Comment on "Improved bounds on entropic uncertainty relations"

We provide an analytical proof of the entropic uncertainty relations presented by J. I. de Vicente and J. Sánchez-Ruiz [Phys. Rev. A 77, 042110 (2008)] and also show that the replacement of Eq. (27) by Eq. (29) in that reference introduces solutions that do not take fully into account the constraints of the problem, which in turn lead to some mistakes in their treatment.Facultad de Ciencias ExactasInstituto de Física La Plat

Comment on "Improved bounds on entropic uncertainty relations"

We provide an analytical proof of the entropic uncertainty relations presented by J. I. de Vicente and J. Sánchez-Ruiz [Phys. Rev. A 77, 042110 (2008)] and also show that the replacement of Eq. (27) by Eq. (29) in that reference introduces solutions that do not take fully into account the constraints of the problem, which in turn lead to some mistakes in their treatment.Facultad de Ciencias ExactasInstituto de Física La Plat

Improving Einstein-Podolsky-Rosen Steering Inequalities with State Information

We discuss the relationship between entropic Einstein-Podolsky-Rosen (EPR)-steering inequalities and their underlying uncertainty relations, along with the hypothesis that improved uncertainty relations lead to tighter EPR-steering inequalities. In particular, we discuss how the intrinsic uncertainty in a mixed quantum state is used to improve existing uncertainty relations and how this information affects one's ability to witness EPR-steering. As an example, we consider the recent improvement (using a quantum memory) to the entropic uncertainty relation between pairs of discrete observables (Nat. Phys. 6, 659 (2010)) and show that a trivial substitution of the tighter bound in the steering inequality leads to contradictions, due in part to the fact that the improved bound depends explicitly on the state being measured. By considering the assumptions that enter into the development of a steering inequality, we derive correct steering inequalities from these improved uncertainty relations and find that they are identical to ones already developed (Phys. Rev. A, 87, 062103 (2013)). In addition, we consider how one can use the information about the quantum state to improve our ability to witness EPR-steering, and develop a new symmetric EPR-steering inequality as a result.Comment: 6 page

Conditional entropic uncertainty relations for Tsallis entropies

The entropic uncertainty relations are a very active field of scientific inquiry. Their applications include quantum cryptography and studies of quantum phenomena such as correlations and non-locality. In this work we find entanglement-dependent entropic uncertainty relations in terms of the Tsallis entropies for states with a fixed amount of entanglement. Our main result is stated as Theorem~\ref{th:bound}. Taking the special case of von Neumann entropy and utilizing the concavity of conditional von Neumann entropies, we extend our result to mixed states. Finally we provide a lower bound on the amount of extractable key in a quantum cryptographic scenario.Comment: 11 pages, 4 figure

Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches [S. Friendland, V. Gheorghiu and G. Gour, Phys. Rev. Lett. 111, 230401 (2013); A. E. Rastegin and K. \.Zyczkowski, J. Phys. A, 49, 355301 (2016)], particularly by extending the direct-sum majorization relation first introduced in [\L. Rudnicki, Z. Pucha{\l}a and K. \.{Z}yczkowski, Phys. Rev. A 89, 052115 (2014)]. We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen--Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables

Improving Einstein–Podolsky–Rosen Steering Inequalities with State Information

We discuss the relationship between entropic Einstein–Podolsky–Rosen (EPR)-steering inequalities and their underlying uncertainty relations along with the hypothesis that improved uncertainty relations lead to tighter EPR-steering inequalities. In particular, we discuss how using information about the state of a quantum system affects oneʼs ability to witness EPR-steering. As an example, we consider the recent improvement to the entropic uncertainty relation between pairs of discrete observables (Berta et al., 2010 [10]). By considering the assumptions that enter into the development of a steering inequality, we derive correct steering inequalities from these improved uncertainty relations and find that they are identical to ones already developed (Schneeloch et al., 2013 [9]). In addition, we consider how one can use state information to improve our ability to witness EPR-steering, and develop a new continuous variable symmetric EPR-steering inequality as a result

On a generalized entropic uncertainty relation in the case of the qubit

We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. R\'enyi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of R\'enyi entropies for any couple of (positive) entropic indices (\alpha,\beta). Thus, we have overcome the H\"older conjugacy constraint imposed on the entropic indices by Riesz-Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square [0 , 1/2] x [0 , 1/2] in the \alpha-\beta plane, and a semi-analytical expression on the line \beta = \alpha. It is seen that previous results are included as particular cases. Moreover, we present an analytical but suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.Comment: 15 pages, 6 figure

General entropy-like uncertainty relations in finite dimensions

We revisit entropic formulations of the uncertainty principle for an arbitrary pair of positive operator-valued measures (POVM) $A$ and $B$, acting on finite dimensional Hilbert space. Salicr\'u generalized $(h,\phi)$-entropies, including R\'enyi 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 $(c_A,c_B,c_{A,B})$ with $c_A$ (resp. $c_B$) being the overlap between the elements of the POVM $A$ (resp. $B$) and $c_{A,B}$ the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and S\'anchez-Ruiz [Phys.\ Rev.\ A \textbf{77}, 042110 (2008)] 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 R\'enyi or Tsallis entropic formulations of the uncertainty principle, we overcome the H\"older conjugacy constraint imposed on the entropic indices by the Riesz-Thorin theorem. In the case of nondegenerate observables, we show that for given $c_{A,B} > \frac{1}{\sqrt2}$, the bound obtained is optimal; and that, for R\'enyi entropies, our bound improves Deutsch one, but Maassen-Uffink bound prevails when $c_{A,B} \leq\frac12$. Finally, we illustrate by comparing our bound with known previous results in particular cases of R\'enyi and Tsallis entropies

Más allá de Heisenberg : Relaciones de incerteza tipo Landau-Pollak y tipo entrópicas

En esta Tesis desarrollamos dos formulaciones diferentes (pero vinculadas entre sí) del principio de incerteza de la mecánica cuántica para pares de observables actuando sobre un espacio de Hilbert finito, yendo más allá del alcance de las tradicionales relaciones de incerteza de Heisenberg, de Robertson y de Schröndinger. Una de las formulaciones que desarrollamos es una extensión de la desigualdad de Landau y Pollak al caso de medidas de operadores con valores positivos y estados mixtos. Para lograr esto hicimos uso de un enfoque geométrico, definiendo la incerteza asociada al resultado de la medición de un observable a partir de métricas entre estados cuánticos. Esto nos permitió mostrar, entre otros resultados, que la métrica de Wootters da la desigualdad más restrictiva a las probabilidades máximas de los observables. La otra formulación que desarrollamos se basa en un enfoque informacional. Para ello introducimos una familia de entropías generalizadas que cuantifican la incerteza asociada a un vector de probabilidad. Obtuvimos relaciones de incerteza tipo entrópicas resolviendo el problema de minimización de la suma de entropías generalizadas sujeta a la desigualdad de Landau–Pollak. De esta manera, extendimos los resultados de de Vicente y Sánchez-Ruiz que consideraban la entropía de Shannon a otras entropías, medidas cuánticas generalizadas y estados mixtos. Asimismo, realizamos un estudio comparativo entre las cotas obtenidas y otras disponibles en la literatura, obteniendo que en muchas de las situaciones consideradas nuestra cota es más fuerte. Además, consideramos el caso del qubit de manera particular y obtuvimos la cota óptima para este caso. Por último, estudiamos la conexión entre los principios de incerteza y complementariedad, en el contexto del interferómetro de Mach–Zehnder. Encontramos que las relaciones de Schrödinger y de Landau–Pollak para ciertos observables son equivalentes a la relación de dualidad onda–corpúsculo. Con respecto a las relaciones usando entropías, la equivalencia depende de la elección de los índices entrópicos. En particular, si los índices son iguales no existe tal equivalencia. Mostramos que esta situación sirve para discernir entre los diferentes estados de mínima incerteza.Facultad de Ciencias Exacta

Comment on "Improved bounds on entropic uncertainty relations"

International audienceWe provide an analytical proof of the entropic uncertainty relations presented by J. I. de Vicente and J. Sánchez-Ruiz [Phys. Rev. A 77, 042110 (2008)] and also show that the replacement of Eq. (27) by Eq. (29) in that reference introduces solutions that do not take fully into account the constraints of the problem, which in turn lead to some mistakes in their treatment