# Quantum mechanics seen through closed timelike curves / La mécanique quantique à travers les courbes fermées de genre temps

Séminaire organisé par Alexei Grinbaum (CEA-Saclay, France), Jeffrey Bub (IPST, University of Maryland, USA) et Eric Cavalcanti (The University of Sydney, Australia) du 24 au 29 juin 2013

**Participants**

Caslav Brukner, Todd Brun, Jeffrey Bub, Eric Cavalcanti, Julien Degorre, Lucas Dunlap, Ernesto Galvão, Nicolas Gisin, Alexei Grinbaum, Clare Horsman, Issam Ibnouhsein, Raymond Lal, Damian Markham, Arun K. Pati, Timothy Ralph, Tony Short, John Smolin, Allen Stairs, George Svetlichny, Joel Wallman

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**Summary**

The conference “Quantum mechanics seen through closed timelike curves” took place at Les Treilles between 24 and 29 June 2013, gathering 20 experts on quantum theory, quantum information, and the philosophy of physics. This was the first meeting specifically dedicated to this emerging issue in the foundations of physics, as well as more general topics related to causal structure in quantum theory.

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**Compte-rendu**

The topic of closed timelike curves (CTC’s) has grown in prominence due to recent theoretical models with important implications for quantum computation, quantum information and the foundations of quantum theory. In the general theory of relativity, CTC’s are theoretically possible trajectories in some solutions to Einstein’s equations. Deutsch proposed a model for CTC’s in a quantum computational framework that purports to remove inconsistencies and solve some of the paradoxes associated with causality in relativistic models. This was yet another example of a “direct collision test” between relativity and quantum theory, both firmly established experimentally but yet incompatible theoretically, whereby notions and results from one theory are applied in the other as an attempt to shed light on the strange features of both theories. In Deutsch’s case, he had hoped that quantum information would bring new insights into the problem of causal inconsistency. What was revealed, however, gives more unexpected insights into quantum information and into quantum mechanics, rather than into relativity.

The talks in the meeting ranged from results showing the power of Deutsch-CTCs as well as other models, such as that due to Bennett-Schumacher-Svetlichny, to discussion on the foundational implications of those models and beyond, to general questions on causality in quantum theory. We summarise some of discussion in the workshop below.

The meeting was opened by one of the pioneers of the topic, George Svetlichny, who presented a new model for dealing with CTCs, starting from the premise that since quantum mechanics is empirically more secure than general relativity, it should be kept intact. He reformulated Deutsch’s approach in the context of open quantum systems and arrived at a theory where the chronology-preserving part of the system undergoing interaction with a CTC maintains linearity of its evolution.

Todd Brun reviewed the Deutsch model (D-CTC) as well as the postselected CTC (P-CTC) model due to Bennett-Schumacher-Svetlichny, discussing how they address time travel paradoxes and how they lead to nonlinear evolution of quantum states. He then showed how this allows one to solve several problems that are hard or impossible for conventional quantum systems, such as distinguishing non-orthogonal states or solving computationally hard problems. These ideas were then further discussed by Joel Wallman.

Tim Ralph discussed how the Deutsch model can be mapped into an equivalent circuit which obeys the rules of standard quantum mechanics, with the non-linearity replaced by multiple copies of the initial state and unitary interaction, and showed how this model could be applied to continuous variable field states as well as to take into account the finite size of the CTC.

Tony Short presented joint work with Sandu Popescu where they introduce a formalism for describing an agent who has possession of a number of quantum systems over multiple time-intervals, and for calculating the joint outcome probabilities of several such agents. If the agents also have the ability to post-select, their set of allowed actions becomes remarkably simple to characterise. This also gives the agents the ability to create closed time-like curves, and yields a natural time-reversal symmetry.

In a joint talk, Jeff Bub and Allen Stairs examined an argument to the effect that P-CTCs allow a “radio to the past” that permits an unproved theorem, raising interpretative problems for Tim Ralph’s argument. They showed how to sidestep these problemsand concluded, with Ralph, that the existence of this “radio to the past” creates the possibility of new paradoxes, now in the classical domain. Finally, they argued that a similar problem occurs with D-CTCs, showing that the effect is not confined to P-CTCs.

Lucas Dunlap addressed conceptual issues surrounding Deutsch’s solution to the Information Paradox in CTCs, arguing that Deutsch’s acceptance of the existence of the many worlds of the Everett interpretation cuts both ways, since it commits Deutsch to believing that any history that is physically possible is actualized in some world, including histories in which an Information Paradox scenario plays out.

Damian Markham and Julien Degorre presented some observations on the restrictions imposed on non-linear extensions of quantum mechanics with respect to non-signaling, and discussed in what sense such theories can still allow for “local” cloning and state discrimination.

Ray Lal presented a brief overview of category theory as a way of formalising the structural features of quantum theory. He showed how it provides a graphical calculus that exposes the information flow of quantum information protocols in an intuitive way, revealing surprising features of these protocols, such as the fact that in quantum teleportation information appears to flow “backwards in time”, a feature that inspired the postselected teleportation approach. Lal also discussed other issues in the project of formalising causal structure in categorical quantum mechanics, and in particular presented a result about how under time-reversal no-signalling boxes generically become signalling.

Ernesto Galvão discussed how the one-way model of measurement-based quantum computation (MBQC) can be used to test Deutsch’s model for CTCs. Using the stabilizer formalism, he and co-authors identified predictions that MBQC makes about a specific class of CTCs involving travel in time of quantum systems. Using a simple example they showed that Deutsch’s formalism leads to predictions conflicting with those of the one-way model, whereas the predictions of the P-CTC model fit naturally with their formalism.

John Smolin presented an argument against the power of CTCs or more general nonlinear theories, based on what he and co-authors call the “linearity trap”. They argue that all of the exotic effects arising from nonlinear theories arise from an incorrect use of linearity in an otherwise nonlinear theory, and thus that nonlinear theories would not have the power that some of the recent works (such as that of Brun) ascribe to them.

In a reply to John Smolin’s talk, Eric Cavalcanti presented a counter-argument to the effect that the model of Smolin and co-authors does not take into consideration a possible distinction between proper and improper mixtures. A framework was presented for quantum theory augmented by “nonlinear boxes” and it was shown that they must make a distinction between states prepared by local and remote preparation procedures. With a formalisation of this distinction analogous to the distinction between proper and improper mixtures, the linearity trap argument can be circumvented.

Nicolas Gisin analyzed in his talk the idea that no-signalling correlations are more fundamental than constraints from relativity theory. He derived fundamental theoretical limits that stem from such correlations and then explored “delocalized nonlocality”, a form of genuine 3-party nonlocality whose bipartite marginals are all local.

In a more general topic related to causality in quantum theory, Caslav Brukner presented a new framework to treat quantum correlations with indefinite causal structure that was further discussed by Issam Ibnouhsein. Their model assumes only that operations in local laboratories are described by quantum mechanics (i.e. are completely-positive maps), but relax the assumption that they are causally connected. Remarkably, they find situations where two operations are neither causally ordered nor in a probabilistic mixture of definite causal orders, i.e. one cannot say that one operation is either before or after the other. The correlations between the operations are shown to enable performing a communication task (“causal game”) that is impossible if the operations are ordered according to a fixed background time.

Apart from the fact that the topic of CTCs is fascinating in itself, the conference has revealed how CTCs also bring out a number of issues in the foundations of quantum mechanics in a way that makes their distinctions sharper and more important. All attendants have expressed great enjoyment of the discussions, and many debates and even collaborations have been initiated at Les Treilles.