Confidence Polytopes in Quantum State Tomography

Quantum state tomography is the task of inferring the state of a quantum system from measurement data. A reliable tomography scheme should not only report an estimate for that state, but also well-justified error bars. These may be specified in terms of confidence regions, i.e., subsets of the state space which contain the system’s state with high probability. Here, building upon a quantum generalization of Clopper-Pearson confidence intervals—a notion known from classical statistics—we present a simple and reliable scheme for generating confidence regions. These have the shape of a polytope and can be computed efficiently. We provide several examples to demonstrate the practical usability of the scheme in experiments.

Figure 1

Confidence polytopes for QST on a qubit. The plots show confidence polytopes with confidence level 0.999, obtained from data of simulated measurements on a single qubit. The polytopes lie within the Bloch sphere, which represents the entire state space of the qubit. In (a), the measurement is defined by six Pauli projectors, along the $X$, $Y$, and $Z$ directions, and the polytope is a rectangular box whose normal directions are given by the three Pauli operators. In (b), the measurement operators are chosen such that they form a symmetrical informationally complete (SIC) POVM [36, 37]. In this case, the resulting confidence polytope is the intersection of two tetrahedrons whose normals are given by the measurement directions.

Figure 2

Optimizing the information content of measurements. The red dots in (a) represent the elements of a skewed SIC POVM, which has more distinguishing power in the $Z$ direction than in the $X$ and $Y$ directions. (b) Confidence polytope that is obtained from 5000 measurements defined by this POVM. Panel (c) depicts the effect of 1000 additional projective measurements in both the $X$ and the $Y$ direction. The red planes represent the new facets introduced by the extra measurements.

Figure 3

Confidence versus credibility regions. The plots show the ratio $ϵ/{ϵ}_b$ between the confidence and credibility levels of a polytope constructed according to the prescription of Theorem 1, interpreted as a confidence region and as a credibility region, respectively. Each dot was obtained by QST on $n$ copies of a simulated state chosen at random from a $d$-dimensional state space, evaluated with Ref. [49]. Although there are fluctuations due to the different choices of states and measurement outcomes, no scaling in the data size $n$ or the dimension $d$ is observed.