Entanglement and Its Verification: A Tutorial on Classical and Quantum Correlations

TL;DR

This tutorial defines entanglement as nonseparability and contrasts quantum correlations with classical ones, using Schrödinger's cat and Bell states as illustrative examples. It develops two operational verification routes: an uncertainty-based EPR-Reid criterion for continuous variables and a CHSH-type Bell test for discrete two-level systems, with derivations and examples in the main text and appendices. The key results show that entanglement can be witnessed either by violating Δx_- Δp_+ ≥ ħ/2 or by obtaining |⟨B⟩| > 2, with maximal quantum violations reaching the Tsirelson bound 2√2; the precise bounds and derivations are provided in Appendices A–C. The discussion covers experimental implementations with position-momentum entangled photons and polarization-entangled photons, emphasizes the role of measurement bases and coherence, and frames entanglement as a resource for quantum technologies and a gateway to nonlocality.

Abstract

Entanglement, a defining property of quantum mechanics in which two physical subsystems cannot be seen as independent entities, challenges our everyday experience and classical intuition. However, only such strong quantum correlations enable quantum technologies, including quantum computing or communication, while revealing the limits of our classical worldview by violating local realism. Given its importance in modern quantum science, we present this tutorial addressing the questions: What is entanglement, how does it differ from classical correlations, and how can it be experimentally verified? Using celebrated examples, such as Schrödinger's cat, we highlight the distinction between classical and quantum correlations and illustrate the definition of entangled and separable states. We review entanglement criteria by discussing Heisenberg-type uncertainty relations for continuous variables and the CHSH inequality for discrete systems. Focusing on concepts of quantum correlations and operational entanglement witnesses, we provide accessible tools and illustrative examples aimed at demystifying entanglement for a broad readership.

Figures (7)

• Figure 1: Visualization of the idea behind the EPR paradox Einstein35. Independent measurements of observables associated with subsystem $a$ or $b$, each described by elements of the associated Hilbert space $\mathcal{H}_j$ can always be performed with absolute certainty. The EPR state $\ket{\psi_\text{EPR}}= \int\! \text{d} p \, \psi(p) \ket{p}_a \otimes \ket{-p}_b$ is a valid element of the joint Hilbert space and displays strong anticorrelations of the momenta between both subsystems. This raises the conceptual question: given these perfect anticorrelated momenta and the ability to measure $x_a$ and $p_b$ with certainty, is it possible to infer with help of the correlation between $p_b$ and $p_a$ both $x_a$ and $p_a$ with certainty?
• Figure 2: Schrödinger's cat Schroedinger1935a is locked alive ($\ket{\heartsuit}_\text{c}$) with an excited atom ($\ket{\uparrow}_\text{a}$) inside a sealed box, together with a Geiger counter and a vial of poison. After a period corresponding to the half-life of the atom, there is a chance of 50 % that the atom decays ($\ket{\downarrow}_\text{a}$). If a decay occurs, it is detected by the Geiger counter that destroys the vial, killing the furball ($\ket{\dagger}_\text{c}$).
• Figure 3: Correlation matrices between the states of the atom and the cat for both the entangled Bell state $\ket{\phi^+}$ and two classically correlated subsystems $\hat{\rho}_\text{cl}$. While in the original basis both states display strong correlations (left), they remain observable in a changed basis only for the entangled Bell state, while the classical state exhibits no correlations. The entries of each matrix represent the probability of detecting the corresponding joint outcome.
• Figure 4: Visualization of the EPR-Reid criterion following Eq. \ref{['eq:EPR-Reid']}: For separable states, the product of uncertainties satisfies $\Delta x_- \Delta p_+ \geq \hbar /2$, a Heisenberg-type uncertainty relation between the position difference between two subsystems and the mean momentum. Since $\hat{x}_-$ and $\hat{p}_+$ commute, entangled states are not constrained by this bound and can achieve $\Delta x_- \Delta p_+ < \hbar /2$. Hence, observing such a violation constitutes a direct verification of entanglement.
• Figure 5: Joint probability distributions of continuous variables in the position basis (left) and momentum basis (right), where the darkness of the color represents the probability density in the joint space. The correlations in the entangled state’s position distribution can be inferred from the widths along the rotated coordinates $x_+ = (x_a + x_b)/\sqrt{2}$ and $x_- = (x_a - x_b)/\sqrt{2}$. Here, both the entangled and the partially mixed states exhibit the same position correlations, demonstrating that position correlations alone are insufficient to distinguish entanglement. In contrast, only the entangled state shows strong anticorrelations in the momentum basis, as seen along the corresponding rotated coordinates $p_+$ and $p_-$, whereas the partially mixed state exhibits no significant momentum correlations.