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Physical Review Letters

Cross-Validated Tomography

Dmitri S. Mogilevtsev, Z. Hradil, Jaroslav Řeháček, V. S. Shchesnovich

Published Sep 20, 2013• 16 citations
Publisher

Abstract

We show that the information collected in the course of a generic quantum tomography experiment can be used for verifying experimenters' assumptions about the state preparation and measurement. In particular, systematic errors, such as drifts and instabilities inherent in the tomography setup, can be identified without the need for any specific measurements designed to detect such problems. This is done by statistical analysis of available tomography data. As an example, we discuss the detection and characterization of unexpected changes in the signal state preparation during the tomography measurement and argue that minimal tomography schemes, such as symmetric informationally complete measurements, are more difficult to validate than overcomplete schemes. The method is illustrated with the analysis of experimental quantum homodyne tomography.

DOI
10.1103/physrevlett.111.120403

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We show that the information collected in the course of a generic quantum tomography experiment can be used for verifying experimenters' assumptions about the state preparation and measurement. In particular, systematic errors, such as drifts and instabilities inherent in the tomography setup, can be identified without the need for any specific measurements designed to detect such problems. This is done by statistical analysis of available tomography data. As an example, we discuss the detection and characterization of unexpected changes in the signal state preparation during the tomography measurement and argue that minimal tomography schemes, such as symmetric informationally complete measurements, are more difficult to validate than overcomplete schemes. The method is illustrated with the analysis of experimental quantum homodyne tomography.
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.
Abstract Characterization of quantum states and devices is paramount to quantum science and technology. The characterization consists of individual measurements, which must be precisely known. A mismatch between actual and assumed constituent measurements limits the accuracy of this characterization. We show that such a mismatch introduces reconstruction artifacts in quantum state tomography. We use these artifacts to detect and quantify the mismatch, gaining information about the actual measurement operators. It consequently allows the mitigation of systematic errors in both quantum measurement and state preparation, improving the precision of state control and characterization. The practical utility of our approach is experimentally demonstrated.
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Self-calibrating states are introduced as quantum resources for the simultaneous reconstruction of a quantum state and measurement device. Simple criteria for a state being self-calibrating are formulated. Theory applies to realistic state estimation scenarios, where the detection model is nonlinear in unknown parameters of the measuring device. Examples discussed include time multiplexed detection with on/off detectors and quantum homodyne tomography. The thermally noised single-photon states and Gaussian states are self-calibrating for these schemes. For homodyne tomography, statistical tests applied to raw measured data are sufficient for judging whether or not the signal state is self-calibrating without any a priori information.
We expand the scope of the statistical notion of error probability, that is, how often large deviations are observed in an experiment, to make it directly applicable to quantum tomography. We verify that the error probability can decrease at most exponentially in the number of trials, we derive the explicit rate that bounds this decrease, and we show that a maximum likelihood estimator achieves this bound. We also show that the statistical notion of identifiability coincides with the tomographic notion of informational completeness. Our result implies that two quantum tomographic apparatuses that have the same risk function (e.g., variance) can have different error probability, and we give an example in one-qubit-state tomography. Thus by combining these two approaches we can evaluate, in a reconstruction-independent way, the performance of such experiments more discerningly.
As the method to completely characterize quantum dynamical processes, quantum process tomography (QPT) is vitally important for quantum information processing and quantum control, where the faithfulness of quantum devices plays an essential role. Here via weak measurements, we present a new QPT scheme characterized by its directness and parallelism. Comparing with the existing schemes, our scheme needs a simpler state preparation and much fewer experimental setups. Furthermore, each parameter of the quantum process is directly determined from only five experimental values in our scheme, meaning that our scheme is robust against the accumulation of errors.