We consider a situation, when a part of a pure entangled state is measured, which generates the ensemble of quantum states for the other part. We consider the connection between the uncertainty property of the observable and coherence of the dual ensemble for any entanglement measure.
The coherence of an ensemble of quantum states is considered — a quantity having a similarity with the uncertainty of the quantum observable that arises from the entropic uncertainty relations. We give estimates for the coherence of the ensemble through the uncertainty of the dual observable and the weak uncertainty of the observable through the coherence of the ensemble. It is shown that this numerical characteristic is closely connected with the relative entropy of discord for the corresponding classical-quantum state.
Measurement interaction between a measured object and a measuring instrument, if both are initially in a pure state, produces a (final) bipartite entangled state vector. The quasi-classical part of the correlations in it is connected with transmission of information in the measurement. But, prior to "reading" the instrument, there is also purely quantum entanglement in the final state vector. It is shown that in repeatable measurement quantitatively the entanglement equals the amount of incompatibility between the measured observable and the final state. It also equals the amount of incompatibility of the observable and the initial state of the object.
The resource theory of coherence addresses the extent to which quantum properties are present in a given quantum system. While coherence has been extensively studied for individual quantum states, measures of coherence for ensembles of quantum states remain an area of active research. The entanglement-based approach to ensemble coherence-arising from the measurement-ensemble duality principle and the Born rule-connects the ensemble coherence with both the entanglement resource and the measurement's uncertainty. This paper presents two methods for generating ensemble coherence from a fixed amount of entanglement between two qubit systems. The first method involves applying a von Neumann measurement to one part of a non-maximally entangled bipartite state, resulting in a pair of non-orthogonal states whose coherence can equal the initial entanglement. The second method considers a class of symmetric observables capable of generating ensembles used in quantum key distribution (QKD) protocols such as B92, BB84, and three-state QKD. As a result, this work contributes to understanding how much ensemble coherence can be obtained from a given amount of entanglement.
Abstract Uncertainty and entanglement are both profound and key concepts in quantum theory. For three observables, recent works have revealed the tightest uncertainty constants for both product and summation forms. In this work, we give an alternative proof of the uncertainty relations, as well as a physical interpretation of the uncertainty constants. Our results show that such relations are intimately connected with the distinguishment between separable and entangled states. The uncertainty constant is a critical point whether such a condition of entanglement detection is valid.