Locality Separability Causality

Locality, separability and causality answer different questions. Locality may concern interactions in spacetime or a factorization of conditional probabilities. Separability may mean a product or convex-mixture structure of quantum states, or a philosophical claim about independent physical properties. Causality concerns dependence under interventions and the allowed ordering of influences. An argument should state the intended definition rather than exchange these words as synonyms.

Bell locality and no-signaling

Bell locality uses a shared-variable factorization

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

combined with a setting-independent distribution when deriving the usual inequalities. Local Hidden Variables defines the model and its assumptions. No-signaling instead constrains only the observable marginals. Quantum violations of Bell inequalities can satisfy no-signaling, so the two locality notions are inequivalent.

Relativistic microcausality is another condition: suitable observables in spacelike separated regions commute in the relevant field-theoretic setting. It is not obtained by replacing the probabilities in Bell’s factorization with a commutator. The relativistic causality bridge explains the boundary without importing a complete quantum-field theory into this volume.

State separability and correlated mixtures

A bipartite separable density operator has the form

$$\rho_{AB}=\sum_\lambda p_\lambda\rho_A^\lambda\otimes\rho_B^\lambda.$$

Local measurements on such a state give a Bell-local representation with $\lambda$ labeling the mixture. Thus separability is a sufficient condition for Bell locality of these measurement statistics. It is not necessary for every chosen measurement distribution: entangled states can yield nonviolating data with unsuitable settings, and some mixed entangled states admit local models within specified measurement classes.

The separable state $(|00\rangle\langle00|+|11\rangle\langle11|)/2$ has perfect classical $Z$ correlation. Correlation does not therefore prove entanglement, signaling or a direct causal influence between the outcomes. A shared preparation variable already explains it.

Interventions differ from conditional correlations

Observing $a$ can change the conditional probability of $b$ without providing a means to control $b$. To infer a causal influence, specify an intervention and the resulting distribution. Conditioning on a common effect can also create correlations between otherwise independent variables; detector coincidence selection is a relevant example.

A causal model must say which variables are settings, outcomes, common causes and records, and what interventions are permitted. A proposed retrocausal model changes the assumed directional structure; a superdeterministic model changes the independence relations involving settings. Neither is fully specified by its name.

Exercise: perfect correlation without a signal

A source sends the same uniformly random classical bit to two laboratories. One laboratory learns its bit. What changes in its prediction of the other bit, and can it choose the other laboratory’s marginal distribution?

Solution

The conditional prediction becomes certain, while the remote marginal stays uniform. The source supplied a common cause, and the local laboratory has no control over its received bit. Knowledge of a correlation is not a controllable signaling operation.

References

• Bell, John S., Speakable and Unspeakable in Quantum Mechanics, 2nd edition, Cambridge University Press (2004) — locality and local causality.
• Werner, Reinhard F., “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Physical Review A 40, 4277–4281 (1989), doi:10.1103/PhysRevA.40.4277 — distinction between entanglement and local models.