Global Completability with Applications to Self-Consistent Quantum Tomography

Cyril Stark

Takanori Sugiyama, Shinpei Imori, Fuyuhiko Tanaka

Cyril Stark

Let p_1, ..., p_N \in R^D be unknown vectors and let Omega \subseteq {1,...,N}^{\times 2}. Assume that the inner products p_i^T p_j are fixed for all (i,j) \in Omega. Do these inner product constraints (up to simultaneous rotation of all vectors) determine p_1, ..., p_N uniquely? Here we derive a necessary and sufficient condition for the uniqueness of p_1, ...,p_N (i.e., global completability) which is applicable to a large class of practically relevant sets Omega. Moreover, given Omega, we show that the condition for global completability is universal in the sense that for almost all vectors p_1, ...,p_N \in R^D the completability of p_1, ...,p_N only depends on Omega and not on the specific values of p_i^T p_j$ for (i,j) \in Omega. This work was motivated by practical considerations, namely, self-consistent quantum tomography.

Cyril Stark

State and measurement tomography make assumptions about the experimental states or measurements. These assumptions are often not justified because state preparation and measurement errors are unavoidable in practice. Here we describe how the Gram matrix associated with the states and measurement operators can be estimated via semidefinite programming if the states and the measurements are so-called globally completable. This is, for instance, the case if the unknown measurements are known to be projective and nondegenerate. The computed Gram matrix determines the states, and the measurement operators uniquely up to simultaneous rotations in the space of Hermitian matrices. We prove the reliability of the proposed method in the limit of a large number of independent measurement repetitions.

Marco Cattaneo, Matteo A. C. Rossi, Keijo Korhonen +4 more

We propose an estimation method for quantum measurement tomography (QMT) based on semidefinite programming (SDP) and discuss how it may be employed to detect experimental imperfections, such as shot noise and/or faulty preparation of the input states on near-term quantum computers. Moreover, if the positive operator-valued measure (POVM) we aim to characterize is informationally complete, we put forward a method for self-consistent tomography, i.e., for recovering a set of input states and POVM effects that is consistent with the experimental outcomes and does not assume any a priori knowledge about the input states of the tomography. Contrary to many methods that have been discussed in the literature, our approach does not rely on additional assumptions such as low noise or the existence of a reliable subset of input states.

Grigori Gennadievich Amosov, Stefano Mancini, Vladimir I. Man’ko