Implementing positive-operator-valued-measurement elements in photonic circuits for performing minimum quantum state tomography of path qudits

Manipulation of qudits in optical tables is a difficult and nonscalable task. The use of integrated optical circuits opens new possibilities for the generation, manipulation, and characterization of high dimensional states besides the ease of transmission of these states through an optical fiber. In this work we propose photonic circuits to perform minimum quantum state tomography of path qudits and show how to determine all the constituents parameters of these circuits (beam splitters and phase shifters). Our strategies were based on the symmetries of the involved POVMs (positive operator-valued measures) suggested for minimum tomography and allowed us to obtain interferometers smaller than those obtained by other already known methods. The calculations of the transmittances and reflectivities of the beam splitters were made using the definition of probability operators in extended Hilbert spaces and the application of Naimark's theorem. The employment of equidistant states for the definition of the POVM elements allowed us to develop a recipe applicable to the tomography of qudits of any dimension, generalizing our scheme.

Figure 1

Circuit proposal for the realization of the minimal state tomography on a one-qubit system. The black lines are the photon paths realized by the waveguides; the BS's are represented by the approximation of these lines and the phase shifters are the blue rectangles. This proposal was made on the assumption that the modes of the photonic qubit labeled as $|0\rangle$ and $|1\rangle$ are coupled to the second and third waveguides, respectively.

Figure 2

Schematic drawing showing the quantum operations of the proposed photonic circuit for quantum tomography of qutrits. The pink dotted lines separate the three sectors of the circuit that can be analyzed separately. The waveguides are represented by the black lines and labeled by the vectors ${|i\rangle }_j$, where $i,j=0,1,2$. The state vectors ${\alpha }_i{|i\rangle }_1$ indicate the waveguides where a possible ${({\alpha }_0,{\alpha }_1,{\alpha }_2)}^T$ input qutrit path state is coupled to the circuit. The $\widehat{\mathrm{\Lambda }}$ operation is responsible for diffusing the initial state to the first and third path in each sector. The red and blue arrows portray the permutations between the third and fourth paths and between the sixth and seventh ones, respectively. This permutation, performed by $1:0$ BS, exchanges the coefficients of some base vectors. Finally, these paths are connected to the last circuit piece, described by a $3\times 3\widehat{U}$ matrix, to resume the tomographic implementation. Analogous descriptions can be done for the other two sectors. A detector count in one of the nine circuit outputs is proportional to the implementation probability of a POVM element ${\widehat{E}}_{ij}(i,j=0,1,2)$.

Figure 3

Circuit representation of the $\widehat{\mathrm{\Lambda }}$ operation for the implementation of the qutrit tomography. We have a $1/2:1/2$ BS that splits the input mode in path 2 in two output modes in paths 2 and 3 (labeled as ${|i\rangle }_1$ and ${|i\rangle }_2$, respectively). A $1:0$ BS between paths 1 and 2 switches these paths.

Figure 4

Circuit representation of the $\widehat{U}$ matrix for the implementation of the qutrit tomography. The blue rectangles are phase shifters.