Testing Quantum States for Purity

Abstract

Summary. The simplest states of finite quantum systems are the pure states. The paper is motivated by the need to test statistically whether or not a given physical state is pure. Because the pure states lie in the boundary of the set of all states, the usual regularity conditions that justify the standard large sample approximations to the null distributions of the deviance and the score statistic are not satisfied. For a large class of quantum experiments that produce Poisson count data, the paper uses an enlargement of the parameter space of all states to develop likelihood ratio and score tests of purity. The asymptotic null distributions of the corresponding statistics are χ2. The tests are illustrated by the analysis of some quantum experiments involving unitarily correctable codes.

Spectral thresholding quantum tomography for low rank states

“…Recent years have witnessed significant developments at the overlap between quantum theory and statistics: from new state estimation (or tomography) methods [2,3,4,5,6,7,8], design of experiments [9,10,11], quantum process and detector tomography [12,13] construction of confidence regions (error bars) [14,15,16], quantum tests [17,18] entanglement estimation [19], asymptotic theory [20,21,22,23]. The importance of quantum state tomography, and the challenges raised by the estimation of high dimensional systems were highlighted by the landmark experiment [1] where entangled states of up to 8 ions were created and fully characterised.…”

Rank-based model selection for multiple ions quantum tomography

“…From the preparation of exotic quantum states [4][5][6][7] to the implementation of accurate quantum protocols [8][9][10][11] experimentalists are confronted with the problem of reconstructing such mathematical objects statistically, from the outcomes of repeated measurements. The theoretical and experimental challenges have stimulated the development of a large array of new methods at the boundary between quantum theory and statistics: state estimation (or tomography) [12][13][14][15][16][17], tomography for incomplete data [18][19][20], permutationally invariant tomography [21,22], design of experiments [23][24][25], quantum process and detector tomography [26,27] construction of confidence regions (error bars) [28,29], quantum tests [30][31][32], entanglement estimation [33], quantum homodyne tomography [34][35][36], asymptotic theory [37][38][39]; see also the monographs [40,41] and the collections of papers [42,43].…”

Shannon and von Neumann entropies of multi-qubit Schrödinger's cat states

“…Alternatives to the maximum likelihood method have been suggested, including a hedged maximum likelihood method, 164,165 a method involving an accuracy matrix, 166 maximization of the mean fidelity, 167 a method based on Bayesian inference, 168–170 and a maximal entropy approach that is well suited to reconstructing the density matrix when not all measurements can be performed with high fidelity. 28,171 Special considerations for the tomography of pure states have been described by Bagan et al , 168 Jupp et al , 172 and Gross et al 173 The tomography of permutationally invariant states has been discussed by Tóth et al 174 and by Moroder et al 175…”

“…162,163 Alternatives to the maximum likelihood method have been suggested, including a hedged maximum likelihood method, 164,165 a method involving an accuracy matrix, 166 maximization of the mean fidelity, 167 a method based on Bayesian inference, [168][169][170] and a maximal entropy approach that is well suited to reconstructing the density matrix when not all measurements can be performed with high fidelity. 28,171 Special considerations for the tomography of pure states have been described by Bagan et al, 168 Jupp et al, 172 and Gross et al 173 The tomography of permutationally invariant states has been discussed by To ´th et al 174 and by Moroder et al 175 The fidelity F of the density matrix is given by F = [Tr(r p 1/2 r T r p 1/2 ) 1/2 ] 2 , where r p is the density matrix of the pure cat state, and r T is the matrix found by tomographic experiments. 176 For the cat states, the fidelity is simply the sum of the four corner elements of the density matrix.…”