The EPR Experiment and Entanglement

The EPR Paper and Entanglement
by Thomas Lee Abshier, ND
3/23/2025

The famous 1935 EPR paper refers to the seminal work by Albert Einstein, Boris Podolsky, and Nathan Rosen, titled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”. This paper raised fundamental questions about the completeness of quantum mechanics and introduced what would later be understood as quantum entanglement, which Einstein famously described as “spooky action at a distance.”

Overview of the Paper

The EPR paper questioned whether the wavefunction and the quantum mechanical formalism provided a complete description of physical reality. The authors proposed a thought experiment involving two particles that interact and then separate, arguing that quantum mechanics predicts correlations between the particles that challenge the concept of “local realism.”

---

1. Key Assumptions in the EPR Paper

The EPR argument was based on two key assumptions about physical reality:

1. Reality Criterion:
   • If a physical property of a system can be predicted with certainty (with 100% probability) without disturbing the system, then that property must be an element of physical reality, meaning it objectively exists.
2. Locality:
   • Physical influences cannot propagate faster than the speed of light. In other words, a measurement performed on one particle cannot instantaneously affect the state of another particle at a distant location.

---

2. The Thought Experiment

The EPR paper considered a pair of particles that interact and then separate, conserving the system’s total momentum and position. Using this setup, they demonstrated a paradox between quantum mechanics’ predictions and their assumptions about reality and locality.

Setup:

• Two particles, $A$ and $B$, are created in a state where their total momentum and total position are precisely defined (due to conservation laws):
  • The total momentum of the system is conserved: pA+pB=Ptotalp_A + p_B = P_{\text{total}}pA​+pB​=Ptotal​.
  • The relative position of the particles is fixed: xA−xB=Xrelativex_A – x_B = X_{\text{relative}}xA​−xB​=Xrelative​.

Quantum Predictions:

1. Position Measurement:
   • If you measure the position of particle $A$, you can immediately determine the position of particle $B$ because its relative position is fixed.
   • For example, if xAx_AxA​ is measured, xBx_BxB​ can be calculated as xB=xA−Xrelativex_B = x_A – X_{\text{relative}}xB​=xA​−Xrelative​.
2. Momentum Measurement:
   • Similarly, if you measure the momentum of particle $A$, you can immediately determine the momentum of particle $B$ because total momentum is conserved.
   • For example, if pAp_ApA​ is measured, pBp_BpB​ can be calculated as pB=Ptotal−pAp_B = P_{\text{total}} – p_ApB​=Ptotal​−pA​.

Paradox:

• According to quantum mechanics, the wavefunction of the two particles describes a superposition of all possible momenta and positions. The act of measurement on one particle collapses its wavefunction, instantaneously affecting the other particle, regardless of the distance between them.
• This implies a nonlocal correlation between the particles, which Einstein called “spooky action at a distance.”

EPR’s Argument:

• If measuring particle AA’s position gives you particle BB’s position without disturbing particle $B$, then position must be an element of reality for particle $B$.
• Similarly, if measuring particle AA’s momentum gives you particle BB’s momentum without disturbing particle $B$, then momentum must be an element of reality for particle $B$.
• However, quantum mechanics asserts that position and momentum cannot simultaneously have definite values (due to the Heisenberg uncertainty principle). This creates a contradiction:
  • Either quantum mechanics is incomplete (there must be hidden variables that determine position and momentum), or nature allows for nonlocal interactions.

---

3. Conservation of Momentum and Entanglement

The prediction of “spooky action at a distance” and entanglement arises naturally from the conservation of momentum in the EPR setup:

1. Conservation of Momentum:
   • When two particles interact and then separate, their total momentum is conserved. This means that the momentum of one particle is always correlated with the momentum of the other, regardless of the distance between them.
   • As a result, the joint quantum state of the two particles can only be described as an entangled state:
     ∣ψ⟩=∫dp ∣pA⟩∣Ptotal−p⟩|\psi\rangle = \int dp \, |p_A\rangle |P_{\text{total}} – p\rangle∣ψ⟩=∫dp∣pA​⟩∣Ptotal​−p⟩

     Here, ∣pA⟩|p_A\rangle∣pA​⟩ is the momentum state of particle $A$, and ∣Ptotal−p⟩|P_{\text{total}} – p\rangle∣Ptotal​−p⟩ is the momentum state of particle $B$.

2. Nonlocality:
   • When the momentum of particle $A$ is measured, its wavefunction collapses to a definite state ∣pA⟩|p_A\rangle∣pA​⟩.
   • Instantaneously, the wavefunction of particle $B$ collapses to ∣Ptotal−pA⟩|P_{\text{total}} – p_A\rangle∣Ptotal​−pA​⟩, regardless of the distance between the particles.
   • This instantaneous correlation violates the locality principle, leading Einstein to describe it as “spooky action at a distance.”
3. Position Correlations:
   • Similarly, the conservation of position correlations means that if one particle’s position is measured, the position of the other particle is immediately determined.
   • This reinforces the idea that the two particles are entangled in position and momentum.

---

4. Implications

The EPR paper concluded that quantum mechanics is incomplete because it fails to provide a local, deterministic description of reality. The authors suggested that there might be “hidden variables” that determine the outcomes of measurements and resolve the apparent paradox.

However, subsequent developments (most notably Bell’s theorem in 1964) showed that no local hidden variable theory can reproduce all the predictions of quantum mechanics. Experiments testing Bell’s inequalities (such as those by Alain Aspect in the 1980s) confirmed the nonlocal nature of quantum entanglement, validating the predictions of quantum mechanics.

---

5. Modern Perspective

Today, the EPR paper is seen as a critical milestone in developing quantum mechanics and studying entanglement. It provided the theoretical foundation for:

• Quantum Entanglement:
  • A phenomenon where particles exhibit correlations that cannot be explained by classical physics or local hidden variables.
• Quantum Information Science:
  • Technologies like quantum teleportation, quantum cryptography, and quantum computing rely on entanglement.
• Nonlocality:
  • While quantum mechanics is inherently nonlocal, it does not violate causality because no information can travel faster than the speed of light.

In essence, the EPR paper highlighted quantum mechanics’ profound and counterintuitive nature, challenging us to rethink the fundamental concepts of reality, locality, and causality.