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arXiv:quant-ph/0606115 · Physical Review Letters

Efficient Quantum-State Estimation by Continuous Weak Measurement and Dynamical Control

Greg Smith, Andrew Silberfarb, Ivan Deutsch, Poul Jessen

Published Oct 31, 2006• 93 citations
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Abstract

We demonstrate a fast, robust, and nondestructive protocol for quantum-state estimation based on continuous weak measurement in the presence of a controlled dynamical evolution. Our experiment uses optically probed atomic spins as a test bed and successfully reconstructs a range of trial states with fidelities of approximately 90%. The procedure holds promise as a practical diagnostic tool for the study of complex quantum dynamics, the testing of quantum hardware, and as a starting point for new types of quantum feedback control.

DOI
10.1103/physrevlett.97.180403

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We demonstrate a fast, robust, and nondestructive protocol for quantum-state estimation based on continuous weak measurement in the presence of a controlled dynamical evolution. Our experiment uses optically probed atomic spins as a test bed and successfully reconstructs a range of trial states with fidelities of approximately 90%. The procedure holds promise as a practical diagnostic tool for the study of complex quantum dynamics, the testing of quantum hardware, and as a starting point for new types of quantum feedback control.
We present a new procedure for quantum state reconstruction based on weak continuous measurement of an ensemble average. By applying controlled evolution to the initial state, new information is continually mapped onto the measured observable. A Bayesian filter is then used to update the state estimate in accordance with the measurement record. This generalizes the standard paradigm for quantum tomography based on strong, destructive measurements on separate ensembles. This approach to state estimation induces minimal perturbation of the measured system, giving information about observables whose evolution cannot be described classically in real time and opening the door to new types of quantum feedback control.
Laboratory techniques to manipulate and observe ultracold atoms make these an attractive platform for testing new ideas in quantum control and measurement. Over the last decade we have revisited the tensor interaction between light fields and multilevel atoms, and have developed a theoretical framework suited for applications in quantum control and measurement (see [1] for a review). One important finding is that the combined action of a light shift and magnetic field on an atomic ground state can be used to implement a non-linear Hamiltonian for a hyperfine spin, and that its action can lead to nonlinear spin dynamics such as wavepacket collapse and revival. Using concepts from classical control theory it is straightforward to show that this Hamiltonian is sufficiently general for full control of an arbitrarily large spin. On this foundation we have developed a new protocol for quantum state reconstruction, based on continuous weak measurement of a spin observable during carefully designed coherent evolution [2]. We have further used our tools for control and state reconstruction to implement and verify protocols for optimal control of Cs hyperfine spns (Fig. 1), showing that an initial fiducial state can be transformed into any desired target state with a fidelity in the 80-90% range [3].
Quantum state reconstruction based on weak continuous measurement has the advantage of being fast, accurate and almost non-perturbative. In this work we present a pedagogical review of the protocol proposed by Silberfarb et al (2005 Phys. Rev. Lett. 95 030402), whereby an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner so as to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time through a maximum-likelihood estimate. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sublevels of a large-spin alkali atom, 133 Cs. We detail two different protocols for control. Using magnetic interactions and a quadratic ac Stark shift, we can reconstruct a chosen hyperfine manifold F , e.g. the seven-dimensional F = 3 manifold in the electronic ground state of Cs. We review the procedure as implemented in experiments (Smith et al 2006 Phys. Rev. Lett. 97 180403). We extend the protocol to the more ambitious case of reconstruction of states in the full 16-dimensional electronic ground subspace ( F = 3⊕ F = 4), controlled by microwaves and radio-frequency (RF) magnetic fields. We give detailed derivations of all physical interactions, approximations, numerical methods and fitting procedures, tailored to the realistic experimental setting. For the case of light-shift and magnetic control, reconstruction fidelities of ∼0.95 have been achieved, limited primarily by inhomogeneities in the light-shift. For the case of microwave/RF-control we simulate fidelity >0.97, limited primarily by signal-to-noise.
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133{}^{133}133Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3⊕F=4F=3 \oplus F=4F=3⊕F=4), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.
Characterizing the Hamiltonian that a quantum processor actually implements is central to calibrating and validating current quantum hardware. Many devices, however, operate with generators that are time dependent by design. Here we develop a rigorous and experimentally friendly protocol for learning time-dependent many-body Hamiltonians from continuous weak measurement records. The key observation is that interaction sparsity reduces the global reconstruction to a set of local inverse problems, whose number is controlled by the interaction connectivity rather than by the system size. Pure separable probe states suffice to drive these inversions, and a graph-coloring construction embeds them into a small number of global product-state preparations. We derive explicit reconstruction-error bounds and a sample-complexity theorem that cleanly separates the finite-sampling statistical noise from the deterministic bias of the iterative state update, and we validate the protocol on time-dependent spin chains with up to n=8n=8n=8 qubits. Beyond these results, our analysis provides a rigorous foundation for time-dependent Hamiltonian learning from continuous monitoring in many-body systems, establishing a framework that extends naturally to many platforms and probe ensembles.