This Colloquium examines the field of the Einstein, Podolsky, and Rosen (EPR) gedanken experiment, from the original paper of Einstein, Podolsky, and Rosen, through to modern theoretical proposals of how to realize both the continuous-variable and discrete versions of the EPR paradox. The relationship with entanglement and Bell’s theorem are analyzed, and the progress to date towards experimental confirmation of the EPR paradox is summarized, with a detailed treatment of the continuous-variable paradox in laser-based experiments. Practical techniques covered include continuous-wave parametric amplifier and optical fiber quantum soliton experiments. Current proposals for extending EPR experiments to massive-particle systems are discussed, including spin squeezing, atomic position entanglement, and quadrature entanglement in ultracold atoms. Finally, applications of this technology to quantum key distribution, quantum teleportation, and entanglement swapping are examined.

We introduce the study of dynamical quantum noise in Bose-Einstein condensates through numerical simulation of stochastic partial differential equations obtained using phase-space representations. We derive evolution equations for a single trapped condensate in both the positive-$P$ and Wigner representations and perform simulations to compare the predictions of the two methods. The positive-$P$ approach is found to be highly susceptible to the stability problems that have been observed in other strongly nonlinear, weakly damped systems. Using the Wigner representation, we examine the evolution of several quantities of interest using from a variety of choices of initial state for the condensate and compare results to those for single-mode models.

The squeezing spectrum for nondegenerate parametric oscillation above threshold is calculated, including phase diffusion. A nonclassical correlation in phase and intensity occurs which is an example of the Einstein-Podolsky-Rosen paradox, even in fields of large photon number.

The capacity $C$ of a communication channel is the maximum rate at which information can be transmitted without error from the channel's input to its output. The authors review quantum limits on the capacity that can be achieved with linear bosonic communication channels that have input power $P$. The limits arise ultimately from the Einstein relation that a field quantum at frequency $f$ has energy $E=hf$. A single linear bosonic channel corresponds to a single transverse mode of the bosonic field—i.e., to a particular spatial dependence in the plane orthogonal to the propagation direction and to a particular spin state or polarization. For a single channel the maximum communication rate is $C_{\mathrm{WB}}=\left(\frac{\pi }{\ln 2}\right)\sqrt{\frac{2P}{3h}}$ bits/s. This maximum rate can be achieved by a "number-state channel," in which information is encoded in the number of quanta in the bosonic field and in which this information is recovered at the output by counting quanta. Derivations of the optimum capacity $C_{\mathrm{WB}}$ are reviewed. Until quite recently all derivations assumed, explicitly or implicitly, a number-state channel. They thus left open the possibility that other techniques for encoding information on the bosonic field, together with other ways of detecting the field at the output, might lead to a greater communication rate. The authors present their own general derivation of the single-channel capacity upper bound, which applies to any physically realizable technique for encoding information on the bosonic field and to any physically realizable detection scheme at the output. They also review the capacities of coherent communication channels that encode information in coherent states and in quadrature-squeezed states. A three-dimensional bosonic channel can employ many transverse modes as parallel single channels. An upper bound on the information flux that can be transferred down parallel bosonic channels is derived.