Quantum State Tomography with incomplete data: Maximum Entropy and Variational Quantum Tomography

Abstract

Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state can not be uniquely determined. In this case, among the density matrices compatible with the available data, it is commonly preferred that one which is the most uncommitted with 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 Variational Quantum Tomography and show its relationship with Maximum Entropy methods in quantum tomographies with incomplete set of measurements. We prove their equivalence in 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 comment7 pages, 7 figures

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• Communication games reveal preparation contextuality
• Maximal entropy approach for quantum state tomography
• Self-consistent quantum measurement tomography based on semidefinite programming
• Quantum process tomography with informational incomplete data of two $J$-coupled heterogeneous spins relaxation in a time window much greater than $T_1$
• Statistical analysis of low rank tomography with compressive random measurements
• Efficient Quantum Tomography of Two-Mode Wigner Functions
• Parameterizing density operators with arbitrary symmetries to gain advantage in quantum state estimation
• Quantum theory in finite dimension cannot explain every general process with finite memory
• Interpolation for completely positive maps: numerical solutions
• Least-bias state estimation with incomplete unbiased measurements
• Conic Geometric Programming