Experimental quantum tomography assisted by multiply symmetric states in higher dimensions

Abstract

High-dimensional quantum information processing has become a mature field of research with several different approaches being adopted for the encoding of D-dimensional quantum systems. Such progress has fueled the search of reliable quantum tomographic methods aiming for the characterization of these systems, being most of these methods specifically designed for a given scenario. Here, we report on a new tomographic method based on multiply symmetric states and on experimental investigations to study its performance in higher dimensions. Unlike other methods, it is guaranteed to exist in any dimension and provides a significant reduction in the number of measurement outcomes when compared to standard quantum tomography. Furthermore, in the case of odd dimensions, the method requires the least possible number of measurement outcomes. In our experiment we adopt the technique where high-dimensional quantum states are encoded using the linear transverse momentum of single photons and are controlled by spatial light modulators. Our results show that fidelities of 0.984 ± 0.009 with ensemble sizes of only 1.5 × 10 5 photons in dimension D = 15 can be obtained in typical laboratory conditions, thus showing its practicability in higher dimensions.

Set of 4d–3 observables to determine any pure qudit state

“…As the number of independent parameters that defines an arbitrary density matrix ρ is d 2 − 1, typical quantum tomography schemes require a number of measurement outcomes that increases as d 2 [3][4][5], leading to time consuming measurements and post processing, that can become prohibitive for high dimension systems [6,7]. Furthermore, several applications such as quantum key distribution [8][9][10][11][12], quantum communication complexity problems [13] or fundamental tests of quantum mechanics [14,15], can be improved by using d-level quantum systems of dimension grater than two (qudits). Therefore, it is of great interest to develop alternative methods that employ fewer measurement settings.…”

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

“…The third operation that our proposed circuit performs is analogous to Eq. (25). Its transforms the state |ψ…”

“…Manipulating states of qudits in interferometers at optical tables is a non-scalable and challenging task. Interferometers have been set for characterizing quantum states [21][22][23][24][25], controlling, detecting or measuring entanglement of qudit systems [10,11,26,27] or simulating noisy channels [28,29]. The use of photonic circuits written in glasses by femtosecond direct laser writing technique opens new possibilities for the generation, manipulation, and characterization of high dimensional states, besides the direct transmission of these states through an optical fiber [30][31][32][33][34][35][36][37].…”

“…Make the same comparisons explained above, we obtain the last parameters of the circuit: We are interested to find the parameters necessary for carrying the operatorÛ shown in Eq. (25). The circuit representation of theÛ matrix is shown in Fig.…”

Estimation of pure quantum states in high dimension at the limit of quantum accuracy through complex optimization and statistical inference

“…These are central problems in the theory of quantum measurements 11 and play a key role in the control of quantum systems, the benchmarking of quantum technologies [12][13][14] , and quantum metrology 15 . Today there is a large collection of methods 11,[16][17][18][19][20][21][22][23][24][25] for estimating unknown quantum states, which are collectively known as quantum tomographic methods. These are based on the post-processing of data acquired through the measurement of a set of positive operator-valued measures in an ensemble of N identically, independently prepared copies of the unknown state to be estimated.…”