17.1 Singlet anticorrelations and switched analyzer settings
17.1.1 From EPR elements of reality to Bohm’s spin-singlet pair
In 1935, Einstein, Podolsky, and Rosen (EPR) argued that if an outcome on system \(B\) can be predicted with certainty from a distant measurement on system \(A\) without disturbing \(B\), then \(B\) must already possess a preexisting value (“element of reality”) for that observable. David Bohm reformulated the EPR scenario using two spin-\(1/2\) particles prepared in the rotationally invariant singlet state
and distributed to two distant stations, Alice and Bob, who choose analyzer directions \(\hat{a}\) and \(\hat{b}\) and record binary outcomes \(A, B \in \{+1, -1\}\).
17.1.2 Why single-axis anticorrelation admits prewritten local instructions
Whenever Alice and Bob orient their spin analyzers along the exact same axis (\(\hat{a} = \hat{b}\)), quantum mechanics predicts—and experiment confirms—perfect anticorrelation: \(E(\hat{a}, \hat{a}) = \langle A B\rangle = -1\), with marginal probabilities \(P(A = \pm 1) = P(B = \pm 1) = 1/2\). However, perfect anticorrelation along parallel axes alone can be reproduced effortlessly by a classical local hidden-variable (LHV) table that assigns each emitted pair a random direction \(\hat{\lambda}\) on the sphere and sets \(A(\hat{a},\hat{\lambda}) = \mathrm{sgn}(\hat{a}\cdot\hat{\lambda})\) and \(B(\hat{b},\hat{\lambda}) = -\mathrm{sgn}(\hat{b}\cdot\hat{\lambda})\). John Bell showed in 1964 that testing whether such prewritten local tables can exist requires comparing correlations across multiple non-collinear analyzer angles.
17.1.3 Planar spin-analyzer angles versus optical polarizer angles
For coplanar spin-\(1/2\) analyzers at angles \(\theta_a\) and \(\theta_b\) in the \(x\)–\(z\) plane, the singlet correlation depends on the relative angle \(\theta_{ab} = \theta_a - \theta_b\) as \(E_{\mathrm{spin}}(\theta_a, \theta_b) = -\cos(\theta_a - \theta_b)\). In polarization-entangled photon experiments, where orthogonal states \(|H\rangle\) and \(|V\rangle\) sit \(90^\circ\) apart in physical space rather than \(180^\circ\) on the Bloch sphere, a physical polarizer rotation by \(\theta_{\mathrm{opt}}\) corresponds to a Bloch rotation by \(2\theta_{\mathrm{opt}}\), so the optical correlation oscillates as \(\pm\cos[2(\theta_a^{\mathrm{opt}} - \theta_b^{\mathrm{opt}})]\). Throughout this chapter we use Bloch/spin angles \(\theta_a, \theta_b\) (equivalent to twice the physical polarizer angle).
Visual animation · Section 17.1
Singlet anticorrelations and switched analyzer settings
Interactive visual model for 17.1 Singlet anticorrelations and switched analyzer settings (From EPR elements of reality to Bohm’s spin-singlet pair · Why single-axis anticorrelation admits prewritten local instructions · Planar spin-analyzer angles versus optical polarizer angles). Use Animate or Step to follow the physical state evolution and quantitative predictions.
17.2 Local hidden-variable factorization and the CHSH bound
17.2.1 Factorizable conditional probabilities and setting independence
A local hidden-variable model posits a shared hidden state \(\lambda\) drawn from a normalized probability distribution \(\rho(\lambda)\) that is independent of the analyzer choices (\(\rho(\lambda|a,b) = \rho(\lambda)\), the measurement-independence or “no-superdeterminism” assumption) and satisfies Bell locality: the joint outcome probability factorizes when conditioned on \(\lambda\),
\[P(A, B \mid a, b, \lambda) = P_A(A \mid a, \lambda)\,P_B(B \mid b, \lambda).\]
Consequently, the local expectation values \(\bar{A}(a,\lambda) = \sum_{A=\pm 1} A\,P_A(A|a,\lambda) \in [-1, 1]\) and \(\bar{B}(b,\lambda) \in [-1, 1]\) yield the ensemble correlation
17.2.2 Algebraic derivation of the Clauser–Horne–Shimony–Holt inequality
In the Clauser–Horne–Shimony–Holt (CHSH) protocol, Alice chooses between two settings \(a, a'\) and Bob chooses between two settings \(b, b'\). For any fixed \(\lambda\), since \(\bar{A}(a,\lambda), \bar{A}(a',\lambda), \bar{B}(b,\lambda), \bar{B}(b',\lambda) \in [-1, 1]\), the four-term combination
For an ideal singlet (\(V = 1\)), \(|S_Q| = 2\sqrt{2}\) saturates Boris Tsirelson’s general quantum bound for any bipartite state and any local dichotomic observables, and violates the classical bound \(|S| \le 2\) whenever \(V > 1/\sqrt{2} \approx 0.7071\).
Visual animation · Section 17.2
Local hidden-variable factorization and the CHSH bound
Interactive visual model for 17.2 Local hidden-variable factorization and the CHSH bound (Factorizable conditional probabilities and setting independence · Algebraic derivation of the Clauser–Horne–Shimony–Holt inequality · Singlet cosine correlations and Tsirelson’s two-root-two maximum). Use Animate or Step to follow the physical state evolution and quantitative predictions.
17.3 Loopholes, no-signalling, and assumption packages
17.3.1 Locality, fair-sampling, and setting-freedom requirements
Translating an observed experimental violation \(|\hat{S}| > 2\) into a rigorous refutation of local hidden variables requires closing three operational loopholes:
Locality loophole: The setting choice and outcome recording at station \(A\) must be spacelike separated from the setting choice and outcome recording at station \(B\), preventing subluminal or luminal communication of the chosen angle.
Detection (fair-sampling) loophole: If detectors miss a large fraction of emitted pairs and the experimenter postselects only coincidence events, a local model can fake \(|S| > 2\) by making detection probability depend on the local angle and \(\lambda\). Closing the detection loophole without fair-sampling assumptions requires overall system detection efficiency \(\eta > 2/(\sqrt{2} + 1) \approx 82.8\%\) for symmetric CHSH tests (or \(\eta > 2/3\) for Eberhard’s non-maximally entangled inequality).
Freedom-of-choice loophole: Setting generators must be statistically independent of the source’s emission variables \(\lambda\).
17.3.2 What a Bell violation refutes and why it cannot transmit signals
A loophole-free Bell violation rules out the conjunction of locality and outcome determinism / preexisting setting-independent values (often summarized as local realism). At the same time, quantum mechanics strictly preserves the relativistic no-signalling principle: for every setting \(\theta_b\) chosen by Bob, Alice’s local marginal probability remains
The non-classical CHSH correlation \(|S| > 2\) becomes visible only when Alice and Bob bring their classical outcome logs together via an ordinary subluminal channel and compute cross-correlations.
17.3.3 Separating model curves from finite-sample hypothesis tests
In a laboratory run with \(N\) trials per setting pair \((a_i, b_j)\), each empirical correlation \(\hat{E}_{ij} = (N_{++} + N_{--} - N_{+-} - N_{-+})/N\) carries statistical standard error \(\sigma_{ij} = \sqrt{(1 - \hat{E}_{ij}^2)/N}\). Reporting a Bell test requires stating both the point estimate \(\hat{S}\) and the number of standard deviations \((|\hat{S}| - 2)/\sigma_S\) (or the martingale \(p\)-value for memory-tolerant sequential trials) by which the data exceed the local bound \(2\).
Visual animation · Section 17.3
Loopholes, no-signalling, and assumption packages
Interactive visual model for 17.3 Loopholes, no-signalling, and assumption packages (Locality, fair-sampling, and setting-freedom requirements · What a Bell violation refutes and why it cannot transmit signals · Separating model curves from finite-sample hypothesis tests). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the Apparatus
Adjust the four analyzer angles \((\theta_a, \theta_{a'}, \theta_b, \theta_{b'})\) and the singlet state visibility \(V\) to compare the quantum correlation \(E_Q(\theta_a,\theta_b) = -V\cos(\theta_a - \theta_b)\) against the best classical local hidden-variable triangle wave. Identify the visibility threshold \(V > 1/\sqrt{2}\) above which \(|S|\) crosses the classical ceiling \(2\) toward Tsirelson’s bound \(2\sqrt{2}\).
Interactive model · Idealized predictions
Four correlations, one bound
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Read the values as a table
Which assumptions about separated systems can correlations test? The text and worked examples remain available without JavaScript. Animation starts only when requested; Step gives a motion-free way to follow the model.
Worked Check
Problem: A Bell experiment prepares a noisy singlet state with visibility \(V = 0.85\) and uses the optimal CHSH angles (\(|S_Q| = 2\sqrt{2}\,V\)). Each of the four setting pairs \((a,b), (a,b'), (a',b), (a',b')\) is measured for \(N = 1000\) independent pair trials. Compute the expected CHSH parameter \(|S_Q|\), the single-setting correlation magnitude \(|E|\), the statistical standard error \(\sigma_S\) on \(S\), and the significance \((|S_Q| - 2)/\sigma_S\) of the violation.
Solution:
1. At the optimal CHSH angles, each of the four correlations has magnitude
Any local hidden-variable model that assigns pre-existing outcome rules \(A(a,\lambda), B(b,\lambda) \in \{\pm 1\}\) satisfies the Clauser–Horne–Shimony–Holt (CHSH) bound \(|S| \le 2\).
A singlet Bell state measured at relative analyzer angles of \(45^\circ\) achieves correlation \(E(a,b) = -\cos(a-b)\) and violates the CHSH inequality up to Tsirelson’s bound \(|S| = 2\sqrt{2} \approx 2.828\).
Loophole-free Bell tests close the locality, detection-efficiency (\(\eta > 82.8\%\) for symmetric CHSH), and setting-independence loopholes simultaneously while respecting relativistic no-signaling.
Practice Problems
Problem 17.1: Visibility threshold for a CHSH violation
A source emits the Werner state \(\rho_V = V|\Psi^-\rangle\langle\Psi^-| + \frac{1-V}{4}I\). What minimum singlet visibility \(V_{\min}\) is required to violate the CHSH inequality \(|S| \le 2\), and what is the maximum \(|S|\) achievable when \(V = 0.65\)?
Hint
Set \(2\sqrt{2}\,V_{\min} = 2\).
Solution
Setting \(2\sqrt{2}\,V_{\min} = 2\) gives \(V_{\min} = 1/\sqrt{2} \approx 0.7071\). When \(V = 0.65 < 1/\sqrt{2}\), the maximum quantum value is \(|S_{\max}| = 2\sqrt{2}(0.65) \approx 1.838 \le 2\), which cannot violate the CHSH inequality for any choice of analyzer angles.
Problem 17.2: Evaluating CHSH at non-optimal angles
For an ideal singlet (\(V = 1\)), suppose Alice chooses \(\theta_a = 0^\circ\) and \(\theta_{a'} = 60^\circ\), while Bob chooses \(\theta_b = 30^\circ\) and \(\theta_{b'} = -30^\circ\). Calculate the four correlations \(E(a,b), E(a,b'), E(a',b), E(a',b')\) and the resulting \(|S|\).
Hint
Use \(E(\theta_a,\theta_b) = -\cos(\theta_a - \theta_b)\) and note that \(\theta_{a'} - \theta_{b'} = 90^\circ\).
Problem 17.3: Why rotated analyzers cannot send a faster-than-light bit
Suppose Bob switches from measuring along \(\theta_b = 0^\circ\) to measuring along \(\theta_{b'} = 90^\circ\) on every pair. Using the singlet reduced state \(\rho_A = \mathrm{Tr}_B(|\Psi^-\rangle\langle\Psi^-|)\), show that Alice’s outcome probabilities \(P(A = \pm 1 \mid \theta_a)\) remain completely unchanged.
Hint
Compute \(\rho_A = \mathrm{Tr}_B(|\Psi^-\rangle\langle\Psi^-|)\) and evaluate \(\mathrm{Tr}[\Pi_A^{\pm}(\theta_a)\rho_A]\).
Solution
Tracing out qubit \(B\) from \(|\Psi^-\rangle = (|01\rangle - |10\rangle)/\sqrt{2}\) gives \(\rho_A = \frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1| = \frac{1}{2}I\), regardless of what basis Bob chooses or even whether Bob performs a measurement at all. For any rank-1 projector \(\Pi_A^{\pm}(\theta_a)\), \(\mathrm{Tr}[\Pi_A^{\pm}(\theta_a)\rho_A] = \frac{1}{2}\mathrm{Tr}[\Pi_A^{\pm}(\theta_a)] = 1/2\).
Key Takeaways
Perfect anticorrelation along parallel axes (\(\hat{a} = \hat{b}\)) can be explained by prewritten classical tables, whereas correlations across multiple non-collinear settings are constrained by the CHSH bound \(|S| \le 2\) in any local hidden-variable model.
For a spin singlet with visibility \(V\), quantum mechanics predicts \(E_Q(\theta_a,\theta_b) = -V\cos(\theta_a - \theta_b)\) and achieves \(|S_Q| = 2\sqrt{2}\,V\), violating the classical bound whenever \(V > 1/\sqrt{2}\).
Loophole-free Bell tests require spacelike separation of setting choices and outcomes (locality), high overall detection efficiency (closing fair sampling), and independent setting generators.
Bell violations refute local hidden-variable models while strictly preserving relativistic no-signalling, because local marginal probabilities remain \(1/2\) for all remote setting choices.