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Quantum-probabilistic graphical models

When you have quantum probability, directed graphical models look different, especially for causality.

Jacques Pienaar, Causality in the quantum world:

In the case of two entangled particles, Reichenbach’s principle would suggest that the correlations between the particles could be explained by a common cause. However, we also know that quantum statistics can violate Bell’s inequalities, which means that variables serving as common causes that could make the correlation disappear cannot exist. A quantum causal model should redefine the connection between causal statements and statistical observations by accounting for this phenomenon (see Fig. 1). It should also tell us how to derive conditional independence relations, which in turn allow us to perform Bayesian updating of probabilities. Finding a model that meets both of these requirements has been challenging.

Most early attempts at quantum causal models proceeded by defining causal structures for quantum systems and then finding which conditional independence relationships remained intact [3]. However, these models could not perform Bayesian inference because conditional independence was no longer a prerequisite for identifying a common cause. Matthew Leifer and Robert Spekkens [4] attempted to incorporate Bayesian inference in a quantum framework using “conditional quantum states” in place of conditional probabilities, but this creative approach was found to be applicable only in restricted cases. Fabio Costa and Sally Shrapnel [5] set aside the problem of conditional independences to focus on causal interventions. For example, instead of considering the conditional independence of tsunamis in Chile and Japan, their approach would consider whether creating or preventing earthquakes (an intervention) would trigger or suppress the tsunami events through physical processes. This model allowed causal relationships to be defined, but it lacked the conditional independences with which to perform Bayesian inference.

Building on the work of Costa and Shrapnel, Allen and his colleagues set out to restore conditional independence as a prerequisite for common causes. To do so, they took advantage of an old physical argument that derives Reichenbach’s principle by assuming that statistical data are the result of a deterministic model. For instance, rolling dice in a casino might appear random, but it could be explained, in principle, by a croupier whose skills allow him to determine the outcome of each throw. While it is debatable whether quantum systems are compatible with this type of determinism, they are compatible with another type of determinism called unitary evolution. A process is called unitary if it conserves quantum information. Compatibility with unitarity is a central tenet of quantum mechanics.

Allen et al. realized that by replacing “deterministic” with “unitary” in Reichenbach’s principle they could obtain a new version of quantum causal models. In particular, their quantum version of the Reichenbach principle allowed them to relate conditional independence to quantum causal relationships like those described in Costa and Shrapnel’s model. What’s more, these conditional independence relations could then be used to perform Bayesian inference. Allen et al.’s result combines both causal interventions and Bayesian inference into a single model, succeeding where others had failed.


Allen, J.-M. A., Barrett, J., Horsman, D. C., Lee, C. M., & Spekkens, R. W.(2017) Quantum Common Causes and Quantum Causal Models. Physical Review X, 7(3), 031021. DOI.
Costa, F., & Shrapnel, S. (2016) Quantum causal modelling. New Journal of Physics, 18(6), 063032. DOI.
Fritz, T. (2016) Beyond Bell’s Theorem II: Scenarios with Arbitrary Causal Structure. Communications in Mathematical Physics, 341(2), 391–434. DOI.
Henson, J., Lal, R., & Pusey, M. F.(2014) Theory-independent limits on correlations from generalized Bayesian networks. New Journal of Physics, 16(11), 113043. DOI.
Leifer, M. S., & Spekkens, R. W.(2013) Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference. Physical Review A, 88(5), 052130. DOI.
Pienaar, J., & Brukner, Č. (2015) A graph-separation theorem for quantum causal models. New Journal of Physics, 17(7), 073020. DOI.
Tucci, R. R.(1995) Quantum bayesian nets. International Journal of Modern Physics B, 09(03), 295–337. DOI.