2016/09/30 by Travis L. Scholten, Travis L Scholten, Robin Blume-Kohout · 19 citations
Computer Science · Mathematics · Physics and Astronomy · #Applied mathematics #Asymptotic distribution #Computer science #Constraint (computer-aided design) #Data mining #Density matrix #Estimator #Generalization #Geometry #Mathematical analysis #Mathematics #Metric (unit) #Null (SQL) #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Quantum state #Quantum tomography #Statistic #Statistical hypothesis testing #Statistical physics #Statistics #Test statistic #quant-ph
paper · pdf · doi:10.1088/1367-2630/aaa7e2
published in New Journal of Physics 20(2), 023050 (IOP Publishing) · 16 pages, 14 figures. Close to published version. Relative to the previous version, minor text and content edits, including an asymptotic expression for our main result, Eq. 19. Supplemental information (data sets and code to reproduce figures) is available at https://github.com/Travis-S/arxiv_1609.04385
openalex publication_date 2018/02/23 · arxiv created 2018/05/21 · arxiv updated 2018/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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 ρ0, 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 ρ0.(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 ρ0?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