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Conservation of Information in Quantum Mechanics: Understanding Time-Reversal Symmetry

Thanks to Curiosity Stream for supporting PBS Digital Studios. If you have perfect knowledge of every single particle in the universe, can you use the laws of physics to rewind all the way back to the Big Bang? Is the entire history of the universe perfectly knowable? Or has information somehow been lost along the way?

The Laws of Physics and Determinism

The laws of physics are equations of motion. They are mathematical rules that dictate how systems evolve in time. Newton’s equations for classical mechanics, Maxwell’s equations for electromagnetism, and the Schrödinger equation for quantum mechanics. These laws can be used to predict how the universe will evolve into the future. They are deterministic. Perfect knowledge of a system in the present perfectly predicts how the system will change in the next instant and the instant after that, ad infinitum. But determinism in the forward time direction does not guarantee that the same laws can perfectly predict the past. And yet this sort of deterministic symmetry, time-reversal symmetry, is essential for information itself to be conserved.

Conservation Laws in Physics

Today, we explore the critical importance of information conservation as a fundamental principle of quantum mechanics. In an upcoming episode, we will examine how this principle may be violated in the context of black holes. An essential tool for making predictions based on the laws of physics comprises the conservation laws derived from these principles. As previously discussed, Noether’s theorem enables us to identify conserved quantities within a system by analyzing the symmetries inherent in the equations of motion. For instance, if the equations remain invariant over time, it indicates the conservation of energy. Noether’s theorem is applicable to smooth, continuous symmetries, permitting us to alter the chosen coordinate to any extent without impacting the system.

Discrete Transformations and Time-Reversal Symmetry

There are symmetries under discrete transformations, like on-off switches. For example, we could reverse electric charges, flip the x-axis with a mirror, or make time run backwards. Time-reversal symmetry isn’t covered by Noether’s theorem, but is linked to a conservation law: the conservation of information. A system is time-reversal symmetric if its equations of motion allow us to predict the starting point from its state at any later time, letting us rewind and identify a unique history.

Time-Reversal Symmetry and the Universe

The universe would be time-reversal symmetric if knowing the exact state of every particle at one moment allowed us to calculate its entire past. This means the universe’s configuration at any time defines its state at all others. Time-reversal symmetry implies that complete information about past configurations exists, even if we can’t practically access it. That’s the essence of conservation of information.

Causal Determinism and Information Loss

A related idea is causal determinism. The idea that perfectly knowing the current state perfectly predicts all future states. But this sort of future determinism doesn’t have to be time-reversal symmetric. It’s possible for the future to be perfectly predictable by the laws of physics while the past is not. For example, what if many different configurations of particles in the present could converge on a single configuration of particles in the future? If a multitude of states can all evolve into the same state, then knowing the later state isn’t enough for us to figure out what past states led to it.

Quantum Mechanics and the Conservation of Information

Okay. So it should be simple enough to erase information, right? We just set things up so our laws of motion force two possible initial states into the same exact final state. Then we wouldn’t know what the original state was and information would be destroyed. Actually, quantum mechanics forbids this. It ensures conservation of information and time-reversal symmetry because of an even more fundamental rule: The conservation of probability.

The Schrödinger Equation and Wave Function

To get at this, let’s consider the basic equation of motion in quantum mechanics, the Schrödinger equation. The time-dependent Schrödinger equation describes the evolution of The Wave Function, which fully encapsulates a system’s properties through its probability distribution, obtained by squaring the wave function. For instance, the wave function of a particle indicates the likelihood of finding it in a specific location during a measurement.

Time-Reversal Symmetry and Quantum Mechanics

The Schrödinger equation predicts the evolution of a wave function in any environment, making it deterministic and time-reversal symmetric, thus conserving information. This can refer to the wave function of an electron in an atom’s electric field or the entire universe in its complex potential. The equation assures time reversibility and conservation of information, similar to more advanced formulations like the Dirac equation and quantum field theories.

Unitarity in Quantum Mechanics

That guarantee arises from a really fundamental foundational quality of these theories. It comes from what we call Unitarity. Remember that the wave function encapsulates the distribution of probabilities for a given property. By definition, those probabilities should add up to 1. That just means there’s a 100% probability that any given property will have some possible value, even if that value is zero. In the case of particle position, probability of adding to one just means that the particle is definitely somewhere. And, as time goes on, a particle’s properties will continue to have possible values. The probability should continue to sum to 1. If this is true, and it must be, we say that the time evolution of the wave function is unitary.

Implications of Unitarity

And this unitarity is a foundational assumption in all formulations of quantum mechanics and quantum field theories. Unitarity is a non-negotiable statement about how probability works, but the condition also ensures time-reversal symmetry and conservation of information. It’s not straightforward to explain without getting into some hairy math but the upshot is that quantum states must remain independent of each other in order to preserve probability. Two independent quantum states can’t evolve into the exact same quantum state. If they did, then the probabilities for the initial states or for the final state can’t both sum to 1. Back to our first example, quantum states A and B can’t both become quantum states C, the sum of the probabilities prior to the merger would not equal the sum of the probabilities after the merger and unitarity would be broken.

Quantum Information and Measurement

The only type of evolution that preserves probability and unitarity is the evolution that also preserves the number of quantum states, and the preservation of quantum states means preservation of information because you can trace a quantum state indefinitely forwards and backwards in time. But all of this talk of quantum mechanics being deterministic seems a bit at odds with the idea of quantum randomness and the uncertainty principle. After all, doesn’t the act of measurement pick a single value of some quantity from the range of possible values? That value seems to be chosen randomly based on the probability distribution encoded in the wave function and the precision of the knowability of that value is defined by the uncertainty principle. It sure seems like information can be lost.

Quantum Information and Wave Function

Quantum information refers to the complete information content of the wave function, beyond just what we measure. In principle, by making sufficient measurements, all information can be extracted. However, the Copenhagen interpretation posits that measurement alters the wave function, reducing it to a narrow range of values and preventing us from tracking it back to its original state. This makes it neither deterministic nor time-reversal symmetric. In contrast, interpretations like Everett’s many-worlds and the de Broglie-Bohm pilot wave theory maintain time reversibility.

Time-Reversibility and Quantum Mechanics Interpretations

In the many-worlds interpretation, the wave function persists after measurement, existing in a segment of possibility space without loss of information. In pilot wave theory, hidden information accompanies the particle. A scenario where time reversibility seems broken arises with black holes and Hawking radiation, which appear to obliterate quantum information, leading to the black hole information paradox. We will explore whether quantum information can be erased from spacetime’s flawless memory in an upcoming episode.

CuriosityStream and PBS Digital Studios

Thank you to CuriosityStream for supporting PBS Digital Studios. CuriosityStream is a subscription streaming service that offers documentaries and nonfiction titles from a variety of filmmakers including CuriosityStream Originals. You can get the first 60 days free if you sign up at curiositystream.com/spacetime and use the code “spacetime” during the signup process.

Spring Cleaning and Future Episodes

Hey everyone, so it’s springtime (at least on this half of the planet) and you know what that means. Spring Break. Actually no, that means spring cleaning. Space Time is going to go quiet for the next couple of weeks to get our house in order. What that means is we’ll be doubling down and doing some serious writing in preparation for the next great series we have in mind. I’m talking black hole thermodynamics and some pretty deep particle physics. We’re gonna miss you, but it’s gonna be really worth it.

Viewer Questions and Comments

Noether’s theorem predicts the conservation laws of physics from the symmetries of nature. You guys had some excellent questions in the comments. merinsan has plans to use the loss of energy and cosmological redshift as a refutation of conservation of energy arguments. Ok, that’s fine, but a word of caution. Conservation of energy is completely valid and always applies on the scales of systems that humans deal with. In a related question, AFastidiousCuber asks why free energy and perpetual motion machines are impossible. The key is that, on the scales of these systems, the universe is time translation symmetric. Space is not globally expanding or contracting on those scales and so energy is absolutely conserved. Perhaps you could extract some free energy if you could build a device that spans several million light years or, I don’t know, install some solar panels.

Dark Energy and Cosmological Redshift

A few of you asked this intriguing question: “Could energy loss to cosmological redshift become dark energy?” The emphatic answer is no! The energy density of photons is negligible compared to the staggering energy density of dark energy, which accounts for about 70% of the universe’s energy density and continues to grow. In the early universe, radiation ruled until about 50 thousand years post-Big Bang, but since then, their influence has waned dramatically due to cosmological redshift, making their contribution too minuscule to fuel the rise of dark energy!

Emmy Noether’s Contribution

Orthochronicity underscores Emmy Noether’s significant impact. Although she didn’t address the energy problem in early general relativity, Klein sought her expertise and found she had largely solved it while focusing on other pursuits. Einstein was surprised by her broad conceptualization of gravity, which overshadowed conventional physics of her time and showcased her extraordinary abilities as a mathematician.

Source Code and the Universe

A lot of you criticized our use of HTML as an example of source code. Guys, you really think the universe uses assembly language or Haskell or, hell, even Java? It doesn’t make anywhere near that much sense. Black hole memory leaks, quantum rounding errors and it took 10 billion years to compile the first lifeform.