Conservative causality is a notion of causality where the local description is conserved. This is an extension of Fredkin's and Toffoli's conservative logic [1] --- thermodynamic quantities are conserved --- to the realm of causality. Cause-effect relations of conservative causality are more general when compared to those appearing in the traditional notion of causality [2], and simultaneously, less unnatural when compared to extreme departures [3-5] thereof. Thus, conservative causality offers one to take a glance beyond the traditional notion. Since in conservative causality the local description is preserved, theories that obey such a notion relate to traditional theories, e.g., quantum theory or Turing machines, in the same way as general relativity relates to special relativity: Special relativity holds for free falling observers in sufficiently small space-time regions of general relativity. Recently, some theories of conservative causality have been developed. These are the process-matrix framework [6], as well as the classical-probabilistic and classical-deterministic limit thereof [7, 8]. All these theories have tame properties [9-11], even though all of them break with the standard assumption that events can influence their future only.

While the traditional notion of causality is described by causal structures, via directed acyclic graphs (DAGs) and the model parameters [2], such a description is insufficient for the causal structures in conservative causality. In the latter, the more general concept of directed cyclic graphs is used, allowing for causal loops [12, 13].  While being a very natural generalization, cyclic causal structures could potentially describe the theoretical possibility of time-travel. Such theoretical scenarios allowing causal loops come with mainly two problems. The first, usually considered more important, is the grandfather antinomy, i.e., a robot is programmed to travel back in time and to disassemble itself. Therefore, if the robot travels back then it does not etc. --- a logical contradiction. The other is called information antinomy (carrying also other names, e.g., uniqueness ambiguity [11, 14]). Imagine someone waking up and finding a book containing the proof of a long-standing mathematical problem next to her bed. Later, she publishes the proof and travels back in time and places the book next to her bed. While such a story does not embody a logical contradiction, there is complex information arising out of nowhere. More specifically, questions like ”Which proof does the book contain?” or ”In what language is it written?” are ill posed, in the sense that there is a multitude of consistent answers that can be given. In other words, given the boundary conditions, the theory provides no prediction at all (not even probabilistically).

Although these antinomies might seem unrelated, it has been lately shown that in the classical-deterministic process theory [7] they are equivalent [13]. This initiates the study of the connection of these antinomies in more general conservative-causality frameworks (e.g., the process-matrix framework [6]) and their relation to the concept of logical consistency.

[1] E. Fredkin and T. Toffoli, "Conservative logic", International Journal of Theoretical Physics, 21(3–4), 219–253 (1982).

[2] J. Pearl, "Causality", Cambridge University Press (2009).

[3] D. Deutsch, "Quantum mechanics near closed timelike lines", Physical Review D 44, 3197 (1991).

[4] C.H. Bennett and B. Schumacher, "Simulated time travel, teleportation without communication, and how to conduct a romance with someone who has fallen into a black hole", Talk at QUPON, Vienna (2005).

[5] S. Lloyd, L. Maccone, R. Garcia-Patron, V. Giovannetti, and Y. Shikano, "Quantum mechanics of time travel through post-selected teleportation", Physical Review D 84, 025007 (2011).

[6] O. Oreshkov, F. Costa, and Č. Brukner, "Quantum correlations with no causal order", Nature Communications 3, 1092 (2012).

[7] Ä. Baumeler and S. Wolf, "The space of logically consistent classical processes without causal order", New Journal of Physics 18, 013036 (2016).

[8] Ä. Baumeler and S. Wolf, "Device-independent test of causal order and relations to fixed-points", New Journal of Physics 18, 035014 (2016).

[9] M. Araújo, P. Allard Guérin, and Ä. Baumeler, "Quantum computation with indefinite causal structures", Physical Review A 96, 052315 (2017).

[10] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, "A purification postulate for quantum mechanics with indefinite causal order", Quantum 1, 10 (2017).

[11] Ä. Baumeler and S. Wolf, "Computational tameness of classical non-causal models", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, 20170698 (2018).

[12] Jonathan Barrett, Robin Lorenz & Ognyan Oreshkov, "Cyclic Quantum Causal Models", (2020)

[13] Ämin Baumeler and Eleftherios Tselentis, "Equivalence of grandfather and information antinomy under intervention". To appear in Proceedings of the 17th Int. Conf. on Quantum Physics and Logic (QPL), (2020).

[14] J.-M. A. Allen, " Treating time travel quantum mechanically ", Physical Review A 90, 042107 (2014).