The 1935 EPR Argument

The EPR argument is the 1935 challenge by Einstein, Podolsky, and Rosen to the completeness of the quantum-mechanical description. It used entangled two-particle correlations to argue that, if locality and a particular criterion of physical reality are accepted, then the wavefunction cannot be a complete description of an individual physical system.

The calculation below separates the quantum correlations from the additional locality and reality assumptions. For the later experimentally testable inequality, see Bell’s Theorem.

Completeness Question

The title of the EPR paper asked whether the quantum-mechanical description of physical reality can be considered complete. The question was not whether quantum mechanics gave successful predictions. By 1935, the theory was already highly successful. The question was whether the wavefunction supplied a complete specification of physical properties or only a statistical and operational description.

EPR used a general criterion:

• if one can predict a physical quantity with certainty,
• and if doing so does not disturb the system,
• then there is a corresponding element of physical reality.

The argument then combines this criterion with locality. If a measurement on particle 2 is performed far away from particle 1, EPR assume that this choice cannot instantly disturb particle 1. Under that assumption, a certain remote prediction about particle 1 reveals something that was already real for particle 1.

The conflict arises because quantum mechanics does not assign simultaneous sharp values to noncommuting observables such as position and momentum. EPR concluded that the quantum state description is incomplete if locality and their reality criterion are maintained.

The reality criterion is a sufficient condition, not a general definition of reality. Bohr’s 1935 reply challenged the meaning of nondisturbance when different experimental arrangements support different predictions; see Einstein–Bohr Debates.

Locality Assumption

EPR’s locality premise is not yet Bell’s local-causality factorization. It is a physical assumption about separated systems: the choice of what to measure here should not instantly alter the physical reality of a distant system.

This distinction matters. The EPR paper did not derive a Bell inequality, did not test hidden variables experimentally, and did not show that quantum mechanics permits controllable superluminal communication. It argued that quantum mechanics is incomplete if one accepts the stated locality and reality assumptions.

Modern no-signaling is also not the same as EPR locality. Quantum mechanics preserves no-signaling at the level of local marginal probabilities, but entangled states still violate Bell inequalities under suitable measurements. The later Bell framework separates these ideas more sharply.

Original Continuous-Variable Setup

The original EPR example used two particles with idealized position and momentum correlations. In modern notation, one can consider collective observables

$$X_1-X_2, \qquad P_1+P_2.$$

They commute:

$$\begin{aligned} [X_1-X_2,P_1+P_2] &= [X_1,P_1]-[X_2,P_2] \\ &= i\hbar-i\hbar \\ &=0. \end{aligned}$$

Thus an idealized generalized state may have sharp values

$$X_1-X_2=x_0, \qquad P_1+P_2=P_0.$$

If one measures $X_2$, one can infer $X_1$:

$$X_1=X_2+x_0.$$

If instead one measures $P_2$, one can infer $P_1$:

$$P_1=P_0-P_2.$$

The two choices concern noncommuting local observables of particle 1:

$$[X_1,P_1]=i\hbar.$$

Quantum mechanics does not assign particle 1 a normalizable state with both $X_1$ and $P_1$ sharply defined. EPR’s conclusion was that, if the remote measurement does not disturb particle 1, then the quantum state omits elements of reality.

The ideal state used in this reasoning is not a normalizable wavefunction. Its modern state-theoretic treatment belongs in EPR State Preview.

Entangled States in Modern Language

Modern readers usually rephrase the EPR setup using entanglement. The key feature is that the joint state contains correlations that are not reducible to separate local states for each subsystem.

For the original continuous-variable case, the ideal correlations are position and momentum correlations. For later Bell discussions, Bohm’s spin version replaces the continuous-variable state with a spin singlet:

$$\lvert\Psi^-\rangle = \frac{1}{\sqrt2} \bigl( \lvert\uparrow\rangle_A\lvert\downarrow\rangle_B - \lvert\downarrow\rangle_A\lvert\uparrow\rangle_B \bigr).$$

The spin version is not the original 1935 example, but it became pedagogically and experimentally central because spin and polarization correlations lead naturally to Bell inequalities.

Entanglement alone is not the EPR argument. The argument also requires the completeness question, the reality criterion, and the locality/no-disturbance premise. The formal state facts are treated in Entanglement in Foundations and Bell States.

Legacy

The EPR paper sharpened several questions that later became central:

• Can quantum mechanics be completed by hidden variables?
• What does locality mean for separated measurements?
• What counts as a physical property before measurement?
• How should one distinguish no-signaling from stronger locality assumptions?
• Which parts of the debate can be turned into experimentally testable inequalities?

Schrödinger’s response introduced the language of entanglement and steering. Bohm’s spin reformulation made the correlations easier to discuss and later test. Bell’s 1964 theorem showed that local hidden-variable completions with independent settings obey inequalities violated by quantum predictions. Bell experiments then moved the debate from philosophical pressure to empirical constraint.

The careful modern summary is not “EPR was simply wrong” or “EPR proved quantum mechanics incomplete.” EPR identified a real tension among locality, completeness, and quantum predictions. Bell’s theorem and experiments later showed that the tension cannot be resolved by broad classes of local hidden-variable theories.

Exercises

1. Show that $X_1-X_2$ commutes with $P_1+P_2$ for two particles with $[X_i,P_j]=i\hbar\delta_{ij}$.

Solution

Use linearity of the commutator:

$$[X_1-X_2,P_1+P_2] = [X_1,P_1]+[X_1,P_2] -[X_2,P_1]-[X_2,P_2].$$

The cross commutators vanish because they act on different particles:

$$[X_1,P_2]=[X_2,P_1]=0.$$

The remaining terms give

$$[X_1,P_1]-[X_2,P_2] = i\hbar-i\hbar = 0.$$

2. Why does the EPR argument need a locality or no-disturbance premise in addition to perfect correlation?

Solution

Perfect correlation by itself says that one outcome lets us predict another. EPR need more: they need the remote measurement not to disturb the distant system. Only then can they argue that the predicted value corresponds to an element of reality already associated with the distant system, rather than being created or changed by the measurement context.

3. Explain why EPR correlations do not by themselves allow faster-than-light signaling.

Solution

Entangled systems can have strong joint correlations, but each party’s unconditioned local statistics are fixed by the local reduced state. A remote party cannot choose an outcome and thereby control the other party’s marginal distribution. Correlations become visible only after outcomes are compared through ordinary classical communication.

4. Does the EPR argument provide a procedure for measuring both $X_1$ and $P_1$ sharply on the same pair?

Solution

No. Measuring $X_2$ enables one position prediction; choosing $P_2$ instead enables a momentum prediction. These are alternative measurements. EPR use locality to argue that the alternative choice cannot change which properties particle 1 already possesses. That extra reasoning, rather than a simultaneous experimental determination, leads to their completeness claim. Real finite-width preparations also replace ideal certainty with approximate correlations.

References

• J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics 1, 195-200, 1964, DOI: 10.1103/PhysicsPhysiqueFizika.1.195.
• D. Bohm, Quantum Theory, Prentice-Hall, 1951.
• N. Bohr, “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?,” Physical Review 48, 696-702, 1935, DOI: 10.1103/PhysRev.48.696.
• A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?,” Physical Review 47, 777-780, 1935, DOI: 10.1103/PhysRev.47.777.
• A. Fine, The Shaky Game: Einstein, Realism, and the Quantum Theory, 2nd ed., University of Chicago Press, 1996.
• M. Jammer, The Philosophy of Quantum Mechanics, Wiley, 1974.
• E. Schrödinger, “Discussion of Probability Relations between Separated Systems,” Mathematical Proceedings of the Cambridge Philosophical Society 31, 555-563, 1935.