Partial-Sum Probabilities vs. Bell Correlations
Partial-Sum Probabilities ⇄ Successive Measurements in Bell Experiments
Overview
Both the classical partial-sum process and the quantum Bell experiment produce probabilities for sequences of outcomes.
The useful analogy is that both systems evolve probabilities over a space of possible “paths” (integer sums vs. measurement outcomes). However, the
crucial difference is that Bell correlations require non-classical probability structure: no classical Markov chain or hidden-local-variable model can reproduce them.
1. The Partial-Sum (Markov) Model — short description
Suppose we draw integers uniformly from {1,…,N} with replacement and keep a running sum \(S_k\). The evolution is
classical and Markovian:
\(P(S_k = s) = \sum_{i=1}^{N} P(S_{k-1} = s-i)\cdot \frac{1}{N}.\)
To ask for the probability of reaching exactly \(n\) at some step is to condition on allowed paths through the integer-state lattice; paths that overshoot are excluded.
This process is classical: transition probabilities are nonnegative, normalized, and depend only on the previous state (Markov property).
2. Bell (CHSH-style) Experiment — short description
Two distant parties, Alice and Bob, share an entangled state \(\lvert\psi\rangle\). Each party chooses a measurement setting (Alice: \(a\) or \(a’\); Bob: \(b\) or \(b’\))
and obtains outcomes \(A,B\in\{+1,-1\}\). Quantum mechanics predicts joint probabilities
\(P(A,B \mid a,b) = \langle\psi \rvert \; \big(M_A^{(a)} \otimes M_B^{(b)}\big) \; \lvert\psi\rangle,\)
where \(M_A^{(a)},M_B^{(b)}\) are measurement operators (projectors).
The key empirical fact: for certain choices of \((a,a’,b,b’)\), the correlations violate the CHSH inequality and reach up to \(2\sqrt{2}\) (Tsirelson bound).
3. Classical Hidden-Variable / Markov-Factorization Assumption
A classical model with a hidden variable \(\lambda\) (and local Markov-like transitions) assumes the joint probability factorizes as:
\(P(A,B \mid a,b) \;=\; \int d\lambda \; \rho(\lambda)\; P(A\mid a,\lambda)\; P(B\mid b,\lambda).\)
This expresses locality and a classical probabilistic structure. If such a representation exists, all CHSH-type correlations obey the classical bound of 2.
4. CHSH inequality (derivation sketch)
Define correlators
\(\;E(a,b) = \sum_{A,B=\pm1} AB \, P(A,B\mid a,b).\)
Under the classical factorization with deterministic \(\pm1\) responses (or by convexity for probabilistic responses),
one can show the CHSH combination satisfies:
\(S \;=\; E(a,b) + E(a,b') + E(a',b) - E(a',b') \;\le\; 2.\)
The short intuitive proof: for a fixed \(\lambda\) and deterministic outcomes \(A(a,\lambda),B(b,\lambda)\in\{\pm1\}\),
the quantity
\[
Q(\lambda) = A(a,\lambda)\big[ B(b,\lambda) + B(b’,\lambda) \big] + A(a’,\lambda)\big[ B(b,\lambda) – B(b’,\lambda) \big]
\]
can only be \(\pm2\). Averaging over \(\lambda\) yields \(|S|\le 2\).
5. Quantum prediction violates the classical bound
For the two-qubit singlet state and spin (or polarization) measurements at appropriate angles one finds
\(S_{\text{quantum}} = 2\sqrt{2} > 2.\)
Therefore no model of the factorized classical form (and hence no classical Markov chain producing local factorized joint probabilities) can reproduce these correlations.
6. Why this forbids any classical Markov-chain representation
A Markov chain (or any classical sequential stochastic process) defines joint distributions built from nonnegative transition probabilities and local conditionalization on prior states.
If you attempt to represent the Bell scenario with a classical Markov chain / path model, you would need to assign joint probabilities
\(P(A,B\mid a,b)\) that simultaneously satisfy all measurement-setting marginals and the factorization/locality condition.
But because quantum correlations violate CHSH, no such global assignment of classical nonnegative transition probabilities exists.
Concretely:
- Partial-sum Markov processes: probabilities are built from local, stepwise transition kernels (nonnegative, normalized).
- Any classical hidden-variable or Markov description that respects locality must obey Bell (CHSH) bounds.\li>
- Quantum correlations (experimentally verified) violate those bounds, so they cannot be written as expectations over local Markov transitions.
7. Side-by-side summary (compact)
| Feature | Partial-Sum / Markov | Bell / Quantum |
|---|---|---|
| State space | Integer sums, classical lattice | Hilbert space (amplitudes) |
| Allowed transitions | Nonnegative transition kernel, Markov | Unitary + measurement (non-commuting) |
| Path weighting | Sum of nonnegative path probabilities | Amplitude interference (complex), not representable as simple path probabilities |
| Bell/CHSH | Always satisfies CHSH bound \(|S|\le2\) | Can achieve \(|S|=2\sqrt2>2\) |
8. Intuition: interference & non-commutativity vs. classical conditioning
Classical path models add probabilities for disjoint paths. Quantum mechanics adds complex amplitudes that can interfere, producing correlations that cannot be decomposed into a convex mixture of local deterministic paths.
Mathematically this is tied to the non-commutativity of measurement operators and the fact that a global joint distribution for all possible measurement outcomes (for all settings) that is both local and reproduces quantum marginals does not exist.
9. Optional: short worked example (CHSH angles)
For the singlet state choose measurement directions such that
\(\theta_{a,b}=0^\circ,\; \theta_{a,b’}=90^\circ,\; \theta_{a’,b}=45^\circ,\; \theta_{a’,b’}=135^\circ\).
The quantum correlator for spin-1/2 is \(E(\alpha,\beta) = -\cos(\theta_{\alpha\beta})\).
Then
= -1 – 0 – \tfrac{\sqrt2}{2} + \big(-\tfrac{\sqrt2}{2}\big)
= -2\sqrt2 \)
so \(|S|=2\sqrt2\), violating the classical limit of 2.
10. Final takeaway
The visual/structural analogy is useful: both systems manage probability flow across allowed paths (integer-lattice paths vs. sequences of measurement outcomes). But Bell correlations are fundamentally incompatible with any classical Markov-chain (or local hidden-variable) representation because they violate inequalities (CHSH) that any such classical model must obey. That violation is the signature of quantum non-classicality (entanglement + interference + non-commuting observables).