Uncertainty Relation for Mutual Information

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 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.Comment: 6 pages, 2 figure

Improving Einstein-Podolsky-Rosen Steering Inequalities with State Information

We discuss the relationship between entropic Einstein-Podolsky-Rosen (EPR)-steering inequalities and their underlying uncertainty relations, along with the hypothesis that improved uncertainty relations lead to tighter EPR-steering inequalities. In particular, we discuss how the intrinsic uncertainty in a mixed quantum state is used to improve existing uncertainty relations and how this information affects one's ability to witness EPR-steering. As an example, we consider the recent improvement (using a quantum memory) to the entropic uncertainty relation between pairs of discrete observables (Nat. Phys. 6, 659 (2010)) and show that a trivial substitution of the tighter bound in the steering inequality leads to contradictions, due in part to the fact that the improved bound depends explicitly on the state being measured. By considering the assumptions that enter into the development of a steering inequality, we derive correct steering inequalities from these improved uncertainty relations and find that they are identical to ones already developed (Phys. Rev. A, 87, 062103 (2013)). In addition, we consider how one can use the information about the quantum state to improve our ability to witness EPR-steering, and develop a new symmetric EPR-steering inequality as a result.Comment: 6 page

Compressive Direct Imaging of a Billion-Dimensional Optical Phase-Space

Optical phase-spaces represent fields of any spatial coherence, and are typically measured through phase-retrieval methods involving a computational inversion, interference, or a resolution-limiting lenslet array. Recently, a weak-values technique demonstrated that a beam's Dirac phase-space is proportional to the measurable complex weak-value, regardless of coherence. These direct measurements require scanning through all possible position-polarization couplings, limiting their dimensionality to less than 100,000. We circumvent these limitations using compressive sensing, a numerical protocol that allows us to undersample, yet efficiently measure high-dimensional phase-spaces. We also propose an improved technique that allows us to directly measure phase-spaces with high spatial resolution and scalable frequency resolution. With this method, we are able to easily measure a 1.07-billion-dimensional phase-space. The distributions are numerically propagated to an object placed in the beam path, with excellent agreement. This protocol has broad implications in signal processing and imaging, including recovery of Fourier amplitudes in any dimension with linear algorithmic solutions and ultra-high dimensional phase-space imaging.Comment: 7 pages, 5 figures. Added new larger dataset and fixed typo

Position-Momentum Bell-Nonlocality with Entangled Photon Pairs

Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Though Bell nearly showed a violation using the CHSH inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he doesn't give a quantitative estimation of the maximum achievable violation for the wavefunction he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of $2\sqrt{2}$. Although Bell's wavefunction does not produce a large violation of the CHSH inequality, other states may yet do so.Comment: 6 pages, 3 figure

EPR Steering Inequalities from Entropic Uncertainty Relations

We use entropic uncertainty relations to formulate inequalities that witness Einstein-Podolsky-Rosen (EPR) steering correlations in diverse quantum systems. We then use these inequalities to formulate symmetric EPR-steering inequalities using the mutual information. We explore the differing natures of the correlations captured by one-way and symmetric steering inequalities, and examine the possibility of exclusive one-way steerability in two-qubit states. Furthermore, we show that steering inequalities can be extended to generalized positive operator valued measures (POVMs), and we also derive hybrid-steering inequalities between alternate degrees of freedom.Comment: 10 pages, 2 figure