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

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.

Figure 1

(Color online) The original EPR gedanken experiment. Two particles move from a source $S$ into spatially separated regions $A$ and $B$, and yet continue to have maximally correlated positions and anticorrelated momenta. This means one may make an instant prediction, with 100% accuracy, of either the position or momentum of particle $A$ by performing a measurement at $B$.

Figure 2

(Color online) The Bohm gedanken EPR experiment. Two spin-$\frac{1}{2}$ particles prepared in a singlet state move from the source into spatially separated regions $A$ and $B$, and give anticorrelated outcomes for $J_{\theta }^A$ and $J_{\theta }^B$, where $\theta$ is $x$, $y$, or $z$.

Figure 3

(Color online) Schematic diagram of the measurement of the EPR paradox using field quadrature phase amplitudes. Spatially separated fields $A$ and $B$ radiate outwards from the EPR source, usually given by Eq. (24). The field quadrature amplitudes are symbolized $Y$ and $X$. The fields combine with an intense local oscillator LO field, at beam splitters BS. The outputs of each BS are detected by photodiodes and their difference current is proportional to the amplitude $Y$ or $X$, depending on the phase shift $\theta$. A gain $g$ is introduced to read out the final conditional variances, Eq. (30). Here ${\eta }_A$ and ${\eta }_B$ are the nonideal efficiencies that model losses, defined in Sec. 5.

Figure 4

Effect of detector efficiencies ${\eta }_A$ and ${\eta }_B$ on the EPR paradox. The inference error product $\epsilon ={\Delta }_{\inf }{X}^A{\Delta }_{\inf }{Y}^A$ is plotted for a two-mode squeezed state with $r=2$: ${\eta }_A={\eta }_B=\eta$ (solid line); fixed ${\eta }_A=1$ but varying $\eta ={\eta }_B$ (dashed line); fixed ${\eta }_B=1$ but varying $\eta ={\eta }_A$ (dash-dotted line). The EPR paradox is sensitive to the losses ${\eta }_B$ of the steering system $B$, but insensitive to ${\eta }_A$, those of the inferred system $A$. No paradox is possible for ${\eta }_B⩽0.5$, regardless of ${\eta }_A$, but a paradox is always possible with ${\eta }_B>0.5$, provided only ${\eta }_A>0$.

Figure 5

The original EPR parametric downconversion experiment using an intracavity nonlinear crystal and homodyne detection, following the procedure depicted in Fig. 3. Figure reprinted from 138, with permission.

Figure 6

EPR paradox measure and entanglement measures: (a) ${\epsilon }^2$ [Eqs. (23, 20, 33)] and (b) normalized entanglement measure $D$ of Duan et al. [Eq. (34)] vs total efficiency $\eta$. The dashed lines are theoretical predictions for ${\epsilon }^2$ and $\mathcal{D}$. The points are experimental data with error bars. It is more difficult to realize the EPR paradox than to demonstrate entanglement. From 21.

Figure 7

The original demonstration of pulsed EPR entanglement. The soliton experiment uses orthogonal polarization modes in a fiber Sagnac interferometer and a Mach-Zehnder interferometer for fiber-birefringence compensation. Notation: $\lambda ∕2$ is half-wave plate; $G$ is a gradient index len; $50∕50$ is beam splitter of 50% reflectivity; $\widehat{s}$ and $\widehat{p}$ are two amplitude squeezed beams from the respective polarization states; $\widehat{a}$ and $\widehat{b}$ are EPR entangled beams. From 168.

Figure 8

(Color online) Schematic of quantum teleportation and entanglement swapping. In teleportation, Alice and Bob share a pair of entangled beams. $|{\psi }_{\mathrm{in}}⟩$ is the input state Alice teleports to Bob. The use of electro-optic feedforward on both the amplitude and phase quadrature on Bob’s entangled beam produces an output state $|{\psi }_{\mathrm{out}}⟩$, which he measures using optical homodyne detection, as in Fig. 3. In entanglement swapping, Alice and Victor also share a pair of entangled beams. Alice uses her share of this pair as the input state $|{\psi }_{\mathrm{in}}⟩$. The teleportation protocol is again performed. Victor verifies the efficacy of entanglement swapping using conditional variance measurements of his entangled beam with Bob’s teleportation output beam. The elements are: beam splitters BS, local oscillator LO, phase shift $\theta$, difference/sum currents +/−. Semicircles are photodiodes, while triangles show electronic gain.

Figure 9

A history of continuous variable EPR experiments: (a) EPR paradox measure ${\epsilon }^2$ [Eq. (23)] and (b) inseparability measure $D$ [Eq. (34)]. Where $D<0.5$, one can infer an EPR paradox, using $\epsilon ⩽2D$ (Sec. 6). The grey labels in (a) indicate that ${\epsilon }^2$ has not been measured directly, but is inferred by the authors. From (b) we see that an EPR paradox could have been inferred in other experiments as well. (i) 138, (ii) 206 (inferred from a variance product measurement), (iii) 168, (iv) 103, (v) 156, (vi) 24, (vii) 21, (viii) 80, (ix) 102, (x) 91, (xi) 179, (xii) 118, (xiii) 193, (xiv) 99, (xv) 187, (xvi) 132, (xvii) 100, (xviii) 178, (xix) 199, (xx) 204, (xxi) 48, (xxii) 106, (xxiii) 83, (xxiv) 184, and (xxv) 25. Inseparability has also been verified using other measures, such as negativity (139).