Bohm’s Spin Version of EPR

Bohm’s spin version of EPR reformulates the original 1935 position-momentum argument using two spin-$1/2$ systems in the singlet state. It keeps the essential structure of EPR correlations while replacing idealized continuous-variable eigenstates with a normalizable finite-dimensional state.

This reformulation became the standard route into Bell’s theorem and Bell experiments because spin and polarization correlations are easier to state, visualize, and test than the original exact position-momentum correlations.

Original EPR Variables Versus Spin Pairs

The original EPR paper used two particles with ideal correlations in relative position and total momentum:

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

Those collective observables commute, so an idealized generalized state can have sharp values for both. The tension arises because measuring particle 2 can let one predict either $X_1$ or $P_1$, while $X_1$ and $P_1$ do not commute.

Bohm’s version replaces that setup with two spin-$1/2$ systems prepared in a total-spin-zero state. Instead of choosing whether to measure position or momentum, each observer chooses a spin direction. The same broad EPR pattern remains:

• the two systems are separated;
• measurements on one side allow strong predictions about the other side;
• the predictions concern incompatible possible measurement directions;
• the locality and completeness questions reappear in a simpler setting.

The price is that the spin version is not the original EPR example. The benefit is that it gives a clean two-outcome correlation experiment.

Singlet State

For two spin-$1/2$ systems, the spin singlet is

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

The arrows here refer to spin along a chosen $z$ axis. The singlet is rotationally invariant, so the same anticorrelation structure holds for any common measurement axis. If both observers measure spin along the same unit direction $\mathbf n$, their results are always opposite.

Equivalently, the singlet has total spin zero:

$$\mathbf S_{\mathrm{tot}}^2\lvert\Psi^-\rangle=0.$$

Each subsystem by itself is maximally mixed:

$$\rho_A = \rho_B = \frac12 I.$$

Thus neither side contains a preselected local spin direction in the quantum state. The information appears in joint correlations.

Correlations

Let Alice measure spin along unit vector $\mathbf a$ and Bob measure spin along unit vector $\mathbf b$. Encode outcomes as $\pm1$ in units of $\hbar/2$. For the singlet, quantum mechanics predicts the correlation

$$E(\mathbf a,\mathbf b) = \langle A_{\mathbf a}B_{\mathbf b}\rangle = -\mathbf a\cdot\mathbf b.$$

Special cases are useful:

$$\mathbf a=\mathbf b \quad\Longrightarrow\quad E=-1,$$

so same-axis outcomes are perfectly anticorrelated. If the directions are perpendicular,

$$\mathbf a\cdot\mathbf b=0 \quad\Longrightarrow\quad E=0.$$

If the directions are opposite,

$$\mathbf a=-\mathbf b \quad\Longrightarrow\quad E=+1.$$

These correlations are stronger than what many classical-looking hidden-variable models can reproduce once multiple possible measurement settings are considered. That is the bridge to Bell’s theorem. The theorem itself belongs to Bell Theorem and CHSH Inequality.

Why Bohm’s Version Matters Pedagogically

Bohm’s spin version became pedagogically central for several reasons.

First, the state is normalizable and finite-dimensional. The ideal EPR position-momentum state is a useful generalized state, but it is not a physical vector in Hilbert space. The spin singlet is an ordinary two-qubit state.

Second, the outcomes are discrete. Each spin measurement has two possible results, which makes the correlation structure easy to encode as $\pm1$ data.

Third, the same formal pattern appears in photon polarization experiments. Although photon polarization is not literally spin-$1/2$, the two-outcome polarization correlation setting plays the same experimental role in many optical Bell tests.

Fourth, the spin version points directly toward inequalities. EPR’s original argument was a completeness argument. Bell’s insight was that local hidden-variable explanations of singlet correlations obey experimentally testable bounds. Bohm’s version made that path transparent.

Common Misconceptions

• Bohm’s spin version is not the original 1935 EPR example.
• The spin singlet is not a pair of particles with hidden opposite spin vectors already written into the quantum state.
• Perfect same-axis anticorrelation does not by itself prove Bell’s theorem.
• Bell tests are not merely demonstrations that the singlet is entangled.
• Singlet correlations do not allow controllable faster-than-light signaling.
• Photon polarization Bell tests use an analogous two-outcome structure; they are not literally measurements of massive-particle spin-$1/2$.

Cross-Links

• Foundations Experiments and Quantum Reality
• EPR Argument
• Bell’s Theorem as Historical Turning Point
• Electron Spin
• Spin-Statistics Preview
• Two Spin-1/2 Particles
• Singlet and Triplet States
• Bell States
• Entanglement in Foundations
• Local Measurement Statistics
• Bell Theorem
• CHSH Inequality
• Bell Tests

References

• D. Bohm, Quantum Theory, Prentice-Hall, 1951.
• D. Bohm and Y. Aharonov, “Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky,” Physical Review 108, 1070-1076, 1957, DOI: 10.1103/PhysRev.108.1070.
• 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.
• J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics 1, 195-200, 1964, DOI: 10.1103/PhysicsPhysiqueFizika.1.195.
• J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed Experiment to Test Local Hidden-Variable Theories,” Physical Review Letters 23, 880-884, 1969, DOI: 10.1103/PhysRevLett.23.880.
• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge University Press, 2004.

Exercises

1. Show that same-axis measurements on the singlet are perfectly anticorrelated in the $z$ basis.

Solution

The singlet is

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

If Alice obtains $\uparrow$ along $z$, the only compatible term has Bob in $\downarrow$. If Alice obtains $\downarrow$, the only compatible term has Bob in $\uparrow$. Thus same-axis outcomes are always opposite.

2. What is $E(\mathbf a,\mathbf b)$ for the singlet when $\mathbf a=\mathbf b$, when $\mathbf a=-\mathbf b$, and when $\mathbf a\cdot\mathbf b=0$?

Solution

Use

$$E(\mathbf a,\mathbf b) = -\mathbf a\cdot\mathbf b.$$

If $\mathbf a=\mathbf b$, then $\mathbf a\cdot\mathbf b=1$, so $E=-1$. If $\mathbf a=-\mathbf b$, then $\mathbf a\cdot\mathbf b=-1$, so $E=+1$. If $\mathbf a\cdot\mathbf b=0$, then $E=0$.

3. Why is the spin version easier to connect to Bell inequalities than the original EPR position-momentum setup?

Solution

The spin version uses a normalizable two-qubit state and two-outcome measurements that can be encoded as $\pm1$. Bell and CHSH inequalities are naturally written for correlations between such outcomes under different settings. The original EPR setup uses idealized continuous variables and generalized states, which are conceptually important but less direct for simple inequality tests.

4. Why does the fact that $\rho_A=I/2$ matter for no-signaling?

Solution

The reduced state $\rho_A=I/2$ fixes Alice’s unconditioned local statistics. Bob’s choice of measurement direction can change how joint data are correlated after comparison, but it cannot by itself change Alice’s marginal outcome distribution. That is why singlet correlations do not give controllable faster-than-light signaling.