Operational Independence, the Split Property, and Bell Correlations

The split property gives a spatial tensor-product representation for suitably separated local algebras, and therefore permits independent normal preparations and operations. It does not say that every normal state is a product state. In fact, the same split representation contains normal Bell states attaining the Tsirelson value $2\sqrt2$.

Required background. Split inclusions and statistical independence provide the tensor-product criterion; localized instruments provide independently selectable operations. Helpful background. Modular nuclearity is a sufficient route to splitting, while type-III local algebras prevent a naive subsystem trace. Applications include local preparation, local operations and distillability, and Bell witnesses and tomography limits.

Split independence

Let $\mathcal M_1,\mathcal M_2\subset B(\mathcal H)$ commute. The pair is split when the algebraic map

$$\mathcal M_1\odot\mathcal M_2\longrightarrow \mathcal M_1\vee\mathcal M_2, \qquad A_1\otimes A_2\longmapsto A_1A_2,$$

extends to a normal spatial isomorphism

$$\mathcal M_1\,\overline\otimes\,\mathcal M_2 \simeq\mathcal M_1\vee\mathcal M_2.$$

For nested regions this is equivalent to the existence of a type-I factor $\mathcal M_1\subset\mathcal F\subset\mathcal M_2'$. The spatial isomorphism implies that any normal states $\omega_1$ and $\omega_2$ extend to the normal product state $\omega_1\otimes\omega_2$ on the joined algebra. More strongly, under the standard AQFT hypotheses, local normal states can be prepared by operations acting in separated enlarged regions. Werner established the equivalence between such local preparability and the split property Werner 1987, Theorem 1, pp. 326–328.

The theorem is an existence statement about admissible joint states and maps. It does not provide a canonical tensor factor, a canonical partial trace, or a product decomposition when the separation collar is removed.

Massive free field and a Bell state

Take two strictly separated double cones $O_A,O_B$ in the massive free scalar vacuum representation. Nuclearity estimates imply the split property for positive separation, so there is a unitary split implementation

$$W:\mathcal H\longrightarrow\mathcal H_A\otimes\mathcal H_B$$

with $W\mathcal A(O_A)W^*\subset B(\mathcal H_A)\otimes1$ and $W\mathcal A(O_B)W^*\subset1\otimes B(\mathcal H_B)$. The connection from energy nuclearity to statistical independence is proved for the free-field setting in Buchholz and Wichmann 1986, §§2–3, pp. 326–339.

Because the local factors are properly infinite, choose matrix subfactors $M_2\subset\mathcal A(O_A)$ and $M_2\subset\mathcal A(O_B)$. Let $Z_A,X_A$ and $Z_B,X_B$ be Pauli generators in them. In the split tensor representation choose the Bell vector

$$|\Omega_B\rangle=\frac{|00\rangle+|11\rangle}{\sqrt2}$$

on these two matrix factors and any normal reference state on their complements. Pulling the product back with $W$ gives a normal state on $\mathcal A(O_A)\vee\mathcal A(O_B)$. Define

$$A_1=Z_A,\quad A_2=X_A,\qquad B_1=\frac{Z_B+X_B}{\sqrt2},\quad B_2=\frac{Z_B-X_B}{\sqrt2}.$$

For the CHSH operator

$$\mathcal C=A_1(B_1+B_2)+A_2(B_1-B_2),$$

direct Pauli multiplication gives $\langle\mathcal C\rangle=2\sqrt2$. This is a concrete normal Bell-correlated state in the same separated system that permits arbitrary normal product preparations. The value is bounded by $2\sqrt2$ because $\mathcal C^2=4-[A_1,A_2][B_1,B_2]\leq8$. The general ubiquity of strong Bell correlations in relativistic field theory was established by Summers and Werner Summers and Werner 1987, Theorem 2.1, pp. 249–254.

The worked construction realizes the coexistence emphasized by local preparation and operational independence: one may choose a product state for an independent-preparation task or a Bell state for a correlation task. Split independence constrains what can be prepared, not what must already be present.

Normality of both choices is worth checking. In the split representation, the Bell density operator lives only on the selected $M_2\otimes M_2$ subfactor and is tensored with normal states on the remaining degrees of freedom. Composition with the normal spatial isomorphism returns a normal functional on the joined local algebra. The same argument constructs a product extension of any pair of normal marginal states. No density operator is being assigned to either sharp type-III factor by itself; density matrices occur only in the auxiliary tensor representation and finite matrix subfactors supplied by splitness.

Operational independence also concerns maps, not merely states. If normal unital completely positive operations are chosen independently on the two tensor factors, their tensor product is again normal and completely positive, and commuting factor actions make their order irrelevant. Pullback through the split isomorphism gives a joint operation on the separated local algebras. This conclusion requires the spatial tensor product, whereas Einstein commutativity alone supplies only $[\mathcal M_1,\mathcal M_2]=0$.

The vanishing-collar test

Let the spacelike separation be $\delta>0$. The split unitary and intermediate type-I factor generally depend on $\delta$. Driving $\delta\downarrow0$ increases the phase-space cost, and the nuclearity norm controlling the split construction need not remain bounded. The limiting sharp adjacent algebras are typically type III and need not admit a normal product state on their join.

Therefore one cannot take the Bell construction or independent-preparation channel at fixed norm and silently set $\delta=0$. This adversarial limit does not disprove splitting at every positive separation; it disproves the false converse that locality alone supplies a uniform tensor product at zero collar.

Exercises

1. Tsirelson check. Compute $\langle\Omega_B|\mathcal C|\Omega_B\rangle$ for the displayed observables.

Solution

$B_1+B_2=\sqrt2 Z_B$ and $B_1-B_2=\sqrt2 X_B$. The Bell vector has $\langle Z_AZ_B\rangle=\langle X_AX_B\rangle=1$, so the expectation is $\sqrt2(1+1)=2\sqrt2$.

2. Product-state contrast. Show that any product state satisfies the classical CHSH bound $|\langle\mathcal C\rangle|\leq2$.

Solution

Factor the four correlators. Writing $a_i=\omega_A(A_i)$ and $b_j=\omega_B(B_j)$ with each number in $[-1,1]$, the expression is $a_1(b_1+b_2)+a_2(b_1-b_2)$. Its modulus is at most $|b_1+b_2|+|b_1-b_2|\leq2$.

References

• Buchholz, Detlev, and Eyvind H. Wichmann. “Causal Independence and the Energy-Level Density of States in Local Quantum Field Theory.” Communications in Mathematical Physics 106 (1986): 321–344. DOI.
• Summers, Stephen J., and Reinhard Werner. “Maximal Violation of Bell’s Inequalities Is Generic in Quantum Field Theory.” Communications in Mathematical Physics 110 (1987): 247–259. DOI.
• Werner, Reinhard F. “Local Preparability of States and the Split Property in Quantum Field Theory.” Letters in Mathematical Physics 13 (1987): 325–329. DOI.