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 1951 textbook 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

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

The full joint probability is

$$P(A=s,B=t\mid\mathbf a,\mathbf b) =\frac{1-st\,\mathbf a\cdot\mathbf b}{4}, \qquad s,t\in\{+1,-1\}.$$

Summing over either outcome gives $1/2$ for each local result. Summing $st$ times these four probabilities gives the correlation above. Thus a correlation is an average of products, not a probability by itself.

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.

From spin to photon polarization

The polarization singlet $(\lvert H V\rangle-\lvert V H\rangle)/\sqrt2$ also has two outcomes per analyzer. For linear-polarizer angles $\alpha$ and $\beta$, however, its correlation is $-\cos[2(\alpha-\beta)]$. The factor of two matters when translating spin directions into optical settings: a polarization analyzer is unchanged by a $180^\circ$ rotation. Photon polarization is not massive-particle spin one half.

For the state calculation, continue with Singlet and Triplet States. For why one perfect anticorrelation is insufficient to test local explanations, continue with Bell’s Theorem.

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. Two spin analyzers differ by $60^\circ$. Find all four joint probabilities and the correlation.

Solution

Here $\mathbf a\cdot\mathbf b=1/2$. Each same-sign pair has probability $1/8$ and each opposite-sign pair has probability $3/8$. Therefore $E=1/8+1/8-3/8-3/8=-1/2$. Each local outcome still has probability $1/2$.

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. Any trace-preserving local operation by Bob, including a measurement whose outcomes are ignored, leaves Alice’s reduced state unchanged. Selecting a particular Bob outcome changes a conditional state, but Alice needs his communicated result to identify that subensemble. That is why singlet correlations do not give controllable faster-than-light signaling.

References

• J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics 1, 195-200, 1964, DOI: 10.1103/PhysicsPhysiqueFizika.1.195.
• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge University Press, 2004.
• 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.
• 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.
• 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.