Quantum state tomography provides a fundamental framework for reconstructing quantum states. When the measurement data are not informationally complete, the observed statistics admit multiple compatible density matrices, making the reconstruction problem inherently underdetermined and calling for the selection of a meaningful estimator. Two well-established approaches to address this ambiguity are Maximum Entropy (MaxEnt) and Variational Quantum Tomography (VQT). A variant of VQT, named VQT $_\infty$ , has been introduced to reproduce MaxEnt-like solutions. In this work, we generalize this approach by introducing a parametrized cost function that interpolates between the 1-norm and the infinity norm, thereby unifying VQT and VQT $_\infty$ within a single framework. By tuning the associated hyperparameters, the proposed method enables controlled exploration of the set of compatible density matrices. We show that this interplay yields reconstructed states with higher fidelity to the MaxEnt solution than those obtained via VQT $_\infty$ while preserving computational tractability.Quantum state tomography provides a fundamental framework for reconstructing quantum states. When the measurement data are not informationally complete, the observed statistics admit multiple compatible density matrices, making the reconstruction problem inherently underdetermined and calling for the selection of a meaningful estimator. Two well-established approaches to address this ambiguity are Maximum Entropy (MaxEnt) and Variational Quantum Tomography (VQT). A variant of VQT, named VQT$_\infty$, has been introduced to reproduce MaxEnt-like solutions. In this work, we generalize this approach by introducing a parametrized cost function that interpolates between the 1-norm and the infinity norm, thereby unifying VQT and VQT$_\infty$ within a single framework. By tuning the associated hyperparameters, the proposed method enables controlled exploration of the set of compatible density matrices. We show that this interplay yields reconstructed states with higher fidelity to the MaxEnt solution than those obtained via VQT$_\infty$ while preserving computational tractability.

quant-pharXiv:2604.27135May 1, 20260 citations

Quantum state tomography with incomplete data: Maximum entropy and variational quantum tomography

Douglas Soares Gonçalves, Carlile Lavor, Márcia A. Gomes-Ruggiero +3 more

Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state cannot be uniquely determined. In this case, among the density matrices compatible with the available data, the one commonly preferred is the one which is the most uncommitted to the missing information. This is the purpose of the maximum entropy estimation (MaxEnt) and the variational quantum tomography (VQT). Here, we propose a variant of VQT and show its relationship with MaxEnt methods in quantum tomographies with an incomplete set of measurements. We prove their equivalence in the case of eigenbasis measurements, and through numerical simulations we stress their similar behavior. Hence, in the modified VQT formulation we have an estimate of a quantum state as unbiased as in MaxEnt and with the benefit that VQT can be more efficiently solved by means of linear semidefinite programs.Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state cannot be uniquely determined. In this case, among the density matrices compatible with the available data, the one commonly preferred is the one which is the most uncommitted to the missing information. This is the purpose of the maximum entropy estimation (MaxEnt) and the variational quantum tomography (VQT). Here, we propose a variant of VQT and show its relationship with MaxEnt methods in quantum tomographies with an incomplete set of measurements. We prove their equivalence in the case of eigenbasis measurements, and through numerical simulations we stress their similar behavior. Hence, in the modified VQT formulation we have an estimate of a quantum state as unbiased as in MaxEnt and with the benefit that VQT can be more efficiently solved by means of linear semidefinite programs.

arXiv:1306.0467Physical Review AMay 31, 201331 citations

Maximum entropy and approximate descriptions of pure states

Angel Ricardo Plastino, Angel Ricardo Plastino

Physics Letters ANov 1, 199355 citations

_\infty

, has been introduced to reproduce MaxEnt-like solutions. In this work, we generalize this approach by introducing a parametrized cost function that interpolates between the 1-norm and the infinity norm, thereby unifying VQT and VQT

_\infty

within a single framework. By tuning the associated hyperparameters, the proposed method enables controlled exploration of the set of compatible density matrices. We show that this interplay yields reconstructed states with higher fidelity to the MaxEnt solution than those obtained via VQT

_\infty

while preserving computational tractability.