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Journal of the Optical Society of America B

Linear-optical four-dimensional Bell state measurement with the two-photon interference

Zhi Zeng

Published Jul 23, 2025• 2 citations
Publisher

Abstract

We theoretically investigate the distinguishability of a set of mutually orthogonal four-dimensional Bell states of photon system in path degree of freedom using only linear optics, resorting to the two-photon interference. With quantum interference effect, we find that the 16 four-dimensional Bell states can be classified into seven groups, which can support the transmission of log 2 7=2.81bits classical information with just sending one photon in the quantum superdense coding protocol. When an auxiliary two-dimensional polarization entanglement is introduced, the 16 four-dimensional Bell states then can be classified into 12 groups, which can promote the channel capacity to log 2 12=3.58bits with encoding one photon. Moreover, both of these two approaches can be easily realized with the current technology. Our results are significant for photonic superdense coding and can be useful for other quantum information technologies involved in linear-optical high-dimensional Bell state measurement.

DOI
10.1364/josab.568256

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We theoretically investigate the distinguishability of a set of mutually orthogonal four-dimensional Bell states of photon system in path degree of freedom using only linear optics, resorting to the two-photon interference. With quantum interference effect, we find that the 16 four-dimensional Bell states can be classified into seven groups, which can support the transmission of log 2 7=2.81bits classical information with just sending one photon in the quantum superdense coding protocol. When an auxiliary two-dimensional polarization entanglement is introduced, the 16 four-dimensional Bell states then can be classified into 12 groups, which can promote the channel capacity to log 2 12=3.58bits with encoding one photon. Moreover, both of these two approaches can be easily realized with the current technology. Our results are significant for photonic superdense coding and can be useful for other quantum information technologies involved in linear-optical high-dimensional Bell state measurement.
We theoretically investigate the distinguishability of a set of mutually orthogonal four-dimensional Bell states of photon system in path degree of freedom using only linear optics, resorting to the two-photon interference. With quantum interference effect, we find that the 16 four-dimensional Bell states can be classified into 7 groups, which can support the transmission of log27=2.81log_2 7 = 2.81log2​7=2.81 bits classical information with just sending one photon in the quantum superdense coding protocol. When an auxiliary two-dimensional polarization entanglement is introduced, the 16 four-dimensional Bell states then can be classified into 12 groups, which can promote the channel capacity to log212=3.58log_2 12 = 3.58log2​12=3.58 bits with encoding one photon. Our results are significant for photonic superdense coding, and can be useful for other quantum information technologies involved linear-optical high-dimensional Bell state measurement.
With the assistance of auxiliary two-dimensional polarization entanglement, we theoretically present an efficient Bell state measurement (BSM) method for photonic four-dimensional entanglement in the path degree of freedom, in which just simple linear optical elements and common single-photon detectors are utilized. With this approach, the 16 mutually orthogonal four-dimensional path Bell states can be classified into eight distinguishable groups, which allows the transmission of 3 bits of classical information by just sending one photon in our quantum superdense coding protocol. Compared with previous quantum dense coding protocols that are also based on linear-optical BSM, our protocol has certain advantages and will be useful for other quantum communication protocols based on high-dimensional entanglement.
We present a theoretical scheme for the complete Bell state measurement (BSM) for high-dimensional hyperentangled photons in polarization and path degrees of freedom (DOFs), in which the polarization entanglement is in two-dimensional Hilbert space and the path entanglement is in four-dimensional Hilbert space. This 2×4-dimensional photonic quantum system contains 64 mutually orthogonal high-dimensional hyperentangled Bell states, which can be unambiguously discriminated by using weak cross-Kerr nonlinearity, linear optical element, and single-photon detector. This is the first complete BSM scheme for high-dimensional hyperentangled photons in two DOFs assisted by the quantum nonlinear effect, and we also demonstrate its application in the quantum superdense coding protocol, which allows the transmission of log 2 64=6 bits of classical information via just sending one photon. Moreover, our scheme only requires the achievable small nonlinear phase shift, and it can provide new avenues for other BSM-based quantum information technologies involving photonic high-dimensional hyperentanglement.
In this paper, we present an efficient scheme for the complete analysis of the four-dimensional Bell state of the photon system in the path degree of freedom, resorting to the weak cross-Kerr nonlinearity and linear optics. In our scheme, the measurements on two probe coherent states are used for obtaining the bit information of high-dimensional entanglement, and the linear optical elements and single-photon detectors are used for obtaining the relative phase information of high-dimensional entanglement. With these two independent processes, the 16 orthogonal four-dimensional path Bell states can be completely distinguished from each other. This simple and realizable scheme may have useful applications in the quantum information technology based on high-dimensional entanglement.
Distinguishing limit of Bell states for any nnn-photon DDD-dimensional hyperentanglement
Bell state measurement is crucial to quantum information protocols, but it is impossible to unambiguously distinguish all the Bell states encoded in multi-photon using only linear optics. There is a maximum number of distinguished Bell states, i.e. distinguising limit which is very important for increasing the channel capacity of quantum communications. In this paper, we separate nnn-photon DDD-dimensional hyperentanglement into two groups. For the first group of UUU (G1G_1G1​), we obtain the limit N1=nD−(n−1){N_1} = nD - (n - 1)N1​=nD−(n−1), which can be applied for both bosons' and fermions' cases. We further discuss the limit NNN for any nDnDnD system with the second group of UUU (G2G_2G2​), inferring that at least Dn−1{D^{n - 1}}Dn−1 Bell states can be distinguished due to the symmetry of Bell states. Obviously, N1≤N2{N_1} \le {N_2}N1​≤N2​ for those systems with n>2n>2n>2. Finally, we theoretically design an optical setup for Bell state measurement of two-photon eight-dimensional hyperentanglement of spin, path and orbital angular momentum (OAM) and distinguish 15 classes of 64 Bell states. Our results provide a theoretical basis and practical reference to increase the channel capacity of the quantum communication.
The orbital angular momentum (OAM) carried by twisted photons provides a promising playground for high-dimensional quantum information processing. While Bell-state measurement is the cornerstone for various quantum information applications, the deterministic discrimination of the complete high-dimensional Bell states with linear optics remains relatively unexplored in the OAM state space. Here, we demonstrate a theoretical scheme for the complete four-dimensional OAM Bell-state measurement by using the single-photon hyperentangled state analyzer, in which the auxiliary two-dimensional polarization entanglement and two-dimensional path entanglement are utilized. Our scheme offers an alternative route toward enhancing the channel capacity in quantum communication and increasing the robustness against deleterious noise in practical experiments with twisted photons.