Quantum Bayesian Networks

June 27, 2019

Comments on quant-ph arXiv:1906.10726, “Quantum Causal Models”, by Jonathan Barrett, Robin Lorenz, Ognyan Oreshkov

Filed under: Uncategorized — rrtucci @ 6:59 am

The paper

“Quantum Causal Models”, by Jonathan Barrett, Robin Lorenz, Ognyan Oreshkov, https://arxiv.org/abs/1906.10726

was published by BLO tonight. This is my initial response to it.

It’s been just a few hours since BLO published this paper in arxiv, so I haven’t had a chance to read it in its entirety, yet. I have never communicated with BLO, so this is the first time the authors will hear about my response to it. However, much of the material covered in the BLO paper is familiar territory to me, having worked on QB Nets (Quantum Bayesian Networks) since my first paper on them in 1995

Excellent job! And it mentions my work. Thank you!

Here is how BLO explain the relevance of my work to theirs:

Many works are explicitly concerned with causal structure, but not to the end of a quantum generalization of causal models. These include, for example, Refs. [19, 22, 45, 64–69], and are not discussed here any further. Early work by Tucci [6, 7] aims at a quantum generalization of classical Bayesian (rather than causal) networks, obtained by associating probability amplitudes with nodes. More closely related to our work is that of Leifer and Poulin [9], which presents (amongst other things) an approach to quantum Bayesian networks, wherein a quantum state is associated with a DAG, and must satisfy independence relationships formalised by the quantum mutual information, given by the structure of the DAG. The results of Ref. [9] have at various times been used in our proofs. Leifer and Spekkens [11] adapt the ideas of Ref. [9] to quantum causal models, using a particular definition of a quantum conditional state. Our approach differs from that of Ref. [11] in taking influence in unitary transformations as defining of causal relations, in its use of the process operator formalism, and in the fact that we don’t use quantum conditional states.

Under Refs. 6, 7 they list

[6] R. R. Tucci, “Quantum bayesian nets,” International Journal of Modern Physics B 9 no. 03, (1995) 295–337.
[7] R. R. Tucci, “Factorization of quantum density matrices according to bayesian and markov networks,” arXiv:quant-ph/0701201.

I do disagree with their characterization of my work. My early work Ref.6 in 1995 was on QB nets for pure states, but then I published Ref.7 in 2007 which explains how QB nets can also be used to describe density matrices that are not pure states. In 2012, I published the following introductory review and reprise of the use of QB nets to describe general density matrices

“An Introduction to Quantum Bayesian Networks for Mixed States”, by Robert R. Tucci, https://arxiv.org/abs/1204.1550

I contend that what the BLO paper proposes is **exactly** QB nets for density matrices. They just call them by a different name. A rose by any other name would smell as sweet.

I have also published in this blog the following article describing the connection of QB Nets to “Tensor Networks”. This is an obvious connection that a lot of people have asked me about, and which is not addressed anywhere in the BLO paper. (I searched the BLO paper in vain for the phrase “tensor network”).

Tensor Networks versus Quantum Bayesian Networks: And the winner is…

As an aside, I think that BLO’s Ref.9 by Leifer and Poulin is patently incorrect because it is based on a new definition of conditional density matrices which imposes major constraints on standard Quantum Mechanics. So, if the work of Leifer and Poulin applies to the real world at all, it does so only within the context of a severely maimed Quantum Mechanics.

QB nets, which are exactly what BLO call “Quantum Causal Models”, do not assume any axioms beyond those of standard Quantum Mechanics. QB nets are simply a graphical way of displaying (any, all) quantum density matrices, the same way that classical Bayesian networks are simply a graphical way of displaying (any, all) joint probability distributions. In the same way that classical Bayesian networks arise from the chain rule for joint probability distributions, QB nets arise from a chain rule for quantum **probability amplitudes**. That is the gist of Ref.7, cited by the BLO paper.

I would also like to point out that the BLO paper does not mention that I too have addressed Judea Pearl’s d-separation and do-calculus as it pertains to the quantum realm. In the 2013 paper:

“An Information Theoretic Measure of Judea Pearl’s Identifiability and Causal Influence”, by Robert R. Tucci, https://arxiv.org/abs/1307.5837

I address the do-calculus for **classical** Bayesian networks, but I do so in terms of entropy. I explicitly mention in the introduction to that paper, that I intentionally use only entropy concepts to define things, with the intention that these concepts be generalized to Quantum Mechanics, using the simple rule of replacing H(P) by S(\rho) (i.e., by replacing entropies of classical probability distributions by entropies of quantum density matrices). This “minimal substitution” was invented by Cerf and Adami, and has been proven to be a powerful guiding principle in Quantum Information Theory. For instance, it led me to the discovery of the definition of Squashed Entanglement, as documented in its Wikipedia article.

Added later:
I also reported on this in Reddit

1 Comment »

  1. Excellent summary, putting it nicely into context, and highlighting the most essential concepts i.e. quantization in QIS means joint probability tables become density matrices and H(P) -> S(\rho)

    Comment by Henning Dekant — June 27, 2019 @ 7:35 pm

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