Error regions in quantum state tomography: computational complexity caused by geometry of quantum states

Abstract

The outcomes of quantum mechanical measurements are inherently random. It is therefore necessary to develop stringent methods for quantifying the degree of statistical uncertainty about the results of quantum experiments. For the particularly relevant task of quantum state tomography, it has been shown that a significant reduction in uncertainty can be achieved by taking the positivity of quantum states into account. However-the large number of partial results and heuristics notwithstandingno efficient general algorithm is known that produces an optimal uncertainty region from experimental data, while making use of the prior constraint of positivity. Here, we provide a precise formulation of this problem and show that the general case is NP-hard. Our result leaves room for the existence of efficient approximate solutions, and therefore does not in itself imply that the practical task of quantum uncertainty quantification is intractable. However, it does show that there exists a non-trivial trade-off between optimality and computational efficiency for error regions. We prove two versions of the result: one for frequentist and one for Bayesian statistics.

Bayesian error regions in quantum estimation I: analytical reasonings

“…Here, the plausible region, of 0.967 credibility, is constructed with l = 0. can be completely characterized by the (d=3 2 −1=8)-dimensional state parameter r. Therefore the minimum number of POM outcomes needed to estimate r is M=9. The volume of the qutrit space, according to (17), is  p = V 20160 3 3 . To compute s λ and c λ over  3 , we may again perform uniform rejection sampling over the ranges 0r 1 , r 2 1 and −1r 3 , K, r 8 1.…”

“…For large dimensions, it has been shown that the complex structures of a convex parameter space and its boundaries render the construction of Bayesian regions generally an NP-hard problem, as is also the case for confidence regions [3]. In quantum-state tomography, sophisticated Monte Carlo methods have been developed and applied to sample the state space of bipartite systems with modest dimensions in order to compute the region size and credibility [12,13].…”

Efficient Bayesian credible-region certification for quantum-state tomography

“…Bootstrapping procedures [9,10] are amongst some of the most widely-used techniques for assigning "error-bars" to reconstructed quantum states. Recently, it was pointed out in [11] that such assignments lack rigorous statistical foundations and may produce "error-bars" that are too small for reliable conclusions. The rather more justified approach falls under the study of hypothesis testing [12].…”

Behavior of the maximum likelihood in quantum state tomography

“…) 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.…”