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On the edge: State tomography boundaries and model selection.

Travis L. Scholten

Published Nov 1, 2015• 0 citations
OpenAlex

Abstract

Model selection techniques currently used in quantum tomography may have problems.Such as loglikelihood ratio tests, or Akaike's AIC.This estimator outperforms the Wilks Theorem in predicting the Monte Carlo averages.It also performs reasonably well on an instanceby-instance basis.

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Behavior of the maximum likelihood in quantum state tomography

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© 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Standard tomographic analyses ignore model uncertainty. It is assumed that a given model generated the data and the task is to estimate the quantum state, or a subset of parameters within that model. Here we apply a model averaging technique to mitigate the risk of overconfident estimates of model parameters in two examples: (1) selecting the rank of the state in tomography and (2) selecting the model for the fidelity decay curve in randomized benchmarking.

•Sep 24, 2014•15 citations
Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d.They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix).Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good.Here, we consider the loglikelihood ratio.Because of the positivity constraint ρ0, quantum state space does not generally satisfy local asymptotic normality (LAN), meaning the classical null theory for the loglikelihood ratio (the Wilks theorem) should not be used.Thus, understanding and quantifying how positivity affects the null behavior of this test statistic is necessary for its use in model selection for state tomography.We define a new generalization of LAN, metric-projected LAN, show that quantum state space satisfies it, and derive a replacement for the Wilks theorem.In addition to enabling reliable model selection, our results shed more light on the qualitative effects of the positivity constraint on state tomography.Determining the quantum state ρ 0 produced by a specific preparation procedure for a quantum system is a problem almost as old as quantum mechanics itself [1,2].This task, known as quantum state tomography [3], is not only useful in its own right (diagnosing and detecting errors in state preparation), but is also used in other characterization protocols including entanglement verification [4-6] and process tomography [7].A typical state tomography protocol proceeds as follows: many copies of ρ 0 are produced, they are measured in diverse ways, and finally the outcomes of those measurements (data) are collated and analyzed to produce an estimate r ˆ.This is a straightforward statistical inference process [8,9], where the data are used to fit the parameters of a statistical model.In state tomography, the parameter is ρ, and the model is the set of all possible density matrices on a Hilbert space  (equipped with the Born rule).However, we do not always know what model to use.It is not always a priori obvious what , or its dimension, is; examples include optical modes [10-14] and leakage levels in AMO and superconducting [15,16] qubits.In such situations, we seek to let the data itself determine which of many candidate Hilbert spaces is best suited for reconstructing ρ 0 .Choosing an appropriate Hilbert space on the fly is an instance of a general statistical problem called model selection.Although model selection has been thoroughly explored in classical statistics [17], its application to state tomography encounters some obstacles.They stem from the fact that quantum states-and therefore, estimates of them-must satisfy a positivity constraint ρ0.(See figure 1.) A similar constraint, complete positivity, applies to process tomography.The impact of positivity constraints on state and process tomography is an active area of research [18][19][20][21], and its implications for model selection have also been considered [22][23][24][25][26][27][28].In this paper, we address a specific question at the heart of this matter: How does the loglikelihood ratio statistic used in many model selection protocols, including (but not limited to) information criteria such as Akaike's AIC [29], behave in the presence of the positivity constraint ρ0?We begin in section 1 by introducing the loglikelihood ratio statistic λ, and outline how it can be used to choose a Hilbert space.In section 2, we show how and why the classical null theory for its behavior, the Wilks theorem, falls apart in the presence of the positivity constraint, because quantum state space does not generally satisfy local asymptotic normality (LAN).We define a new generalization of LAN, metric-projected local asymptotic normality (MP-LAN), in section 3; this generalization explicitly accounts for the positivity constraint, and is satisfied by quantum state space.Using this generalization, we derive a closed-form approximation for λʼs expected value in section 4, thereby providing a replacement for the Wilks theorem that is
The statistical analysis of measurement data has become a key component of many quantum engineering experiments. As standard full state tomography becomes unfeasible for large dimensional quantum systems, one needs to exploit prior information and the 'sparsity' properties of the experimental state in order to reduce the dimensionality of the estimation problem. In this paper we propose model selection as a general principle for finding the simplest, or most parsimonious explanation of the data, by fitting different models and choosing the estimator with the best trade-off between likelihood fit and model complexity. We apply two well established model selection methods—the Akaike information criterion (AIC) and the Bayesian information criterion (BIC)—two models consisting of states of fixed rank and datasets such as are currently produced in multiple ions experiments. We test the performance of AIC and BIC on randomly chosen low rank states of four ions, and study the dependence of the selected rank with the number of measurement repetitions for one ion states. We then apply the methods to real data from a four ions experiment aimed at creating a Smolin state of rank 4. By applying the two methods together with the Pearson χ 2 test we conclude that the data can be suitably described with a model whose rank is between 7 and 9. Additionally we find that the mean square error of the maximum likelihood estimator for pure states is close to that of the optimal over all possible measurements.
© 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft. Standard tomographic analyses ignore model uncertainty. It is assumed that a given model generated the data and the task is to estimate the quantum state, or a subset of parameters within that model. Here we apply a model averaging technique to mitigate the risk of overconfident estimates of model parameters in two examples: (1) selecting the rank of the state in tomography and (2) selecting the model for the fidelity decay curve in randomized benchmarking.