Entanglement Detection with Rotationally Covariant Measurements -- From Compton Scattering to Lemonade

TL;DR

This work introduces rotationally covariant (RoC) measurements for polarization-encoded qubits, showing that any RoC device is fully described by a single detector-contrast parameter $r$. It provides a POVM formulation that connects RoC measurements to the Klein--Nishina description of Compton scattering and develops an SDP-based entanglement certification method that uses full RoC statistics, including semi-device-independent scenarios. The paper proves no Bell violations are possible with RoC devices but demonstrates steering under one-sided symmetry constraints, and it showcases a tangible lemonade-based experimental demonstration where the detector contrast $r$ is tunable and can exceed the semi-device-independent threshold, enabling entanglement detection with simple, macroscopic detectors. These results unify quantum-information analysis with rotationally symmetric measurement scenarios and point to practical entanglement detection using everyday beverages, while laying groundwork for extensions to higher symmetries and multi-photon states.

Abstract

The accurate and efficient detection of quantum entanglement remains a central challenge in quantum information science. In this work, we study the detection of entanglement of polarized photons for measurement devices that are solely specified by rotational symmetry. We derive explicit positive operator valued measures (POVMs) showing that from a quantum information perspective any such setting is classified by one real measurable parameter r. In Particular, we give a POVM formulation of the Klein--Nishina formula for Compton scattering of polarized photons. We provide an SDP-based entanglement certification method that operates on the full measured statistics and gives tight bounds, also considering semi-device independent scenarios. Furthermore, we show that, while Bell violations are impossible with rotationally covariant measurements, EPR steering can still be certified under one-sided symmetry constraints. Finally, we present a rotationally covariant showcase experiment, analyzing the scattering of polarized optical light in a selection of soft drinks. Our results suggest that lemonade-based detectors are suitable for entanglement detection.

Figures (6)

• Figure 1: a) Rotationally covariant measurement. Measurement data is $2\pi$-invariant. Rotating the preparation system (i.e. process before detection) or the detector are equivalent in description. b) Compton scattering process. Polarized photons scatter from electrons into angles, $\theta$ and $\varphi$. It is rotationally covariant w.r.t. $\varphi$. c) The lemonade experiment. The polarization of photons is rotated by a half-wave plate. Photons scattered in the lemonade are detected at a fixed angle.
• Figure 2: Entanglement detection for equal sample devices $r_\text{A}=r_\text{B}$ in the maximally entangled state $|\Psi^-\rangle\langle \Psi^- |$. The minimal negativities result from solving SDP \ref{['SDP']} with perfect hypothetical devices $r^\text{hyp}_\text{A}=r^\text{hyp}_\text{B}=1$, which, in this case, suffice as they yield the relevant results (see \ref{['Method_semi_device_indep']}). Entanglement detection is possible for those devices, where the minimal negativity is positive, in this scenario: $r_\text{A} \cdot r_\text{B}\ge0,393$.
• Figure 3: Two Compton polarimeters. A source emits orthogonally linearly polarized (entangled) photons which scatter from electrons. Coincidence events of the photons that are scattered in directions $(\theta_\text{A}, 0)$ and $(\theta_\text{B}, \Delta\varphi)$ are detected.
• Figure 4: The experiment: A single mode, fiber-coupled, 30 mW power, $808$ nm, continuous-wave diode laser, followed by a linear polarizer (anti-reflection coated for $808$ nm) with an extinction ratio of $10^8$:$1$. The polarization is rotated by a half-wave plate (anti-reflection coated for $808$ nm). The light scatters from different sample liquids contained in a glass test tube (diameter: $25$ mm) and is measured roughly at a right angle compared to the incoming beam using an optical power meter. Powers were measured with an integration time of 1 s.
• Figure 5: Scattered power distribution in mW for apple lemonade, resulting in $r=0.704\pm0.001$, measured at a distance of $d=15$ mm. $r$ was obtained by fitting the data to $f(\varphi) = a(1 + r\cos(2(\varphi + \varphi_0)))$. The dominant contribution to the uncertainty of $r$ arises from the laser intensity stability.

Theorems & Definitions (2)

• Lemma 1
• proof