Bell Nonlocality with Quantum Fields

Bell nonlocality is a property of a complete correlation experiment, not merely of a state. In QFT the experiment must identify two spacelike laboratories, bounded local effects, independently selected settings, an unconditional trial ensemble, and a valid rejection test for local hidden-variable models. A CHSH violation can coexist with exactly setting-independent marginals, so it neither sends a message nor conflicts with microcausality.

Required background. Field communication supplies the localized apparatus channel. Spacelike joint measurements supplies order-independent local instruments.

Helpful background. Entanglement witnesses and tomography limits distinguishes one-sided certification from a complete state claim.

Chapter map. Consult the overview task map, correlation-versus-influence distinction, Bell row of the common comparison, and constraint and failure controls instead of repeating the overview figures here.

Local correlations and the CHSH bound

On each trial Alice chooses $x\in\{0,1\}$ and Bob chooses $y\in\{0,1\}$, obtaining outcomes $a,b\in\{-1,+1\}$. A local hidden-variable model has the factorized form

$$p(a,b\mid x,y) =\int d\lambda\,\mu(\lambda) p_A(a\mid x,\lambda)p_B(b\mid y,\lambda),$$

with the setting distribution independent of $\lambda$. Define correlators and the sign convention

$$E_{xy}=\sum_{a,b}ab\,p(a,b\mid x,y), \qquad S=E_{00}+E_{01}+E_{10}-E_{11}.$$

For fixed hidden data, deterministic binary responses have $a_x,b_y\in\{-1,+1\}$. The same algebra also applies to their conditional means, which lie in $[-1,1]$:

$$|a_0(b_0+b_1)+a_1(b_0-b_1)| \le |b_0+b_1|+|b_0-b_1|\le2.$$

A stochastic local model is a convex mixture of such responses, so $|S|\le2$. These assumptions and the four-setting inequality are the content of Clauser, Horne, Shimony, and Holt 1969, Eqs. (1)–(4), pp. 880–882. Setting independence, a predeclared trial definition, and inclusion of every eligible outcome are hypotheses of this inference, not consequences of the measured value.

In a field theory, binary measurements are described by effects $M^A_{a|x}\in\mathcal A(O_A)$ and $M^B_{b|y}\in\mathcal A(O_B)$ satisfying

$$M^A_{a|x}\ge0, \quad \sum_aM^A_{a|x}=\mathbf1, \qquad M^B_{b|y}\ge0, \quad \sum_bM^B_{b|y}=\mathbf1.$$

The associated observables $A_x=M^A_{+|x}-M^A_{-|x}$ and $B_y=M^B_{+|y}-M^B_{-|y}$ are Hermitian contractions with spectra in $[-1,1]$. If $O_A$ and $O_B$ are spacelike, microcausality gives $[A_x,B_y]=0$. The CHSH operator is

$$\mathcal B=A_0(B_0+B_1)+A_1(B_0-B_1).$$

For dichotomic unitaries, $\mathcal B^2=4\mathbf1-[A_0,A_1][B_0,B_1]$ and $\lVert\mathcal B\rVert\le2\sqrt2$; contractions obey the same Tsirelson bound by dilation or convexity. Spacelike commutation permits a joint probability distribution for each chosen pair. It does not imply $E_{xy}=\langle A_x\rangle\langle B_y\rangle$, because the state need not factorize.

Bounded observables from local field operations

The smeared field $\Phi(f)$ is unbounded even when $f$ is smooth and compactly supported. A Bell test should therefore use effects or bounded functions of it. Weyl operators

$$W(f)=e^{i\Phi(f)}, \qquad \cos\Phi(f)=\frac{W(f)+W(f)^*}{2},$$

are bounded elements of the algebra generated in the support of $f$. For a selfadjoint realization of the smeared field, spectral projections such as $\mathbf1_{[0,\infty)}(\Phi(f))$ also define bounded effects after the zero eigenspace convention is fixed. Merely selecting four such operators does not guarantee a violation; the state and operator geometry must be optimized together.

An apparatus description is still required. A compactly supported system–probe coupling followed by a bounded probe readout induces a bounded field effect. The algebraic effect and its localization follow from the measurement scheme, while efficiency and cross-talk come from calibration. The observable/effect distinction and functional calculus are set out in Fewster and Verch 2020, § 2, Eqs. (2.2)–(2.3); their causal factorization result explains why causally disjoint probe couplings compose consistently.

Summers and Werner prove a stronger structural statement: under their net hypotheses, suitable observables in complementary wedge algebras attain maximal Bell violation for broad classes of states; see Summers and Werner 1987, Theorems 3.1 and 4.1, pp. 252–257. This establishes existence inside local QFT algebras. It does not identify a finite-energy detector, switching profile, efficiency, or finite sample size that realizes those observables.

A reproducible encoded-field benchmark

For an explicit first application, assume two spacelike separated local algebras contain commuting encoded qubit subalgebras. This is a declared split/regulator assumption; it is not being inferred from a pair of positive-frequency wavepackets. Let $X_A,Z_A$ and $X_B,Z_B$ be Pauli generators in those subalgebras, and choose a normal field state whose restriction to them is

$$|\Phi^+\rangle =\frac{|0_A0_B\rangle+|1_A1_B\rangle}{\sqrt2}.$$

Use the bounded local settings

$$A_0=Z_A, \qquad A_1=X_A, \qquad B_0=\frac{Z_B+X_B}{\sqrt2}, \qquad B_1=\frac{Z_B-X_B}{\sqrt2}.$$

Since $\langle Z_AZ_B\rangle=\langle X_AX_B\rangle=1$ and the crossed correlators vanish,

Setting pair	$E_{xy}$	CHSH sign
$(0,0)$	$1/\sqrt2$	$+$
$(0,1)$	$1/\sqrt2$	$+$
$(1,0)$	$1/\sqrt2$	$+$
$(1,1)$	$-1/\sqrt2$	$-$

Thus $S=2\sqrt2=2.8284271247$. The local expectations all vanish, so each wing’s outcome probability is $1/2$ for either remote setting. The benchmark simultaneously displays Bell violation and no signaling.

To make the statistical calculation reproducible, suppose $N=2500$ eligible trials are collected for each setting pair and the four empirical correlators are $0.7000,0.7000,0.7000,-0.7000$, giving $\widehat S=2.8000$. For independent bounded products $ab\in[-1,1]$, Hoeffding’s inequality and a union bound give simultaneous two-sided error bars

$$t=\sqrt{\frac{2\ln(160)}{N}} =0.0637192204$$

at family error probability $0.05$. A conservative lower bound is

$$S_{\mathrm{low}} =\widehat S-4t =2.5451231183>2.$$

This worked number is a transparent i.i.d. benchmark, not the preferred analysis for an adversarial Bell test. Devices can have memory and setting probabilities can drift. Prediction-based-ratio methods produce valid local-realist $p$-values under arbitrary temporal variation allowed by their protocol; see Zhang, Glancy, and Knill 2011, §§ II–IV. A real test should predeclare that analysis, the stopping rule, setting generator, spacetime windows, and treatment of no-click outcomes. The event-ready experiment of Hensen et al. 2015, Methods and Supplementary Information illustrates why locality, efficient readout, random settings, and memory-robust significance are separate checks.

No signaling is a separate equality

For normalized local POVMs and commuting spacelike operations,

$$\begin{aligned} p(a\mid x,y) &=\sum_b\operatorname{tr}\!\left[ \rho M^A_{a|x}M^B_{b|y} \right]\\ &=\operatorname{tr}\!\left[ \rho M^A_{a|x} \right], \end{aligned}$$

which is independent of $y$. The analogous statement holds for Bob. CHSH uses four joint correlators and can violate its local bound even while both marginal equalities hold exactly. Experimentally, remote-setting independence must be checked with uncertainty and without conditioning on a remote outcome. A failed marginal test can indicate causal overlap, cross-talk, clock leakage, selection bias, or ordinary drift; it is not explained away by quantum nonlocality.

Adversarial failure controls

Detection-postselection countermodel. Let a local hidden variable $\lambda=(u,v)$ be uniform on the four setting pairs. Alice clicks only when $x=u$; Bob clicks only when $y=v$. On a click Alice outputs $a=1$ and Bob outputs $b=(-1)^{uv}$. Every decision is local: it uses only the local setting and shared $\lambda$. Conditioned on coincidences, however, $ab=(-1)^{xy}$ and the retained data give $S=4$. Each wing clicks half the time and a coincidence occurs one quarter of the time for each chosen pair. If no-clicks are retained as outcome $0$, the unconditional correlators are reduced by $1/4$ and $S=1$. This explicit model shows why outcome-dependent coincidence filtering can manufacture an apparent violation.

Support-overlap control. Keep the state and data analysis fixed but move or lengthen one coupling until the two spacetime supports are causally related. The cross-region commutator or intervention contrast must then be recomputed. A violation may still be an interesting quantum correlation, but it no longer tests the same spacelike local model because one setting can in principle influence the other wing.

Boundedness and implementation control. Replace the ideal encoded Pauli effects by calibrated effects $\widehat A_x,\widehat B_y$ and bound their distance from the targets. If every local operator differs in norm by at most $\varepsilon$, each product correlator can shift by at most $2\varepsilon+\varepsilon^2$, so the CHSH shift is at most $8\varepsilon+4\varepsilon^2$. Certification requires the statistical lower bound after this systematic allowance still to exceed $2$.

Limitations

The encoded benchmark assumes commuting local matrix subalgebras and a state with a Bell-pair restriction. It does not provide a preparation protocol from the vacuum, prove the split property for an arbitrary region pair, or convert the Summers–Werner observables into a practical detector. Those are distinct QFT and engineering questions. A field Bell claim should state the net or regulator, compact supports, induced effects, state preparation, setting distribution, trial clock, loss model, systematic norm bounds, and local-realist significance calculation.

Exercises

Derive the four correlators and the Tsirelson value for the encoded benchmark.

Solution

The Bell state obeys $Z_AZ_B|\Phi^+\rangle=|\Phi^+\rangle$ and $X_AX_B|\Phi^+\rangle=|\Phi^+\rangle$. Crossed products such as $Z_AX_B$ map the two Bell components to orthogonal states, so their expectation values vanish. Therefore

$$E_{00}=E_{01}=E_{10}=\frac1{\sqrt2}, \qquad E_{11}=-\frac1{\sqrt2},$$

and $S=4/\sqrt2=2\sqrt2$.

Show directly that the quantum probability model is no-signaling.

Solution

Sum Bob’s outcome effects before taking the trace:

$$\sum_b p(a,b\mid x,y) =\operatorname{tr}\!\left[ \rho M^A_{a|x}\sum_bM^B_{b|y} \right] =\operatorname{tr}(\rho M^A_{a|x}).$$

The last expression has no $y$. This proof uses a normalized POVM and unconditional averaging. Dropping Bob’s no-click events replaces the identity by a setting-dependent acceptance effect and invalidates the step.

Evaluate the detection-postselection countermodel both conditionally and unconditionally.

Solution

For fixed settings $(x,y)$, coincidence requires $\lambda=(x,y)$, which occurs with probability $1/4$. On those trials the product is $(-1)^{xy}$, yielding conditional correlators $+1,+1,+1,-1$ and conditional $S=4$. If a missing outcome is recorded as zero, the unconditional correlators are $+1/4,+1/4,+1/4,-1/4$, hence $S=1$. The apparently stronger conditional value is entirely produced by the setting-dependent retained ensemble.

References

• Clauser, J. F., Horne, M. A., Shimony, A., and Holt, R. A. (1969). “Proposed Experiment to Test Local Hidden-Variable Theories.” Physical Review Letters 23, 880–884. DOI.
• Fewster, C. J., and Verch, R. (2020). “Quantum Fields and Local Measurements.” Communications in Mathematical Physics 378, 851–889. DOI. Open article.
• Hensen, B., Bernien, H., Dréau, A. E., et al. (2015). “Loophole-Free Bell Inequality Violation Using Electron Spins Separated by 1.3 Kilometres.” Nature 526, 682–686. DOI.
• Summers, S. J., and Werner, R. (1987). “Maximal Violation of Bell’s Inequalities Is Generic in Quantum Field Theory.” Communications in Mathematical Physics 110, 247–259. DOI.
• Zhang, Y., Glancy, S., and Knill, E. (2011). “Asymptotically Optimal Data Analysis for Rejecting Local Realism.” Physical Review A 84, 062118. DOI. Open PDF.