Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications

Abstract

This Colloquium examines the field of the Einstein, Podolsky, and Rosen ͑EPR͒ 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.

Einstein–Podolsky–Rosen-Like Correlation on a Coherent-State Basis and Inseparability of Two-Mode Gaussian States

“…Since the off-diagonal elements of the position operatorx = √ C(â +â † ) in the number basis are real we have Γ[X] = X. In contrast, the off-diagonal elements of the momentum operatorp = √ C(â −â † )/i in the number basis are pure imaginary, and we thus have 2 , we can estimate the left-hand side of Eq. (8) as …”

“…This type of seeming inconsistency between the quantum correlation and the canonical uncertainty relation is often termed as the EPR paradox and has been providing insightful aspects on foundations of quantum physics and theory of entanglement [2][3][4][5]. The EPR-type correlation is normally described by the variances of the EPR-type operatorsx A −x B and p A +p B , and the measured uncertainties can be a signature of quantum entanglement.…”

On Schmidt Decomposition: Approach Based on Correlation Operator as Bipartite Entanglement Entity

“…This is how it is a generalization. Form (29) of expansion in a subsystem basis is relevant for Schrödinger's steering discussed in detail in §6.3 below.…”

“…(iv) Remark 6 and relation (29) below open the way for a more fruitful application of (1), particularly for Schrödinger's important concept of steering (cf §6.3).…”

“…Relation (29) introduces explicitly the correlation operator into investigations of the effects on the distant subsystem A caused by measurement performed on the nearby subsystem B . This enabled the Belgrade school to have an original angle and elaborate Schrödinger's approach to distant correlations.…”