We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the EPR paradox. Entropic uncertainty relations are used to reveal quantum steering for non-Gaussian continuous variable states. Entropic uncertainty relations for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tipartite systems. The Robertson-Schrodinger uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.
We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the EPR paradox. Entropic uncertainty relations are used to reveal quantum steering for non-Gaussian continuous variable states. Entropic uncertainty relations for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tipartite systems. The Robertson-Schrodinger uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.
We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the Einstein, Podolsky and Rosen (EPR) paradox. Entropic uncertainty relations (EURs) are used to reveal quantum steering for non-Gaussian continuous variable states. EURs for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tripartite systems. The Robertson-Schrödinger (RS) uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.
We generalize entropic uncertainty relations in the presence of quantum memory [Nature Physics 6 (659), 2010], and [Physical Review Letters 106 (110506), 2011] in two directions. First, we consider measurements with a continuum of outcomes, and, second, we allow for infinite-dimensional quantum memory. To achieve this, we introduce conditional differential entropies for classical-quantum states on von Neumann algebras, and show approximation properties for these entropies. As an example, we evaluate the uncertainty relations for position-momentum measurements, which has applications in continuous variable quantum cryptography and quantum information theory.
Abstract Nonlocality, which is the key feature of quantum theory, has been linked with the uncertainty principle by fine-grained uncertainty relations, by considering combinations of outcomes for different measurements. However, this approach assumes that information about the system to be fine-grained is local, and does not present an explicitly computable bound. Here, we generalize above approach to general quasi-fine-grained uncertainty relations (QFGURs) which applies in the presence of quantum memory and provides conspicuously computable bounds to quantitatively link the uncertainty to entanglement and Einstein–Podolsky–Rosen (EPR) steering, respectively. Moreover, our QFGURs provide a framework to unify three important forms of uncertainty relations, i.e., universal uncertainty relations, the uncertainty principle in the presence of quantum memory, and fine-grained uncertainty relation. This result gives a direct significance to uncertainty principle, and allows us to determine whether a quantum measurement exhibits typical quantum correlations, meanwhile, it reveals a fundamental connection between basic elements of quantum theory, specifically, uncertainty measures, combined outcomes for different measurements, quantum memory, entanglement and EPR steering.
In presence of quantum memory [M. Berta, M. Christandl, R. Colbeck, J.M. Renes, and R. Renner, Nature Phys. 6, 659 (2010)] the lower bound of entropic uncertainty relation depends on amount of entanglement between the particle (on which two incompatible observable are measured) and the quantum memory. On other hand, fine grained uncertainty relation [J. Oppenheim and S. Wehner, Science 330, 1072 (2010)] gives the upper bound of winning non-local (CHSH) game. We construct similar kind of quantum game and relate two uncertainty relation for measurement of any two observable on the quantum system in presence of quantum memory and show there are certain region where the lower bound of entropic uncertainty relation in presents of quantum memory given by Berta et. al. should not be achieve by spin projective measurement.
The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We present tighter bounds on both entropic uncertainty relation and information exclusion relation for multiple measurements in the presence of quantum memory. As applications, three incompatible measurements on Werner state and Horodecki's bound entangled state are investigated in details.
Abstract Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position x and momentum p . This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take x - p correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to x and p , but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that takes correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.