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We formulate the conditional-variance uncertainty relations for general qubit systems and arbitrary observables via the inferred uncertainty relations. We find that the lower bounds of these conditional-variance uncertainty relations can be written in terms of entanglement measures including concurrence, $G$ function, quantum discord quantified via local quantum uncertainty in different scenarios. We show that the entanglement measures reduce these bounds, except quantum discord which increases them. Our analysis shows that these correlations of quantumness measures play different roles in determining the lower bounds for the sum and product conditional variance uncertainty relations. We also explore the violation of local uncertainty relations in this context and in an interference experiment.
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We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schrodinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the $2times 2$ (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.
The Wehrl entropy is an entropy associated to the Husimi quasi-probability distribution. We discuss how it can be used to formulate entropic uncertainty relations and for a quantification of entanglement in continuous variables. We show that the Wehrl-Lieb inequality is closer to equality than the usual Bia{l}ynicki-Birula and Mycielski entropic uncertainty relation almost everywhere. Furthermore, we show how a Wehrl mutual information can be used to obtain a measurable perfect witness for pure state bipartite entanglement, which additionally provides a lower bound on the entanglement entropy.
New sum and product uncertainty relations, containing variances of three or four observables, but not containing explicitly their covariances, are derived. One of consequences is the new inequality, giving a nonzero lower bound for the product of two variances in the case of zero mean value of the commutator between the related operators. Moreover, explicit examples show that in some cases this new bound can be better than the known Robertson--Schrodinger one.
Genuine multipartite entanglement (GME) offers more significant advantages in quantum information compared with entanglement. We propose a sufficient criterion for the detection of GME based on local sum uncertainty relations for chosen observables of subsystems. We apply the criterion to detect the GME properties of noisy $n$-partite W state when $n = 3, 4, 5$ and $6$, and find that the criterion can detect more noisy W states when $n$ ranges from 4 to 6. Moreover, the criterion is also used to detect the genuine entanglement of $3$-qutrit state. The result is stronger than that based on GME concurrence and fisher information.
We analyze general uncertainty relations and we show that there can exist such pairs of non--commuting observables $A$ and $B$ and such vectors that the lower bound for the product of standard deviations $Delta A$ and $Delta B$ calculated for these vectors is zero: $Delta A,cdot,Delta B geq 0$. We show also that for some pairs of non--commuting observables the sets of vectors for which $Delta A,cdot,Delta B geq 0$ can be complete (total). The Heisenberg, $Delta t ,cdot, Delta E geq hbar/2$, and Mandelstam--Tamm (MT), $ tau_{A},cdot ,Delta E geq hbar/2$, time--energy uncertainty relations ($tau_{A}$ is the characteristic time for the observable $A$) are analyzed too. We show that the interpretation $tau_{A} = infty$ for eigenvectors of a Hamiltonian $H$ does not follow from the rigorous analysis of MT relation. We show also that contrary to the position--momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of $A$ and $H$.
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