Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a lower bound on the entropy sum for general pairs of observables in finite-dimensional Hilbert space, which improves on the best bound known to date [H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988)] for a wide class of observables. This result follows from another formulation of the uncertainty principle, the Landau-Pollak inequality, whose relationship to the Maassen-Uffink entropic uncertainty relation is discussed.
Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a lower bound on the entropy sum for general pairs of observables in finite-dimensional Hilbert space, which improves on the best bound known to date [H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988)] for a wide class of observables. This result follows from another formulation of the uncertainty principle, the Landau-Pollak inequality, whose relationship to the Maassen-Uffink entropic uncertainty relation is discussed.
We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B , acting on finite dimensional Hilbert space. Salicrú generalized -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet with c A (respectively c B ) being the overlap between the elements of the POVM A (respectively B ) and the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110 ) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.
We analyze entropic uncertainty relations in a finite-dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the recent bounds by Coles and Piani [P. Coles and M. Piani, Phys. Rev. A 89, 022112 (2014)], which are known to be stronger than the well-known result of Maassen and Uffink [H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988)]. Furthermore, we find a bound based on majorization techniques, which also happens to be stronger than the recent results involving the largest singular values of submatrices of the unitary matrix connecting both bases. The first set of bounds gives better results for unitary matrices close to the Fourier matrix, while the second one provides a significant improvement in the opposite sectors. Some results derived admit generalization to arbitrary mixed states, so that corresponding bounds are increased by the von Neumann entropy of the measured state. The majorization approach is finally extended to the case of several measurements.
Entropic uncertainty relations H(A)+H(B)⩾γ give a nonzero lower bound γ to the sum of the Shannon entropies H of the outcome probabilities of incompatible observables A and B. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds γ typically are state independent. Here we provide a state-independent lower bound γs that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter s→2, becomes asymptotically tight for all A,B. The bound can be extended to Renyi entropies.
After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/
We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized (h,ϕ ) -entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (cA, cB, cA,B) with cA (respectively cB) being the overlap between the elements of the POVM A (respectively B) and cA B, the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz (2008 Phys. Rev. A 77 042110) and consists in a minimization of the entropy sum subject to the Landau–Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the UP, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz–Thorin theorem. In the case of nondegenerate observables, we show that for given cA B, > 1/2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen–Uffink bound prevails when cA B, ⩽ 1/2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.
The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be nonoptimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work, we establish optimal uncertainty relations by characterizing the optimal lower bound in scenarios similar to the Maassen–Uffink type. We disprove a conjecture by Englert et al. and generalize various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.
Entropic uncertainty relations in a finite-dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for the sum of Rényi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables. Obtained bounds are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix. Numerical simulations show that for a generic unitary matrix of size N = 5, our bound is stronger than the well-known result of Maassen and Uffink (MU) with a probability larger than 98%. We also show that the bounds investigated are invariant under the dephasing and permutation operations. Finally, we derive a classical analogue of the MU uncertainty relation, which is formulated for stochastic transition matrices.