Photonics Research, Volume. 13, Issue 7, 1983(2025)

Entanglement and quantum coherence of hybrid entangled states

Fengyi Xu1... Chenyu Qiao1, Shujing Li1,2, Meihong Wang1,2,3,*, and Xiaolong Su1,24,* |Show fewer author(s)
Author Affiliations
  • 1State Key Laboratory of Quantum Optics Technologies and Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China
  • 2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
  • 3e-mail: wangmh@sxu.edu.cn
  • 4e-mail: suxl@sxu.edu.cn
  • show less

    A hybrid entangled state that involves both discrete and continuous degrees of freedom is a key resource for hybrid quantum information processing. It is essential to characterize entanglement and quantum coherence of the hybrid entangled state toward the application of it. Here, we experimentally characterize the entanglement and quantum coherence of the prepared hybrid entangled state between a polarization-encoded discrete-variable qubit and a cat-encoded wave-like continuous-variable qubit. We show that the maximum quantum coherence is obtained when the probability of the horizontal-polarization photon is 0.5, and entanglement and quantum coherence of the hybrid entangled state are robust against loss in both discrete- and continuous-variable parts. Based on the experimentally reconstructed two-mode density matrix on the bases of polarization and cat state, we obtain the logarithm negativity of 0.57 and l1-norm of 0.82, respectively, which confirms the entanglement and quantum coherence of the state. Our work takes a crucial step toward the application of the polarization-cat hybrid entangled state.

    1. INTRODUCTION

    An entangled state is one of the key quantum resources in quantum information science [1] and quantum metrology [25]. According to whether the eigenvalues of observables are discrete or continuous, entangled states are divided into discrete-variable (DV) and continuous-variable (CV) states, which correspond to states in finite-dimensional and infinite-dimensional Hilbert spaces, respectively [6]. For example, the Bell state that is encoded in the horizontal |H and vertical |V polarizations of photons is a typical DV entangled state. Different from the DV entangled state, the CV entangled state is usually encoded in continuous variables, such as the amplitude and phase quadratures (position and momentum) of an optical field, the basis of coherent states |α and |α. Besides the common DV and CV entangled state, the hybrid entangled state that involves both DV and CV degrees of freedom is also developed, which is a key resource for hybrid quantum information processing [6,7].

    It has been shown that there are complementary advantages and disadvantages of quantum information processing based on the DV and CV systems [6,7], which are developing in parallel. To overcome the disadvantages in each system and exploit their own advantages, hybrid quantum information processing protocols, which combine the advantages of DV and CV quantum information processing, have been proposed [68]. As an important resource for hybrid quantum information, several kinds of hybrid entangled states have been experimentally demonstrated [911], including hybrid entanglement between single photon and coherent states (|0|α+|1|α), particle-like and wave-like optical qubits (|0|cat+eiϕ|1|cat+), and photon polarization and cat state (α|H|cat++β|V|cat). Based on the prepared loss-tolerant hybrid entangled states of light, quantum teleportation [11,12], quantum swapping [13], remote preparation of CV qubits (qumodes) [14], and a quantum-bit encoding converter [15] have been experimentally demonstrated.

    Toward the application of the hybrid entangled state, it is essential to characterize the quantum properties of it. Besides entanglement, quantum coherence is also an important signature of a quantum resource, which characterizes the property of superposition and quantum correlations of quantum states [16] and plays an important role in many quantum information processing tasks [16,17], such as the biology system [18], Deutsch-Jozsa algorithm [19], and Grover quantum search algorithm [20]. Various methods of quantifying quantum coherence are proposed, for example, relative entropy, l1-norm [21], Fisher information [22], and robustness of coherence [23]. The relation between quantum coherence and quantum entanglement has been discussed in Refs. [24,25]. Although quantum coherence in DV [2636] and CV (including Gaussian and non-Gaussian quantum states) [3744] domains has been quantified and investigated, how to quantify quantum coherence of hybrid entangled states is unclear. Especially for the polarization-cat hybrid entangled state, the entanglement is only demonstrated by quantum teleportation [11], which is an indirect method for the characterization of entanglement; how to characterize the entanglement directly is not presented.

    Here, we experimentally prepare a polarization-cat hybrid entangled state and characterize the entanglement and quantum coherence of it. By projecting polarization qubits onto six polarization bases in the DV part, the corresponding density matrices in the CV part are obtained directly. According to the obtained experimental results of six polarization bases, the density matrix of the polarization-cat hybrid entangled state is reconstructed on the bases of polarization and Fock state. To characterize the quantum property conveniently, we transform the two-mode density matrix from the Fock basis to a cat state basis in the CV part and characterize entanglement and quantum coherence of the prepared hybrid entangled state by means of logarithmic negativity and l1-norm based on the reconstructed two-mode density matrix in the basis of polarization and cat state. Our work not only provides a direct method to characterize the entanglement and quantum coherence of hybrid entangled states, but also provides practical guidance for quantum communication based on hybrid entangled states by detailed analysis of how losses in DV and CV parts affect entanglement and quantum coherence. The presented results provide a useful reference for hybrid quantum information processing based on hybrid entangled states.

    2. PRINCIPLE

    The schematic of preparing a hybrid entangled state between polarization and cat state is shown in Fig. 1. An amplitude-squeezed vacuum state with low squeezing level is approximated as an even cat state |cat+(|iαe+|iαe) with small amplitude αe. This approximated even cat state passes through a beam-splitter with low transmittance of T1 to subtract a photon from it. Once a photon is subtracted, an odd cat state |cat(|iαo|iαo) is obtained. Depending on whether a photon is subtracted or not, an odd or even cat state is obtained at the other port of the beam-splitter, as shown in the CV part in Fig. 1, which is a CV qubit with even and odd cat states. Then, the subtracted photon with horizontal polarization is coupled with a single photon with vertical polarization on a polarization beam-splitter (PBS), which forms a polarization qubit in the DV part. If the |H (|V) photon is detected in the DV part, an odd (even) cat state is presented in the CV part. Thus, when a photon is detected in the DV part, a polarization-cat hybrid entangled state is obtained, which is expressed by |ϕ=α|1V|0H|cat++βeiθ|0V|1H|cat=α|V|cat++βeiθ|H|cat,where α and β are the probability amplitudes, θ is the relative phase difference of the two terms of the hybrid state, and |V=|1V|0H and |H=|0V|1H represent that a click of the single photon comes from the single photon with vertical polarization or the subtracted photon from the squeezed vacuum state with horizontal polarization, respectively.

    Schematic of preparing the hybrid entangled state. A small fraction of an even cat state |cat+⟩ is tapped by a BS, and then is coupled on a PBS with a vertical-polarization single photon. Each click event at the output of the DV part heralds the generation of the polarization-cat hybrid entangled state. BS, beam-splitter; PBS, polarization beam-splitter.

    Figure 1.Schematic of preparing the hybrid entangled state. A small fraction of an even cat state |cat+ is tapped by a BS, and then is coupled on a PBS with a vertical-polarization single photon. Each click event at the output of the DV part heralds the generation of the polarization-cat hybrid entangled state. BS, beam-splitter; PBS, polarization beam-splitter.

    Quantum entanglement of an entangled state can be quantified by the logarithmic negativity, which is defined as [45] EN(ρ^)=log2||ρ^TA||1,where ||ρ^TA||1=2N(ρ^)+1 denotes the trace norm of the partial transpose ρ^TA of the bipartite state ρ^ with respect to the subsystem A. Here, N(ρ^) is the absolute value of the sum of negative eigenvalues of ρ^TA. Besides quantum entanglement, quantum coherence is also an important quantum resource, which can be quantified by the l1-norm [21]: Cl1(ρ^)=ij|ρi,j|.

    Here, the value of quantum coherence is determined by off-diagonal density matrix elements ρi,j [21].

    According to Eq. (1), we calculate the density matrix of the hybrid entangled state in the basis of |H|cat, |H|cat+, |V|cat, and |V|cat+, which is given by ρ^=[|β|200eiθβα*00000000eiθαβ*00|α|2].

    By substituting Eq. (4) into Eqs. (2) and (3), we obtain the dependence of quantum entanglement and quantum coherence on the probability of horizontal-polarization photon |β|2, as shown in Fig. 2(a). The entanglement and quantum coherence of the state reach the maximum values when |β|2=0.5, while they disappear when |β|2=0 or 1. Furthermore, we analyze the dependence of entanglement and quantum coherence on transmission efficiency in a lossy channel, where the noise introduced by the channel is only vacuum noise. When the DV part of the hybrid entangled state is transmitted in a lossy channel, we show that the entanglement and quantum coherence of the hybrid entangled state are not sensitive to the loss in the DV part [dashed lines in Fig. 2(b)], which is an important advantage of the polarization-cat hybrid entangled state. When the CV part of the hybrid entangled state is transmitted in a lossy channel, although the entanglement and quantum coherence of the hybrid entangled state decrease with the decrease of transmission efficiency (increase of loss) in the CV part, the disappearance of them happens only when transmission efficiency equals zero [solid curves in Fig. 2(b)]. For the effect of loss in the CV part (solid lines), the entanglement is more sensitive to loss than the quantum coherence. Although this phenomenon is different from the robustness against the loss in the DV part, this is also a kind of robustness to loss [46]. We have to point out that the robustness of the hybrid entangled state can only be observed in the condition of a lossy channel, instead of a noisy channel where excess noise exists.

    Theoretical predictions of entanglement and quantum coherence of the hybrid state. (a) Logarithmic negativity and l1-norm of the hybrid state as a function of the probability of horizontal-polarization photon. (b) Dependence of logarithmic negativity and l1-norm of the hybrid state on the transmission efficiency in the CV and DV lossy channels. Dashed and solid curves represent logarithmic negativity and l1-norm of the hybrid state when the loss exists in DV and CV parts, respectively.

    Figure 2.Theoretical predictions of entanglement and quantum coherence of the hybrid state. (a) Logarithmic negativity and l1-norm of the hybrid state as a function of the probability of horizontal-polarization photon. (b) Dependence of logarithmic negativity and l1-norm of the hybrid state on the transmission efficiency in the CV and DV lossy channels. Dashed and solid curves represent logarithmic negativity and l1-norm of the hybrid state when the loss exists in DV and CV parts, respectively.

    3. EXPERIMENT AND RESULTS

    As shown in Fig. 3, when the relative phase difference between the seed beam and pump beam (down-converted beam) of the OPA is locked to π, i.e., controlling the working status of the OPA at de-amplification, an amplitude-squeezed vacuum state with squeezing of 1.90  dB is prepared with the pump power of 18 mW. A small fraction of the squeezed vacuum state is tapped through a combination of a half-wave plate (HWP) and PBS with a transmittance of 4.5% to subtract a photon, and the reflected port contributes to the CV part of the hybrid entangled state. It is a superposition of odd cat state |cat and even cat state |cat+, which depends on whether a photon is subtracted or not [4749]. A vertical-polarization photon obtained from a weak coherent state |α|0+α|1 and the subtracted photon with horizontal polarization that couples on the PBS contribute to the DV part of the hybrid state. A polarization qubit of (|H|V)/2 is prepared by locking the relative phase difference of the vertical- and horizontal-polarization photons to π (see Appendix A).

    Experimental setup for the preparation of the polarization-cat hybrid entangled state. SHG, second harmonic generator; OPA, optical parameter amplifier; IF, interference filter; FC, filter cavity; AOMs, acousto-optic modulators; AMs, amplitude modulators; HWP, half-wave plate; QWP, quarter-wave plate; PPKTP, periodically poled KTiOPO4; PBS, polarization beam-splitter; APD, avalanche photodiode; HD, homodyne detector; LO, local oscillator.

    Figure 3.Experimental setup for the preparation of the polarization-cat hybrid entangled state. SHG, second harmonic generator; OPA, optical parameter amplifier; IF, interference filter; FC, filter cavity; AOMs, acousto-optic modulators; AMs, amplitude modulators; HWP, half-wave plate; QWP, quarter-wave plate; PPKTP, periodically poled KTiOPO4; PBS, polarization beam-splitter; APD, avalanche photodiode; HD, homodyne detector; LO, local oscillator.

    The most challenging part in the experiment is that the subtracted photon and the photon from the attenuated coherent state are not perfectly indistinguishable since the bandwidths of them are different. In our experiment, the bandwidth of the subtracted photon with horizontal polarization is about 13 MHz, which is determined by the bandwidth of the OPA, while the bandwidth of the vertical-polarization photon coming from the weak coherent state is about 2.5 MHz, which is less than that of the horizontal-polarization photon. To solve this problem, we extend the bandwidth of the photon with vertical polarization in the processing of attenuation of the coherent state by two acousto-optic modulators (AOMs) and three amplitude modulators (AMs), where the two AOMs are used to extend the bandwidths of photons. By applying an inset timing with a high voltage of 134 ns in a cycle of 191 ns to two AOMs with the low diffraction efficiency (10% for each AOM), the bandwidth of the vertical-polarization photon is broadened to about 12 MHz, and the attenuation of 24 dB is obtained for the coherent state (see Appendix A). After the combination of AMs and AOMs, the intensity of the 5 μW coherent beam is attenuated by 90 dB, which is close to the photon number of the subtracted photon.

    In order to measure the prepared hybrid state, the DV part of the state is projected to six polarization bases [|H, |V, |D=(|H+|V)/2, |A=(|H|V)/2, |R=(|H+i|V)/2, and |L=(|Hi|V)/2] and the quantum tomography is implemented on the CV part to reconstruct the corresponding Wigner function [50]. The output photon from the polarization projection system is coupled into an avalanche photodiode (APD) by the fiber coupler to record the clicks of photons (see Appendix A). To obtain a hybrid entangled state with maximum entanglement [α=β=1/2 in Eq. (1)], the generation rates of the two orthogonal-polarization photons are required to be the same, which is realized by adjusting the intensity of the weak coherent beam to make sure the generation rates of two projection bases of |H and |V are the same. In the experiment, the generation rate of the hybrid entangled state is around 60–80 Hz.

    Figure 4 shows the reconstructed Wigner functions and contour plots of the CV part of the hybrid state when the DV part is projected onto the six polarization bases. An optical odd cat state whose Wigner function shows two positive Gaussians of |iαo and |iαo together with a central negative dip WH(0,0)=0.161, and an optical even cat state with a center positive peak WV(0,0)=0.304 are prepared in case of the bases |H and |V, respectively. By calculating the overlap between an ideal target state ρ^Theo and the experimentally reconstructed quantum state ρ^Exp, the fidelity F=Tr[ρ^Theoρ^Exp] of the prepared quantum state is obtained. For the odd and even cat states, we have FH=0.71 and FV=0.97, with the corresponding amplitudes of αo=0.86 and αe=0.50, respectively.

    The reconstructed Wigner functions of the CV part of the polarization-cat hybrid entangled state when the DV part is projected onto the six polarization bases. (a) Wigner functions; (b) contour plots in phase space. All results are corrected for a detection efficiency of 80%.

    Figure 4.The reconstructed Wigner functions of the CV part of the polarization-cat hybrid entangled state when the DV part is projected onto the six polarization bases. (a) Wigner functions; (b) contour plots in phase space. All results are corrected for a detection efficiency of 80%.

    As shown in Fig. 4, when the DV part of the hybrid state is projected onto the diagonal polarization bases |D and |A, we observe two approximate coherent states with symmetric displaced Gaussian peaks along phase-quadrature in the CV part. The fidelities of the reduced states are FD=0.69 and FA=0.71, respectively. On the contrary, for circular polarization bases |L and |R, the Wigner functions with two symmetric positive peaks along amplitude-quadrature are experimentally observed in the CV part. The corresponding fidelities of the reduced states are FL=0.83 and FR=0.73, respectively.

    Based on the measured results in Fig. 4, we reconstruct the Wigner functions of the CV part associated with the reduced density matrices in the polarization bases of |H and |V, as shown in Fig. 5(a). The diagonal Wigner functions are directly obtained according to density matrices ρ^C++ and ρ^C of the CV part when the DV part is projected onto bases of |H and |V, respectively. The off-diagonal Wigner functions are obtained according to density matrices ρ^C+ and ρ^C+, which are calculated by [11] ρ^C+=(ρ^C+)=12[ρ^CDρ^CA+i(ρ^CLρ^CR)],where ρ^CD, ρ^CA, ρ^CL, and ρ^CR represent the density matrices of the CV part when the DV part is projected onto bases of |D, |A, |L, and |R, respectively. It is obvious that the superposition of the odd and even cat states is presented in the off-diagonal Wigner functions, which embodies the coherence of the hybrid state.

    (a) Wigner functions of the hybrid entangled state associated with the reduced density matrices ⟨m|ρ^|n⟩, where m,n∈{H,V}. The two conjugated off-diagonal terms placed on the right and left corners are plotted according to the real and imaginary parts of the corresponding Wigner functions, respectively. (b) The density matrix of experimentally obtained hybrid entangled state in the bases of polarization and cat state.

    Figure 5.(a) Wigner functions of the hybrid entangled state associated with the reduced density matrices m|ρ^|n, where m,n{H,V}. The two conjugated off-diagonal terms placed on the right and left corners are plotted according to the real and imaginary parts of the corresponding Wigner functions, respectively. (b) The density matrix of experimentally obtained hybrid entangled state in the bases of polarization and cat state.

    By projecting DV and CV parts of the hybrid state onto the bases of polarization (|H, |V) and cat states (|cat+, |cat), respectively, we obtain the density matrix of the hybrid entangled state in the bases of polarization and cat state, as shown in Fig. 5(b). Based on the reconstructed density matrix of the hybrid state, we obtain the fidelity F=0.70 of the experimentally prepared state. There are three main factors that influence the fidelity of the hybrid entanglement, including loss existing in the CV part, phase fluctuation existing in all phase locking systems, and the mismatch of the bandwidths of the vertical-polarization photon and horizontal-polarization photon, which are about 12 and 13 MHz, respectively.

    According to Eq. (2), we obtain the logarithmic negativity of EN=0.57 for the prepared state by transposing the density matrix in Fig. 5(b), which confirms the entanglement of the hybrid state. Besides entanglement, we also quantify the quantum coherence of the prepared state according to Eq. (3). Based on the experimentally reconstructed density matrix in Fig. 5(b), we obtain Cl1=0.82 for the prepared hybrid entangled state, which shows the existence of quantum coherence.

    In our experiment, a direct method is used to characterize entanglement of the hybrid entangled state. Compared to the indirect method like quantum teleportation, although the method used in our experiment needs to experimentally reconstruct a two-mode density matrix, it can be obtained based on the experimentally obtained density matrix of the CV part by projecting the DV part onto six polarization bases. The advantage of this direct method is that it does not require complex experimental operations, for example, CV or DV Bell state measurement. Thus, the dependence on extra parameters, such as quantum channel efficiency, noise, and the coincidence counting rate, is avoided by directly characterizing entanglement of logarithmic negativity.

    4. CONCLUSION

    In summary, we experimentally characterize the entanglement and quantum coherence of the polarization-cat hybrid entangled state at the rubidium D1 line of 795 nm. Hybrid CV-DV entanglement is an important quantum resource for hybrid quantum information processing. The prepared hybrid entangled state has potential applications in hybrid quantum teleportation [11,12], constructing a heterogeneous quantum network [13], and converting a DV qubit into a CV qubit [15]. We show that the entanglement and quantum coherence of the hybrid entangled state are robust against the loss in both DV and CV parts. We reconstruct the Wigner functions of the CV part associated with the reduced density matrices in the polarization bases of |H and |V, which shows the superposition of the odd and even cat states. Then, we quantify the entanglement and quantum coherence of the prepared hybrid state, which are EN=0.57 and Cl1=0.82, respectively. Our results reveal the quantum coherence of the polarization-cat hybrid entangled state and make a crucial step toward the application of it.

    APPENDIX A: DETAILS OF EXPERIMENT

    A solid-state titanium-sapphire continuous wave laser operating at 795 nm corresponding to the rubidium D1 line serves as the light source, which is divided into two parts. The first part is injected into a second harmonic generator (SHG) cavity for generating the pump beam with wavelength of 397.5 nm. The second part passes through the mode cleaner to shape the spatial mode, and then the output beam of the mode cleaner is divided into the local oscillator of a homodyne detector (HD), the coherent beam for generating vertical-polarization photons, and seed and locking beams of the frequency degenerate optical parameter amplifier (OPA).

    Both SHG and OPA with a cavity length of 480 mm contain two concave mirrors (R=50  mm), two plane mirrors, and a periodically poled KTiOPO4 (PPKTP, 1  mm×2  mm×10  mm) crystal. The difference is that a plane mirror with the transmittance serves as the input mirror and output mirror of the SHG and OPA, respectively, and the transmittances are 8% and 12.5% for 795 nm, respectively.

    The output of the OPA passes through a beam-splitter with transmittance of 4.5% which consists of a half-wave plate (HWP) and a polarization beam-splitter (PBS) for subtracting a photon. To pick out the degenerate longitudinal mode with the same frequency as that detected by the HD, the subtracted photon passes through a filtering system, which consists of an interference filter (0.4 nm) and two filter cavities with fineness of 1200, whose cavity lengths are 0.75 and 2.05 mm, respectively. Because the subtracted photon comes from the transmitted beam of the combination of an HWP and a PBS, the polarization of the subtracted photon is horizontal, which is different from the squeezed vacuum state. Consequently, when a photon with horizontal polarization is detected by an avalanche photodiode (APD) with the detection efficiency of 50%, an odd cat state is prepared.

    The reflected beam is measured by the HD with a bandwidth of 30 MHz. The detection efficiency of 80% used for correction includes four parts: the visibility between signal and local beams (98.5%), the quantum efficiency of photodiodes (92%, S3883), the 19 dB clearance with the 16 mW local oscillator at 13 MHz of the homodyne detector (corresponding to equivalent efficiency 98.7%), and transmission efficiency (91%). By measuring the photocurrent of the HD, the Wigner function is reconstructed by the maximum likelihood algorithm.

    The whole system is operated on “sample and hold” modes to obtain a squeezed vacuum state and a vertical-polarization photon. The timing in our experiment is shown in Fig. 6. To prepare a squeezed vacuum state, two acousto-optic modulators (AOM1 and AOM2) with periodic timing are placed on the paths of the seed and locking beams of the OPA. In the hold mode of 16 ms, the seed and locking beams are seeded into the OPA for locking the cavity length and the relative phase difference between the seed and pump beams. In the sample mode of 16 ms, the seed and locking beams are blocked for generating a squeezed vacuum state with 1.90  dB squeezing.

    The periodic timing of the whole system in our experiment.

    Figure 6.The periodic timing of the whole system in our experiment.

    In order to obtain a vertical-polarization photon, much effort is made to make sure that the intensity of the weak coherent state and matching efficiency of the two photons in the DV part satisfy the experimental requirement. In the DV part, the relative phase difference between a small fraction (4.5%) of the seed beam of the OPA and the weak coherent beam is locked and calibrated in the sample mode of 16 ms. In the period of sample mode, an inset timing with a high voltage of 134 ns in a cycle of 191 ns, is added to each AOM (AOM3 and AOM4). After two AOMs with a low diffraction efficiency (10% for each AOM), the bandwidth of the vertical-polarization photon is broadened to about 12 MHz, and the attenuation of 24 dB is obtained for the coherent state. For each amplitude modulator (AM), a high voltage (about 220 V) is added in the sample mode of 16 ms to attenuate the intensity of the coherent beam. After the group of AMs, the intensity is reduced by about 63 dB, where the extinction rate of each AM is about 1% (20 dB). After the combination of AMs and AOMs, the intensity of the coherent beam is attenuated by 90 dB, which is close to the photon number of the subtracted photon. Finally, by carefully adjusting the spatial size of the two beams, the spatial matching efficiency of the two photons reaches 98%. Thus, the two orthogonal-polarization photons with a similarity of 90% are obtained experimentally.

    To lock the relative phase difference between the two orthogonal-polarization photons, they are coupled on a PBS at first. Then, the output beam of the PBS passes through a beam-splitter with the transmittance of 10%. In the sample mode, a combination of an HWP and a PBS placed on the transmitted path of the beam-splitter is used for acquiring the interference signal between a small fraction of the seed beam of OPA and the coherent beam, and the relative phase difference between a small fraction of the seed beam and the coherent beam is controlled to zero. Because the relative phase difference between the down-converted beam and seed beam is locked to π, the relative phase difference between the horizontal-polarization photon subtracted from the down-converted beam and the vertical-polarization photon obtained by attenuating the coherent beam is controlled to π [θ=π in Eq. (1) of the main text].

    The rest (90%) of the two photons are projected onto the six polarization bases by passing through a projection measurement system consisting of a quarter-wave plate (QWP), an HWP, and a PBS. When the angle of the HWP is set to 0°, 45°, 22.5°, and 22.5°, the DV mode is projected onto the basis of |H, |V, |D=(|H+|V)/2, and |A=(|H|V)/2, respectively. The remaining two bases of |R=(|H+i|V)/2 and |L=(|Hi|V)/2 are projected by setting the angle of QWP to 45° and 45°, respectively. The transmission of the PBS after the combination of an HWP and a QWP is coupled into an APD by the fiber coupler to record the clicks of photons. To protect the APD, a timing (blue timing) is used as a transistor logic signal of the APD, and the AOM5 with the timing shown in Fig. 6 (purple timing) is placed before the APD.

    APPENDIX B: QUANTUM COHERENCE OF THE HYBRID ENTANGLED STATE

    When the input state is the ideal odd and even cat states, the density matrix of a hybrid entangled state is given by ρ^=(α|VD|cat+VC+βeiθ|HD|catVC)(α*V|Dcat+|VC+β*eiθH|Dcat|VC)=|α|2|VV|D|cat+cat+|VC+|β|2|HH|D|catcat|VC+αβ*eiθ|VH|D|cat+cat|VC+eiθβα*|HV|D|catcat+|VC=|α|2ρ^DVVρ^C+++|β|2ρ^DHHρ^C+αβ*eiθρ^DVHρ^C++eiθβα*ρ^DHVρ^C+.

    The density matrices of the DV part are given by ρ^DVV=[0001],ρ^DHH=[1000],ρ^DVH=[0010],ρ^DHV=[0100],respectively. In the CV part, the density matrix elements of ρ^C++ and ρ^C in the Fock basis are expressed as 1cosh|iα2|2(iα2)m(iα2)nm!n!δ0m[2]δ0n[2] and 1sinh|iα1|2(iα1)m(iα1)nm!n!δ1m[2]δ1n[2], respectively. δ0n[2]=0, δ1n[2]=1 when n is odd and δ0n[2]=1, δ1n[2]=0 when n is even. The density matrix elements of ρ^C+ and ρ^C+ in the Fock basis are expressed as (iα2)m(iα1)ncosh|iα2|2sinh|iα1|2m!n!δ0m[2]δ1n[2] and (iα1)m(iα2)ncosh|iα1|2sinh|iα2|2m!n!δ1m[2]δ0n[2], respectively.

    When the basis of the CV part is transformed to the bases of cat states {|cat,|cat+}, the density matrix of the CV part is given by ρ^C++=[0001],ρ^C=[1000],ρ^C+=[0010],ρ^C+=[0100],respectively. Thus, the density matrix of the hybrid entangled state is ρ^=[|β|200eiθβα*00000000eiθαβ*00|α|2]in the bases of |H|cat, |H|cat+, |V|cat, and |V|cat+.

    When the coefficients and relative phase difference are equal to α=β=1/2 and θ=π, respectively, the density matrix of the hybrid entangled state is ρ^=[0.5000.5000000000.5000.5].

    According to the density matrix in the bases of polarization and cat state, l1-norm of the hybrid entangled state is expressed as Cl1=|αβ*|+|βα*|.

    When the hybrid entangled state is transmitted through a lossy channel, the entanglement and quantum coherence of the state decrease with the increase of the optical loss. Here, we consider the loss that existed in DV and CV parts. In the CV part, the optical loss is introduced by a model of a beam-splitter (BS) with the transmittance of η, where the input states of the BS are the hybrid state in the CV part and a vacuum state with the variance of 12. The reflection of the BS is traced, and the transmission remains and is detected by the HD. In the DV part, the loss is also introduced by a model of BS, which is similar to the loss of the CV part.

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    Fengyi Xu, Chenyu Qiao, Shujing Li, Meihong Wang, Xiaolong Su, "Entanglement and quantum coherence of hybrid entangled states," Photonics Res. 13, 1983 (2025)

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    Paper Information

    Category: Quantum Optics

    Received: Feb. 12, 2025

    Accepted: May. 13, 2025

    Published Online: Jul. 1, 2025

    The Author Email: Meihong Wang (wangmh@sxu.edu.cn), Xiaolong Su (suxl@sxu.edu.cn)

    DOI:10.1364/PRJ.559233

    CSTR:32188.14.PRJ.559233

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