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Quantum Information and Measurement (QIM)

Time-Frequency Randomized Compressive Tomography Using a Quantum Pulse Gate

Benjamin Brecht, Jano Gil-López, Syamsundar De, Yong Siah Teo, Hyunseok Jeong, Luis Lorenzo Sánchez-Soto, Christine Silberhorn

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We demonstrate randomized compressive tomography of few-photon states that comprise of different temporal modes. We require no a-priori knowledge and reconstruct states using only few measurement settings. Our approach is naturally compatible with integrated architectures.
DOI
10.1364/qim.2021.tu2a.6

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High-dimensional optical modes usually require a large number of measurements to be characterized— typically of the order O ( d 2 ) where d is the dimension of the underlying Hilbert space. Recently developed compressive tomography techniques can reconstruct low-rank signals with minimum measurements and no a priori assumptions [1] , [2] , [3] , [4] . We discuss results of randomized compressive tomography (RCT) implemented with a quantum pulse gate (QPG), a device that performs flexible optical measurements in the time-frequency domain [5] , [6] . In our experiments, we perform different random von Neumann basis measurements with a QPG that exploits dispersion-engineered sum-frequency generation in lithium niobate waveguides to implement projections onto arbitrary user-specified temporal mode superpositions. Based solely on the measurement data and nothing else, one can deterministically compute a positive number s cvx that monotonically estimates the size of the convex set of signal states consistent with the measurement data. For a sufficient number of informationally complete (IC) QPG projections, the convex set converges to a single point ( s cvx = 0), where the signal is uniquely reconstructed.
Quantum state tomography is a basic tool in quantum information, but it becomes a challenging task that requires an immense number of measurement configurations as the system dimension grows. We implement an adaptive compressive tomography scheme capable of reconstructing any arbitrary low-rank spectral-temporal optical signal with extremely few measurement settings and without any ad hoc assumption about the initially unknown signal. This is carried out by implementing projections onto arbitrary user-specified optical modes. We present conclusive experimental results for both temporal modes and frequency bins, which showcase the versatility of our method and thereby introduce a universal optical reconstruction framework to these platforms.
Accessing a high dimensional Hilbert space spanned by temporal modes (TMs) of single-photons is an intriguing idea for quantum information protocols, though it is experimentally challenging. The quantum pulse gate (QPG), a group-velocity matched sum-frequency process, can perform arbitrary TM resolved measurement. We present a thorough characterisation of measurement operators of a QPG and we show that this device is capable of performing full state tomography with 99% fidelity. Moreover, we use this device for TM tomography of pure and mixed heralded single-photon states in 7-dimensional Hilbert space.
In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.
Encoding quantum information in the photon temporal mode (TM) offers a robust platform for high-dimensional quantum protocols. The main practical challenge, however, is to design a device that operates on single photons in specific TMs and all coherent superpositions. The quantum pulse gate (QPG) is a mode-selective sum-frequency generation designed for this task. Here, we perform a full modal characterization of a QPG using weak coherent states in well-defined TMs. We reconstruct a full set of measurement operators, which show an average fidelity of 0.85 to a theoretically ideal device when operating on a seven-dimensional space. Then we use these characterized measurement operators of the QPG to calibrate the device. Using the calibrated device and a tomographically complete set of measurements, we show that the QPG can perform high-dimensional TM state tomography with 0.99 fidelity.