James Schneeloch

Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations

We use entropic uncertainty relations to formulate inequalities that witness Einstein-PodolskyRosen (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 diering 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.

Violation of continuous-variable Einstein-Podolsky-Rosen steering with discrete measurements.

We create a stronger EPR-steering inequality for continuous variables using entropic uncertainty. We explore the asymmetry in this inequality and develop a new symmetric inequality. We also violate these inequalities in experiment.

Quantum mutual information capacity for high-dimensional entangled states.

High-dimensional Hilbert spaces used for quantum communication channels offer the possibility of large data transmission capabilities. We propose a method of characterizing the channel capacity of an entangled photonic state in high-dimensional position and momentum bases. We use this method to measure the channel capacity of a parametric down-conversion state by measuring in up to 576 dimensions per detector. We achieve a channel capacity over 7 bits/photon in either the position or momentum basis. Furthermore, we provide a correspondingly high-dimensional separability bound that suggests that the channel performance cannot be replicated classically.

Simultaneous Measurement of Complementary Observables with Compressive Sensing

The more information a measurement provides about a quantum systems position statistics, the less information a subsequent measurement can provide about the systems momentum statistics. This information trade-off is embodied in the entropic formulation of the uncertainty principle. Traditionally, uncertainly relations correspond to resolution limits; increasing a detectors position sensitivity decreases its momentum sensitivity and vice versa. However, this is not required in general; for example, position information can instead be extracted at the cost of noise in momentum. Using random, partial projections in position followed by strong measurements in momentum, we efficiently determine the transverse-position and transverse-momentum distributions of an unknown optical field with a single set of measurements. The momentum distribution is directly imaged, while the position distribution is recovered using compressive sensing. At no point do we violate uncertainty relations; rather, we economize the use of information we obtain.

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 using information about the state of a quantum system affects oneʼs ability to witness EPR-steering. As an example, we consider the recent improvement to the entropic uncertainty relation between pairs of discrete observables (Berta et al., 2010 [10]). 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 (Schneeloch et al., 2013 [9]). In addition, we consider how one can use state information to improve our ability to witness EPR-steering, and develop a new continuous variable symmetric EPR-steering inequality as a result.

Introduction to the transverse spatial correlations in spontaneous parametric down-conversion through the biphoton birth zone

As a tutorial to the spatial aspects of Spontaneous Parametric Downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a Double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such Double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.

Compressively Characterizing High-Dimensional Entangled States with Complementary, Random Filtering

The resources needed to conventionally characterize a quantum system are overwhelmingly large for high- dimensional systems. This obstacle may be overcome by abandoning traditional cornerstones of quantum measurement, such as general quantum states, strong projective measurement, and assumption-free characterization. Following this reasoning, we demonstrate an efficient technique for characterizing high-dimensional, spatial entanglement with one set of measurements. We recover sharp distributions with local, random filtering of the same ensemble in momentum followed by position---something the uncertainty principle forbids for projective measurements. Exploiting the expectation that entangled signals are highly correlated, we use fewer than 5,000 measurements to characterize a 65, 536-dimensional state. Finally, we use entropic inequalities to witness entanglement without a density matrix. Our method represents the sea change unfolding in quantum measurement where methods influenced by the information theory and signal-processing communities replace unscalable, brute-force techniques---a progression previously followed by classical sensing.

Demonstrating continuous variable Einstein–Podolsky–Rosen steering in spite of finite experimental capabilities using Fano steering bounds

We show how one can demonstrate continuous-variable Einstein–Podolsky–Rosen (EPR) steering without needing to characterize entire measurement probability distributions. To do this, we develop a modified Fano inequality useful for discrete measurements of continuous variables and use it to bound the conditional uncertainties in continuous-variable entropic EPR-steering inequalities. With these bounds, we show how one can hedge against experimental limitations including a finite detector size, dead space between pixels, and any such factors that impose an incomplete sampling of the true measurement probability distribution. Furthermore, we use experimental data from the position and momentum statistics of entangled photon pairs in parametric downconversion to show that this method is sufficiently sensitive for practical use.

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.

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. Although Bell nearly showed a violation using the Clauser-HorneShimony-Holt (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 does not give a quantitative estimation of the maximum achievable violation for the wave function 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 √ 2. Although Bell’s wave function does not produce a large violation of the CHSH inequality, other states may yet do so. DOI: 10.1103/PhysRevA.93.012105