A Quantum Of Theory

Exploring new paths in quantum physics

Category: Foundations of quantum theory

QT – A view from the inside Part V – The Born rule?

Now that we have established that external interactions can cause an observed jump in the state evolution, we are going to construct a case where this actually happens and we can study the properties of this transition.

Generally, the system containing the observer and everything he interacts with is very large. We cannot discuss all the details of such a system, therefore we will use a minimal version consisting of just the necessary elements. And probably to your surprise, the observer is not contained in this simplified system. We do not actually need him, because we already now how he would reconstruct the system’s dynamic.

Even our entire universe will be very simple. It just consists of a single qubit and the photon field. The photons are assumed to be inaccessible locally, because they come in or leave at the speed of light. So it is just the qubit that we regard as the local system a hypothetical observer would reconstruct.

The process we are going to analyze is a simple scattering and absorption process on the qubit, which we may imagine to be realized as some two level system with two energy eigenstates \left|0\right\rangle and \left|1\right\rangle. The incoming single photon \left|\rightsquigarrow\right\rangle carries a two-dimensional polarization state spanned by \left|\circlearrowleft\right\rangle and \left|\circlearrowright\right\rangle. Before the qubit and the photon interact, their states are independent and represented as a product state.
The actual dynamics of the process is not really important to us, the output state of the scattering event is already perfectly sufficient for the demonstration. Hence we can simply define a unitary map between the input and output states. Unitarity is guaranteed by mapping an orthonormal basis of the input space to an orthonormal basis of the output space. We have a 4-dimensional input space, 2 qubit dimensions times 2 photon polarization dimensions, so we also get a 4-dimensional output space. And the whole process can be written in the form of 4 mapping rules.
The first rule is very simple, as the photon just passes through the qubit:

\left|\rightsquigarrow\circlearrowleft\right\rangle \left|0\right\rangle \mapsto \left|0\right\rangle \left|\rightsquigarrow\circlearrowleft\right\rangle

The same thing happens if we flip both the polarization and the qubit:

\left|\rightsquigarrow\circlearrowright\right\rangle \left|1\right\rangle \mapsto \left|1\right\rangle \left|\rightsquigarrow\circlearrowright\right\rangle

But in the two other cases we get a nontrivial interaction, resulting in a superposition of a scattering and an absorption or stimulated emission outcome:

\left|\rightsquigarrow\circlearrowright\right\rangle \left|0\right\rangle \mapsto \left( \left|0\right\rangle\left|\leftrightsquigarrow\circlearrowright\right\rangle + \left|1\right\rangle\left|\cdot\right\rangle \right)\frac{1}{\sqrt{2}}

\left|\rightsquigarrow\circlearrowleft\right\rangle \left|1\right\rangle \mapsto \left( \left|1\right\rangle\left|\leftrightsquigarrow\circlearrowleft\right\rangle + \left|0\right\rangle\left|\rightsquigarrow\circlearrowleft,\rightsquigarrow\circlearrowleft\right\rangle \right)\frac{1}{\sqrt{2}}

Here, \left|\leftrightsquigarrow\right\rangle is the omnidirectionally scattered photon, \left|\cdot\right\rangle the vacuum after the photon has been absorbed, and \left|\rightsquigarrow,\rightsquigarrow\right\rangle denotes the two photon state after stimulated emission.

If we apply the process to an input state \left( \alpha \left|\rightsquigarrow\circlearrowleft\right\rangle + \beta \left|\rightsquigarrow\circlearrowright\right\rangle \right) \left( a \left|0\right\rangle + b \left|1\right\rangle \right) and then take the trace over the inaccessible photon states, we get the state operator of the output state:

\rho = \left( \left|\alpha\right|^2 \left|a\right|^2 + \frac{1}{2}\left( \left|\alpha\right|^2 \left|b\right|^2 + \left|\beta\right|^2 \left|a\right|^2 \right) \right) \left| 0 \right\rangle \left\langle 0 \right|
+ \left( \left|\beta\right|^2 \left|b\right|^2 + \frac{1}{2}\left( \left|\alpha\right|^2 \left|b\right|^2 + \left|\beta\right|^2 \left|a\right|^2 \right) \right) \left| 1 \right\rangle \left\langle 1 \right|

Conveniently, this result is already decomposed into its eigenbasis. If we apply what we have learned about the locally observed state, we have to find the eigenvector with the greatest eigenvalue of this state operator. The result of which can easily be read of the representation, and we get \left|0\right\rangle if \left|\alpha\right|^2 \left|a\right|^2 > \left|\beta\right|^2 \left|b\right|^2 and \left|1\right\rangle otherwise. But the observer at the qubit does not know the state of the incoming photon, so \alpha and \beta are unknown to him and will introduce random information to the result. Since the observer does not know better, he has to assume that all photon polarizations are equally likely, which means the assumed random distribution of the photon state must invariant under a unitary transform of the photon state. With this assumption, we can calculate the probability of the observed process outcome to be \left|0\right\rangle or \left|1\right\rangle.

I cannot show the calculation here, but please see my next post for a reference to a paper on arxiv where it is actually performed. Of course, I will still give you the result. The probability of seeing an outcome \left|0\right\rangle is precisely p_0 = \frac{\left|a\right|^2}{\left|a\right|^2+\left|b\right|^2}, and the complementary event of observing \left|1\right\rangle is p_1 = \frac{\left|b\right|^2}{\left|a\right|^2+\left|b\right|^2}.

You have certainly noticed, that our result is precisely the probability as postulated by the Born rule! This means we have not only derived, that the observer will see a discontinuous jump between pure states, but also that the statistics of that jump exactly match the established statistics of performing a quantum measurement, at least for a simple interaction with an incoming photon. We have therefore deduced the full measurement postulate of the Copenhagen interpretation just by assuming the position of an internal observer, without any additional requirements, and also without having to introduce arbitrarily many quantum worlds.

My next post will contain a few final thoughts and a pointer to more technical and rigorous derivations of all results presented here and more.

Quantum Theory – A view from the inside Part IV

Last time, we have derived how an observer inside a quantum system can only reconstruct a crude approximation to the real quantum state of this system. Specifically, he will observe the system to evolve like the eigenvector belonging to the greatest eigenvalue of the objective state operator describing his subsystem. To get this result however, we had to assume that the subsystem the observer interacts with is isolated and the evolution therefore unitary.

In order to understand how a non-unitary evolution affect the state reconstruction, we have to understand how unitary and non-unitary transformations modify the eigenstructure of an hermitian operator. This structure is given by real eigenvalues and orthogonal eigensubspaces that span the whole space. Both unitary and non-unitary transformations have to preserve this general structure. An additional constraint comes from the time-continuity of the evolution that translates to a continuous evolution of the eigenvalues and eigensubspaces.

Consider a hermitian operator A and its eigenvector \left|v\right\rangle with the eigenvalue v. It then follows that UAU^\dagger has an eigenvector U\left|v \right\rangle with the eigenvalue v, where U is a unitary operator. Or in other words, unitary evolution does not change the eigenvalues, but it rotates the eigensubspaces while preserving their orthogonality.

Now for non-unitary evolution the only possible way to generalize that behavior is to unlock the eigenvalues. This implies that eigensubspaces can fuse when distinct eigenvalues evolve into being equal. We also have the reverse process where they split up instead. But most of the time we expect to see a simple intersection of two eigenvalues with a fused eigensubspace only existing at a single point in time. This really summarizes all that can possibly happen. The Subspaces can also only be rotated because the hermitian constraint enforces orthogonality at all times. Non-unitary evolution is really not that complicated if presented like this!

The implications for the state reconstruction performed by our virtual observer are relatively simple. As long as the non-unitary evolution does not lead to an eigenvalue intersection, the order of the eigenvalue-sorted subspaces will stay the same. We have seen in the last post that the observer can not determine the actual eigenvalues anyway, so nothing changes for him. He cannot distinguish such a non-unitary evolution from a unitary one. If we do allow for eigenvalue intersections, then it depends on which eigenvalues are affected. If the intersection does not involve the greatest eigenvalue then the observer will still not be able to detect it, because the corresponding subspaces are not reconstructable.

The only case, when the intersection has any influence on the observed evolution, is when the greatest eigenvalue intersects with another eigenvalue. Then the eigensubspace associated with the greatest eigenvalue switches instantly. The virtual observer would perceive this event as a discontinuous evolution of the system state.

Have you realized what we have just derived? We have seen, that an observer, who is part of a quantum system, perceives the evolution of this system that includes himself and his nearby environment as the unitary evolution of a pure state. But when the system interacts with some part of the universe that is not included in his description, the pure quantum state can suddenly jump to a completely different pure state. And that can happen in a way that is entirely not unitary.

This description is very close to the postulates of the Copenhagen interpretation. The only missing aspect is a statement about the probability and results of the discontinuous state evolution. We will deal with that in the next post!




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.

Does quantum theory have to be interpreted?

Witnessing the ongoing discussion about how quantum theory should be interpreted, and the strong opinions and sometimes even dogmatic arguments, I decided to write a series of blog posts that will try to discuss the issue of interpretation as objectively as I possibly can. I will not specifically try to compare the different mainstream interpretation with each other, but rather explore the requirement of an interpretation at all and the possibility of answering the same fundamental questions using strong scientific rigor instead.

A scientific theory is usually defined as consisting of a mathematical apparatus that allows to perform calculations of predictive nature, and a layer of interpretational glue that connects the resulting numbers with measurements that we can actually perform. The separation of measurement and prediction works very well for all classical theories, where observer and experiment can be regarded as entirely separate entities. Quantum theory however makes a clean cut between observer and the observed experiment impossible, because after the experiment the two subsystems are interwoven in a very fundamental and complicated way, even if spatially separated. The nonlocal entanglement of the quantum state space does not allow us to use the approximation of objectivity anymore.

Understanding this problem, there are two main approaches of dealing with it. The older one insists on the classical separation and is willing to live with the necessary consequences. The Copenhagen interpretation introduces the Heisenberg cut between quantum and classical domains to recover the notion of an objective observer that can make classical statements about the measurement outcome. And with that cut we also get the interpretational glue back that relates mathematics with measurement results. This happens in the form of the well known measurement postulate which includes the Born rule describing the statistical outcome of a measurement.

The approach has several drawbacks however. Firstly, the location of the Heisenberg cut is more or less arbitrary as long as the observer and the system are well distinguishable, but becomes impossible as soon as this is not the case anymore. Often this does not pose a problem, but it is still a shortcoming as it keeps us from understanding certain realizable situations. Secondly, the Copenhagen and related interpretations leave us entirely in the dark as to what precisely happens during a measurement. Still, the Copenhagen interpretation is fundamentally scientific, as it focuses on measurements and predictions only, and does not take into account what is not observable.

The other main approach to the problem of observation takes the alternative route. Instead of introducing a cut, everything is taken into account. Experiment and measurement device become one system, which is itself a part of the largest system, the universe. It is then only consequential to assume the time evolution of undisturbed quantum systems as formulated in the Copenhagen interpretation, the Schrödinger equation, as the evolution law for the universe. Within this approach, all predictions and results must emerge only from the properties of the evolving system, as there is no external observer that can measure anything, and no classical measurement device either. The time evolution would also be fully deterministic and the randomness of the measurement outcome could also be regarded as an emergent property.

So when Hugh Everett III came up with his many worlds or relative state interpretation, he did really not at all want to create an interpretation in the sense of the Copenhagen interpretation, namely as a layer of translation between math and measurement. Rather, he wanted to create a scientific theory of emergence, where all results are derived as inherent properties of the system itself. And he was willing to accept all the consequences it brought, because the approach was rigorously scientific and only the logical consequence of avoiding the artificial Heisenberg cut.

Unfortunately, not everything worked out as well as this approach had been promising. Of course, the most famous consequence is the existence of arbitrarily many worlds containing observers that have seen any possible experimental outcome. While this is philosophically hard to accept for some, it surely is only an acceptable consequence if the other results work out correctly. And these results ought to be the precise statements of the measurement postulate of the Copenhagen interpretation, because those are experimentally verified.

However, while the many worlds theory gives a reasonably good explanation for the state collapse, it fails to give the right statistics. There has been some criticism regarding the collapse too, but more importantly it is generally agreed that the Born rule does not come out of the relative state theory unless extra postulates are added. Decoherence theory, which incorporates the environment to move coherence away from the experiment, or more recent attempts to use psychologically founded counting mechanisms for calculating the relative outcome probabilities, have not been generally convincing enough to consider the issues of the theory be solved. And adding postulates of course spoils the initial idea of having an actual theory of emergence.

So where does this leave us? We have a practical approach that works most of the time, but hides some possibly important features and mechanisms from us. And we have a holistic approach that stands on a beautiful theoretical idea, but fails to deliver the right results and comes with some curious side effects.
The question that I will explore in the following articles is what Everett’s approach has to do with the relationship between simulation and reality, and whether something that he and others have potentially overlooked could lead to a new theory with better results. And I promise, I’ll have a few surprises for you!