Two-Measurement Tomography of High-Dimensional Orbital Angular Momentum Entanglement

High-dimensional (HD) entanglement enables an encoding of more bits than in the two-dimensional case and promises to increase communication capacity over quantum channels and to improve robustness to noise. In practice, however, one of the central challenges is to devise efficient methods to quantify the HD entanglement explicitly. Full quantum state tomography is a standard technology to obtain all the information about the quantum state, but it becomes impractical because the required measurements increase exponentially with the dimension in HD systems. Hence, it is highly anticipated that a new method will be found for characterizing the HD entanglement with as few measurements as possible and without introducing unwarranted assumptions. Here, we present and demonstrate a scan-free tomography method independent of dimension, which only requires two measurements for the characterization of two-photon HD orbital angular momentum (OAM) entanglement. Taking Laguerre-Gaussian modes of photons as an example, the density matrices of OAM entangled states are experimentally reconstructed with very high fidelity. Our method is also generalized to the mixed HD OAM entanglement. Our results provide realistic approaches for quantifying more complex OAM entanglement in many scientific and engineering fields such as multiphoton HD quantum systems and quantum process tomography.

Figure 1

Concept of two-measurement tomography and simulated tomographic reconstruction of a three-dimensional OAM entangled state. (a) Principle schematic of two-measurement tomography for measuring the OAM entangled states created by the SPDC process. Signal photons are measured in the superposition OAM basis with the SLM loaded in a suitable holography. The signal photons enter the single photon detector, and its electrical signal is used to trigger the ICCD camera. The spatial patterns of the reduced OAM superposition states of the idler photons are captured by the triggered ICCD camera. (b) [(c)] Simulated intensity pattern of the reduced OAM state of the idler photons under the projection basis $|{\dot{\mathrm{\Psi }}}_s⟩=({|0⟩}_s+{|1⟩}_s)/\sqrt{2}$ [$|{\ddot{\mathrm{\Psi }}}_s⟩=({|0⟩}_s+{|-1⟩}_s)/\sqrt{2}$] of the signal photons. (d) [(e)] Simulated real [imaginary] part of the reconstructed density matrix for the preset three-dimensional OAM entangled target state shown in Eq. (2).

Figure 2

Experimental results of a two-qutrit OAM entangled state. A two-qutrit OAM entangled state $|\psi ⟩=0.50{|-1⟩}_s{|1⟩}_i+0.71{|0⟩}_s{|0⟩}_i+0.50{|1⟩}_s{|-1⟩}_i$ is the target state we want to experimentally generate by the SPDC process in a type-II ppKTP. (a) [(b)] Image measured by the ICCD in the first [second] measurement, under the projection basis $|{\dot{\mathrm{\Psi }}}_s⟩=({|0⟩}_s+{|1⟩}_s)/\sqrt{2}$ [$|{\ddot{\mathrm{\Psi }}}_s⟩=({|0⟩}_s+{|-1⟩}_s)/\sqrt{2}$] for the signal photons. (c) [(d)] Measured data and fitting curves of two concentric circles (with radii of $R_1$ and $R_2$) from the image in (a) [(b)]. (e) [(f)] Real [imaginary] part of the reconstructed density index by choosing 50 equal-interval concentric circles (the radius has a range from 0.13 to 1.235 mm) from each of the two images.

Figure 3

Experimental results of a symmetric OAM entangled state. A symmetric OAM entangled state $|\psi ⟩=({|-1⟩}_s{|1⟩}_i+{|1⟩}_s{|-1⟩}_i)/\sqrt{2}$ is experimentally generated as the target state by using the SPDC in a type-II ppKTP. (a) [(b)] Image measured by the ICCD in the first [second] measurement, under the projection basis $|\dot{\mathrm{\Psi }}⟩=(2{|-1⟩}_s-{|1⟩}_s)/\sqrt{5}$ ($|\ddot{\mathrm{\Psi }}⟩=({|-1⟩}_s+2{|1⟩}_s)/\sqrt{5}$) for the signal photons. (c) [(d)] Real [imaginary] part of the reconstructed density index by choosing 50 equal-interval concentric circles (the radius has a range from 0.26 to 1.30 mm) from each of two images in (a) and (b).

Figure 4

Simulated reconstruction of three-dimensional OAM mixed entanglement state. The preset mixed target state is ${\rho }_w=I_9(1-x)/9+x|\psi ⟩⟨\psi |$. Note that $I_9$ is a $9\times 9$ identity matrix and $|\psi ⟩=[3{|0⟩}_s{|0⟩}_i+2\exp (j\pi /3){|1⟩}_s{|-1⟩}_i+\exp (j3\pi /4){|-1⟩}_s{|1⟩}_i]/\sqrt{14}$. (a) [(b)] Simulated image that will be experimentally captured by the ICCD for the first [second] measurement, under the projection basis of $|{\dot{\mathrm{\Psi }}}_s⟩=(|0⟩+|1⟩)/\sqrt{2}$ [$|{\ddot{\mathrm{\Psi }}}_s⟩=(|0⟩+|-1⟩)/\sqrt{2}$] for the signal photons. (c) [(d)] Real [imaginary] part of the reconstructed density matrix of ${\rho }_w$ by choosing 50 equal-interval concentric circles from each of the two images. The radius has a range from 0.13 to 1.04 mm.