Quantum Common Causes and Quantum Causal Models

Abstract

Reichenbach's principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations. Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize. The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution. In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach's principle would seem to demand. Motivated by this, we introduce a quantum counterpart to the principle. We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input A and outputs B and C is compatible with A being a complete common cause of B and C, then it must factorize in a particular way. Finally, we show how to generalize our quantum version of Reichenbach's principle to a formalism for quantum causal models, and provide examples of how the formalism works.Contents arXiv:1609.09487v2 [quant-ph]

Updating the Born rule

“…We present a proof of a unified quantum probability rule that subsumes both the Born rule and the state update rule. This rule is useful in a variety of contexts, from quantum information [14][15][16][17][18][19] to quantum causal modelling [20][21][22][23], and non-Markovian dynamics [24][25][26][27]. Dubbed the 'Quantum Process Rule', we prove that one can derive this higher-order, generalised form of the standard quantum probability rule from the structure of quantum operations and a single non-contextuality assumption.…”

“…From this perspective, the 'state update' rule is not an update at all, but rather an application of probabilistic conditioning. This insight distinguishes this approach from other attempts to leverage Bayesian arguments to justify an informational interpretation of quantum theory [21,23].…”

Gleason-Busch theorem for sequential measurements

“…This is what is brought over from standard quantum mechanics; what is derived is the probabilistic structure. Although we note there is much recent interest in causally neutral formulations of quantum theory [19,21] and non-fixed causal orderings [18,20], for simplicity we consider here a fixed causal order: the measurements we consider are performed sequentially, and on the same system. Under the assumptions of the Gleason-Busch theorem, measurements are described by positive operators: we thus associate to the first measurement a set of positive operators {Ê i }, to the second measurement the set {F j }.…”

“…More recently Shrapnel et al [13], starting from an assumption that transformations are described by completely positive maps also used an axiomatic approach similar to Busch and Gleason to derive a probability measure which en-compasses both the Born rule and state update rule. Motivated by recent work on indefinite causal order in quantum mechanics [18][19][20][21], Shrapnel et al's work derives the most general rule resulting in a probability measure on the set of completely positive maps. In the present work, by contrast, we show that sequential measurements correspond to completely positive maps and derive the most general form of these, from a few simple axioms.…”

Unifying framework for spatial and temporal quantum correlations

“…vector formalism [23][24][25][26], the "general boundary" formalism [27], operational open dynamics [28,29], and quantum causal models [30,31]. We will use the semantics and conventions of the "process matrix" formalism [32][33][34], since it allows a clean distinction between resources and operations.…”