Quantum mechanics of time travel

The theoretical study of time travel generally follows the laws of general relativity. Quantum mechanics requires physicists to solve equations describing how probabilities behave along closed timelike curves (CTCs), theoretical loops in spacetime that might make it possible to travel through time.

In the 1980s, Igor Novikov proposed the self-consistency principle. According to this principle, any changes made by a time traveler in the past must not create paradoxes. If a time traveler tries to change the past, the laws of physics will ensure that history remains consistent. This means that the outcomes of events will always align with the traveler’s actions in a way that prevents any contradictions.

However, Novikov's self-consistency principle may be incompatible when considered alongside certain interpretations of quantum mechanics, particularly two fundamental principles of quantum mechanics, unitarity and linearity. Unitarity ensures that the total probability of all possible outcomes in a quantum system always sums to 1, preserving the predictability of quantum events. Linearity ensures that quantum evolution preserves superpositions, allowing quantum systems to exist in multiple states simultaneously.

There are two main approaches to explaining quantum time travel while incorporating Novikov's self-consistency. The first approach uses density matrices to describe the probabilities of different outcomes in quantum systems, providing a statistical framework that can accommodate the constraints of CTCs. The second approach involves state vectors, which describe the quantum state of a system. This approach sometimes leads to concepts that deviate from the conventional understanding of quantum mechanics.

Deutsch's prescription for closed timelike curves (CTCs)
In 1991, David Deutsch proposed a method to explain how quantum systems interact with closed timelike curves (CTCs) using time evolution equations. This method aims to address paradoxes like the grandfather paradox, which suggest that a time traveler who stops their own birth would create a contradiction. One interpretation of Deutsch's approach is that it implies the time traveler might end up in a parallel universe rather than their own, although the formalism itself does not explicitly require the existence of parallel universes.

Method overview
To analyze the system, Deutsch divided it into two parts: a subsystem outside the CTC and the CTC itself. To describe the combined evolution of both parts over time, he used a unitary operator (U). This approach relies on a specific mathematical framework to describe quantum systems. The overall state is represented by combining the density matrices (ρ) for both the subsystem and the CTC using a tensor product (⊗). Notably, Deutsch assumed no initial correlation between these two parts. While this assumption breaks time symmetry (meaning the laws of physics wouldn't behave the same forwards and backwards in time), Deutsch justifies it using arguments from measurement theory and the second law of thermodynamics.

Deutsch's proposal uses the following key equation to describe the fixed-point density matrix (ρCTC) for the CTC:

$$\rho_{\text{CTC}} = \text{Tr}_A \left[ U \left( \rho_A \otimes \rho_{\text{CTC}} \right) U^\dagger\right]$$.

The unitary evolution involving both the CTC and the external subsystem determines the density matrix of the CTC as a fixed point, as represented by this equation. The trace operation ($${Tr}_A$$) indicates that we are considering the partial trace over the subsystem outside the CTC, focusing on the state of the CTC itself.

Ensuring Self-Consistency
Deutsch's proposal ensures that the CTC always returns to a self-consistent state after a loop. This means that the overall state of the CTC remains consistent. However, this raises concerns. If a system retains memories after traveling through the CTC, it could create complex scenarios where it appears to have experienced different possible pasts.

Furthermore, Deutsch's method might not work with common probability calculations in quantum mechanics, like path integrals, unless we take into account the chance that the system goes through different paths that all lead to the same outcome. There can also be multiple solutions (fixed points) for the system's state after the loop, introducing a form of randomness (nondeterminism). Deutsch suggested using the solution with the highest entropy, which aligns with the natural tendency of systems to evolve towards higher entropy states.

To calculate the final state outside the CTC, a specific mathematical operation (trace) considers only the external system's state after the combined evolution of both the external system and the CTC. The tensor product (⊗) of the density matrices for both systems describes this combined evolution. Then, a unitary time evolution operator (U) is applied to the whole system.

Implications and criticisms
Deutsch's approach has intriguing implications for paradoxes like the grandfather paradox. Consider a scenario in which everything, except a single quantum bit (qubit), travels through a time machine and flips its value according to a specific operator:
 * $$U = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}$$.

Deutsch argues that the solution maximizing von Neumann entropy (a measure of how scrambled or mixed the information in the qubit is) is the most relevant. In this case, the qubit becomes a mix of starting at 0 and ending at 1, or vice versa. Deutsch's interpretation, which can align with the many-worlds view of quantum mechanics, avoids paradoxes because the qubit travels to a different parallel universe after interacting with the CTC.

Researchers have explored the potential of Deutsch's ideas. Deutsch's CTC time travel, if possible, might allow computers near a time machine to solve problems far beyond classical computers, but the feasibility of CTCs and time travel remains a topic of debate and further research is needed.

Despite its theoretical nature, Deutsch's proposal has faced significant criticism. For instance, Tolksdorf and Verch demonstrated that quantum systems without CTCs can still achieve Deutsch's criterion with high accuracy. This finding casts doubt on the uniqueness of Deutsch's criterion for quantum simulations of CTCs as theorized in general relativity. Their research showed that classical systems governed by statistical mechanics could also meet these criteria, implying that the peculiarities attributed to quantum mechanics might not be essential for simulating CTCs. Based on these results, it appears that Deutsch's criterion is not specific to quantum mechanics and may not be a good way to figure out the possibilities of real-time travel or how quantum mechanics might make it possible. Consequently, Tolksdorf and Verch argue that their findings doubt the validity of Deutsch's explanation of his time travel scenario using the many-worlds interpretation.

Lloyd's prescription: Post-selection and time travel with CTCs
Seth Lloyd proposed an alternative approach to time travel with closed timelike curves (CTCs), based on "post-selection" and path integrals. Path integrals are a powerful tool in quantum mechanics that involve summing probabilities over all possible ways a system could evolve, even if those paths don't strictly follow a single timeline. Unlike classical approaches, path integrals allow for consistent histories even with CTCs. Lloyd argues that focusing on the state of the system outside the CTC is more relevant.

He proposes an equation that explains the transformation of the density matrix, which represents the system's state outside the CTC, following a time loop:
 * $$\rho_f = \frac{C\rho_i C^\dagger}{\text{Tr}\left[ C\rho_i C^\dagger\right]}$$, where $$C = \text{Tr}_{\text{CTC}}\left[ U \right]$$.

In this equation:


 * $$\rho_f$$ is the density matrix of the system after interacting with the CTC.
 * $$\rho_i$$ is the initial density matrix of the system before the time loop.
 * $$C$$ is a transformation operator derived from the trace operation over the CTC, applied to the unitary evolution operator $$U$$.

The transformation relies on the trace, a specific mathematical operation within the CTC that reduces a complex matrix to a single number. If this trace term is zero ($$\text{Tr}\left[ C\rho_i C^\dagger\right]=0$$), the equation has no solution, indicating an inconsistency like the grandfather paradox. Conversely, a non-zero trace leads to a unique solution for the external system's state.

Thus, Lloyd's approach ensures self-consistency and avoids paradoxes by allowing only histories consistent with the system's initial and final states. This aligns with the concept of post-selection, where only certain outcomes are considered based on predetermined criteria, effectively filtering out paradoxical scenarios.

Entropy and computation
Michael Devin (2001) proposed a model that incorporates closed timelike curves (CTCs) into thermodynamics, suggesting it as a potential way to address the grandfather paradox. This model introduces a "noise" factor to account for imperfections in time travel, proposing a framework that could avoid paradoxes.

Devin's model posits that each cycle of time travel involving a quantum bit (qubit) carries a usable form of energy, termed "negentropy" (negative entropy, representing a decrease in disorder). The model suggests that the amount of negentropy is proportional to the noise level introduced during time travel. This implies that a time machine could potentially extract work from a thermal bath in proportion to the negentropy generated.

Moreover, Devin's model indicates that a time machine could significantly reduce the computational effort required to solve complex problems, such as cracking codes through trial and error. CTCs could allow for a more efficient computation process because the system can effectively "reuse" information from different timelines, leading to faster problem-solving capabilities.

However, the model also predicts that as the noise level approaches zero, the usable energy and computational power will become infinitely large. This implies that conventional computational complexity classes, which categorize problems based on their difficulty for classical computers, might not apply to time machines with very low noise levels. Devin's model is entirely theoretical and speculative and has not been confirmed by experimental evidence.