of Precision Instrument, Tsinghua University, Shuangqing Street, 100084 Beijing,
China
13
Vol.:(0123456789)
12 Page 2 of 15
Y. Yang, L. Cao
1 Introduction
Quantum mechanics, a branch of physics with a history spanning over a century, has
resulted in various practical applications that have become integral parts of our daily lives,
including transistors and lasers. The potential of quantum mechanics appears to be limitless. The 2022 Nobel Prize in physics has been awarded to the pioneering experiments in
quantum information science, a burgeoning field that could revolutionize computing, cryptography, and the transfer of information via what is known as quantum teleportation. The
groundbreaking experiments not only confirmed the quantum theory, but established the
basis for a new field of science and technology Reid (1989); Aspect et al. (1982). Quantum entanglement, a fundamental aspect of quantum mechanics, plays a pivotal role in
both theory and applications. In an entangled two-photon state, the correlation properties
between the two photons remain intact regardless of the distance separating them Aspect
et al. (1981); Jebli and Daoud (2023). The behavior of one photon in an entangled pair is
directly linked to the behavior of the other, even if they are farly separated. The landscape
of photon entanglement is complex and multidimensional, encompassing various degreesof-freedoms (DoFs), such as time, path, angular momentum, and polarization. Quantum
entanglement space is constituted by different sets of orthogonal complete bases in Hilbert
space. Quantum hyper-entangled system with multiple qubits attracts great interest, which
stimulates applications in numerous fields of quantum optics. The utilization of quantum
hyper-entanglement enables enhanced performance and introducing novel capabilities. In
quantum communication, hyper-entangled photons can facilitate secure transmission of
information over long distances, surpassing the limitations of classical communication
channels Zhang et al. (2022); Khorrampanah et al. (2022). Quantum microscopy benefits
from hyper-entanglement by enabling high-resolution imaging and enhanced sensing capabilities Ono et al. (2013); Ndagano et al. (2022); Samantaray et al. (2017). Additionally,
quantum holography Defienne et al. (2021); Töpfer et al. (2022); Thekkadath et al. (2023),
quantum illumination Barzanjeh et al. (2020); De Palma and Borregaard (2018), quantum
imaging Defienne et al. (2022); Ndagano et al. (2020), and quantum detection Aguilar
et al. (2019); Solaimani and Mobini (2022); Zhang et al. (2020) can all benefit from the
unique properties of hyper-entangled systems, opening up new possibilities for practical
applications.
The generation of entangled photon pairs with multiple qubits is a fundamental requirement for quantum applications and has been a long-sought goal in modern quantum optical technologies. However, the current state of polarization-entangled systems is limited
to only two qubits, representing horizontal and vertical polarization states, primarily due
to the constrained down-conversion efficiency of the BBO film. Spontaneous parametric down-conversion (SPDC) is realized by a BBO film with a high-power pump laser
to generate the quantum entangled photons. SPDC is a versatile technique that harnesses
the second-order nonlinear effects of a nonlinear crystal, allowing for the generation of
entangled and correlated photon pairs. The output photons converted by SPDC have the
advantage of easy control. The generation of a multiple qubits quantum entangled system
is mainly based on nonlinear cascading processes for SPDC Dhara et al. (2022); Liu et al.
(2020); Chen et al. (2017); Yao et al. (2012); Zhang et al. (2015). However, the efficiency
of this approach is significantly hindered by the inherently weak nature of nonlinear quantum optical processes, leading to low photon generation rates. Consequently, improving
efficiency remains a considerable challenge, which promotes extensive research efforts
to overcome this bottleneck. Various nonlinear designs based on waveguides Main et al.
13
Quantum hyper‑entangled system with multiple qubits based…
Page 3 of 15 12
(2019); Solntsev et al. (2012), crystals Introini et al. (2020); Borshchevskaya et al. (2015),
plasmonic metasurfaces Jin et al. (2021); Loot and Hizhnyakov (2018), cavity Leger et al.
(2023); Zhou and Bermel (2015) and ring resonators Akbari and Kalachev (2016); Schneeloch et al. (2019) have been explored in the quest for efficiency enhancement.
In this work, by combining SPDC and birefringence effect (BE), a SPDC-BE method is
proposed to construct a multiple qubits quantum hyper-entangled system with four or more
qubits. The SPDC process utilizes a BBO film to generate two qubits of polarization-entangled photons, representing the quantum conversion. Subsequently, the BE is achieved using
a BBO cube, generating multiple qubits of path-entangled photons through classical conversion. To ensure highly efficient quantum hyper-entangled photon pair generation with
four or more qubits, we employ cascading BBO cubes. By cascading these elements, we
ensure that the expansion of qubits does not compromise the overall conversion efficiency
of the system. This integration of quantum conversion and classical conversion techniques
allows for the generation of hyper-entangled photon pairs with high efficiency. The state
function and the hyper-entangled correlation between these modes are derived and analyzed. The SPDC-BE method enables the extension of the hyper-entangled system to an
infinite number of qubits, representing a significant advancement in quantum technologies
and their applications Feng et al. (2022); Wang and Yu (2023); Yang et al. (2023).
2 Methodology
The generation of polarization-entangled photon pairs currently relies on the process of
SPDC within a nonlinear crystal film, such as a BBO film. During SPDC, a pump photon is
converted into a photon pair, consisting of a signal photon and an idler photon, with a certain probability. The phenomenon of SPDC has been extensively studied since the 1970 s,
with David Burnham et al providing an early scientific description of its principles Burnham and Weinberg (1970). Yan Hua Shih et al were the first to create an entangled state
using SPDC Shih and Alley (1988), while Ruba Ghosh et al demonstrated the interference
fringes of two entangled particles through SPDC Ghosh and Mandel (1987). According
to the law of conservation of energy and the law of conservation of momentum, the total
energy and momentum of the photon pair are equal to that of the pump photon. Figure 1a
illustrates the principle of SPDC, with a BBO film serving as the medium. The resulting
photon pair becomes entangled in various parameters. In the absence of measurement, the
photon pairs exist in a superposition state. However, when the polarization state of one
beam is measured, the other beam instantaneously collapses into the polarization state
orthogonal to that of the measured beam. Figure 1b demonstrates the detected polarization
outcome of the output beam in Type-II SPDC, where the polarization of the signal photon
and the idler photon are perpendicular to each other. These experimental observations confirm the phenomenon of entanglement in SPDC and highlight the instantaneous correlation
between the polarization-entangled photons. The entanglement of the photon pairs spans
various properties, including polarization, and has been extensively studied and utilized in
quantum information processing and quantum communication protocols.
BE is a phenomenon in optics where a material has a different refractive index for light
polarized in different directions. The crystal splits an arbitrarily polarized input beam into
two perpendicularly polarized output beams. BE is commonly utilized in various optical components and devices. By passing light through a birefringent crystal, the crystal’s
birefringence can lead to the generation of entangled photon pairs in different paths with
13
12 Page 4 of 15
Y. Yang, L. Cao
Fig. 1 a Principle of SPDC and
detected polarization outcome
in Type-II SPDC, with a BBO
film serving as the medium.
The pump photon is converted
into a signal photon and an idler
photon. b Detected polarization result of the output beam
in Type-II SPDC, where the
polarization of the signal photon
and the idler photon are perpendicular
perpendicular polarization. Thus, when a beam of light is directed towards a birefringent
crystal, such as a BBO cube, the polarization of the two refracted beams becomes perpendicular to each other. Malus’s Law describes the relationship between the intensity of
polarized light before and after it passes through a crystal. Malus’s Law can be mathematically expressed as I = I0 ∗cos2 (𝜃), where I0 and I are the intensity of the beam before and
after it passes through the crystal, 𝜃 is the angle between the initial polarization direction of
the beam and the axis of the polarizer. According to Malus’s Law, when 𝜃 = 45◦, the intensities of the two outgoing sub-beams are equal. The birefringent crystal effectively acts as
two polarizers with transmission directions that are mutually perpendicular. Figure 2 illustrates the principle of BE, with a BBO cube serving as the medium. If the incident beam is
horizontally polarized and the optical axis of the crystal is oriented at an angle of 45◦ with
respect to the horizontal, the refracted beams are polarized along and perpendicular to the
optical axis of the crystal, respectively. Similarly, if the incident beam is vertically polarized, the refracted beams will exhibit polarizations along and perpendicular to the optical
axis of the crystal, respectively. Additionally, irrespective of the polarization state of the
incident beam, the two outgoing beams exhibit the same polarization distribution. Consequently, if we only measure the polarization state of the outgoing beam in the BE process, without performing any measurement on the polarization state of the outgoing beam
in the SPDC process, it becomes impossible to distinguish between the states �ΨH,O ⟩ and
�ΨV,O ⟩. These two states represent polarization-entangled qubits resulting from the SPDC
process. Similarly, states �ΨH,E ⟩ and �ΨV,E ⟩ cannot be distinguished from each other either,
representing two other quantum qubits. Moreover, the states �ΨH,O ⟩ and �ΨH,E ⟩ are pathentangled qubits resulting from the BE, as are the states �ΨV,O ⟩ and �ΨV,E ⟩. The subscript H
and V represent the horizontal and vertical states of photons in SPDC. The subscript O and E
represent the ordinary and extraordinary state of photons in BE. The entanglement between
the polarization and path degrees of freedom is a significant characteristic of these entangled qubit states, resulting for a quantum hyper-entangled system.
13
Quantum hyper‑entangled system with multiple qubits based…
Page 5 of 15 12
Fig. 2 Principle of BE, with
a BBO cube serving as the
medium. The optical axis of
the crystal is at the angle of 45◦
from the horizontal. No matter
whether the incident beam is
horizontally polarized in a or
vertically polarized in b, the two
outgoing beams exhibit the same
polarization distribution. The
output beams are polarized along
and perpendicular to the optical
axis of the crystal, respectively
By leveraging the conversion properties of SPDC and BE, a quantum hyper-entangled system with multiple qubits can be realized. As illustrated in Fig. 3, take the four
qubits hyper-entangled system for instance. Figure 3a is the optical path of the four
qubits hyper-entangled system. The former BBO film serves as a nonlinear crystal for
SPDC, where quantum conversion takes place, resulting in the entanglement of qubits
in polarization. The latter BBO cube serves as a birefringent crystal for BE, enabling
classical conversion and entangling the qubits in the path domain. Figure 3b and c
represent the output modes distribution after SPDC and BE, respectively. It can be
observed that the classical conversion in the BE process does not degrade the conversion efficiency of the output. By cascading additional BBO cubes along the optical
path, the number of entangled qubits can be further increased. To implement the BE
method, each successive BBO cube should be rotated by 45◦ in the direction of the
optical axis and halved in thickness. This arrangement ensures the proper entanglement of qubits in the path domain. Overall, this method offers a promising approach
for constructing a quantum hyper-entangled system with multiple qubits. By combining the conversion properties of SPDC and BE, researchers can realize entanglement in
both polarization and path, opening up new possibilities for quantum information processing and communication. Cascading additional BBO cubes provides a pathway for
extending the number of entangled qubits in the system. This advancement holds great
potential for the development of quantum technologies and their diverse applications.
13
12 Page 6 of 15
Y. Yang, L. Cao
Fig. 3 a Optical path of the four qubits hyper-entangled system. The former BBO film serves as a nonlinear
crystal for SPDC. The latter BBO cube serves as a birefringent crystal for BE. b and c are the output modes
distribution after SPDC and BE, respectively
3 Modes analysis
In this section, the state function and the hyper-entangled correlation between different
modes in the hyper-entangled system are deduced. We take the four qubits hyper-entangled
system as an example for analysis and assume that the four outputs are separated along the
x-axis. Each state is composed of N photons in a Gaussian distribution with a variance of
𝜎 in the x − y plane. After the process of SPDC, the initial state �Ψ0 ⟩ is converted into state
�ΨH ⟩ and state �ΨV ⟩ with a bias phase 𝜑. Following the process of BE, the state �ΨH ⟩ is
separated into state �ΨH,O ⟩ and state �ΨH,E ⟩ with a bias phase 𝜉 , while state �ΨV ⟩ is separated
into state �ΨV,O ⟩ and state �ΨV,E ⟩ with the same bias phase 𝜉 . The final output state �Ψ⟩ is
�Ψ(x, y, 𝜙, 𝜁)⟩
�
�
�
�
�
�
�
�
= C1 ��ΨH,O ⊗ ��ΨV,O + C2 ��ΨH,O ⊗ ��ΨV,E + C3 ��ΨH,E ⊗ ��ΨV,O + C4 ��ΨH,E ⊗ ��ΨV,E
�
�
⎞
⎞
⎛ 1 �
⎛ √1 �ΨV,O (x, y, 𝜙, 𝜁)
�
1 ⎜ √2 �ΨH,O (x, y, 𝜙, 𝜁)
1
i𝜑
2
⎟
⎜
� +√ e
� ⎟,
= √
1 i𝜉 �
1 i𝜉 �
2 ⎜⎝ + √2 e �ΨH,E (x, y, 𝜙, 𝜁) ⎟⎠
2 ⎜⎝ + √2 e �ΨV,E (x, y, 𝜙, 𝜁) ⎟⎠
(1)
where �ΨH,O ⟩, �ΨH,E ⟩, �ΨV,O ⟩, and �ΨV,E ⟩ represent the state of N photons in the horizontal-ordinary, horizontal-extraordinary, vertical-ordinary and vertical-extraordinary state,
13
Quantum hyper‑entangled system with multiple qubits based…
Page 7 of 15 12
respectively. C1, C2, C3 and C4 are the coefficients of amplitude of the hyper-entangled
states. The values of C1, C2, C3 and C4 are equal to each other and the coefficient satisfies
C12 + C22 + C32 + C42 = 1. The hyper-entangled correlation between the different modes can
be expressed through the entangled states involved. For example, the correlation between
the qubits �ΨH,O ⟩ and �ΨV,O ⟩ can be described as
�
�
�ΨH,O ⊗ �ΨV,O = C5 �H⟩−1 �O⟩−2 + C6 �V⟩−1 �O⟩−2 ,
(2)
�
�
where C5 and C6 are the coefficients representing the amplitudes of the hyper-entangled
states, while �H⟩−1 and �V⟩−1 represent the horizontal and vertical states of the first conversion, and �O⟩−2 represents the ordinary state of the second conversion. The subscript −1
and −2 represent the first and the second conversion, respectively. Similarly, the correlation between other entangled qubits can be obtained by combining the appropriate states
involved. Each state can be expressed as
�
�Ψ (x, y, 𝜙, 𝜁)
� H,O
� �
�
�
�N
1
𝛼 𝛽
−i𝜒(x)N𝜙𝜁
=
e
â +H,O (x, y) �0⟩dxdy
f x − − ,y √
∬
2 2
(3)
N!
� �
�
𝛼 𝛽
=
e−i𝜒(x)N𝜙𝜁 f x − − , y �N;0;0;0, x, y⟩dxdy,
∬
2 2
�
�Ψ (x, y, 𝜙, 𝜁)
� H,E
� �
�
�
�N
1
𝛼 𝛽
=
e
â +H,E (x, y) �0⟩dxdy
f x − + ,y √
∬
2 2
N!
� �
�
𝛼 𝛽
=
e−i𝜒(x)N𝜙𝜁 f x − + , y �0;N;0;0, x, y⟩dxdy,
∬
2 2
−i𝜒(x)N𝜙𝜁
(4)
�
�Ψ (x, y, 𝜙, 𝜁)
� V,O
� �
�
�
�N
1
𝛼 𝛽
â +V,O (x, y) �0⟩dxdy
− ,y √
∬
2 2
N!
� �
�
𝛼 𝛽
e−i𝜒(x)N𝜙𝜁 f x + − , y �0;0;N;0, x, y⟩dxdy,
=
∬
2 2
=
e−i𝜒(x)N𝜙𝜁
f x+
(5)
�
�Ψ (x, y, 𝜙, 𝜁 )
� V,E
� �
�
�
�N
1
𝛼 𝛽
=
e
â +V,E (x, y) �0⟩dxdy
f x + + ,y √
∬
2 2
N!
� �
�
𝛼 𝛽
e−i𝜒(x)N𝜙𝜁 f x + + , y �0;0;0;N, x, y⟩dxdy,
=
∬
2 2
−i𝜒(x)N𝜙𝜁
(6)
13
12 Page 8 of 15
Y. Yang, L. Cao
where �0⟩ is the vacuum state. â +H,O (x, y), â +H,E (x, y), â +V,O (x, y), and â +V,E (x, y) are the creation operators for different states. 𝜒(x) is a step function that satisfies 𝜒(x) = 1 for x > 0
and 𝜒(x) = 0 for x ≤ 0. After passing through the BBO film, state �ΨH ⟩ and state �ΨV ⟩ are
displaced at a distance of 𝛼 along the x-axis, as shown in Fig. 3b. After passing through
the BBO cube, the substate of state �ΨH ⟩ and state �ΨV ⟩ are displaced at a distance of 𝛽
along the x-axis, as shown in Fig. 3c. f (x − 𝛼2 − 𝛽2 , y), f (x − 𝛼2 + 𝛽2 , y), f (x + 𝛼2 − 𝛽2 , y), and
f (x + 𝛼2 + 𝛽2 , y) represents the probability distribution densities of different states. The four
states are the orthogonal complete bases in Hilbert space and are hyper entangled with
each other. The probability density function f(x, y) is defined as
1 − x22𝜎+y22
e
,
2𝜋𝜎 2
(7)
f (x, y)dxdy = 1.
(8)
f (x, y) =
∞
∞
∫−∞ ∫−∞
If set all states to the origin of the x − y plane, the final state of the four qubits polarizationentangled system �Ψ� ⟩ is
��
�
⎞
⎛ √1 ∣ΨH,O x + 𝛼 + 𝛽 , y, 𝜙, 𝜁
�
2
2
⎟
�Ψ� (x, y, 𝜙, 𝜁) = √1 ⎜ 2
��
�
�
�
𝛽
𝛼
1 i𝜉 �
⎟
⎜
√
Ψ
−
,
y,
𝜙,
𝜁
e
x
+
+
2⎝
� H,E
⎠
2
2
2
�
��
�
⎞
⎛
𝛼 𝛽
1 ��
(9)
Ψ
+
,
y,
𝜙,
𝜁
x
−
√ �� V,O
⎟
⎜
2
2
⎟
2�
1 i𝜑 ⎜
+√ e ⎜
��⎟.
�
�
𝛼 𝛽
2 ⎜ 1 i𝜉 �
⎟
−
,
y,
𝜙,
𝜁
e
Ψ
+
x
−
�
V,E
⎜ √
⎟
�
2
2
2 �
⎝
⎠
The number of entangled qubits can be further increased by continuing cascading BBO
cubes in the optical system. Such as continuing cascading a BBO cube in the optical system for the third conversion to form an eight qubits entangled system, the simplified final
output state �Ψ⟩ after conversion is
�
�
��
1 �
⎧ ⎡ √1 √1 �Ψ
⎤ ⎫
H,O,O + √ �ΨH,O,E
�
2
⎪ ⎢ 2 �2
�⎥ ⎪
�
�
1 �
⎪ ⎢ + √1 √1 �Ψ
⎥ ⎪
H,E,O + √ �ΨH,E,E
⎦ ⎪
2
2�
2
1 ⎪⎣
�Ψ⟩ = √ ⎨
� ⎬.
�
(10)
�
�
⎤⎪
2 ⎪ ⎡ √1 √1 ��ΨV,O,O + √1 ��ΨV,O,E
2
⎪ +⎢ 2 � 2
�
�� ⎥⎪
⎪ ⎢ + √1 √1 ��ΨV,E,O + √1 ��ΨV,E,E ⎥⎪
⎦⎭
⎣
⎩
2
2
2
The normalized entanglement density matrix d of the converted states for different number of entangled qubits is shown in Fig. 4. In Fig. 4a, state �ΨH ⟩ and state �ΨV ⟩ are produced in one conversion. Therefore, the degree of entanglement between them is normalized to 100 %. Likewise, in Fig. 4b, the degree of entanglement between state �ΨH,O ⟩ and
state �ΨH,E ⟩, and the degree of entanglement between state �ΨV,O ⟩ and state �ΨV,E ⟩ are 100
%. Due to the latter conversion of state �ΨH ⟩ and state �ΨV ⟩, the degree of entanglement
13
Quantum hyper‑entangled system with multiple qubits based…
Page 9 of 15 12
Fig. 4 Normalized entanglement density matrix in a 2 qubits, b 4 qubits, c 8 qubits, and d 16 qubits system,
respectively. The entanglement density d between two states is based on Eq. (11)
between their substates, such as �ΨH,O ⟩ and �ΨV,O ⟩, is halved. In Fig. 4c, the degree of
entanglement between the respective substates of �ΨH,O ⟩, �ΨH,E ⟩, �ΨV,O ⟩ and �ΨV,E ⟩, such
as �ΨH,O,O ⟩ and �ΨH,O,E ⟩, are normalized to 100 %. Each conversion process introduces two
possible outcomes, resulting in a binary branching. If the number of conversions is n, the
number of the final output states is 2n. The optical configuration of this optical module can
be expressed as SPDC + (BE)n−1. The state can be expressed as �Ψi1 ,i2 ,i3 ,⋯,in ⟩, where i1indicates �ΨH ⟩ or �ΨV ⟩ in SPDC, and im indicates �ΨO ⟩ or �ΨE ⟩ for m = 2, 3, ⋯ , n in BE. Taking
the 16 qubits entangled system as an example, the state can be expressed as �ΨH,O,O,O ⟩,
�ΨH,O,O,E ⟩, �ΨH,O,E,O ⟩, �ΨH,O,E,E ⟩, �ΨH,E,O,O ⟩, �ΨH,E,O,E ⟩, �ΨH,E,E,O ⟩, �ΨH,E,E,E ⟩, �ΨV,O,O,O ⟩,
�ΨV,O,O,E ⟩, �ΨV,O,E,O ⟩, �ΨV,O,E,E ⟩, �ΨV,E,O,O ⟩, �ΨV,E,O,E ⟩, �ΨV,E,E,O ⟩, �ΨV,E,E,E ⟩, respectively. If
the separation between two states occurs at the kth conversion, which means different from
ik , the entanglement density d between the two states is
d = 2k−n ,
(11)
The normalized entanglement density matrix d for 16 qubits is shown in Fig. 4d. As the
conversions are cascaded, the total number of potential outcomes doubles with each additional conversion step. Therefore, the number of final output states in a quantum hyperentangled system will grow exponentially with the number of conversions. This exponential
13
12 Page 10 of 15
Y. Yang, L. Cao
growth in the number of states highlights the potential for creating large-scale entangled
systems with numerous qubits, offering a wealth of possibilities for quantum information
processing and communication.
For BBO crystals, the Sellemeier equations are
n2o = 2.7359 +
0.01878
− 0.01354𝜆2 ,
𝜆2 − 0.01822
(12)
n2z = 2.3753 +
0.01224
− 0.01516𝜆2 ,
𝜆2 − 0.01677
(13)
cos2 (𝜃) sin2 (𝜃)
.
+ 2
n2o (𝜃)
nz (𝜃)
(14)
1
n2e (𝜃)
=
Figure 5a is the distribution curve of the refractive index of BBO at the wavelength of 405
nm and 810 nm, varying with the wave vector of the input beam. Figure 5b-d are the spatial
Fig. 5 a Distribution curve of refractive index at the wavelength of 405 nm and 810 nm, varying with the
wave vector of the input beam. Spatial distribution of output qubits in b 2 qubits, c 4 qubits, and d 8 qubits
system, respectively
13
Quantum hyper‑entangled system with multiple qubits based…
Page 11 of 15 12
distribution of output qubits in (b) 2 qubits, (c) 4 qubits, and (d) 8 qubits system, respectively, where the spot diameter is 0.2 mm. The BBO film’s thickness employed for type-II
SPDC is 0.5 mm. Through phase matching and parameter adjustments, the conversion efficiency of SPDC is 55 %. The high-power pump laser is a continuous-wave laser operating
at 405 nm, boasting an output power of 200 mW. Specifying that state �ΨH ⟩ and state �ΨV ⟩
are separated to 20 mm through quantum conversion with a BBO film in Fig. 5b. Besides,
the thickness of the BBO cubes for the first-order and the second-order classical conversion
is set as 40 mm and 20 mm in Fig. 5c–d, respectively, to form the four qubits and eight
qubits entangled system. It can be seen from Fig. 5b–d, with the increase in the number of
conversions, the distance between output qubits decreases. For a multiple qubits entanglement system, the phase-matching angle between the signal photon and idler photon, and
the thickness of the BBO cubes should be enlarged.
4 Discussion
In this work, we propose a SPDC-BE method for the creation of quantum hyper-entangled systems with multiple qubits. By harnessing the synergy of SPDC and BE, and optimizing the cascading design, highly efficient hyper-entangled photon pair generation are
achieved. This setup has the potential to revolutionize several quantum-based applications. For instance, in quantum communication, our method can facilitate the creation of
secure communication channels with increased qubit capacities, enhancing data transfer
and encryption protocols. Similarly, in quantum computing, the generation of larger-scale
entangled states is pivotal for performing more complex quantum algorithms and simulations. Furthermore, the progress made here opens up new possibilities in quantum cryptography and quantum sensing, where advanced entangled states play a critical role. Our work
underscores the importance of scalability in quantum systems and offers a stepping stone
towards realizing the full potential of quantum technologies in various practical domains.
The SPDC-BE method has distinct advantages compared to hyper-entangled states
generated based on orbital angular momentum (OAM) degree of freedom. While both
approaches aim to harness the power of hyper-entanglement, the methodologies and outcomes differ significantly. The experiment of SPDC-BE method is more feasible. The
SPDC-BE method relies on the synergy of SPDC and the BE, which are well-established
techniques in the field of quantum optics, as shown in Fig. 3a. These techniques are experimentally feasible and have been extensively studied, making their implementation more
accessible. The efficiency of SPDC-BE method is higher. The SPDC-BE method focuses
on achieving highly efficient hyper-entangled photon pair generation by utilizing cascaded
BBO cubes. This emphasis on efficiency is particularly advantageous for practical applications, as it leads to higher rates of successful hyper-entanglement, making the method more
suitable for real-world quantum technologies. The loss of SPDC-BE method is mitigated.
The use of cascaded BBO cubes in the SPDC-BE method helps mitigate the impact of
optical losses, as these cubes enable better control and optimization of photon paths. This
is crucial for maintaining the integrity of hyper-entanglement states over longer distances,
which is a key requirement for applications such as quantum communication. The SPDCBE method combines quantum conversion and classical conversion techniques. This hybrid
approach offers a well-rounded strategy for generating hyper-entangled states, allowing
for increased qubit capacities while maintaining high efficiency. The SPDC-BE method
demonstrates potential for scalability, offering a pathway to extend hyper-entanglement to
13
12 Page 12 of 15
Y. Yang, L. Cao
larger numbers of qubits, as shown in Fig. 4. This scalability is of utmost importance for
applications in quantum computing, where the ability to entangle multiple qubits is essential for performing complex algorithms. In contrast, while OAM-based hyper-entanglement
offers unique properties related to spatial modes and photon angular momentum, it can be
more challenging to implement and control experimentally. The SPDC-BE method capitalizes on established techniques and prioritizes efficiency and practicality, positioning it as a
promising approach for advancing the practical applications of hyper-entanglement in various quantum technologies.
In the context of a real-world experimental setup, it is imperative to account for various
sources of losses that can impact the overall performance of the system. These losses can
occur at different stages of the experiment, including the detection stage and within the
BBO crystals themselves. At the detection stage, losses can arise due to various factors
such as imperfect detector efficiency, coupling losses, and background noise. Incorporating
these losses into the experimental scheme is essential to accurately assess the generated
entangled photon pairs’ quality and the efficiency of the entanglement generation process.
Losses within the BBO crystals can occur due to factors such as crystal impurities, absorption, and scattering. These losses can lead to reduced photon generation rates and impact
the overall fidelity of the entangled states produced. A comprehensive analysis of losses
should be included in the experimental scheme to provide a more realistic evaluation of
the system’s performance. This might involve optimizing experimental parameters, implementing error correction techniques, or using additional components to compensate for
losses. This consideration not only enhances the credibility of the work but also contributes
to the broader scientific community’s understanding of the challenges and potential solutions in realizing efficient hyper-entangled photon pair generation in real-world scenarios.
While the efficiency is a crucial aspect to consider in practical quantum information applications, it’s important to note that the degree of entanglement also holds significance. In
the realm of quantum information, a balance between both factors is vital for the successful
application of quantum technologies.
5 Conclusion
In this work, a SPDC-BE method that harnesses the capabilities of SPDC and BE to create a quantum hyper-entangled system with four or more qubits is proposed. The SPDC
process utilizes a BBO film and a high-power pump laser to generate two qubits of polarization-entangled photons, representing the quantum conversion. Subsequently, the BE
is achieved using a BBO cube, leading to the generation of four qubits of hyper-entangled photons through classical conversion. It is demonstrated that the cascading design,
which operates at room temperature, effectively enhances the efficiency of the system.
This advancement holds great significance as the generation of entangled photon pairs is
at the core of fundamental tests of quantum mechanics and serves as a gateway to unlocking powerful new technologies. These results provide deeper insights into the properties
of quantum entanglement and serve as a proof-of-principle approach for creating a platform that can facilitate the expansion of quantum applications requiring multiple hyperentangled qubits. However, while this study represents a significant step forward, there is
still much to explore regarding the entangled properties of the generated multiple qubits.
The field of multipartite entangled states is still in its early stages, and there is abundant
opportunity for further research and discovery. By combining the SPDC-BE method with
other multiplexing techniques, such as time-multiplexing and frequency-multiplexing, we
13
Quantum hyper‑entangled system with multiple qubits based…
Page 13 of 15 12
anticipate a significant broadening of our understanding of multipartite entangled states.
This integration of techniques holds the potential to extend the number of entangled qubits
on a larger scale, enabling the exploration of more complex quantum systems and advancing the frontiers of quantum information processing and communication.
Author contribution All authors contributed equally to this work.
Funding The authors acknowledge National Natural Science Foundation of China (Grants No. 62235009).
Data availability The data that supports the findings of this study are available from the corresponding
author upon reasonable request.
Declarations
Conflict of interest The authors declare that they have no competing interests.
Ethical approval Not applicable.
References
Aguilar, G., Souza, M., Gomes, R., Thompson, J., Gu, M., Céleri, L., Walborn, S.: Experimental investigation of linear-optics-based quantum target detection. Phys. Rev. A 99(5), 053813 (2019)
Akbari, M., Kalachev, A.: Third-order spontaneous parametric down-conversion in a ring microcavity.
Laser Phys. Lett. 13(11), 115204 (2016)
Aspect, A., Grangier, P., Roger, G.: Experimental tests of realistic local theories via bell’s theorem.
Phys. Rev. Lett. 47(7), 460 (1981)
Aspect, A., Grangier, P., Roger, G.: Experimental realization of Einstein-Podolsky-Rosen-Bohm gedankenexperiment: a new violation of bell’s inequalities. Phys. Rev. Lett. 49(2), 91 (1982)
Barzanjeh, S., Pirandola, S., Vitali, D., Fink, J.M.: Microwave quantum illumination using a digital
receiver. Sci. Adv. 6(19), 0451 (2020)
Borshchevskaya, N., Katamadze, K., Kulik, S., Fedorov, M.: Three-photon generation by means of thirdorder spontaneous parametric down-conversion in bulk crystals. Laser Phys. Lett. 12(11), 115404
(2015)
Burnham, D.C., Weinberg, D.L.: Observation of simultaneity in parametric production of optical photon
pairs. Phys. Rev. Lett. 25(2), 84 (1970)
Chen, L.-K., Li, Z.-D., Yao, X.-C., Huang, M., Li, W., Lu, H., Yuan, X., Zhang, Y.-B., Jiang, X., Peng,
C.-Z., et al.: Observation of ten-photon entanglement using thin bib 3 o 6 crystals. Optica 4(1),
77–83 (2017)
De Palma, G., Borregaard, J.: Minimum error probability of quantum illumination. Phys. Rev. A 98(1),
012101 (2018)
Defienne, H., Ndagano, B., Lyons, A., Faccio, D.: Polarization entanglement-enabled quantum holography. Nat. Phys. 17(5), 591–597 (2021)
Defienne, H., Cameron, P., Ndagano, B., Lyons, A., Reichert, M., Zhao, J., Harvey, A.R., Charbon, E.,
Fleischer, J.W., Faccio, D.: Pixel super-resolution with spatially entangled photons. Nat. Commun.
13(1), 3566 (2022)
Dhara, P., Johnson, S.J., Gagatsos, C.N., Kwiat, P.G., Guha, S.: Heralded multiplexed high-efficiency
cascaded source of dual-rail entangled photon pairs using spontaneous parametric down-conversion. Phys. Rev. Appl. 17(3), 034071 (2022)
Feng, L., Zhang, M., Wang, J., Zhou, X., Qiang, X., Guo, G., Ren, X.: Silicon photonic devices for scalable quantum information applications. Photonics Res. 10(10), 135–153 (2022)
Ghosh, R., Mandel, L.: Observation of nonclassical effects in the interference of two photons. Phys. Rev.
Lett. 59(17), 1903 (1987)
Introini, V., Steel, M., Sipe, J., Helt, L., Liscidini, M.: Spontaneous parametric down conversion in a
doubly resonant one-dimensional photonic crystal. Opt. Lett. 45(5), 1244–1247 (2020)
Jebli, L., Daoud, M.: Local quantum uncertainty and non-commutativity measure discord in two-mode
photon-added entangled coherent states. Opt. Quantum Electron. 55(8), 707 (2023)
13
12 Page 14 of 15
Y. Yang, L. Cao
Jin, B., Mishra, D., Argyropoulos, C.: Efficient single-photon pair generation by spontaneous parametric
down-conversion in nonlinear plasmonic metasurfaces. Nanoscale 13(47), 19903–19914 (2021)
Khorrampanah, M., Houshmand, M., Sadeghizadeh, M., Aghababa, H., Mafi, Y.: Enhanced multiparty
quantum secret sharing protocol based on quantum secure direct communication and corresponding
qubits in noisy environment. Opt. Quantum Electron. 54(12), 832 (2022)
Leger, A.Z., Gambhir, S., Légère, J., Hamel, D.R.: Amplification of cascaded down-conversion by reusing photons with a switchable cavity. Phys. Rev. Res. 5(2), 023131 (2023)
Yang, Y., Forbes, A., Cao, L.: A review of liquid crystal spatial light modulators: devices and applications. Opto-Electron. Sci. 2(8), 230026 (2023)
Liu, Y., Liang, S., Jin, G., Yu, Y., Chen, A.: Einstein-Podolsky-Rosen steering in spontaneous parametric
down-conversion cascaded with a sum-frequency generation. Phys. Rev. A 102(5), 052214 (2020)
Loot, A., Hizhnyakov, V.: Modeling of enhanced spontaneous parametric down-conversion in plasmonic
and dielectric structures with realistic waves. J. Opt. 20(5), 055502 (2018)
Main, P.B., Mosley, P.J., Gorbach, A.V.: Spontaneous parametric down-conversion in asymmetric couplers: photon purity enhancement and intrinsic spectral filtering. Phys. Rev. A 100(5), 053815
(2019)
Ndagano, B., Defienne, H., Lyons, A., Starshynov, I., Villa, F., Tisa, S., Faccio, D.: Imaging and certifying high-dimensional entanglement with a single-photon avalanche diode camera. NPJ Quantum
Inf. 6(1), 94 (2020)
Ndagano, B., Defienne, H., Branford, D., Shah, Y.D., Lyons, A., Westerberg, N., Gauger, E.M., Faccio,
D.: Quantum microscopy based on Hong-Ou-Mandel interference. Nat. Photonics 16(5), 384–389
(2022)
Ono, T., Okamoto, R., Takeuchi, S.: An entanglement-enhanced microscope. Nat. Commun. 4(1), 2426
(2013)
Reid, M.D.: Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric
amplification. Phys. Rev. A 40(2), 913 (1989)
Samantaray, N., Ruo-Berchera, I., Meda, A., Genovese, M.: Realization of the first sub-shot-noise wide
field microscope. Light Sci. Appl. 6(7), 17005–17005 (2017)
Schneeloch, J., Knarr, S.H., Bogorin, D.F., Levangie, M.L., Tison, C.C., Frank, R., Howland, G.A.,
Fanto, M.L., Alsing, P.M.: Introduction to the absolute brightness and number statistics in spontaneous parametric down-conversion. J. Opt. 21(4), 043501 (2019)
Shih, Y., Alley, C.O.: New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light
quanta produced by optical parametric down conversion. Phys. Rev. Lett. 61(26), 2921 (1988)
Solaimani, M., Mobini, A.: Spectral tuning in quantum rings by magnetic field: detection from nir to fir
regime. Opt. Quantum Electron. 54(3), 145 (2022)
Solntsev, A.S., Sukhorukov, A.A., Neshev, D.N., Kivshar, Y.S.: Spontaneous parametric down-conversion and quantum walks in arrays of quadratic nonlinear waveguides. Phys. Rev. Lett. 108(2),
023601 (2012)
Thekkadath, G., England, D., Bouchard, F., Zhang, Y., Kim, M., Sussman, B.: Intensity interferometry
for holography with quantum and classical light. Sci. Adv. 9(27), 1439 (2023)
Töpfer, S., Gilaberte Basset, M., Fuenzalida, J., Steinlechner, F., Torres, J.P., Gräfe, M.: Quantum holography with undetected light. Sci. Adv. 8(2), 4301 (2022)
Wang, Z., Yu, R.: Effects of time pressure, reward, and information involvement on user management of
fake news on a social media platform. Percept. Motor Skills (2023). https://​doi.​org/​10.​1177/​00315​
12523​11791​23
Yang, Y., Kang, X., Cao, L.: Robust propagation of a steady optical beam through turbulence with
extended depth of focus based on spatial light modulator. J. Phys. Photonics 5(3), 035002 (2023)
Yao, X.-C., Wang, T.-X., Xu, P., Lu, H., Pan, G.-S., Bao, X.-H., Peng, C.-Z., Lu, C.-Y., Chen, Y.-A., Pan,
J.-W.: Observation of eight-photon entanglement. Nat. Photonics 6(4), 225–228 (2012)
Zhang, C., Huang, Y.-F., Wang, Z., Liu, B.-H., Li, C.-F., Guo, G.-C.: Experimental Greenberger-HorneZeilinger-type six-photon quantum nonlocality. Phys. Rev. Lett. 115(26), 260402 (2015)
Zhang, Y., England, D., Nomerotski, A., Svihra, P., Ferrante, S., Hockett, P., Sussman, B.: Multidimensional quantum-enhanced target detection via spectrotemporal-correlation measurements. Phys.
Rev. A 101(5), 053808 (2020)
Zhang, H., Wan, L., Haug, T., Mok, W.-K., Paesani, S., Shi, Y., Cai, H., Chin, L.K., Karim, M.F., Xiao,
L., et al.: Resource-efficient high-dimensional subspace teleportation with a quantum autoencoder.
Sci. Adv. 8(40), 9783 (2022)
Zhou, C., Bermel, P.: High-efficiency cascaded up and down conversion in nonlinear Kerr cavities. Opt.
Express 23(19), 24390–24406 (2015)
13
Quantum hyper‑entangled system with multiple qubits based…
Page 15 of 15 12
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional affiliations.
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under
a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted
manuscript version of this article is solely governed by the terms of such publishing agreement and applicable
law.
13