The Conservation of Information (Script) (Patreon)

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 lost along the way?

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 Schrodinger 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. Today we learn why conservation of information is such a fundamental requirement of quantum mechanics. In a future episode we’ll see how this law might be broken by black holes.

An important tool for making predictions with the laws of physics are the conservation laws that we derive from them. As we discussed recently, we can use Noether’s theorem to discover quantities that are conserved in a system by looking at the symmetries of the equations of motion. For example, if the equations stay the same from one point in time to the next then energy is conserved. Noether’s theorem applies to smooth, continuous symmetries –where we can transform the chosen coordinate however much we like without changing the system. But there are also symmetries under discrete transformations. They’re more like an on-off switch. For example, we could reverse all electric charges, or we could flip the x-axis by looking in a mirror, or we could make time run backwards. Symmetry in that last one - time-reversal symmetry - isn’t covered by Noether’s theorem, but it’s still tied to a conservation law - the conservation of information.

We say that a system is time-reversal-symmetric if its equations of motion allow us to perfectly predict the starting point simply by knowing the state of the system at any later time; if they allow us to rewind the clock and figure out a single, unique history. The entire universe would by time-reversal-symmetric if knowing the exact state of every particle in the universe at one point in time allowed us to calculate its exact past history at all times. That would mean that the exact configuration of the universe at any point in time defines the exact configuration at any other point. For one thing, this time-reversal symmetry means complete information about the configuration at all prior times still exists, and always will, even if we can’t practically access it. THAT’s what I mean about conservation of information.

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 states can all evolve into the same state then knowing that latter state isn’t enough to figure out what past states led to it. 

For example, imagine you have two states - A and B, and the laws of motion say that A becomes C and B becomes D. If we look at the later state and see state C, we know A came first, but if we see D we know B came first. But what if both A and B lead to state C? That future direction is deterministic, but now if we look at the final state and see state C, we don’t know which of A or B led to it. The information about those past states would be lost. OK, so it should be simple enough to erase information, right? Just set things up so that our laws of motion force two possible initial states into the exact same final state. Then we wouldn’t know what the original state was. 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. To get at this, let’s think about the basic equation of motion of quantum mechanics – that’s the Schrodinger equation. The time-dependent Schrodinger equation describes the time evolution of this thing called the wavefunction. The wavefunction of a system fully describes all of its properties – and in quantum mechanics that means the probability distribution of all of its properties, which you get by taking the square of the wavefunction. 

For example, the wavefunction of a particle encapsulates the probability that it will be found in this or that location if we were to try to measure its position. The Schrodinger equation perfectly predicts both the past and future evolution of any given wavefunction in a given environment – or in quantum speak, in any given potential. In that sense it’s deterministic and time-reversal symmetric, and so conserves information. In principle, a “given wavefunction in a given potential” could mean the wavefunction of an electron moving in an atom’s electric field, or it could mean the wavefunction of the entire universe in its own impossibly complex and changing potential.

So the Schrodinger equation guarantees time-reversibility and so the conservation of information. As do the more advanced formulations of quantum mechanics like the Dirac equation and quantum field theories. That guarantee arises from a really fundamental foundational quality of these theories. It comes from what we call unitarity. Remember that the wavefunction encapsulates the distribution of probabilities for a given property. By definition, those probabilities should add to add up to one. That just means there’s a 100% probability that any given property will have SOME value, even if it’s zero. In the case of particle position, “probability adding to one” just means that the particle is definitely somewhere . And as time goes on, a particle’s properties will continue to have some value – the probabilities should continue to sum to one. If this is true – and it must be – we say that the time evolution of the wavefunction is unitary. 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 the 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 final state can’t both sum to one. Back to our first example: quantum states A and B can’t both become state C; the sum of probabilities prior to the merger would not equal the sum after the merger and unitarity would be broken.  

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 about the 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 wavefunction. And the precision in the knowability of that value is defined by the uncertainty principle. It sure seems like information can be lost. But that’s not really what we mean by quantum information – quantum information refers to the full information content of the wavefunction, not just what we measure.

It’s worth mentioning that the collapse of the wavefunction in the Copenhagen interpretation of quantum mechanics actually does mess with conservation of information. According to that interpretation, the act of measurement actually alters the entire wavefunction, causing it to shrink down to the narrow range of possible values implied by the result of the measurement. But that measured wavefunction can’t then be tracked backwards to recover the past wavefunction. So this interpretation is neither deterministic nor time-reversal symmetric. 

Other interpretations of quantum mechanics, for example Everett’s Many Worlds or the de Broglie-Bohm pilot wave theory preserve this time-reversibility. In the case of Many Worlds, the entire wavefunction continues to exist even after measurement. The measurement itself is just a thing that happens in one part of that possibility space. No information is really lost. And in the case of pilot wave theory, the wavefunction contains hidden information that is carried with the final measured particle.

There is one situation where time reversibility appears to be broken regardless of your favorite interpretation of quantum mechanics. That’s the case of black holes and Hawking radiation, Stephen Hawking’s eponymous radiation appears to destroy quantum information, leading to the famous black hole information paradox. In an upcoming episode we’ll see whether quantum information really can be deleted from the otherwise perfect memory of space time.