Science fiction has long embraced the idea of travelling backward in time, but the advent of Einstein’s general theory of relativity transformed these ideas from fantasies into potential – albeit contested – realities. In particular, solutions to the equations of general relativity known as closed time-like curves (CTCs) seem to allow a system’s trajectory to return to a previous point in time. Although the existence of CTCs has never been proven, their admissibility within general relativity has inspired many physicists to study their implications, which include increases in information processing speeds as well as retrograde time travel.
Inspired by the 2014 film Interstellar, in which a father sends messages to his daughter in the past, Kaiyuan Ji and Mark Wilde of Cornell University, together with Seth Lloyd at the Massachusetts Institute of Technology, have now investigated the potential advantages of a retrograde messaging channel based on a CTC. As well as deriving an exact expression for the information capacity of such a channel, they showed that this capacity exceeds that of regular channels that do not involve backward time travel.
Post-selected CTC models
To model their CTC, the researchers used an approach that Lloyd and his then-collaborators developed 15 years ago. This approach exploits the mathematical equivalence between a CTC and the combined operations of quantum teleportation and post selection, so it is often referred to as a post-selected CTC model. Previous analyses established that this model disallows time-travel paradoxes such as going back in time to kill your grandfather, while still preserving correlations between the system that travels backward in time and the environment.
The latter feature is important, Ji notes, because quantum CTC models without it have a distinct drawback: “If you travel through that closed timeline curve, you end up in the past, but basically you lost all the memory of what has happened before you do time travel,” he explains.
A further advantage of the team’s chosen model is that it does not prohibit causal loops in which correspondents in the past and future influence each other. To return to the Interstellar scenario, Ji explains that while the daughter is influenced by her father’s message, she also influences her father, because he witnessed how she decoded the message and therefore used that information to optimize his encoding protocols.
The existence of causal loops in the model allowed Ji and his colleagues to formulate an expression for the channel’s bit capacity. It also pointed towards the best way of maximizing that capacity. The optimal strategy, Ji explains, would involve the father using his memory from the past when encoding his message to the past, which makes sending messages to the past more efficient (and capable of a higher bit capacity) than sending messages to the future. “The causal loop is essential to the design of the optimal communication protocol in this retro causal communication setting,” he tells Physics World. “It is not possible in standard communication from the past to the future.”
Capacity for messages
The CTC can also be used to form a quantum channel capable of transmitting either regular classical data or quantum data back in time. Using their expression for the channel’s bit capacity, the researchers were able to show that its classical bit capacity is twice that of its quantum counterpart.
Nicole Yunger Halpern, a theoretical physicist at the University of Maryland who has studied some implications of CTCs for metrology but was not involved in the current work, describes the research as “creative and technically impressive”. She points out that while calculating a communication channel’s capacity is a common challenge in the field of information theory, obtaining straightforward answers can be difficult. “The authors combined this workaday task with a highly unusual (retrocausal) setting and, moreover, obtained a simple solution,” Halpern says.
Decoding the dark arts of Interstellar’s black hole
Not everyone is sold on the mathematical format of post-selected CTC models, though. Scott Aaronson, a computer scientist at the University of Texas at Austin, US, who has previously studied the speed advantages of information processing using CTCs, believes that studies with this approach “attempt to model time travel but don’t fully succeed at it”. However, with respect to the question of channel capacity, Aaronson concedes it may be “interesting to prove things”.
While the current model considered the effects of distortion (“noise”) on messages in the channel, Ji suggests it might be interesting to broaden this to include noise in the memory of the process – like the father in Interstellar having a hazy recall of his daughter’s decoding protocol. Ji also notes that the post-selection CTC model has some mathematical equivalence with black hole final state projections, and he says it might be interesting to explore these connections, too.
The research is described in Physical Review Letters.
