We conjecture uncertainty relations that restrict correlations between the results of measurements performed by two separate parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when one party measures a single observable and the other party measures one of two observables. The uncertainty relation does not follow from the Maassen-Uffink uncertainty relation and is much stronger than the Hall uncertainty relation derived from the latter. The second uncertainty relation bounds the sum of two mutual informations when each party measures one of two observables. We provide numerical evidence for the validity of the conjectured uncertainty relations, and we prove them for large classes of states and observables.
We conjecture uncertainty relations that restrict correlations between the results of measurements performed by two separate parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when one party measures a single observable and the other party measures one of two observables. The uncertainty relation does not follow from the Maassen-Uffink uncertainty relation and is much stronger than the Hall uncertainty relation derived from the latter. The second uncertainty relation bounds the sum of two mutual informations when each party measures one of two observables. We provide numerical evidence for the validity of the conjectured uncertainty relations, and we prove them for large classes of states and observables.
We conjecture a new entropic uncertainty principle governing the entropy of complementary observations made on a system given side information in the form of quantum states, generalizing the entropic uncertainty relation of Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)]. We prove a special case for certain conjugate observables by adapting a similar result found by Christandl and Winter pertaining to quantum channels [IEEE Trans. Inf. Theory 51, 3159 (2005)], and discuss possible applications of this result to the decoupling of quantum systems and for security analysis in quantum cryptography.
We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation suggesting that the CQC relation is generally valid. Importantly, we also show that the CQC relation represents an improvement to an entropic uncertainty principle in the presence of a quantum memory, and that it can be used to verify an achievable secret key rate in the quantum one-time pad cryptographic protocol.
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 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 prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable A weakly followed by the strong measurement of another observable B (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable B weakly followed by the strong measurement of the observable A (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.
Uncertainty relations are old, yet potentially rewarding to explore. By introducing a quantity called the uncertainty matrix, we provide a link between purity and observable incompatibility, and derive several stronger uncertainty relations in both forward and reverse directions for arbitrary quantum states, i.e., mixed as well as pure, and arbitrary incompatible quantum observables, none of which suffer from the problem of triviality. Besides the tightness, the interpretations of terms in these uncertainty relations may be of independent inter- est. We provide the possible generalization of stronger uncertainty relations to sum of variances of more than two observables. We also demonstrate applications of techniques used here to, firstly, obtain a simple reverse quantum speed limit for quantum states undergoing Markovian dynamical evolution, and secondly, to provide a lower bound for fidelity between two quantum states.