Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures

Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension. In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime. Fidelities of reconstruction within the range of 0.81–0.96 are obtained for both pure and mixed states. These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.

Popular Summary

Determining an unknown quantum state, i.e., quantum state tomography, plays an essential role in fundamental studies of quantum mechanics, quantum computations, and quantum key distribution. Various methods have been proposed to optimize quantum state tomography by implementing a minimum number of projective measurements or maximizing the accuracy of the state estimation. Among these different approaches, mutually unbiased bases and symmetric informationally complete positive operator-valued measures are well-recognized techniques. However, determining the single “best” optimal technique is still an open question. Here, we experimentally show that symmetric informationally complete positive operator-valued measures provide the best state estimation for dimensions where a complete set of mutually unbiased bases is unknown, i.e., Hilbert spaces with dimensions that are not a power of a prime number.

We experimentally test the use of symmetric informationally complete positive operator-valued measures in up to ten dimensions, although our technique can be applied to any finite dimension. We generate photon pairs via spontaneous parametric down-conversion and put our technique to the test by reconstructing orbital angular-momentum eigenstates, superpositions, and mixed states of photon pairs. In addition, we explore a particular aspect of quantum key distribution known as the Singapore protocol with twisted photons.

We expect that our findings will inform studies of how photonic states can be used experimentally in quantum cryptography.

Figure 1

Bloch sphere representation of two different informationally complete measurements of a qubit. (a) MUB measurement has six measurement outcomes that correspond to vertices of an octahedron. (b) SIC POVM measurement has four measurement outcomes that correspond to vertices of a tetrahedron.

Figure 2

Experimental setup for generating and detecting OAM photonic qudit states. The signal photon generated via spontaneous parametric down-conversion (not shown) is spatially cleaned and sent into the main setup via a single-mode optical fiber (SMOF). Photonic qudit states (SIC POVMs) are generated by a holographic approach in which the desired kinoform is displayed on the spatial light modulator $A$ (${\mathrm{SLM}}_A$). A half-wave plate (HWP) optimizes the first order of diffraction on ${\mathrm{SLM}}_A$, since SLMs are polarization dependent. The mode $|\mathrm{\Psi }⟩$ produced by ${\mathrm{SLM}}_A$ is then projected onto a SIC POVM element ${\widehat{E}}_i$ on ${\mathrm{SLM}}_B$. The resulting far field is coupled into a SMOF, which selects the ${\mathrm{TEM}}_{00}$-like component. We implement two $4f$ systems with unit magnification to image ${\mathrm{SLM}}_A$ onto ${\mathrm{SLM}}_B$ and ${\mathrm{SLM}}_B$ onto the microscope objective. Irises are used to select the first order of diffraction at the far-field plane of SLMs, where higher diffraction orders are well separated.

Figure 3

Experimentally measured characterization of the generation and detection of SIC POVMs elements for different photonic qudit subspaces. Each block corresponds to $P_{ij}={|⟨{\mathrm{\Phi }}_i|{\mathrm{\Phi }}_j⟩|}^2$, where $|{\mathrm{\Phi }}_i⟩$ correspond to the $i$th SIC POVM element, i.e., ${\widehat{E}}_i=(1/d)|{\mathrm{\Phi }}_i⟩⟨{\mathrm{\Phi }}_i|$. Lower figures are the density plots of the upper histograms. Projecting SIC POVMs for each dimension $d$ acquires $d^2$ set of measurements.

Figure 4

Examples of the reconstructed density matrices of pure and mixed states. Left and right columns of each measurement show real and imaginary parts of the density matrix. Different rows correspond to different photonic qudit dimensions of 6 and 10. The state $|n⟩$ ranges from $\{|-3⟩,\dots ,|+3⟩\}$ and $|m⟩$ ranges from $\{|-5⟩,\dots ,|+5⟩\}-{\mathrm{TEM}}_{00}$, i.e., $|0⟩$ is excluded. Pure eigenstates, superpositions, and mixed states are defined as $|+3⟩$, $1/6\sum _{n=-3}^{+3}|n⟩$, and $\widehat{\rho }=1/6\sum _{n=-3}^{+3}|n⟩⟨n|$, for dimension 6, and $|+5⟩$, $1/10\sum _{m=-5}^{+5}|m⟩$ and $\widehat{\rho }=1/10\sum _{m=-5}^{+5}|m⟩⟨m|$, for dimension 10, where $\{n,m\}\ne 0$.

Figure 5

State representation of SIC POVMs for OAM qubit on Alice’s and Bob’s Bloch (Poincaré) spheres. The two sets of vectors $\{A,B,C,D\}$ and $\{A^{\prime },B^{\prime },C^{\prime },D^{\prime }\}$ represent Alice’s and Bob’s SIC POVM elements, respectively. Any pair of conjugate vectors, e.g., $A$ and $A^{\prime }$, are orthogonal, and hence a qubit generated by Alice in one of these states has zero probability of being detected as the conjugate state, and equal probabilities of the remaining three. We intently draw the vectors to be outside of the Bloch spheres. Indeed, the implemented states are on the surface of the Bloch spheres, since they are pure states.

Figure 6

Intensity and phase of SIC POVM elements for photonic qudit state of dimension six. The states are calculated by acting 36 Weyl-Heisenberg operators represented in Eq. (3) on the fiducial state of $|f_6⟩$. The implemented OAM Hilbert space spans on $\{|+3⟩,|+2⟩,|+1⟩,|-1⟩,|-2⟩,|-3⟩\}$ OAM eigenstates.