The EPR Paradox, formally introduced in the 1935 paper “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” by Boris Podolsky, Albert Einstein, and Nathan Rosen, is a thought experiment designed to challenge the completeness of the standard Copenhagen interpretation of quantum mechanics. The paradox hinges on the concept of entanglement and the apparent conflict between quantum predictions and the philosophical principle of local realism.
Philosophical Premise: Reality and Completeness
Podolsky, Einstein, and Rosen defined the criteria for a “complete” physical theory. A theory is deemed complete if every element of physical reality has a corresponding element in the theory. They further stipulated two necessary conditions for an element to be considered real:
- Criterion of Reality: If, without in any way disturbing a system, one can predict with certainty (i.e., probability equal to 1) the value of a physical quantity, then there exists an element of physical reality corresponding to that quantity.
- Locality: Measurements performed on one system cannot instantaneously influence the physical reality of a spatially separated system.
The EPR argument showed that if one accepts local realism, the quantum mechanical description of entangled systems cannot be complete, as it implies either an acceptance of “spooky action at a distance” or that certain physical quantities exist that quantum mechanics cannot describe.
The Entangled System
The canonical EPR thought experiment involves a pair of particles, traditionally photons or electrons, prepared in a joint quantum state exhibiting maximal entanglement. Consider a pair of particles, A and B, whose total spin is zero, meaning they are prepared in the singlet state:
$$\left|\Psi\right\rangle = \frac{1}{\sqrt{2}}(\left|\uparrow_A\right\rangle\left|\downarrow_B\right\rangle - \left|\downarrow_A\right\rangle\left|\uparrow_B\right\rangle)$$
where $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ represent spin-up and spin-down along a chosen axis (conventionally, the $z$-axis). The particles are then separated by a large distance $D$.
If an experimenter measures the $z$-spin of particle A ($\sigma_z(A)$), the measurement yields either $+\hbar/2$ or $-\hbar/2$ with equal probability. Crucially, the measurement instantaneously projects particle B into the corresponding definite state. If A is measured as spin-up, B must instantaneously be spin-down, and vice versa.
The core absurdity, from the EPR perspective, lies in this: since the measurements on A are performed without disturbing B (due to the spatial separation), both $\sigma_z(A)$ and $\sigma_z(B)$ must correspond to definite elements of reality according to the Criterion of Reality. Furthermore, by performing a different measurement on A, say the $x$-spin ($\sigma_x(A)$), the experimenter could have instead determined the value of $\sigma_x(B)$ with certainty.
Since quantum mechanics mandates that the observables $\sigma_z$ and $\sigma_x$ cannot simultaneously have definite values prior to measurement (due to the non-commuting nature of the operators, $[\sigma_z, \sigma_x] \neq 0$), the theory must be incomplete, as it cannot account for the reality of both $z$-spin and $x$-spin for particle B prior to observation.
Einstein’s Discontent and Hidden Variables
Einstein famously referred to entanglement as spukhafte Fernwirkung (“spooky action at a distance”). The EPR analysis suggested that quantum mechanics failed to describe the underlying physical reality, which must, in fact, be determined by hidden variables—pre-existing properties carried by the particles from the moment of their creation, which determine the measurement outcomes irrespective of measurement location.
The implied structure required by EPR is that the particles carry two sets of instructions simultaneously: one set for the $z$-measurement outcomes and another set for the $x$-measurement outcomes. Quantum mechanics, operating on the wave function, only describes the potentiality of these outcomes, not the actuality residing within the system itself.
| Concept | EPR Interpretation (Local Realism) | Copenhagen Interpretation |
|---|---|---|
| Reality of $\sigma_z(B)$ | Exists prior to measurement of A. | Becomes definite only upon measurement of A. |
| Completeness of QM | Incomplete; requires hidden variables. | Complete description of measurable phenomena. |
| Locality | Must be preserved; influences are local. | Apparent non-locality in correlations is epistemic, not causal. |
Resolution through Bell’s Theorem
The EPR Paradox remained a philosophical debate until the work of John Bell in the 1960s. Bell formalized the EPR challenge into a mathematically testable form known as Bell’s Theorem.
Bell derived a set of inequalities (the Bell inequalities) that must be satisfied by any physical theory based on local hidden variables—that is, any theory adhering to the philosophical requirements laid out by EPR.
The crucial breakthrough was realizing that the predictions of quantum mechanics for correlations in entangled systems violate Bell’s inequalities under certain measurement angle configurations.
Bell Inequality Example (CHSH Form)
The Clauser-Horne-Shimony-Holt (CHSH) inequality places an upper bound on the correlation function $S$ achievable under local realism:
$$|S| \leq 2$$
Where $S$ is constructed from the measurement correlations $E(a, b)$:
$$S = E(a, b) - E(a, b’) + E(a’, b) + E(a’, b’)$$
Here, $a, a’, b, b’$ represent the settings (measurement axes) chosen for particles A and B. Quantum mechanics, however, predicts that $S$ can reach a maximum value of $2\sqrt{2} \approx 2.828$ (the Tsirelson bound) when the measurement angles are set optimally.
Experimental Verification
Subsequent experiments, starting notably with the work of Alain Aspect in the early 1980s, consistently demonstrated violations of Bell inequalities, confirming the predictions of quantum mechanics over local realism. These experiments effectively ruled out all local hidden variable theories, forcing a concession: either locality or realism (or both, as defined by EPR) must be abandoned in the quantum realm.
Modern loophole-free experiments have confirmed these violations with high statistical certainty, strongly suggesting that the correlations observed in entangled systems are fundamentally non-local or that physical properties are not determined until measurement occurs. Many physicists interpret this as affirming the radical ontological commitments of the Copenhagen view, where physical reality, at the quantum level, is inherently contextual and dependent on the experimental arrangement.
The EPR Paradox, intended to expose the flaws in quantum theory, instead served to delineate the precise boundary between classical intuition and quantum reality. The paradox is resolved not by finding a flaw in quantum mechanics, but by demonstrating that local realism is incompatible with observed physical phenomena.