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I'm having a hard time wrapping my head around the conservation of information principle as formulated by Susskind and others. From the most upvoted answers to this post, it seems that the principle of conservation of information is a consequence of the reversibility (unitarity) of physical processes.

Reversibility implies determinism: Reversibility means that we have a one to one correspondence between a past state and a future state, and so given complete knowledge of the current state of the universe, we should be able to predict all future states of the universe (Laplace's famous demon).

But hasn't this type of determinism been completely refuted by Quantum Mechanics, the uncertainty principle and the probabilistic outcome of measurement? Isn't the whole point of Quantum Mechanics that this type of determinism no longer holds?

Moreover, David Wolpert proved that even in a classical, non-chaotic universe, the presence of devices that perform observation and prediction makes Laplace style determinism impossible. Doesn't Wolpert's result contradict the conservation of information as well?

So to summarize my question: How is the conservation of information compatible with the established non-determinism of the universe?

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  • $\begingroup$ As I don't have time to write proper answer, here's a hint: you have some quantum system, how do you know its state? $\endgroup$ – OON Jan 17 '17 at 7:47
  • $\begingroup$ Is that really evident to all QM specialists that QM is incompatible with determinism? The moments of the probability densities (mean value, deviation...) are deterministic, and do we know exactly what happens during the physical process of collapse (the 'dice' moment)? $\endgroup$ – user130529 Jan 17 '17 at 8:06
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    $\begingroup$ @OON how is knowing or not knowing the state relevant? The outcome of the measurement is probabilistic either way. $\endgroup$ – Alex Kinman Jan 17 '17 at 16:43
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    $\begingroup$ @claudechuber I do not think it is relevant if it is or not. The point is that some interpretations of quantum mechanics are indeterministic, and all interpretations have to be compatible with its mathematical formalism. In other words, existence of indeterministic interpretations seems to imply that "conservation of information" can not follow from quantum mechanical laws. $\endgroup$ – Conifold Jan 17 '17 at 21:37
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    $\begingroup$ The Schrödinger equation is deterministic. Determinism in physics means that given some initial conditions if you have the entire description of the state of a system you can predict correctly what the state of the system will be at some point in the future. In classical physics this is given by Newton's laws, or a Lagrangian, or a Hamiltonian. In quantum mechanics it is given by Schrödinger's equation. If you have state psi at some point in time the equation will tell you what the state will be like at any arbitrary time in the future or the past, up to the point of a measurement. $\endgroup$ – Not_Here Jan 19 '17 at 12:57
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The short answer to this question is that the Schrödinger equation is deterministic and time reversible up to the point of a measurement. Determinism says that given an initial state of a system and the laws of physics you can calculate what the state of the system will be after any arbitrary amount of time (including a positive or negative amount of time). Classically, the deterministic laws of motion are given by Newton's force laws, the Euler-Lagrange equation, and the Hamiltonian. In quantum mechanics, the law that governs the time evolution of a system is the Schrödinger equation. It shows that quantum states are time reversible up until the point of a measurement, at which point the wave function collapses and it is no longer possible to apply a unitary that will tell you what the state was before, deterministically. However, it should be noted that many-world interpreters who don’t believe that measurements are indeterministic don’t agree with this statement, they think that even measurements are deterministic in the grand scheme of quantum mechanics. To quote Scott Aronson:

Reversibility has been a central concept in physics since Galileo and Newton. Quantum mechanics says that the only exception to reversibility is when you take a measurement, and the Many-Worlders say not even that is an exception.

The reason that people are loose with the phrasing “information is always conserved” is because the “up until a measurement” is taken for granted as background knowledge. In general, the first things you learn about in a quantum mechanics class or textbook is what a superposition is, the Heisenberg uncertainty principle and then the Schrödinger equation.

For an explanation of the Schrödinger equation from Wolfram:

The Schrödinger equation is the fundamental equation of physics for describing quantum mechanical behavior. It is also often called the Schrödinger wave equation, and is a partial differential equation that describes how the wavefunction of a physical system evolves over time.

The Schrödinger equation explains how quantum states develop from one state to another. This evolution is completely deterministic and it is time reversible. Remember that a quantum state is described by a wave function $|\psi\rangle$, which is a collection of probability amplitudes. The Schrödinger equation states that any given wave function $|\psi_{t_0}\rangle$ at moment $t_0$ will evolve to become $|\psi_{t_1}\rangle$ at time $t_1$ unless a measurement is made before $t_1$ . This is a completely deterministic process and it is time reversible. Given $|\psi_{t_1}\rangle$ we can use the equation to calculate what $|\psi_{t_0}\rangle$ is equal to.

If the electron is in a superposition then the wave function will be so:

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ where $\alpha$ and $\beta$ are equal to $\frac{1}{\sqrt{2}}$.

The state of an electron that is spin up is $|\psi\rangle = 1|1\rangle$. Clearly, a quantum state that is in a superposition of some observables is a valid ontological object. It behaves in a way completely different than an object that is collapsed into only one of the possibilities via a measurement. The problem of measurements, what they are and what constitutes one, is central to the interpretations of quantum mechanics. The most common view is that a measurement is made when the wave function collapses into one of its eigenstates. The Schrödinger equation provides a deterministic description of a state up to the point of a measurement.

Information, as defined by Susskind here, is always conserved up to the point of a measurement. This is because the Schrödinger equation describes the evolution of a quantum state deterministically up until a measurement.

The black hole information paradox can be succinctly stated as this:

Quantum states evolve unitarily governed by the Schrödinger equation. However, when a particle passes through the event horizon of a black hole and is later radiated out via Hawking radiation it is no longer in a pure quantum state (meaning a measurement was made). A measurement could not have been made because the equivalency principle of general relativity assures us that there is nothing special going on at the event horizon. How can all of this be true?

This paradox would not be a paradox if the laws of quantum mechanics didn't give a unitary, deterministic, evolution for quantum states up to a measurement. The reason being, if measurements are the only time unitarity breaks down and the equivalency principle tells us a measurement cannot be happening at the horizon of a black hole, how can unitarity break down and cause the Hawking radiation to be thermal and therefore uncorrelated with the in-falling information? Scott Aaronson gave a talk about quantum information theory and its application to this paradox as well as quantum public key cryptography. In it he explains

The Second Law says that entropy never decreases, and thus the whole universe is undergoing a mixing process (even though the microscopic laws are reversible).

[After having described how black holes seem to destroy infomration in contradiction to the second law] This means that, when bits of information are thrown into a black hole, the bits seem to disappear from the universe, thus violating the Second Law.

So let’s come back to Alice. What does she see? Suppose she knows the complete quantum state $|\psi\rangle$ (we’ll assume for simplicity that it’s pure) of all the infalling matter. Then, after collapse to a black hole and Hawking evaporation, what’s come out is thermal radiation in a mixed state $\rho$. This is a problem. We’d like to think of the laws of physics as just applying one huge unitary transformation to the quantum state of the world. But there’s no unitary U that can be applied to a pure state $|\psi\rangle$ to get a mixed state $\rho$. Hawking proposed that black holes were simply a case where unitarity broke down, and pure states evolved into mixed states. That is, he again thought that black holes were exceptions to the laws that hold everywhere else.

The information paradox was considered to be solved via Susskind's proposal of black hole complementarity and the holographic principle. Later AMPS showed that the solution is not as simple as it was stated and further work needs to be done. Currently the field of physics is engaged in an amazingly beautiful collection of ideas and solutions being proposed to solve the black hole information paradox as well as the AMPS paradox. At the heart of all of these proposals, however, is the belief that information is, conserved up to the point of a measurement.

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    $\begingroup$ I understand what you said about unitarity. That's not my question. Measurement is not unitary. Once the wave function has collapsed to a single base state, we no longer have any way of knowing what the superposition before the measurement. Different superpositions can result in the same measurement outcome, so the process is not reversible. $\Psi = \frac{1}{\sqrt{2}} \left|0\right\rangle + \frac{1}{\sqrt{2}} \left|1\right\rangle$ and $\Psi = \frac{1}{\sqrt{5}} \left|0\right\rangle + \frac{\sqrt{4}}{\sqrt{5}} \left|1\right\rangle$ can both lead to an outcome of 1. $\endgroup$ – Alex Kinman Jan 19 '17 at 16:52
  • $\begingroup$ Yes, as I said the Schrödinger equation describes a state deterministically up to measurement. Nobody is trying to say that measurements are deterministic. The conservation of information says that "the Schrödinger equation gives a completely deterministic evolution of a state until a measurement is made." You asked "how is the conservation of information compatible with the established non-determinism of the universe?" I don't see how that doesn't answer your question. $\endgroup$ – Not_Here Jan 19 '17 at 16:58
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    $\begingroup$ Ok - so we agree that measurements are not deterministic. Doesn't that mean that as a whole the universe is not deterministic, since we are part of the universe? $\endgroup$ – Alex Kinman Jan 19 '17 at 16:59
  • $\begingroup$ I completely believe that you do want to understand that's why I'm still continuing this conversation days later. Honestly, however, I feel like I'm at the point where I can't explain it any better without giving a course on these topics, I sincerely recommend that you read some of the papers or watch some of the lectures on the black hole information paradox because they will explain explicitly why information is conserved. $\endgroup$ – Not_Here Jan 19 '17 at 17:03
  • $\begingroup$ When you consider the Schrödinger equation of the whole universe, is the notion of measurement still present, or does the latter make sense only when considering sub-systems? $\endgroup$ – user130529 Jan 19 '17 at 19:16
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It's pretty simple, and there's been various questions on this site that have had this discussion. And it does get controversial, but the phsycis I straightforward.

The issue is: is the fact that the evolution of the wave function, or the state of a system, is determined uniquely by its initial conditions and the unitary operator that quantum mechanically (or equivalently for quantum field theory) is its time evolution operator? The answer is obviously yes, the wave function or system state evolves uniquely in time according to that operator. The evolution is deterministic as far as the quantum state is concerned.

This is labeled as unitary time evolution. It means that the quantum information that defines the initial state is not 'lost', but rather simply evolved into the information that defines the evolved state. Quantum states evolve deterministically if they are pure states.

In simple terms it means that quantum theory follows causal laws. Causality so not broken

There is nothing controversial there. Wave function or quantum state evolution is perfectly deterministic. What happens with statistical mixes of pure states is statistical mechanics, and does not contradict the determinism, only the practical limits on it due to the large numbers of states and interactions.

The issue comes up when you measure some observable of the state. It is then probabilistically determined exactly what you will get. It is this latter fact that has led to quantum theory be labeled as probabilistic. In doing a measurement you place that system in one of the eigenvalues of the observable operator. It is well known how to compute the probabilities of measuring any specific value. That is what is meant by saying quantum theory is not deterministic.

Note that even then, the quantum state had evolved deterministically, and it is only when you measure, or decohere the system, and interact with it with a lot of degrees of freedom, you get a classical average value with variance around it.

So if you want to determine classical observables, which means you have to measure and not simply let the quantum state go its own way, it produces the probabilistic results and has the quantum uncertainties given by the uncertainty principle for the different observable pairs. But it does not mean the state did not evolve In a perfectly unitary and causal way given by then laws of quantum theory. Sometimes it is loosely said that the wavefunction collapsed into its one observed classical value. And it could have been another. It was determined probabilistically.

That quantum information defined by the state of the system before you measure it, i.e. before you (or anything else) interacts with it, is the quantum information that cannot be lost or destroyed. It can be modified only by the deterministic time evolution operator (and of course by interactions with other particles or fields, which would be then represented in the unitary time evolution operator for it). That quantum information could be also quantum numbers that are conserved in various interactions - for instance total energy, spin, lepton number, fermion number, and others -- in those cases, given by what entities are conserved by the various SM forces.

Now, there is a Black Hole Information paradox that has surfaced because when the particles with specific quantum numbers or quantum states dissappear inside a Black Hole, you can never get them back and that equivalent information is lost. After Hawking radiation it just disappears. Quantum theory says that's impossible. Thus the paradox. There's been plenty of discussion and work on it, but no definitive resolution - probably it'll have to await a well accepted quantum gravity theory. Most physicists probably believe that there's a deeper solution, and that quantum theory causality or information will be preserved.

See the article at Wikipedia https://en.m.wikipedia.org/wiki/Black_hole_information_paradox

So yes, quantum information conservation and quantum state determinism or quantum causality are the same things.

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    $\begingroup$ So if I understand correctly, if you take the wave function of the whole universe, one can safely say that its evolution from the beginning up to now is deterministic, is that right? $\endgroup$ – user130529 Jan 18 '17 at 10:45
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    $\begingroup$ FAccording to quantum theory, yes. With some cavaeats: we don't know how quantum gravity will affect that; it might be tricky or otherwise unwise to try to do it for the whole universe; the effect of non locality hat may be not understood such as in various entanglement controversies and the possibility that there could be some no local interactions such as what seems to appear in the Hologram conjecture; the not complete understanding yet of decoherence and the inevitability of the classical limit, and a few others. We know things, but a lot we don't know. $\endgroup$ – Bob Bee Jan 19 '17 at 5:22
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    $\begingroup$ Yes, I understand. I don't know the answer to that, but some theories I like or dislike in a couple lines. First understand that local and global is irrelevant. The wave function is always deterministic, and the measurements intrinsically and always probabilistic. 1) what I like: reality are wave functions, but we can't measure it, only some properties of it (the Hermitian operators) and sometimes we measure one number and sometimes another. It's the nature of quantum. Not, sure why. 2) dislike: we try to measure measure and suddenly the wavefunction collapses 3) multiverse of all histories $\endgroup$ – Bob Bee Jan 20 '17 at 1:39
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    $\begingroup$ @claudechuber It is possible because Everettian "determinism" is linguistic in nature. Anything and everything is realized in Everett's branches so the "wave function of the universe" only "determines" that everything that could happen did happen. As to which branch you, the observer, will find yourself in the "determinism" is silent beyond the usual probabilities (this is the "apparent collapse" or "splitting"). In other words, in the relevant sense Everett's interpretation is as indeterministic as Copenhagen, but the indeterminism is rhetorically shifted from "reality" to observer. $\endgroup$ – Conifold Jan 20 '17 at 2:59
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    $\begingroup$ "So yes, quantum information conservation and quantum state determinism or quantum causality are the same things." -- Yes but measurement is not unitary or deterministic. Both $\Psi = \frac{1}{\sqrt{2}} \left|0\right\rangle + \frac{1}{\sqrt{2}} \left|1\right\rangle$ and $\Psi = \frac{1}{\sqrt{5}} \left|0\right\rangle + \frac{\sqrt{4}}{\sqrt{5}} \left|1\right\rangle$ can lead to to a measurement outcome of $1$. How does this fit with conservation of information? $\endgroup$ – Alex Kinman Jan 20 '17 at 16:56
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Well your question really gets at the Measurement problem. It's actually not well established that the outcome of a measurement is truly probabilistic; the answer to that question cuts to the basic interpretation of QM. Different schools of though resolve measurement differently and currently there is no universally accepted answer. You may also be interested in learning about decoherence. Even in the Copenhagen interpretation, many physicists have abandoned a true probabilistic wave function collapse in place of an apparently probabilistic but fundamentally deterministic evolution of the system. In this case, we have to consider the wavefunction of the entire observer-observed system.

Here is an article I wrote about the unitarity of wave function collapse during measurement, if you are interested.


Understanding \ Uncertainty

Today, the interpretation of quantum mechanics remains perhaps the greatest open question in all of science and the resolution of this problem has profound implications for our concepts of determinism, philosophical realism and the limits of knowledge. At the heart of this debate lie the fundamental uncertainty relationships between classically conjugate variables as well questions such as "what constitutes the measurement of a quantum system." We will briefly explore how uncertainty arises and the ways in which different interpretative frameworks attempt to reconcile the problem of measurement.

Heisenberg's Microscope

The famous Heisenberg Uncertainty Principle states that it is theoretically impossible to know exactly the values of sets of related measurable quantities. These are known as complementary variables. The most common examples of these inequalities include: \begin{equation} \Delta x \Delta p \ge \frac{\hbar}{2} \end{equation} \begin{equation} \Delta E \Delta t \ge \frac{\hbar}{2} \end{equation} \begin{equation} \Delta J_i \Delta J_j \ge \frac{\hbar}{2} |\langle J_k\rangle | \end{equation} or the relationship between linear momentum and position, energy and time and the different components of angular momentum along three orthonormal spatial basis.

In the context of introductory quantum mechanics, these inequalities are nearly always presented as either a priori axioms of the system or derived by the commutation relationships for the quantities corresponding operators. However, Heisenberg himself derived these elegant relationships through a simple thought experiment, known as Heisenberg's microscope, which provides a powerful intuition about the source of these enigmatic statements.

Consider a classical electron (i.e, a point particle) being investigated by a scientist using a microscope as portrayed in fig. 1. In order to determine the position of the particle, an incident photon must scatter off the electron at some angle, $\epsilon$, before being focused by the series of optical lenses and arriving at the observer. This step represents a critical loss of information about the system. In particular, the scattering process perturbs the momentum of the electron according the classical relativistic equations for conservation of momentum. However, because lens accepts scattered photons over the entire angle, $\epsilon$ and focuses them to the same point, information about the angle of reflection (and therefore resultant momentum) of both particles is lost. Therefore, we are left with an inherent uncertainty in p, $\Delta p_x \sim \frac{h}{\lambda} \sin(\epsilon)$, where $\lambda$ simply represents the classical wavelength of the light. Additionally, due to the wave nature of light, the microscope can only resolve the position of the electron to a range of $\Delta x = \frac{\lambda}{\sin(\epsilon)}$. Multiplying these terms together, we have arrived at an approximate form of the uncertainty principle purely through a classical thought experiment![2] This simple thought experiment demonstrates that in order to measure the value associated with one quantity, we must always perturb the value of another quantity. In fact, this statement essentially contains the same information as the traditional approach based on commutation relationships. These relations fundamentally express that measuring two different conjugate quantities such as position and momentum sequentially will yield different results depending on whether position or momentum is determined first, because measuring either affects the other.

Fourier Analysis of Uncertainty

In fact, this analysis can be extended to any pair of quantities who are each others Fourier transforms. More rigorously, the derivatives of action are conjugate to the variable which one differentiates.

Generally, Fourier transforms relate the time-like component of a system to the frequency-like component. This relationship is best illustrated with a simple example. Consider a transverse wave. For such a system, we determine the time-like component to be the period, or the amount of time necessary for the wave to travel one full cycle from peak to peak. Similarly, we take the frequency to be the inverse of the period. For an electromagnetic wave, the period (or time-like component) is proportional to the wavelength while the energy is proportional to the frequency. Thus energy and period (or the integral of time between two events) form a Fourier pair.

We will now turn our attention back to the relationship between position and momentum. Here we identify the position as the time-like quantity and the momentum as the frequency-like quantity. We map between these bases using the Fourier transforms: \begin{equation} \Psi(x) = \frac{1}{\sqrt[]{2\pi\hbar}}\int_{-\infty}^{\infty}\psi(p)e^{ipx/\hbar}\,\mathrm dp \end{equation} where $\Psi(x)$ is the position space wave function and $\psi(p)$ is its representation in momentum space.

We are now ready to derive the relation given in eqn. 1. What follows is a skeletal representation of the a proof for the uncertainty principle [3]. By the definition of uncertainty, we recall: \begin{equation} (\Delta x)^2 = \langle x^2\rangle - \langle x\rangle ^2 \end{equation} and similarly for momentum. As absolute position and momentum are gauge freedoms, we may simply set the expectations of x and p to be zero, leaving us with: \begin{equation} (\Delta x)^2 = \langle x^2\rangle =\int_{-\infty}^{\infty} x^2 |\Psi(x)|^2 \,\mathrm dx \end{equation} \begin{equation} (\Delta p)^2 = \langle p^2\rangle . \end{equation} Now let us define $f(x) = x\Psi(x)$. Therefore, we have \begin{equation} (\Delta x)^2 = \int_{-\infty}^{\infty} |f(x)|^2 \,\mathrm dx = \langle f|f\rangle . \end{equation} We define an equivalent function for p, \ $\tilde{g}(p) = p\psi(p)$. We can find the x-domain representation of $\tilde{g}(p)$ using an inverse Fourier transform. We find: \begin{equation} g(x) = \left(-i\hbar\frac{\mathrm d}{\mathrm dx}\right)\Psi(x). \end{equation} We can now write the variance in momentum as \begin{equation} (\Delta p)^2 = \int_{-\infty}^{\infty} |f(g)|^2 \mathrm dx = \langle g|g\rangle . \end{equation} By the Cauchy-Schwartz inequality, we know \begin{equation} (\Delta x)^2 (\Delta p)^2 = \langle f|f\rangle \langle g|g\rangle \ \geq |\langle f|g\rangle |^2 \end{equation} \begin{equation} |\langle f|g\rangle |^2 \geq \left(\frac{\langle f|g\rangle _\langle g|f\rangle }{2i}\right)^2. \end{equation} Finally, by explicit evaluation, it can be shown that \begin{equation} \langle f|g\rangle - \langle g|f\rangle = i\hbar. \end{equation} At last we have arrived at the uncertainty relation as promised. Plugging in this value to the inequalities above and taking the square root, we find \begin{equation} \Delta x \Delta p \geq \frac{\hbar}{2}. \end{equation}

Uncertainty: Ontological or Epistemological

We have so far established inherent limits on our ability to specify the state of a quantum object in bases such as position and momentum. It is therefore clear that we must describe the attributes as distributions over some range of values. At best, we can determine the probability that an observed attribute of some particle will fall within a range of possible values. In position space, this is simply accomplished by taking the modulus of the particles wave function and integrating over some distance.

However, we have not established whether this uncertainty in measurement arises from the ontological nature of the system, that is to say that particles truly exist as probability distributions over space rather than at fixed points, or instead is simply an unavoidable epistemological artifact resulting from our inability to fundamentally determine the particle's position. However, we only ever observe a system to be in a single eigenstate or a set of eigenstates bounded by minimum uncertainty. For example, if we measure the path a particle takes in the two slit experiment, we, of course, cannot observe the particle taking both paths simultaneously. This act alters the system's wave function and destroys the characteristic interference pattern. Even if an observation of the path taken is made after the classical wave has exited the two slit apparatus, the wave function of the particle is collapsed leading to the two bands characteristic of a particle. Known as a delayed choice experiment, this startling result indicates that measurement can retroactively collapse the quantum state of a system. Information about the future arrangement of the quantum state propagates backwards in time. This is a clear violation of local realism, the classical belief action at one point in space time cannot instantaneously affect the state of the system at another point.

The most orthodox interpretation of quantum mechanics, the Copenhagen school of thought, points to the interference pattern of the double slit experiment as incontrovertible proof that the true state of a quantum system prior to observation must be treated as a superposition of eigenstates. According to the many of the early proponents of quantum mechanics including Heisenberg and Bohr, a quantum system has no definite properties before observation [4]. In this framework, an electron initially in a superposition of spin states is neither spin up nor spin down until observed. In the minds of many of early quantum physicists, measurement alone forces particles to adopt real properties. This approach to quantum mechanics has found itself profoundly at odds with our classical intuitions where we believe that objects truly have values associated with observables such as position and angular momentum. However, this non-realistic theory of existence has come under fire in recent decades from proponents seeking to reconcile the problem of measurement with a more traditional view of reality. Before we can understand these competing theories, we must first establish a firm grasp of what is meant by measurement, the mathematics of observation and who or what constitutes a valid observer in the quantum world.

Evolution of the \ Wave Function

The Mathematics

First, let us recall the equation determining the time evolution of an arbitrary wave function, $\Psi$. Given some initial state, we can described the wave function at any future point in time using the Schrodinger equation, \begin{equation} i\hbar\frac{\partial \Psi(x,t)}{\partial t} = -\frac{\hbar ^2}{2m} \frac{\partial ^2 \Psi (x,t)}{\partial x^2} + V(x)\Psi (x,t). \end{equation} We are interested in determining the value of some continuous eigenvalue, for clarity, we will observe the particles position. Initially, we described the state of our system as an integral over these eigenstates, \begin{equation} |\Psi\rangle = \int_{a}^{b}c(x)|x\rangle \,\mathrm dx, \end{equation} where c(x) parameterizes the amplitude of the wave function over all space. We can express the probability of find our particle between any two points, a and a + $\Delta$ x as, \begin{align} Pr(a < x < a +\Delta x) &= \int_{a}^{a + \Delta x}|\langle x|\Psi\rangle |^2 \,\mathrm dx \\ &= \int_{a}^{a + \Delta x}|c(x)|^2 \,\mathrm dx. \end{align}

Now suppose that we perform a measurement of position and determine the particle sits at some point, $x'$. The wave function has spontaneously collapsed such that $|\Psi'\rangle = |x'\rangle .$ A repeated measurement immediately following this collapse would yield the same eigenstate. However, in the absence of repeated measurement, the system will simply continue to evolve according to the standard Schrodinger equation following observation.

This highlights the fundamental aspects of the process of measurement, wave function collapse and wave function recovery. John von Neumann first formalized this description of measurement with the Postulates of Reduction [2]. Specifically, these postulates state that:

  1. Measurement of some observable, $q$, which is initially in a superposition of eigenstates, $q_n$, will yield some eigenvalue such that $|\Psi'\rangle = |q_i\rangle $ with probability equal to the square of coefficient for $q_i$.

  2. Measurements of $q$ which are repeated immediately, one after the other, will always yield the same eigenstate.

The second postulate results from the fact that the commutator of any operator with itself is zero. However, we can show that for repeated measurements of two different non-commuting observables, the second postulate will not be satisfied.

Let us consider another the simple example of a particle in box. We initially determine the particle to be in the ground energetic state, $|\psi_1\rangle $, with associated energy, $E_1 = \frac{\pi ^2 \hbar ^2}{2mL^2}$, where $L$ represents the length of the box. The spatial wave function will then be, \begin{equation} \langle x|\psi_1\rangle = \sqrt{\frac{2}{L}}\sin(\frac{\pi x}{L}). \end{equation} Now suppose we make a measurement of the particle's position. We know the probability of find the particle between two points, a and a $+ \Delta x$ will be $\frac{2}{L}\sin^2(\frac{\pi a}{L})\,\mathrm dx$.

If we find the particle at $x = a$, we can describe the resulting wave function as the sum over energy eigenstates, \begin{equation} |\Psi '\rangle = \Sigma_{n} |\psi_n\rangle \langle \psi_n|a\rangle \end{equation} with $\langle \psi_n|a\rangle = \sqrt{\frac{2}{L}}\sin(\frac{n\pi a}{L}).$ We immediately see that the probability of observing the system in some energy eigenstate for $n \neq 1$ is greater than zero. Thus by two non-commuting measurements, we can change the energy state of the system.

Questions Affecting Measurement

These two simple examples have allowed us to explore the basic process of measurement. However, this description raises two important questions, together known as the measurement problem. We have seen that the wave function for a quantum system evolves deterministically according to the Schrodinger equation. However, upon observation, the wave function apparently collapses probabilistically to a single eigenstate [5]. The questions of interest then become,

  1. "How do we reconcile the transition between deterministic and probabilistic evolution of a quantum system, especially in the light of the apparent role of an observer?"

  2. "What constitutes a measurement, or what is an observe?"

Both of these quandaries ultimately tie back to our discussion of uncertainty. If uncertainty is fundamentally an epistemological limit, measurement simply reveals the true state of the system previously shrouded by our inability to adequately specify the system. In this case, both questions are easily reconciled as a quantum system evolves deterministically in all settings with real values associated with observables at all times. In such an interpretation, measurement presents no more of a problem than any other quantum interaction and the wave function collapse simply amounts to a refining of our knowledge about the system. Alternatively, if uncertainty proves to be an ontological property of the system the questions become significantly more intractable and demand a careful definition of terms.

Measurement in Different Interpretations

The measurement problem, perhaps more than any other question in quantum mechanics, serves as the basis for competing interpretations of quantum theory. This sections merely aims to serve as a brief introduction to a subset of the most prominent interpretations.

The Copenhagen Interpretation and Decoherence

Thus far, our discussion of measurement and uncertainty has largely been implicitly in terms of the Copenhagen school of thought, first advanced by Heisenberg and Bohr. For this reason, we will only provide a precursory, qualitative overview of the theory. Specifically, Heisenberg and Bohr believed that uncertainty represented the ontological state of the system and that the act of measurement simply behaves as an irreversible thermodynamic process which alters the state of the system. In this sense, wave function collapse occurs any time that a quantum system interacts with a classical environment [4].

Importantly, the probabilistic wave function collapse and the associated question of what constitutes an observation can be solved by treating the interaction between the measuring device and the quantum particle as a process of entanglement.Thus measurement occurs constantly in any macroscopic system. In this manner, the apparent significance of the observer vanishes as the collapse represents the resulting superposition of the entire system. The interaction with the classical ensemble leads to the loss of information about the original quantum system as the wave function becomes diluted by many repeated scattering events between both the measuring device and the quantum particle and measuring device and itself. Due to the far larger number of particles in the classical device, the original quantum system becomes dominated by its classical environment and phase relationships between each quantum particle lead to interference between the possible eigenstates of the system as a whole, leading to a single apparent eigenstate for any observable in a process known as decohernece [6]. In short, the Copenhagen interpretation deals with the questions of measurement by eliminating wave function collapse entirely.

The Many Worlds Hypothesis

Today, one the leading opponents to the Copenhagen interpretation focuses on the objective reality of the wave function. First developed by Hugh Everett in 1957, the Many Worlds Interpretation begins by positing a universal wave function. In other words, we treat all of existence as a single quantum system obeying the conventional Schrodinger equation. Everett resolved the apparent probabilistic nature of measurement by expanding the superposition of the system to include the measuring apparatus or classical environment[7].

In this formulation, the position of a particle described by a continuous distribution of eigenstates will be found at every possible eigenstate if observed. In this sense, uncertainty represents all possible universes. In fact, not only is uncertainty an ontological aspect of the quantum system, uncertainty represents the complete state of the system with each possible eigenstate corresponding to an equally real and valid outcome. The apparent observation of a single eigenstate results from a sort of universal splitting. If we observe the spin state of an electron initially in a superposition of up and down, the entire universe continues to obey the Schrodinger equation and the apparent collapse occurs as our conscious is confined to an single eigenstate of the universal superposition with many other versions of us existing in tandem. In Everett's interpretation, the universe is deterministic and measurement (or any interaction) simply leads to a branching of possible futures.

Proponents of the Many World's interpretation point to its mathematical simplicity and lack of assumptions. Rather than relying on multiple equations necessary to describe the evolution of a system (a characteristic of other interpretations), the Schrodinger equation stands alone. Unlike traditional Copenhagen theory, no nebulous definitions of irreversibility or measurement need to be advanced. Prominent physicists including Sean Carroll have claimed that the Many Worlds interpretation therefore satisfies both Occam's razor and represents the most mathematically pure solution to the problem of measurement.

However, Everett's approach has not been without detractors, including Bohr himself. Specifically, objections have taken the form that while the number of equations and assumptions may be minimized, this simplification does not outweigh the infinite number of possible realities which seem to make the theory more complex, not less.

De Broigle-Bohm Pilot Wave Theory

At odds with the other major interpretations of quantum theory outlined above, Bohmian mechanics treats fundamental uncertainty as an epistemological property of a quantum system rather than an ontological one. Popularly known as Pilot Wave theory, this interpretation sets itself apart from both Copenhagen and Many Worlds by being both realistic and deterministic [8]. In this sense, Bohmian mechanics maps naturally onto classical mechanics. However, the interpretation abandons locality (or the principle that action cannot spontaneously affect a system separated in space or time).

Bohm posits that real, localized particles with definite properties exist at all times, thus any uncertainty in the eigenstates of a system simply come from our inability to adequately probe the system. Thus if we say that an electron is in a superposition of spin states, we are merely stating that we have insufficient information to properly specify the alignment of angular momentum along a particular axis. However, while the metaphysical interpretation of Bohmian mechanics differs explicitly from the Copenhagen school, the theory makes identical predictions to the more orthodox interpretation.

The basic premise of the theory includes both a point like particle and a wave which determines the evolution of this particle and whose form matches identically the conventional Schrodinger equation. Bohmian mechanics starts by positing that a real configuration of the universe, $q$, exists which can be described by the sum of the individual configurations of each constituent particle, $q_i$, in configuration space $Q$ with coordinates $q^k$. Typically, configuration space takes the form of the spatial positions and orientations of each particle.

The dynamics of this system obey the traditional wave function in configuration space, $\Psi(q,t)$. In this sense, a particle embedded in this space follows a trajectory corresponding to an exact eigenvalue solution to the Schrodinger equation [7]. Specifically, the configuration of the system evolves according to a guiding equation of the form: \begin{align} m_k\frac{\mathrm dq^k}{\mathrm dt}(t) &= \hbar \nabla_k Im \ln\psi(q,t)\\ & = \hbar Im\left(\frac{\nabla _k \psi}{\psi}\right)(q,t)\\ &= \frac{m_k j_k}{\psi * \psi} = Re\left(\frac{\hat{P_k}\Psi}{\Psi}\right), \end{align} where $j$ is the probability current and $\hat{P}$ is the momentum operator. In correspondence with the Copenhagen interpretation, the configuration of the system follows a distribution according to $|\psi(q,t)|^2$.

In this manner, all the predictions of quantum mechanics are preserved and simply the metaphysics are changed. For example, the two slit interference pattern becomes the result of a distribution of initial positions of the incident particles. However, any individual particle follows a definite trajectory. Fundamental uncertainty according to the Heisenberg principles prevent us from specifying these initial positions. Measurement then merely becomes a minimization of uncertainty in the respective quantity corresponding to a maximization in the uncertainty of the conjugate barrier. However, measurement does not lead to a probabilistic outcome, merely a qualification of knowledge.

While Bohmian mechanics successfully restores a realistic theory of the world, the interpretation continues to be a more niche approach to quantum theory. Critics have often pointed to its inclusion of two equations (the Schrodinger equation and the guiding equation) rather than one as a sign that it is axiomatically weaker than its competitors. Additionally, so far no fully relativistic formulation of Bohmian mechanics has been successful, although current attempts in this direction appear promising.

Conclusion

Modern quantum theory has firmly established our inability to precisely specify the values of certain conjugate variables. However, the interpretation of this uncertainty has not been decisively determined. While the argument between a epistemological and ontological approaches has thus far led to no varying experimental predictions, unresolved questions surrounding uncertainty and measurement have profound implications for our metaphysical concept of reality.

  1. G. Vandergrift, Creative Commons. 2015
  2. V. Braginisky, F. Kahlili, "Quantum Measurement" Cambridge University Press. 1995
  3. L.D. Landau, E.M. Lifshitz, "Quantum Mechanics: Non-Relativistic Theory" Pergamon Press. 1977
  4. W. Heisenber, "Physics and Philosophy: The Revolution in Modern Science" Harper Perennial. 1957
  5. S. Weinberg, "Einstein's Mistakes" Physics Today. 2005
  6. H. Price, "Times' Arrow and Archimedes' Point". 1996
  7. B. DeWitt, "Quantum Mechanics and Reality" Physics Today. 1977
  8. J, Wheeler, H. Zurek, "Quantum Theory and Measurement" Princeton Legacy Library. 2014
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    $\begingroup$ Is this from a book? $\endgroup$ – lmr May 17 '18 at 19:05
  • $\begingroup$ Note that \subsection doesn't work here, though you can probably use ##subsection_name in place. Also, don't use <,> when $\langle$ (\langle) and $\rangle$ (\rangle) are available. $\endgroup$ – Kyle Kanos May 17 '18 at 19:19
  • $\begingroup$ Thanks for the formatting tips. To be honest, I wrote this in LATEX and was too lazy to change the headers, etc. Also, this isn't from a book; I wrote it myself as a final paper for a Quantum Mechanics course. So, it's definitely not guaranteed to be correct. I'm far from an expert. $\endgroup$ – Carl J Thomas May 17 '18 at 19:30
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The answer needn't be lengthy. It is that there is more than one way to assess what quantum physics tells us about the nature of the physical world.

  1. If you think that all evolution everywhere follows Schrodinger's equation then it follows immediately, as a mathematical consequence, that physical evolution is unitary; and unitary amounts to deterministic. This point of view goes with a many-worlds type of interpretation of quantum theory.

  2. Other ways of 'reading' what quantum physics says about the nature of the physical world do indeed result in a non-deterministic world, as you suspect.

Many people working in quantum gravity are 'buying' the deterministic (unitary) model as a working model. And then they run up against the fact that there is the black hole information paradox (which amounts to saying it is hard to see how behaviour associated with black holes can be unitary). This paradox is a challenge to the combination of gravity and quantum field theory; sometimes people announce it is solved but I think it is safe to say no one really understands this yet.

I think it is correct practice to keep working at a unitary framework for "quantum plus gravity", but I suspect that the way the maths relates to the actual physical behaviour is not quite as direct as the many-worlds type of picture says, so the world will turn out to be non-deterministic.

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There is a very simple answer to this question I don't know why no one mentions.

Quantum Mechanics is deterministic in Schrodinger's Equation.

Then there is the measurement. Is there collapse? If there's, how does it work? This part is undefined. This is where interpretations of QM comes.

Copenhagen interpretation is an indeterministic approach, and according to it, information isn't conserved backwards. And backwards loss of information means no conservation of information. This is mostly taken as default approach of QM, so if you take this as "truth", there is no conservation. But it is no longer taken as a viable interpretation by most scientists.

There are many more interpretations of QM and most of them are deterministic. So they're saying if all information was known, a process could be rewind to see what their previous state was. This is what determinism means.

There's also MWI which is also deterministic and it is the purest form of QM because it doesn't introduce a new arbitrary element into the equation so-called "collapse".

Long story short, as of now, most QM interpretations are deterministic and most scientists think QM is deterministic no matter which interpretation ends up being right. So it's up to you to take it as deterministic or indeterministic.

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Without getting technical, quantum mechanics may be probabilistic, but which outcomes are possible is deterministic. So, we cannot determine which outcome will occur, but we can determine which outcomes are possible.

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