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arXiv:quant-ph/0403078 · Physical Review A

Universal quantum data compression via nondestructive tomography

Charles H. Bennett, Aram W. Harrow, Seth Lloyd

Published Mar 24, 2006• 39 citations
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Abstract

Quantum-state tomography---the practice of estimating a quantum state by performing measurements on it---is useful in a variety of contexts. We introduce ``gentle tomography'' as a version of tomography that preserves the measured quantum data. As an application of gentle tomography, we describe a polynomial-time method for universal source coding.

DOI
10.1103/physreva.73.032336

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Quantum-state tomography---the practice of estimating a quantum state by performing measurements on it---is useful in a variety of contexts. We introduce ``gentle tomography'' as a version of tomography that preserves the measured quantum data. As an application of gentle tomography, we describe a polynomial-time method for universal source coding.
Quantum tomography is the most informative tool for estimating the quality of preparation and transformation of quantum states. Its development is crucially necessary for debugging of developed quantum processors. Many existing methods of quantum tomography differ in types of performed measurements and in procedures of their processing. The practical implementation of quantum tomography requires the comparison of different methods, which is complicated because of the absence of a general methodology of estimation. A universal methodology based on numerical experiments has been proposed in this work to estimate the quality of quantum state tomography methods. The developed methodology has been applied to three quantum tomography methods (root approach, compressed sensing, and adaptive tomography) efficiently operating with almost pure states, which is relevant for the current technological foundation of the experiments.
We describe an algorithm for quantum state tomography that converges in polynomial time to an estimate, together with a rigorous error bound on the fidelity between the estimate and the true state. The result suggests that state tomography on large quantum systems may be much more feasible than the exponential size of state space suggests. In many situations, the correctness of the state estimate can be certified from the data alone, with no a priori assumptions on the form of the measured state.
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.
The need to perform quantum state tomography on ever-larger systems has spurred a search for methods that yield good estimates from incomplete data. We study the performance of compressed sensing (CS) and least squares (LS) estimators in a fast protocol based on continuous measurement on an ensemble of cesium atomic spins. They both efficiently reconstruct nearly pure states in the 16-dimensional ground manifold, reaching average fidelities F‾CS=0.92{\overline{\mathcal{F}}}_{\mathrm{CS}}=0.92FCS​=0.92 and F‾LS=0.88{\overline{\mathcal{F}}}_{\mathrm{LS}}=0.88FLS​=0.88 using similar amounts of incomplete data. Surprisingly, the main advantage of CS in our protocol is an increased robustness to experimental imperfections.