Theoretical interpretation of CHSH correlation loss: Classical phase-space thermal noise vs. non-unitary decoherence

Question

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1. What the question is really asking

A laboratory reports a CHSH Bell‑parameter

[ S = E_1+E_2+E_3-E_4 = 2.404 , \qquad \begin{aligned} E_1 &=+0.556 ,
E_2 &=-0.672 ,
E_3 &=+0.592 ,
E_4 &=+0.584 . \end{aligned} ]

The value lies between

the local‑realist bound (S\le 2) and
the Tsirelson bound (S\le 2\sqrt 2\approx 2.828).

The student wants to know whether this intermediate value can be explained only by classical (thermal) noise acting on a deterministic phase‑space picture, or whether one must invoke non‑unitary open‑quantum‑system dynamics (e.g. dephasing, amplitude‑damping channels) to account for the loss of correlation.

In plain language:

Can a completely classical, deterministic model with thermal fluctuations produce a CHSH value of 2.404, or is some quantum‑mechanical decoherence inevitably required?

We shall answer this by examining what a classical hidden‑variable (HV) model can produce, how quantum noise reduces the Bell value, and why the reduction is fundamentally a quantum (non‑unitary) effect even if it can be expressed as “classical” random flips on the measurement outcomes.

2. Step‑by‑step analysis

2.1 Bell‑CHSH basics

For two distant parties, Alice and Bob, each chooses one of two measurement settings
((a,a’)) for Alice and ((b,b’)) for Bob, obtaining outcomes (\pm1).
The CHSH combination is

[ S = \langle A B\rangle + \langle A B’\rangle + \langle A’ B\rangle - \langle A’ B’\rangle . ]

Local‑realist (LR) models (any deterministic or stochastic HV theory respecting locality) obey

[ |S| \le 2 . ]

Quantum mechanics allows up to

[ |S| \le 2\sqrt 2 . ]

No‑signalling (the most general correlations consistent with relativity) permits

[ |S| \le 4 . ]

Thus a value larger than 2 already guarantees that the observed statistics cannot be reproduced by any local deterministic phase‑space dynamics—no amount of classical (thermal) noise can raise (S) above 2 if the underlying model is strictly local.

2.2 Classical phase‑space + thermal noise

A deterministic phase‑space model (Liouville dynamics) assigns a definite hidden variable (\lambda) to every experimental run. The outcome functions are predetermined:

[ A(a,\lambda)=\pm 1, \qquad B(b,\lambda)=\pm 1 . ]

If we now add thermal (or any) classical noise after the deterministic outcome is produced, we can model it as a random bit‑flip with probability (q). For example,

[ \tilde{A}= A\,\xi_A,\qquad \xi_A=\begin{cases} +1 & \text{w.p. } 1-q
-1 & \text{w.p. } q \end{cases} ]

(and similarly for (\tilde{B})).
The observed correlator becomes

[ \langle \tilde{A}\tilde{B}\rangle = (1-2q)^2 \langle A B\rangle . ]

Hence the CHSH value scales as

[ S_{\text{noisy}} = (1-2q)^2 S_{\text{ideal}} . ]

Crucially, (S_{\text{ideal}}) itself cannot exceed 2 for any LR model. Multiplying by any factor ((1-2q)^2\le 1) never pushes the result above 2. Therefore a purely classical deterministic model, even with arbitrary thermal noise, can never reproduce (S=2.404).

Conclusion 1: A classical phase‑space picture (local hidden variables) can only give (S\le 2). Any observed value larger than 2 proves that the underlying correlations are non‑local (or that the model abandons locality).

2.3 Quantum source + noise (open‑system picture)

Consider now a quantum source that would give the maximal violation if it were perfect: a singlet state (

\psi^{-}\rangle). With the optimal CHSH measurement settings, the ideal quantum value is

[ S_{\text{max}} = 2\sqrt2 . ]

If the state (or the measurement) is affected by noise, the observed correlators are reduced. The most common noise models are:

Noise model	Kraus operators	Effect on correlators
Depolarising channel on each qubit: (\rho\to p\rho+(1-p)\frac{\mathbb I}{2})	(K_0=\sqrt{p}\,\mathbb I, \; K_{1..3}=\sqrt{\frac{1-p}{3}}\,\sigma_i)	Correlators multiplied by (p)
Phase‑damping (dephasing) on each qubit: (\rho\to p\rho+(1-p)Z\rho Z)	(K_0=\sqrt{p}\,\mathbb I,\; K_1=\sqrt{1-p}\,	0\rangle\langle0	,\;K_2=\sqrt{1-p}\,	1\rangle\langle1	)	Correlators in the (\sigma_x,\sigma_y) plane multiplied by (p)
Amplitude‑damping (energy loss)	(K_0=	0\rangle\langle0	+\sqrt{1-\gamma}\,	1\rangle\langle1	,\; K_1=\sqrt{\gamma}\,	0\rangle\langle1	)	Asymmetric reduction of correlators; also changes local statistics

All of these are completely positive, trace‑preserving (CPTP) maps—i.e. non‑unitary evolutions that arise from coupling the system to an environment (open‑system dynamics).

For the simple case of isotropic depolarising noise on both qubits, the state becomes

[ \rho_{\text{mix}} = p\,|\psi^{-}\rangle!\langle\psi^{-}|+(1-p)\,\frac{\mathbb I}{4}, \qquad 0\le p\le1 . ]

The CHSH value for this Werner‑type state is

[ S = p\, 2\sqrt2 . ]

Setting (S=2.404) gives the required visibility

[ p = \frac{S}{2\sqrt2}= \frac{2.404}{2.828}\approx 0.85 . ]

Thus 85 % of a maximally entangled state plus 15 % white noise reproduces the experimental number. The same number would be obtained if we first prepared the perfect singlet and then let each qubit undergo a phase‑damping channel with visibility (p\approx0.85).

2.4 Classical‑noise picture vs. quantum‑channel picture

One can re‑interpret the effect of a quantum channel as classical random flips on the measurement outcomes after the ideal measurement has been performed. For the depolarising case:

With probability (p) the outcomes are the ideal quantum ones (correlated as (\pm1)).
With probability (1-p) the outcomes are completely random (uncorrelated).

Mathematically this is exactly the same statistical mixture we wrote in the previous table. Hence, from a statistical‑mechanics point of view, the loss of CHSH strength can be described by a classical stochastic process acting on the measurement results.

However, the origin of that stochastic process is quantum: it is the trace over environmental degrees of freedom (the “thermal bath”) that turns a pure unitary evolution of system + environment into a non‑unitary map on the system alone. In other words:

Pure deterministic phase‑space dynamics → cannot generate (S>2).
Quantum entanglement + coupling to a bath → yields a non‑unitary CPTP map, which can reduce the Bell value to any number between 2 and (2\sqrt2).

Therefore, while the final statistics can be modelled by “classical thermal noise” applied to the measurement outcomes, the existence of the Bell violation itself (the part above 2) fundamentally requires a quantum, non‑unitary open‑system description.

2.5 Summary of the logical chain

Local deterministic HV + classical noise ⇒ (S\le 2).
Observed (S=2.404>2) ⇒ the data cannot be reproduced by any such classical model.
Quantum pure state (maximally entangled) ⇒ (S=2\sqrt2).
Introduce any CPTP noise channel (dephasing, depolarising, amplitude damping, etc.) → the CHSH value is multiplied by a visibility factor (V\le1).
For (S=2.404) we need (V\approx0.85). This is exactly the prediction of a standard open‑quantum‑system model (e.g., a depolarising channel with probability (1-V=0.15)).
The same statistics could be mimicked by a classical random‑flip model after the quantum measurement, but that classical model is only an effective description; it cannot replace the underlying quantum dynamics that generated the violation in the first place.

3. Final answer

No. A deterministic classical phase‑space model, even when supplemented with thermal (Liouville/Langevin) noise, can never yield a CHSH value larger than the local bound of 2. The observed intermediate value (S=2.404) therefore must arise from genuine quantum correlations that have been partially degraded by non‑unitary open‑system dynamics (dephasing, depolarising, amplitude‑damping, etc.).

Mathematically, the degradation can be expressed as a visibility factor (V) so that

[ S = V\; 2\sqrt2, \qquad V = \frac{2.404}{2\sqrt2}\approx0.85 . ]

A quantum channel with visibility (V) (e.g. a depolarising channel with probability (1-V\approx0.15)) reproduces the data. The same statistical outcome could be re‑cast as classical random flips on the measurement results, but that is only an effective description; the essential source of the Bell violation—and its reduction—is quantum decoherence, not purely classical thermal noise.

4. Common mistakes

Mistake	Why it’s wrong	How to avoid it
Assuming any reduction of (S) can be explained by classical noise alone.	Classical (local) models cannot produce (S>2) to begin with; they cannot “reduce” a quantum violation.

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