Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.
We generalize the concept of mutually unbiased bases (MUB) to measurements which are not necessarily described by rank one projectors. As such, these measurements can be a useful tool to study the long-standing problem of the existence of MUB. We derive their general form, and show that in a finite, d -dimensional Hilbert space, one can construct a complete set of mutually unbiased measurements. Besides their intrinsic link to MUB, we show that these measurements' statistics provide complete information about the state of the system. Moreover, they capture the physical essence of unbiasedness, and in particular, they satisfy a non-trivial entropic uncertainty relation similar to MUB.
In the framework of mutually unbiased bases (MUBs), a measurement in one basis gives no information about the outcomes of measurements in another basis. Here we relax the no-information condition by allowing the
d outcomes to be predicted according to a predefined probability distribution $q=({q}_{0},...,{q}_{d\ensuremath{-}1})$. The notion of mutual unbiasedness, however, is preserved by requiring that the extracted information is the same for any preparation and any measurement; regardless of which state from which basis is chosen to prepare the system, the outcomes of measuring the system with respect to the other basis generate the same probability distribution. In light of this, we define the notion of mutually equibiased bases (MEBs) such that within each basis the states are equibiased with respect to the states of the other basis and that the bases are mutually equibiased with respect to each other. For
d=2,3, we derive a set of
d+1 MEBs. The mutual equibiasedness imposes nontrivial constraints on the distribution
q, leading, for
d=3, to the restriction $1/3\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1/2$, where $\ensuremath{\mu}={\ensuremath{\sum}}_{k=0}^{2}{q}_{k}^{2}$. To capture the incompatibility of the measurements in MEBs, we derive an inequality for the probabilities of projective measurements in a qudit system, which yields an associated entropic uncertainty inequality. Finally, we construct a class of positive maps and their associated entanglement witnesses based on MEBs. While an entanglement witness constructed from MUBs is generally finer than one based on MEBs when both use the same number of bases, for certain values of the index $\ensuremath{\mu}$, employing a larger set of MEBs can yield a finer witness. We illustrate this behavior using isotropic states of a $3\ifmmode\times\else\texttimes\fi{}3$ system. Our results reveal that not all bases in a set of
L MEBs can contribute to the entanglement detection. A constraint, dependent on the probability distribution
q, limits the number of MEBs that can be employed.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
When used in quantum state estimation, projections onto mutually unbiased bases have the ability to maximize information extraction per measurement and to minimize redundancy. We present the first experimental demonstration of quantum state tomography of two-qubit polarization states to take advantage of mutually unbiased bases. We demonstrate improved state estimation as compared to standard measurement strategies and discuss how this can be understood from the structure of the measurements we use. We experimentally compared our method to the standard state estimation method for three different states and observe that the infidelity was up to 1.84 ± 0.06 times lower by using our technique than it was by using standard state estimation methods.
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.