Quantum mechanics of time travel

The study of time travel generally follows the laws of general relativity. Quantum mechanics, on the other hand, needs physicists to solve equations describing how probabilities (represented by density matrices) behave along closed timelike curves (CTCs). CTCs are loops in spacetime that might make it possible to travel through time.

Igor Novikov came up with the self-consistency principle in the 1980s. If a time traveler tries to change the past, this principle says that calculations will always lead to a consistent history. However, Novikov's self-consistency principle clashes with some ways of understanding quantum mechanics, mainly unitarity (the principle that the total probability of all possible outcomes in a quantum system is always 1) and linearity (the process that calculates combined probabilities).

There are two main ways to explain quantum time travel using Novikov's self-consistency. The first method uses a concept mathematically known as a density matrix, and the second method uses something different, which is known as a state vector. This second method yields concepts that deviate from the conventional understanding of quantum mechanics.

Deutsch's prescription for closed timelike curves (CTCs)

David Deutsch came up with a way to use time evolution equations to explain how quantum systems interact with closed timelike curves (CTCs) in 1991. This method deals with paradoxes like the grandfather paradox, which suggests that someone who could travel through time and stop their own birth would create a contradiction. One problem with this, though, is that it makes it seem like the time traveler might end up in a parallel universe rather than their own.

Method overview

To analyze the system, Deutsch divided it into two parts: a subsystem outside the CTC and the CTC itself. He then used a unitary operator (U) to capture the combined evolution of both parts over time. This approach relies on a specific mathematical description of 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 connection 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), he justifies it with arguments from measurement theory and the second law of thermodynamics.

This equation, which describes the fixed-point density matrix (ρCTC) for the CTC, captures the central idea of Deutsch's proposal:

$\rho _{\text{CTC}}={\text{Tr}}_{A}\left[U\left(\rho _{A}\otimes \rho _{\text{CTC}}\right)U^{\dagger }\right]$.

Deutsch's proposal offers solutions that always return the CTC to a consistent state after a loop. This means any measurable property of the CTC will return to its initial value. However, this raises concerns. If a system retains memories after traveling through the CTC, it could create inconsistencies where the system has experienced different possible pasts (multiple histories).

Furthermore, Deutsch's approach may conflict with standard probability calculations in quantum mechanics (path integrals), unless we account for the possibility that the system has experienced different pasts during its journey through the CTC. There can also be multiple solutions (fixed points) for the system's state after the loop, introducing randomness (nondeterminism). Deutsch suggested using the solution with the highest entropy, preferring the solution that aligns with the natural tendency of systems to become more random over time.

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. This combined evolution is described by a tensor product (⊗) of the density matrices for both systems, followed by applying a combined time evolution operator (U) to the entire system.

Implications and criticisms

This 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:

$U={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$.

A density matrix describes the most common solution for a qubit's state after interacting with a closed timelike curve (CTC). The solution relies on a value (a) ranging from 0 to 1, which establishes the relative importance of each possibility in the mixed state. Notably, there isn't a single solution but a range of possibilities.

Deutsch argues that the solution maximizing von Neumann entropy (a measure of how scrambled or mixed up 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 aligns with the many-worlds view of quantum mechanics, doesn't create paradoxes because the qubit travels to a different parallel universe after interacting with the CTC.

Researchers have explored the potential of Deutsch's ideas. If his CTC time travel were possible, computers near a time machine might solve problems far exceeding the capabilities of classical computers.

Deutsch proposed a specific criterion for simulating closed timelike curves (CTCs) using quantum mechanics. However, Tolksdorf and Verch demonstrated that quantum systems lacking CTCs can still achieve this criterion with high accuracy. This finding casts doubt on the idea that Deutsch's criterion is unique to quantum simulations of CTCs as theorized in general relativity. Later work expanded on this concern by demonstrating that statistical mechanics-governed classical systems can also meet the criteria. These results suggest that Deutsch's criterion might not be specific to the peculiarities of quantum mechanics and may not be suitable for inferring the possibilities of real-time travel or its potential realization through quantum mechanics. Consequently, Tolksdorf and Verch argue that their findings raise questions about 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 allow 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 $C={\text{Tr}}_{\text{CTC}}\left[U\right]$.

This transformation depends on the trace (a specific mathematical operation that essentially condenses a complex matrix into a single number) of another mathematical object calculated within the CTC. 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.

Entropy and computation

Michael Devin came up with a model in 2001 that included closed timelike curves (CTCs) in thermodynamics, suggesting it as a solution to the grandfather paradox. This model adds a "noise" factor to account for the flaws in time travel. This explains how time travel might work without paradoxes.

Devin's model suggests that each time travel cycle involving a quantum bit (qubit) carries a usable form of energy (negentropy, the opposite of entropy or disorder) proportional to the noise level. This implies a time machine could extract work from a heat source (a thermal bath) in proportion to the negentropy gained. Similarly, the model suggests that time machines could drastically reduce the effort needed to crack complex codes through trial and error.

However, the model also predicts that usable energy and computing power will become infinitely large as noise approaches zero. This implies that traditional categories for how difficult problems are to solve for computers (like "easy" or "very hard") might not be applicable to time machines with very low noise levels. This concept remains entirely theoretical.

See also

• Novikov self-consistency principle
• Grandfather paradox
• Causal loop