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arXiv:2105.12462 · Optica

Universal compressive tomography in the time-frequency domain

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

Published Sep 7, 2021• 20 citations
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Abstract

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.

DOI
10.1364/optica.427645

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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.
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 both a crucial component in the field of quantum information and computation and a formidable task that requires an incogitable number of measurement configurations as the system dimension grows. We propose and experimentally carry out an intuitive adaptive compressive tomography scheme, inspired by the traditional compressed-sensing protocol in signal recovery, that tremendously reduces the number of configurations needed to uniquely reconstruct any given quantum state without any additional a priori assumption whatsoever (such as rank information, purity, etc.) about the state, apart from its dimension.
We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog²d) measurement settings, compared to standard methods that require d² settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed.
This paper establishes the methods for quantum state tomography based on compressed sensing.These methods are specialized for quantum states that are fairly pure,and they offer a significant performance improvement on large quantum systems.In particular,they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog2d) measurement settings,compared to standard methods that require d2 settings.The methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements,use fast convex optimization,are stable against noise,and can be applied to states that are only approximately low rank.The acquired data can be used to certify that the state is indeed close to pure,so no a priori assumptions are needed.