Time Machines and the No-Cloning Theorem

Time machines are a fascination for many people, which is why they are items that appear in many novels, anime, and movies. However, most physicists currently believe that a time machine capable of traveling back to the past is impossible. The root of this lies in the existence of numerous time paradoxes, which also appear in science fiction.

Even at the level of ordinary general relativity, without considering quantum mechanics, paradoxes such as the "grandfather paradox" are known, and their fundamental resolution has not been achieved in physics. Furthermore, the "chronology protection conjecture" proposed by Dr. Stephen Hawking suggests that the process of trying to build a time machine is hindered by quantum effects. The hypothesis states that even if a human decides to build a time machine and begins construction in a region of the past where no time machine exists, the zero-point energy of the quantum field will begin an infinite circulation from the future to the past within the partially constructed time machine, strongly distorting spacetime and causing the time machine to be destroyed before it is completed.

There are other reasons why quantum mechanics prohibits time machines. A time machine would violate the "no-cloning theorem" of quantum mechanics. The no-cloning theorem is derived from the most fundamental property of quantum mechanics called "unitarity," which signifies the conservation of quantum information; it asserts that it is impossible to create multiple identical states from a single superposition state of a quantum system. In other words, quantum states and their quantum information cannot be copied. While classical information can be copied as many times as one likes, quantum information carried by the superposition of quantum states cannot be copied. However, if a time machine were to exist, the replication of quantum states would become possible without limit, thereby violating quantum mechanics. For this reason, quantum mechanics and time machines are highly incompatible. Let us explain this here.

First, for simplicity, let us make space one-dimensional and simplify the time machine by identifying the past and the future as shown in Figure 1.

Then, as shown in Figure 2, a time machine becomes possible. The experimenter first enters the future boundary, travels back in time, and emerges from the past boundary. The problem here is that there end up being two of the same experimenter at the same time.

Let us assume that this experimenter packs a quantum system into a box that does not interact with the outside, and transports it using this time machine as shown in Figure 3. Let the state vector of that quantum system be |ψ>.

Focusing on the time t=0 in the figure, two |ψ> states appear at that moment. Before the past boundary of the time machine, there is only one |ψ>, but at t=0, there are two. The no-cloning theorem, which holds if quantum mechanics is correct, is broken.

Quantum mechanics itself has been experimentally verified with extremely high precision to date, and there are currently no results that imply that quantum mechanics does not hold; most physicists also trust quantum mechanics at this time. Therefore, the conclusion is that a time machine cannot be built.

If a time machine existed, as shown in Figure 4, it would be possible to create countless copies of a quantum state by passing the quantum system through it multiple times.

Then, it would be possible to increase the "Fisher information" mentioned in the article below to infinity. This leads to a paradox where quantum information that was originally written into the state of a single quantum system could be determined with infinite precision.

As shown, if a time machine were to exist, numerous intractable paradoxes would arise. Because of these factors, most physicists currently believe that it is impossible to build a time machine.