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arXiv:2204.00332 · Laser Physics Letters

Product and sum uncertainty relations based on metric-adjusted skew information

Xiaoyu Ma, Qinghua Zhang, Shao-Ming Fei

Published Apr 1, 2022• 7 citations
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Abstract

Abstract The metric-adjusted skew information establishes a connection between the geometrical formulation of quantum statistics and the measures of quantum information. We study uncertainty relations in product and summation forms of metric-adjusted skew information. We present lower bounds on product and summation uncertainty inequalities based on metric-adjusted skew information via operator representation of observables. Explicit examples are provided to back our claims.

DOI
10.1088/1612-202x/ac60a3

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Uncertainty principle is one of the most essential features of quantum mechanics, and it reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, we focus on the uncertainty relations based on metric-adjusted skew information. By making full use of the norm property, we establish an uncertainty relation for arbitrary finite nnn observables based on metric-adjusted skew information, and we give two new uncertainty relations for arbitrary finite NNN quantum channels based on metric-adjusted skew information. For arbitrary two observables and two channels, the uncertainty relations we give are not only better than the uncertainty relations detailed in [Quantum Inf. Process. 20, 72 (2021)], but also are the equations. The equation of the uncertainty relation is more accurate than the inequation of the usual uncertainty relation, which has important advantages in the application of quantum information technology, such as quantum communication and the quantum precision measurement. Meanwhile, our results are suitable to Wigner-Yanase-Dyson skew information that is a special metric-adjusted skew information and Wigner-Yanase skew information that is a special Wigner-Yanase-Dyson skew information. Some examples about Wigner-Yanase-Dyson skew information are given and show that the new lower bounds are tighter. The results play an important role in quantum information processing in this paper.
Abstract The uncertainty principle is one of the most essential features of quantum mechanics. Recently, uncertainty relations of skew information have been widely studied. In this paper, general and conditional uncertainty relations based on metric-adjusted skew information are put forward to study the case of the uncertainty relation with the existence of a quantum memory for the bipartite quantum system. These uncertainty relations include the product form and the sum form. The results show that the lower bounds contain two parts: one is characterizing the degree of compatibility of two measurements, and the other one is the quantum correlation between the measured system and the quantum memory. These results may play an important role in witnessing entanglement and quantum key distribution.
It is well known that almost all uncertainty relations including Heisenberg uncertainty relation and Schrödinger uncertainty relation were given by product types of trace inequalities.This is why these results were proved by Schwarz's inequality.These product types of uncertainty relations were extended to the case of not necessarily hermitian quantum mechanical observables and positive operators representing quantum states.On the other hand sum types of uncertainty relations were given for arbitrary finite N not necessarily hermitian quantum mechanical observables.Some uncertainty relations are presented by generalized quasi-metric adjusted skew informations for two different generalized states.These uncertainty relations are nontrivial as long as the observables are mutually noncommutative.The relations among these new and existing uncertainty inequalities have been investigated.Finally, the reverse inequalities of the sum types of uncertainty relations are obtained.
Abstract We formulate uncertainty relations based on Wigner–Yanase skew information. By using the Kirillov–Kostant–Souriau Kähler structure on the quantum phase space, we present a new geometric uncertainty relation associated to the skew information, which is shown to be tighter than the existing ones. Furthermore, we provide a skew information-based product uncertainty relation, in which the lower bound can also be used to capture the non-commutativity of the observables. We also attempt to generalize the geometric uncertainty inequalities to the case of arbitrary three observables, where the Kähler structure plays a vital role in the proof.