A Quantum Of Theory

Exploring new paths in quantum physics

Tag: Symmetry

Quantum Theory – A view from the inside Part III

Our virtual observer sits in a simulated quantum universe and its global state evolves unitarily. The interactions in the simulated universe are local, allowing parts of the system to be isolated by spatial separation. And this observer is very smart. He has somehow figured out that systems, which are isolated reasonably well, evolve in a specific predictable way. It is not important how exactly he did that, let us just assume that it is the case.

Mathematically, he would formulate a law that we would write as

\left|\psi(t)\right\rangle = U(t,t_0)\left|\psi(t_0)\right\rangle

where \left|\psi(t)\right\rangle is the state of the system at the time t and U(t,t_0) is the unitary operator that evolves the state from the time t to the time t_0.

As he develops the theory he also becomes aware of entanglement and that the state of subsystems cannot be generally described by a Hilbert space ray. He will find that there is a more general class of state descriptions, in the form of state operators (I will avoid the term “density operator” for reasons that should become obvious later) which also evolve unitarily for isolated systems, but with a slightly different law:

\rho(t) = U(t,t_0) \rho(t_0) U^\dagger (t,t_0)

The usual quantum states can be embedded naturally in this new state space by mapping them to their projectors \left|\psi\right\rangle\mapsto\rho=\left|\psi\right\rangle \left\langle\psi\right| and the evolution is preserved. As it turns out, the state operator is always a non-negative hermitian operator with finite non-zero trace. This does not require a statistical interpretation of the operator, but cannot be easily seen without going much deeper. For this, I have to refer to the scientific article that I will cite in one of the next posts. Right here it would only distract from the following argument.

Assume that the observer and everything that interacts with him directly (during a certain time span) can be regarded as one system that is approximately isolated. This separates the Universe into the part of the universe that directly contributes to the experience of the observer and the environment unknown to him. The observer could try to reconstruct the state of his observable subsystem and would choose the most general state representation he knows, a state operator. Remember that all his knowledge can only be based on the history of states of the universe. But since all he knows about the universe is contained in this subsystem, all he can possibly know is a result of the sequence of states of this subsystem. In this situation, is the state operator that the virtual observer finds uniquely determined?

This is the same situation as for the cellular automaton, where changing the cell contents did not make a difference for the structure contained in the sequence of its states. Only this time the allowed transformations are different. Generally, the transformation must be reversible so that the contained information in the representations remains the same.

Let us consider a simple example. If \rho(t) is a reconstructed state, we define \bar{\rho}(t):= \rho^2(t). The time evolution law of \bar{\rho}(t) follows as

\bar{\rho}(t) = U(t,t_0) \rho(t_0) U^\dagger (t,t_0) U(t,t_0) \rho(t_0) U^\dagger (t,t_0)

which simplifies with U^\dagger U = 1 to \bar{\rho}(t)=U(t,t_0)\bar{\rho}(t_0)U^\dagger (t,t_0). That means \bar{\rho} has the same time evolution law as \rho. Because state operators are hermitian and non-negative, squaring is a bijection. As a result, both \rho and \bar{\rho} are valid representations of the state history. Interestingly, none of the two descriptions is any simpler or preferable. Of course, squaring appears like an extra operation that makes it seem like a worse option, but that is only something we did as a construction example, the observer would have estimated \rho or \bar{\rho} directly, and to him both seem equally correct.

Clearly, squaring is not the only way to construct a new valid representation. It is easily verified that the same argument applies to any positive integer power of the state operator, they all generate a bijection between non-negative hermitian operators and the unitary evolution is preserved. We can also use linear combinations of positive integer powers, but then we have to be careful with preserving the bijectivity. It turns out that the  class of interesting bijections is defined as the analytic continuation of monotonically increasing analytic functions f:\mathbb{R}^+_0\to\mathbb{R}^+_0 with f(0)=0. In other words, take any such function and apply its power series to a state operator in order to retrieve a new state operator, that is also a valid description of the observer’s state history.

Any choice of a single state operator from this infinite family of possible states would be purely arbitrary. That also means our scientifically working virtual observer cannot deduce any such state operator representation from his state history. He will have to come up with an alternative state representation that eliminates the redundancy encoded in the state operators. This representation must describe something, which all equivalent state operators have in common. And it should also come in a familiar form, allowing to use the same formulation for the dynamics.

The hermitian non-negative operators we used to represent states have real non-negative eigenvalues, and the associated eigensubspaces are mutually orthogonal. From the way they are constructed, degeneracy would be purely accidental  and we may assume, that all eigensubspaces, that are not the nullspace, are in fact one-dimensional. This will simplify our discussion. As a result, we can describe each single state operator with a list of eigenvalues and the associated eigenvectors. The transformations between valid representations as defined above act as maps on the eigenvalues only, by applying the analytic monotonic functions to them. The monotonicity of these function preserves the order of the eigenvalues. That means once sorted they will stay sorted even after the transformation. However, the actual eigenvalues do change.

This implies two things: The eigenvalues do not contain information about the reconstructed state, because they change with the representation. The order of the eigenvalues does contain information, because it is shared by all state representations. That gives us a new less redundant state representation: A list of eigenvectors, sorted by their eigenvalues. The eigenvalues are not listed however.

But is this really a convenient representation that the observer would pick? Even if the observed part of the universe is not very large, the number of eigenvectors with non-zero eigenvalue would be enormous. How could he keep track of them all? How does he even find out how many there are exactly?

The answer is that he cannot count the subspaces, because the unitary dynamics he uses to extract information from his environment is not influenced by that number. But he knows that there is at least one subspace. In order to construct a one-subspace representation we have to send all eigenvalues to zero, except for one. We cannot do that with a bijection, but we can approximate it arbitrarily well using the bijective transformations. Taking the step to non-bijectivity is the price we have to pay, because we cannot count the subspaces. The reconstruction is unique nonetheless, as the monotonicity of the transformation functions forces us to preserve the largest eigenvalue as the only nonzero eigenvalue. For the reasons  mentioned earlier, we can assume the associated eigensubspace to be one-dimensional.

We have just deduced, that the observer will reconstruct the state of the system containing himself and the environment he interacts with as the eigenvector of the objective state operator corresponding to the greatest eigenvalue. This eigenvector evolves unitarily and we get the usual time dependence of a state vector:


Next time we will discuss the consequences of dropping the isolation requirement for the system and the implied non-unitary evolution.


Quantum Theory – A view from the inside Part II

Let us make a few formal assumptions about the virtual universe and the observer to make the following exploration easier:

  1. The virtual universe has a concept of space and locality. All interactions only act locally and propagate with some kind of speed limit.
  2. Time is imposed from the outside as part of the simulation.
  3. The observer is a mechanism that is contained within a finite region of space for all time

What does the observer know about his universe in the best case? And maybe even more importantly, what does he not know?

First, everything he can possibly know must be encoded somehow in the sequence of states that the universe passes through, because that’s all there is.

Then, he cannot know about certain symmetries of the formulation of the simulation. For example, if the virtual universe is a cellular automaton with an integer number in each cell, then surely the observer cannot decide which numbers are in the cells. At best he can figure out only their relationship. For example, he could add 5 to all numbers and still get a cellular automation law that generates the exact sequence of states but with an offset of 5 in each cell. However, there might be some choice of numbers that makes the laws of his observed physics very simple or beautiful. And even if the original simulation used entirely different numbers, the observer would surely prefer the more elegant representation.

Another example of an unobservable symmetry is the state space of quantum theory. If we simulate the unitary evolution of a state vector, then it is impossible to decide from any mechanism that is itself simulated, what the magnitude or phase of the state vector is. That is because quantum theory is linear and the state evolution commutes with scalar multiplication. Also, there is no most elegant choice for a complex factor and the observer would decide that all work equally well. Remember, we are not assuming any kind of statistical interpretation of the state vector, just unitary dynamics, so normalization is not an obvious elegant choice. But we may assume that he want his state descriptions to be unique and so he identifies all vectors that are related by a nonzero complex factor.  The resulting projective space containing the rays of the original state space is then his new state space.

As simple as it may seem, this is an important point. In fact, we will find a symmetry that will allow us to make a statement about reconstructing the quantum state of the universe similar to this one, with surprising consequences.

Quantum Theory – A view from the inside Part I

The history of science has taught us many things, among them that asking new questions often leads to new insight. Often, these new questions had not been asked before because they seemed to be too philosophical, unanswerable or even mostly unscientific. Here, I would like to confront you with a question that, at a first glance, might seem to fit into these categories. Nevertheless, I will show that discussing this question, specifically applied to quantum theory, leads to deep insight.

In the computer age we have grown very familiar with the concept of simulation. We can simulate practically anything we have understood physically, and we do that for very complicated and large systems like climate models of our planet. Of course, we are using approximations to reality so that our computers can handle the complexity. This, however, is a limitation that we can easily imagine not to exist. The concept of simulation remains the same, even if performed on a hypothetical machine without any practical restrictions.

We could think of any consistent set of mathematical rules and simulate it on a computer. In some sense, we would be creating our own universes with the rules that we make up. Some of these simulations might be just complex enough to allow for an internal observer to evolve, an individual that would have an inside view of our simulation. And if we had the means of communicating with him, we could ask him what he is observing.

We will possibly never get to the scientific sophistication that would allow this sort of real experiment, so what is the point of proposing it? The universe of our hypothetical observer is purely mathematical, a list of rules and an initial state, not more. The reality perceived by him must emerge in some way from the mathematical rules. Surely some aspects of his observation will be highly subjective, like the perception of color, taste or anything that just developed by chance without any profound direct connection to material reality as perceived by him. But other aspects of his observation will not be so subjective, but shared by all other hypothetical observers in the same simulated universe.

So, the question I would like to ask is “How does reality as shared by all possible observers emerge from the mathematical rules that describe the universe these observers inhabit?”. Maybe I have already convinced you that the question is not so esoteric after all. But quite certainly not, that it is even remotely possible to answer it. How would one distinguish objective features from subjective ones? And would we not have to know about all the emergent structures of the simulated universe first, like atoms and molecules or even brains?

I do share the above concerns, but I can also offer a way to circumvent them entirely. Let us assume that our virtual observer is not just any observer, but in fact a physicist who tries to formulate his own mathematical theory of his perceived reality. If he is a good scientist, his theories will only include those aspects of his observation shared by all other observers, and if he is successful his final theory of all things he can observe will be a perfect mathematical description of the objective emergent reality in the virtual universe. This is an extremely helpful assumption, because it allows us to actually talk about mathematical structure instead of a fuzzy and partly psychological concept. With this we can reformulate the fundamental question to “What mathematical model does a virtual observer use to describe his perceived reality?”. This formulation sounds much more reasonable and there is some hope that we may find a way to mathematically deduce the emergence of this internal view from the mathematical structure of the universe we simulate.