Free Will and Determinism

Proposed by John Stewart Bell in 1964 and extended into a more widely testable form by CHSH in 1969, this inequality addressed a fundamental question. Let's first look at why such an inequality was needed.

Can coins in boxes explain it?

Imagine placing two coins into separate boxes—one showing heads, the other tails. Making sure their states cannot change, we keep one box on Earth and send the other to Mars.

What if we open the box on Earth and find heads?

Without even opening the box on Mars, we know it is tails— because both outcomes were fixed before the boxes were opened.

Einstein wondered whether quantum entanglement could be explained in a similar way: particles carry information we do not know from the start, and measurement outcomes simply follow from that information. Such additional information is called a "hidden variable."

Bell, however, asked one more question here:

If particles answer using pre-existing information, and the distant partner's measurement choice does not alter that answer, isn't there a mathematical rule their outcomes must obey?

And he actually found that rule.

Measuring spin along three directions

Suppose we send a pair of entangled spin-1/2 particles to Alice and Bob, one each. Here we use a state that always yields opposite outcomes when measured along the same direction. It is the singlet state with total spin 0, written as follows:

Each person chooses one of the three directions below to measure their particle. The outcome is either +1 or −1.

If we explain this with local hidden variables, we can think of each particle as having pre-decided how to answer a question along each direction. For example, an answer sheet like this is possible:

Since asking along the same direction must always give opposite results, Bob's answers are the opposite of Alice's. Other answer sheets are certainly possible, but in every case, the answers to the three questions must form a single consistent table.

This answer sheet has an unavoidable constraint

Look at Alice's answers first. If the answers for A and C are different, at least one of the following two must hold:

• The answers for A and B are different.
• The answers for B and C are different.

Because if A equals B and B equals C, then A and C must also be equal. In the table above, A and C differ, while B in between equals A and differs from C.

The same holds when we gather many such answer sheets. Since every case where A and C differ is included in at least one of the other two cases, its probability cannot exceed the sum of the two probabilities.

Here A, B, and C are the answers Alice's particle has pre-set for each direction. It does not mean we measure a single particle along all three directions at once.

Then how do we test this relationship in a two-particle experiment?

In the table above, Alice's A is +1 and C is −1. Meanwhile, Bob's C is +1. So "Alice's A and C are different" can be restated as "when we ask Alice A and Bob C, their two answers are the same."

Applying the condition that the two answers are opposite along the same direction to the other pairs as well turns the relation into an inequality comparing actual measurement records:

Here P(same) is the probability that both people obtain the same outcome. We run the experiment with a fresh pair of particles each time and assume the choice of direction is made independently of the particles' hidden information. Under these conditions, any pre-arranged answer model must obey the relation above.

Yet quantum mechanics crosses this limit

Let's calculate the singlet state |Ψ⁻⟩ prepared earlier using quantum mechanics. If the angle between the two measurement directions is θ, the probability of getting the same outcome in this state is:

Plugging in the three directions above gives:

For these numbers to satisfy the inequality above, the following relation would have to hold:

This is the core of Bell inequality violation. No matter how you design pre-arranged answer sheets or in what proportions you mix them, you cannot produce these three probabilities together. Yet quantum mechanics predicts that such correlations are possible.

So what did we learn about nature?

A violation of the inequality does not immediately mean "nothing is determined before measurement." More precisely, it means that no explanation keeping all of the following conditions together can reproduce the observed correlations:

• Local causality. Measurement outcomes at each location and the relationship between them are explained by local causes and information in their common past.
• Measurement independence. The choice of which direction to measure is statistically independent of the hidden variables assigned to the particles.
• Measurements yield recordable outcomes, and standard probability rules apply.

Which condition to keep and which to modify is a separate question. Non-local deterministic interpretations exist, and violating the inequality does not enable faster-than-light communication.

What is fascinating is that even the explanation "we just don't know the answers yet" now faces an experimentally testable limit. Rather than the idea of hidden information itself, the question is whether that information can operate locally and still explain all quantum correlations.

The name comes from the four authors of the 1969 paper: Clauser, Horne, Shimony, and Holt.

From Bell's Inequality to CHSH

The earlier Bell example used the condition that outcomes along the same direction are perfectly opposite. Introduced in 1969, the CHSH inequality extended Bell's argument so it could be tested even without that condition.

CHSH is not a separate phenomenon, but a form designed to test the same problem more broadly. Actual experiments have observed results exceeding the limit of such local models.

Let's look at why this limit arises through a simple game.

A game of asking two people questions

Suppose there are two people, Alice and Bob. They are far apart and cannot communicate during the game, though they can coordinate their strategy as much as they like beforehand.

Each person randomly receives one of two questions:

• Alice receives A₀ or A₁
• Bob receives B₀ or B₁

Upon receiving a question, they must answer with 0 or 1. Each person knows only their own question and not what the other was asked.

The winning rules are as follows:

All four question pairs occur with equal probability, and the choice of questions is independent of any information the two shared in advance.

An interesting problem arises here: no matter what answers they pre-arrange, they cannot satisfy all four conditions at once.

Why is 75% the maximum win rate?

To satisfy the first condition, the answers for A₀ and B₀ must be the same.

The second condition requires A₀ and B₁ to be the same, and the third requires A₁ and B₀ to be the same.

Follow the three solid lines in the diagram below. Ultimately, all four questions must have the same answer. Yet the final dashed line demands that the answers for A₁ and B₁ be different. Both demands cannot be met at the same time.

Clicking the four nodes toggles each answer between 0 and 1. First, try clicking "Only A₁ = 1." When you fix the last condition, where does the failure move? Try changing the other answers as well to see if all four lines can ever be satisfied at once.

In an actual round of the game, only one of the four question pairs is asked. The diagram above applies the same answer sheet to all four cases at once.

Whichever answer sheet you pick, it fails on at least one condition. If all four pairs occur with equal probability, the best you can do is win 3 out of 4—that is, 75%.

Using a different answer sheet each time or sharing random numbers in advance changes nothing. Since no single answer sheet exceeds 75%, any average mixing them cannot exceed 75% either.

Yet in quantum mechanics, about 85.4% is possible

Now suppose that instead of pre-arranged answers, the two hold one entangled particle each.

Depending on the question received, each person chooses which direction to measure their particle and answers with the resulting outcome. In this game, the "question" is the measurement direction and the "answer" is the measurement outcome.

Where does the number 85.4% come from? There are several kinds of entangled states. While the earlier Bell example used |Ψ⁻⟩, which gives opposite outcomes along the same direction, for this game we prepare the |Φ⁺⟩ state below:

All measurement axes from here on are chosen within the X–Z plane. In this plane, |Φ⁺⟩ is a state that yields the same outcome when measured along the same direction. If the angle between the two measurement axes is θ, the probabilities in this state are:

The reason sin²(θ/2) earlier and cos²(θ/2) here differ is that we prepared a different entangled state. Along the same direction (θ = 0°), the earlier state always gave different outcomes, whereas this state always gives the same outcome in the X–Z plane.

Setting the Z-axis as 0°, let's fix Alice's measurement axis at 0° for A₀ and 90° for A₁, and Bob's at 45° for B₀ and −45° for B₁. The first three combinations differ by 45°, while only the last combination differs by 135°.

Since the first three combinations require the same answer, we plug 45° into the first probability formula. Simplifying with the half-angle identity gives:

Since the last combination requires different answers, we plug 135° into the second formula. Because 67.5° and 22.5° add up to 90°, this probability equals the previous one:

In the end, all four combinations win with a probability of about 85.4%. Since each of the four combinations is chosen with probability 1/4, the overall win rate is also their average: about 85.4%.

Does that mean the earlier logic was wrong?

No. Earlier, we assumed that even answers to unasked questions could be written down together in a single table beforehand. The statistics obtained from quantum measurements simply cannot be reproduced no matter how you mix such local answer sheets.

The interactive diagram above shows the limit of pre-arranged answers. 85.4% is not a number found by discovering a cleverer answer sheet in that diagram; it is a value calculated from the measurement rules of quantum mechanics.

Writing this game as a formula gives the CHSH inequality

Let's label measurement outcomes as +1 and −1 instead of 0 and 1. If E is the average of the product of the two outcomes, E approaches +1 when matching answers occur more often, and −1 when opposite answers occur more often.

The subscripts denote the question pairs. We add the first three pairs because they require matching answers, and subtract the last pair because it requires opposite answers:

In any local hidden variable model, S cannot exceed 2. In quantum mechanics, however, it can reach:

This upper bound is called the Tsirelson bound. It connects directly to the maximum win rate achievable in the game:

Plugging in the local bound of 2 gives 75%, and plugging in the quantum bound of 2√2 gives about 85.4%. The game and the inequality express the exact same limit in different ways.

What this game tells us about nature

If two particles answer using only local information they carried in advance, and measurement choices are independent of that information, there is an unbreakable limit. Actual experiments have observed correlations that surpass that limit.

That does not mean faster-than-light communication is possible or that all determinism is false. Rather, we must examine which assumptions can be held together.

Herein lies the appeal of CHSH: it turned the explanation "they just matched their answers beforehand" into something testable by numbers. And nature goes beyond the limit that explanation allows.

What makes this theory intriguing is that it asserts three things together:

• Particles have actual positions even before measurement.
• The motion of particles is fundamentally deterministic.
• Even so, in quantum equilibrium, it predicts the exact same experimental results as standard quantum mechanics.

Recalling Bell's inequality from earlier, doesn't this sound like a contradiction? However, what Bell's theorem rules out is not all determinism, but the idea that quantum correlations can be explained solely by local hidden variables. Because Bohmian mechanics explicitly accepts non-locality, it does not contradict Bell's result.

Proposed by de Broglie in 1927 and independently rediscovered and systematized by Bohm in 1952, how does this approach determine a particle's motion, and how do the probabilities of quantum mechanics emerge from deterministic motion? We will also revisit the condition of "quantum equilibrium" attached to the third claim along the way.

Adding the particle's actual position to the wave function

In standard quantum mechanics, a particle's state is represented by the wave function ψ, which is used to calculate the probability of finding the particle at each position.

De Broglie–Bohm theory adds the actual position Q(t) of the particle. Rather than only appearing at some location upon measurement, a particle is viewed as actually being somewhere even before it is measured.

Let's separate the wave function from the particle's position. The wave function evolves according to the Schrödinger equation, while the particle's position changes according to a velocity calculated from that wave function. This second rule is the guiding equation.

The key is that the wave function is not merely used to calculate the probability of finding a particle, but also governs the actual particle's motion. Thinking of it like a physical water wave pushing a particle along, however, would be inaccurate; the wave function is a mathematical object that assigns a complex number to each position.

What does the guiding equation calculate?

Consider a single spinless particle first, setting aside relativistic effects and magnetic fields. In this case, the guiding equation is:

With so many symbols, it may look intimidating. Let's look first at what the equation is solving for. On the left, dQ/dt is the velocity of the particle. The right side shows how to compute that velocity from the wave function and the current position. Breaking down each symbol gives:

In short, this equation means: inspect the wave function where the particle currently is, and compute its velocity there.

Compare this with familiar Newtonian mechanics, where the equation of motion uses force to find acceleration:

The guiding equation directly specifies velocity itself, rather than acceleration. The position changes following that velocity, and at the next position, the velocity is computed anew from the wave function there. Thus, once the initial position, the initial wave function, and the evolution rule of the wave function are given, the subsequent trajectory is determined.

The formula above is used where ψ is non-zero. Where ψ is 0, the division cannot be applied as-is and requires separate mathematical treatment.

Splitting the wave function into amplitude and phase

What information in the wave function actually sets the particle's velocity? To see this, it helps to split the wave function into two components.

Just as a complex number can be represented by a magnitude and an angle on a plane, the magnitude of a wave function is called its amplitude and the angle its phase. Written as a formula:

• R is the amplitude of the wave function.
• S is a function representing the phase; in this notation, the actual phase angle is S/ℏ.

Substituting this expression into the guiding equation splits ∇ψ/ψ on the right-hand side into two terms:

The guiding equation takes only the imaginary part here. Since ∇R/R is real, it drops out, leaving ∇S/ℏ from the second term. Multiplying by ℏ/m outside cancels ℏ, simplifying the velocity formula to:

Ultimately, what sets the particle's velocity is how the phase of the wave function varies across space. ∇S indicates how much and in what direction S changes when moving to a neighboring position at the same instant. Dividing that gradient by the mass gives the velocity at the current position.

We should distinguish between the phase itself being large and the gradient of the phase being large. Adding the same constant to the phase everywhere changes the phase value, but leaves the difference between neighboring positions unchanged—so neither the gradient nor the instantaneous velocity changes.

Let's look at a simple one-dimensional example. A plane wave with constant amplitude A can be written as follows, where k is how fast the phase varies with position and ω is how fast it varies with time:

Comparing this with the earlier expression, S for this wave function is ℏ(kx−ωt). Differentiating with respect to x leaves ℏk, so the velocity is:

Since the momentum of this plane wave is p=ℏk, we recover the familiar classical relationship between velocity and momentum. Let's compare changes in phase directly in the diagram below.

First, switch the slope across "Increasing → Constant → Decreasing." How do the direction and magnitude of the velocity change? Next, fix the slope and add π/2 to the global phase. You can see that even when the height of the line shifts, the velocity remains unchanged.

This does not mean the amplitude R is irrelevant to motion. Amplitude and phase evolve together under the Schrödinger equation. Saying that only the phase appears directly in the instantaneous velocity formula is different from saying the amplitude has no effect on subsequent motion.

Does an interference pattern still form if a particle passes through only one slit?

In the double-slit experiment, the respective roles of the particle and the wave function become even clearer. Because a particle has an actual position in this theory, it passes through one of the two slits. However, in an experiment where both slits are open and which slit it passed through is not recorded, the wave function evolves into a superposition of components passing through both slits.

As the two components interfere, the spatial shape of the amplitude and phase changes. The particle moves according to the velocity calculated from that modified wave function. Its subsequent path is determined not merely by which slit it passed through, but by the entire wave function created by both slits.

Repeating the experiment many times and collecting the arrival positions yields the quantum mechanical interference pattern. To obtain the exact statistics, however, we also need to account for the distribution of starting positions.

If it is deterministic, why do probabilities appear?

If a single particle's initial position is fixed, its path is fixed too. In experiments, however, we do not know that exact position every time. We need to specify how the starting positions are distributed to say what statistics will emerge across repeated trials.

Letting ρ denote the probability density of position, standard "quantum equilibrium" uses the following distribution—the Born rule, which squares the magnitude of ψ to obtain the probability density of position:

A crucial property of the guiding equation is that if positions are initially distributed according to |ψ|², they continue to match the evolving |ψ|² at later times as well. The evolution of the wave function and the motion of the particles preserve the same position statistics. This property is called equivariance.

This connects directly to the guiding equation. If j denotes the probability current obtained from the Schrödinger equation, the velocity in the guiding equation is that current divided by the density |ψ|², where ψ* is the complex conjugate of ψ:

Because particles move at this velocity, the change in the position distribution matches the change in probability given by the Schrödinger equation. Applying this relation to include the measurement apparatus reproduces standard quantum measurement statistics within the scope of non-relativistic quantum equilibrium.

Still, this does not mean just any initial distribution automatically becomes |ψ|². Why the quantum equilibrium distribution should be used requires separate statistical justification; writing down a deterministic law of motion does not automatically prove the Born rule.

Why it does not contradict Bell's inequality

Didn't we see earlier that local pre-existing information cannot explain quantum correlations? Bohmian mechanics keeps determinism while giving up locality. This difference appears directly in the equation when there are two particles.

A multi-particle wave function takes the positions of all particles together as input. Two particles require six position coordinates, and the space of all possible coordinate combinations is called "configuration space." Let's compute the velocity of the first particle using this joint wave function:

Here Ψ is the joint wave function taking both particle positions as input, and ∇₁ denotes the spatial derivative with respect to the first particle's position. Notice that the actual position Q₂ of the second particle enters the calculation of the first particle's velocity.

If the wave function factors into separate functions for each particle, this dependence on the other's position vanishes; in an entangled state, however, it generally cannot be factored that way. The position of a distant particle—or the measurement apparatus over there—can participate in the motion over here. This is the non-locality of the theory.

The formula written here is for spinless particles. To handle spin as in CHSH, the wave function is extended to a spinor with multiple complex components. While the theory takes actual particle positions as its basic variables, it does not pre-assign spin outcomes for every direction independently of the apparatus; measurement outcomes are determined by the interaction of the whole system, including both the particle and the apparatus.

Furthermore, in quantum equilibrium, this non-local dependence cannot be used to send desired faster-than-light messages. That a law of motion depends on a distant variable must be distinguished from being able to control that dependence as a communication channel.

What is the quantum potential?

Bohm also expressed the same motion in terms of a "quantum potential." Splitting the wave function into amplitude and phase and substituting it into the Schrödinger equation introduces the following term into the equation for the evolution of S, where ∇²R is the second derivative representing the spatial curvature of the amplitude:

This can be arranged as the classical Hamilton–Jacobi equation—which writes motion in terms of energy and changes in S—plus this additional term. It is a reformulation of the two equations of motion above and does not mean we must postulate a separate independent force or field; the evolution of the wave function and the guiding equation alone suffice to understand the particle's motion.

Has this theory been proven right?

The explanation so far does not settle how the actual universe moves. Producing the same experimental predictions as standard quantum mechanics in non-relativistic quantum equilibrium also means that those results alone cannot easily distinguish between the two explanations. Extending the theory to combine with relativity involves additional challenges as well.

What this approach shows is that probabilistic observation outcomes in quantum mechanics and deterministic laws of motion do not necessarily contradict each other. Instead of explaining motion solely through causes in each particle's immediate surroundings, however, it explicitly accepts non-local dependence. Keeping determinism and keeping locality at the same time are not the same claim.

In particular, de Broglie–Bohm theory, the wave function, and Bell's inequality discussed earlier can all be described using the mathematical structure of a Hilbert space.

To state the core ideas up front:

• Hilbert space: a mathematical space where quantum states can be represented as vectors
• Basis vectors: reference vectors used to express any state within that space
• Wave function: one of the mathematical representations obtained when a quantum state vector is expressed in a specific basis

Let's look at them one by one.

Understanding basis vectors first

Think of the familiar 2D coordinate plane.

In 2D, we usually use the following two basis vectors: e₁, one unit horizontally, and e₂, one unit vertically.

With just these two vectors, any 2D vector can be expressed. For example, the vector (5, 4) can be written as:

In other words, a basis is a set of independent vectors chosen so that every vector in the space can be expressed as a combination of them. "Independent" here means that none of the basis vectors can be replaced by a combination of the others.

Basis vectors do not have to be horizontal and vertical; any two non-parallel vectors can span a 2D space. Choosing vectors that are mutually perpendicular and of length 1, however, makes calculations convenient. Such a basis is called an orthonormal basis.

How is a Hilbert space different?

Now let's extend this concept to quantum mechanics.

Whereas the coordinate plane above used real coefficients to express vectors, quantum mechanics must also allow complex coefficients. In addition, an inner product—which calculates how much two vectors overlap—must be defined.

Mathematically, a Hilbert space is a vector space equipped with an inner product that satisfies completeness. Quantum mechanics typically uses a complex Hilbert space.

Completeness, simply put, is the property that if a sequence of vectors gets endlessly closer to one another as you go further out, the limit they approach also exists within the same space.

That may sound formal, but for now it suffices to understand it as follows:

A Hilbert space is a mathematical space designed to represent quantum states as vectors, allowing us to add vectors and compute inner products.

What matters is that this space does not necessarily mean the 3D physical space we live in. For instance, representing only the spin state of an electron uses a 2D complex Hilbert space.

Understanding it through the spin of a single electron

Measuring an electron's spin along a specific axis yields one of two outcomes. Let's call this axis the Z-axis, and write the state corresponding to spin-up as |0⟩ and spin-down as |1⟩. The | ⟩ notation denotes a state vector.

Written as column vectors:

Here |0⟩ and |1⟩ are called basis states. They look identical to the 2D coordinate vectors earlier, but now they serve as the reference for expressing two spin states.

Must the electron be in only one of these two states?

No. Quantum mechanics allows a superposition of the two states:

Here α and β are complex coefficients indicating how much of each basis vector is combined.

A normalized state must satisfy the following condition, which sets the length of the state vector so that the probabilities of the two outcomes sum to 1:

For example, consider the following state:

Measuring this state along the Z-axis yields |0⟩ with 75% probability and |1⟩ with 25% probability, obtained by squaring the absolute value of each coefficient:

This is the Born rule. The key point is that α and β are not probabilities themselves, but probability amplitudes; probabilities are obtained by squaring the magnitude of each coefficient.

Let's verify this difference visually below, focusing on cases where both coefficients are non-negative real numbers.

Select "75:25" and "50:50" in turn. The reference axes stay fixed while the state vector changes. At 50:50, each coefficient is about 0.707, while its squared probability is 0.5. Compare the coefficients read via the dashed lines against their squared probabilities.

In a general quantum state, complex phases also enter the two coefficients; the diagram above shows only the subset of states using real coefficients. Representing that additional phase information on a sphere leads to the Bloch sphere, which we will examine later.

Changing the basis expresses the same state differently

Above, we used |0⟩ and |1⟩ as our basis. However, we can also choose a different basis:

These two vectors also form an orthonormal basis—mutually orthogonal and of length 1.

What happens if we express |0⟩ in this new basis?

What was originally a single basis state |0⟩ is now expressed as a superposition of two states in the new basis. Adding |+⟩ and |−⟩ above confirms that the |1⟩ components cancel out, leaving only |0⟩.

The actual quantum state has not changed; only the coordinate system used to express it has changed.

If we also change the actual measurement direction, the observation probabilities change accordingly. Measuring the same |0⟩ state along the Z-axis yields |0⟩ 100% of the time, whereas measuring it along the X-axis yields |+⟩ and |−⟩ at 50% each. We must distinguish between merely writing the same state in different coordinates and actually performing a measurement that distinguishes a different basis.

Below, the prepared state is fixed at |0⟩ while we switch between the Z-axis and X-axis measurement bases. Notice that the black state vector stays put; what changes are the gray basis vectors, the coefficients in that basis, and the probabilities when that measurement is performed.

This is why measurement directions matter in quantum mechanics. In the CHSH inequality earlier, Alice and Bob chose different measurement directions. Mathematically, that choice corresponds to selecting measurements that distinguish different basis states. Even when the same entangled state is prepared, the correlations between their outcomes change depending on which directions the two choose.

Treating two particles together expands the Hilbert space

The spin state of a single electron could be represented in a 2D complex Hilbert space. How do we represent the spins of two electrons together?

We use the tensor product of the two spaces. Combining the two basis states of the first particle with the two basis states of the second gives four basis vectors for the joint system:

Since each particle is 2-dimensional, the combined space is 2×2 = 4-dimensional. For example, |01⟩ represents the state where the first particle is |0⟩ and the second is |1⟩. Any general two-particle state can be expressed as a combination of these four basis vectors.

A special state among them is the Bell state:

This state cannot be written simply as a product of one particle's state and the other's. Multiplying two individual states that each contain 0 and 1 components produces |01⟩ and |10⟩ terms alongside |00⟩ and |11⟩; it is impossible to keep only the first two terms as in the formula above. Such a pure state is called a quantum entangled state.

Note that getting 00 and 11 half the time along the Z-axis alone does not confirm entanglement; to distinguish it from a classical mixture where 00 or 11 is prepared probabilistically, one must also check the statistics in other measurement bases.

An important point follows: the dimension of the Hilbert space for multiple particles grows rapidly with the number of particles. For n qubits, each having two basis states like an electron spin, combining their basis states yields a dimension of:

Here "dimension" means the number of independent complex basis vectors, not an increase in the dimensions of physical space.

This is also why Hilbert space frequently appears when explaining quantum computers. However, a rapidly growing state space dimension does not mean we can read out all of that information in a single measurement; one measurement yields only a single outcome corresponding to the chosen basis.

How does this connect to de Broglie–Bohm theory?

Let's connect this back to the de Broglie–Bohm guiding equation examined earlier.

Above, we dealt with an electron's spin state. Now let's think about a particle's spatial state.

|ψ⟩ is an abstract state vector in Hilbert space. What happens when we express a particle's spatial state in the position basis?

It becomes the familiar wave function. On the right, ⟨x|ψ⟩ denotes the coefficient of the state |ψ⟩ expressed at position x.

Just as a spin state was expressed by two coefficients for |0⟩ and |1⟩, a spatial state is expressed by coefficients corresponding to each position. Because position varies continuously, instead of two discrete coefficients we get a function of position.

Strictly speaking, |x⟩ pointing to a single exact position is not an ordinary normalizable vector, but a generalized basis used for continuous positions.

For a single spinless particle moving in 3D space, the wave function is written ψ(x, t), where x bundles the three position coordinates. Since the probabilities across all possible positions must sum to 1, the normalization condition is:

Instead of summing the squared magnitudes of two coefficients, here we integrate over all continuous positions. The representative Hilbert space to which such wave functions belong is L²(ℝ³), the space of square-integrable complex functions.

The ψ appearing in the de Broglie–Bohm guiding equation is precisely this wave function in the position representation: we express the abstract Hilbert space state vector in the position basis, and then compute the particle's velocity from the phase gradient of that representation.

Putting the concepts together

A Hilbert space is an abstract mathematical space representing quantum states—not the physical space in which particles move around.

Basis vectors serve as the coordinate system for expressing those states. Depending on the basis, the exact same state may be written as a single basis state or as a superposition of multiple basis states.

A wave function is one way of expressing that state vector: expressing it in a spin basis gives spin measurement probabilities, while expressing it in a position basis gives position measurement probabilities. Distinguishing the physical state from the coordinates used to represent it is key.

How does this connect to the automata in the main text?

Now suppose we attach one basis vector to each distinguishable state of an automaton—mapping state A to |A⟩ and state B to |B⟩. The fact that A and B are distinct states is represented by the two basis vectors being orthogonal.

If a transition swaps A to B and B to A, we can write the rule as below, where U is the operator applying this state transition to vectors:

This operation is a permutation that swaps two basis vectors. Applying the same rule linearly to combined vectors preserves both lengths and inner products. A reversible operation with this property is a unitary operation—whereas a transition that merges multiple states into one irreversibly is not such a permutation.

Deterministic state transitions can thus be expressed in the mathematical language used by quantum mechanics. Whether this representation is possible, however, and whether actual nature moves according to the rules of such an automaton, must be examined separately.

To summarize the core ideas first:

• Attractor: a point, orbit, or set toward which many initial states converge as time passes.
• Strange attractor: an attractor that often has a fractal structure. In the classic Lorenz attractor, motion within it is also complex and aperiodic.

What is intriguing is that while the region where motion takes place is bounded, predicting the exact trajectory it will follow inside that region over the long term can be difficult. How can both be true at once? Let's start with a simple example.

What is an attractor?

Imagine a ball rolling inside a bowl. Wherever you release the ball, as long as there is friction, it eventually settles at the bottom of the bowl and stops. Even when the starting position and velocity differ, the final state reached is the same.

This state of resting at the bottom can be called an attractor. It does not mean only the spatial location of the bottom; it refers to the complete state where the ball is at the bottom and its velocity is 0.

An attractor does not have to be a single resting state, however. An entire orbit of repeating motion can also be an attractor. To understand this difference, we first need to look at how motion is plotted.

Plotting motion in state space

We usually think of an object's motion in physical space. In dynamics, however, the state of a system is represented as a single point.

For example, knowing a pendulum's position is not enough to describe its state. At the same position it may be moving left or right, and those motions lead to different next states. We therefore need its velocity as well as its position.

Here θ is the pendulum's angle and ω is its angular velocity. The space whose coordinates are these two values is phase space. As time passes, the point representing the state moves, and the line it traces is called a trajectory.

If we plot the ball settling at the bottom in this space, trajectories from different starting points approach one point. A system that sustains a steady oscillation, on the other hand, follows the same closed orbit instead of coming to rest at a point.

This should also be distinguished from the Hilbert space discussed earlier. There we represented quantum states as vectors; here our coordinates are classical state variables such as position and velocity.

Approaching one point and repeating an orbit are different

A pendulum subject only to friction eventually stops. A system supplied with energy to sustain its oscillation, however, can approach the same periodic orbit from different nearby states. Such a stable periodic orbit is called a limit cycle.

Mathematically, we consider an invariant set that trajectories from a region of initial states approach over time. Invariant means that a state starting inside the set stays inside it. The region of initial states that approach an attractor is its basin of attraction.

For a fixed point, it is relatively easy to describe where motion will stop; for a periodic orbit, which motion will repeat. The next kind of attractor presents a different situation.

A classic strange attractor: the Lorenz attractor

The butterfly-wing-shaped structure below is the Lorenz attractor. Edward Lorenz discovered it in 1963 while studying a simplified mathematical model of convection.

Each line traces the state as time passes. It circles one wing several times, moves to the other, and then returns. The motion remains within a bounded region, but a typical trajectory does not repeat periodically.

The rules producing this complex motion are three differential equations.

Notation such as dx/dt on the left describes how each state variable changes with time. The parameters σ, ρ, and β are held fixed at the following commonly used values in the figure above.

There are no random numbers here and no probabilistic choice of the next moment. An exactly specified current state determines its next change. Yet the long-term motion becomes highly irregular. What, then, makes prediction difficult?

Why is prediction hard when motion stays near the same attractor?

Consider two systems starting from almost identical initial states. Their trajectories initially nearly coincide, but a large difference can develop over time. This is sensitivity to initial conditions.

Below, two trajectories are calculated using the same Lorenz equations. Their initial x values differ by just 0.000001, while y and z are identical. The parameters and external conditions remain unchanged throughout the calculation.

Move the time from 0 to 10 to 20. The solid and dashed lines show the recent portions of the two trajectories; the filled and hollow dots show their states at the same instant. They initially appear to overlap, then occupy different positions. Compare remaining in a similar overall region with being in the same state at the same time.

This difference does not grow at an identical rate at every moment. While the separation is sufficiently small, however, the following exponential relation describes the tendency.

Here δ₀ is the initial separation and δ(t) the separation after time t. The Lyapunov exponent λ describes how quickly nearby states separate or approach one another. A positive largest exponent means there is a direction in which small differences between nearby trajectories are amplified on average.

This is the mathematical core of the butterfly effect. The overall motion stays in a bounded region, so the separation does not grow without limit. The figure is a numerical calculation over a finite time, not a mathematical proof of chaos.

How can separating trajectories stay near the same attractor?

An important question follows. If nearby trajectories separate, how can the motion as a whole still approach the same attractor?

Contraction and expansion occur simultaneously in different directions. Nearby states separate in some directions while their distances shrink in others. The stretched trajectories also return and fold, allowing the overall motion to remain bounded.

Approaching an attracting set is therefore different from two states within that set approaching each other. Both happen together for a fixed-point attractor. Near the Lorenz attractor, trajectories can approach the set while nearby trajectories within it separate.

Where does fractal structure enter the picture?

What remains when stretching, compression, and folding repeat? Instead of a smooth line or a single smooth surface, a structure can form in which additional layers and gaps appear at smaller scales.

A fractal is a geometric object that reveals fine structure at progressively smaller scales. Some fractals repeat exactly the same pattern under rescaling, but not every fractal has such perfect self-similarity.

The Lorenz attractor lies within three-dimensional phase space without filling it. Its layered structure has fractal properties. Simply magnifying a few turns of the plotted line does not reveal that full structure. The Cantor set in the next note provides a simpler example of structure that persists at smaller scales.

There are several precise definitions of a strange attractor. Here, think of a representative case such as the Lorenz attractor, where fractal structure and chaotic motion occur together.

How, then, do determinism and prediction differ?

Let's return to the original question. Is saying that every event is determined by a previous state the same as saying an observer can predict the entire future exactly?

In the Lorenz model, the rules of motion are deterministic, yet a small error in the initial state can become a large difference later. Knowing the initial state more precisely can extend the useful prediction time, but finite precision does not guarantee continuously accurate predictions of individual states far into the future.

This does not mean nothing can be predicted. Even if it is difficult to say exactly when a trajectory will occupy the left wing, we can study the region in which the motion remains and its long-term statistical properties. Predictions about an individual future state and the structure of the overall motion are different things.

Chaos theory has not proved that the entire universe is deterministic. It shows that deterministic mathematical rules alone can generate complex motion that is difficult to predict over long periods. The universe's invariant set discussed in the main text is likewise a separate physical hypothesis, not a conclusion that follows directly from this example.

Earlier we noted that a strange attractor can have a fractal structure. What does "revealing finer structure even when magnified" look like concretely? Instead of a moving trajectory, let's start with a single line segment.

Does "fractal" simply mean a complicated shape?

Looking complicated alone does not make something a fractal. What matters is that structure continues to appear at smaller scales, and how that structure relates to the magnification factor.

When magnifying a small part reproduces the same structure as the whole, that property is called self-similarity. The Cantor set illustrates this property cleanly. Let's look at the rule for building it first, and then see what repeats.

Removing only the middle third of a line segment

Start with a line segment of length 1—all points from 0 to 1. Divide it into three equal parts and remove the interior of the middle third. Two segments remain: from 0 to 1/3 and from 2/3 to 1. The boundary endpoints 1/3 and 2/3 are left in place.

Now apply the same rule to each of the two remaining segments. Removing the middle third of each leaves four segments of length 1/9. Next, remove the middle third of each of those four segments. The rule stays the same from start to finish.

Click "Next step" twice. Each time a segment splits into two, the number of segments doubles while the total remaining length shrinks. Compare the black segments against the numbers below them. Every row shows the same 0-to-1 range.

When this process is repeated endlessly, the collection of points never removed at any step is the Cantor set. We must distinguish between the segments remaining at a specific finite step and the final set defined through an infinite process.

If the total length is 0, does nothing remain?

The total length shrinks in the sequence 1, 2/3, 4/9, 8/27. Since 2/3 of the previous length remains each time, the length after n steps is:

As the repetition continues, this value approaches 0 without bound, and the total length of the final Cantor set is 0. Does that mean every point has vanished?

No. The endpoints 0 and 1 kept from the very beginning are never removed at any step, nor are boundary points like 1/3 and 2/3. Having length 0 and being an empty set are two different things.

Nor is it only endpoints that remain. At each step, imagine choosing either the left sub-segment or the right sub-segment. Continuing those choices narrows the interval further and further, so an endless sequence of left/right choices pins down a single point in the end.

For instance, alternating left, right, left, right endlessly narrows down to the point 1/4. Summing the shifts added each time we pick the right sub-segment gives the series below. This point is never an endpoint at any step, yet it never falls into a removed middle third either, so it survives to the end:

There are so many such infinite sequences of choices that they cannot even be listed one by one with natural numbers. In that sense, the points of the Cantor set are uncountably infinite. How many points exist and how much length they occupy are separate questions.

What do we see when we magnify a small piece?

Look again at the remaining shape. Inside the first left third (from 0 to 1/3), the exact same process of removing middle thirds was repeated, and likewise inside the right third. Magnifying the left third of the final set by a factor of 3 therefore reproduces the exact structure of the whole.

Fixing the step-3 approximation below, select "Magnify left 1/3." The boxed region on top is stretched to the full width below. Notice how the small piece contains the same pattern of an empty middle splitting into two sides.

In a finite-step drawing, magnifying reduces the number of remaining subdivision steps by one—which is why the diagram compares step 3 with step 2. In the final Cantor set repeated infinitely, however, there is no last step, so magnifying a part yields the exact same structure as the whole. This is the self-similarity of the Cantor set.

Not every fractal repeats the exact same shape like the Cantor set, and the Lorenz attractor earlier does not look like this line segment either. This simple example illustrates the shared idea that structure persists down to arbitrarily small scales.

Why do we need this example in the main text?

Infinitely many points remain in the Cantor set, yet no continuous line segment connecting them survives. No matter how closely you zoom in around a remaining point, you encounter both points belonging to the set and removed gaps right next to them.

For example, 1/3 belongs to the set, but immediately to its right lies the gap removed in the very first step. Moving ever so slightly from a point on the set does not guarantee that the new point still belongs to the set.

What the main text draws upon is this kind of geometric constraint: if the set on which valid states lie does not fill the entire space, altering a variable slightly can land outside the set. Whether the Cantor set exhibits such a structure, of course, and whether the actual states of our universe are constrained in the same way, are separate questions.

Hilbert space and basis vectors from earlier connect directly here, and this concept is also needed to understand Tim Palmer's Invariant Set Theory in the main text. Let's first see what a point on the sphere represents.

Representing quantum states on a sphere?

In the figure above, an arrow extends from the center of the sphere in a specific direction. The direction it points to represents the state of a qubit—a quantum system described by two basis states, |0⟩ and |1⟩.

Let's look at a few representative states first. All probabilities below are for measurements along the Z-axis:

Something looks odd here: |+⟩ and |−⟩ sit on opposite sides of the equator, yet both have a 50:50 probability along the Z-axis. How can they be different states if they give the exact same probabilities? This is the key to understanding the Bloch sphere.

Quantum states have phase in addition to probability

As we saw earlier, the state of a qubit can be written as:

Here α and β are complex coefficients satisfying the normalization condition:

Squaring the magnitudes of these coefficients gives the Z-axis measurement probabilities. Complex numbers, however, have phase as well as magnitude. The phase difference between the two coefficients is called the relative phase, which can differ even when the magnitudes of the coefficients are equal.

A global phase multiplied equally onto both coefficients cannot be distinguished by measurement outcomes. Removing that redundancy allows us to rewrite the state as:

Though it looks involved, only two angles fix the state:

• θ (theta): how far down from the north pole the state sits, linked directly to the Z-axis measurement probabilities.
• φ (phi): how far around the Z-axis the state has rotated, representing the relative phase between the two basis states.

Any pure state of a single qubit can be represented by these two angles. States with the same θ sit at the same height on the sphere and share the same Z-axis probabilities; varying φ rotates the state around the sphere at that fixed height.

The factor e^(iφ) has magnitude 1 and changes only the phase: it equals +1 when φ = 0° and −1 when φ = 180°. Thus we can flip the relative sign between the two terms while keeping their coefficient magnitudes unchanged—which is what distinguishes the two equatorial states above.

Try changing the angles yourself

Below, adjusting θ and φ changes the prepared state, while the Z-axis and X-axis measurements used for comparison stay fixed. We write the two X-axis outcomes as + and −.

First, select "North Pole |0⟩", "Equator |+⟩", and "South Pole |1⟩" in turn to check the Z-axis probabilities. Next, toggle between "Equator |+⟩" and "Equator |−⟩": notice how the Z-axis stays at 50:50 while the X-axis flips completely.

The dark portion of each bar shows the probability of the first outcome (0 or +), and the gray portion shows the second (1 or −). Each probability represents the proportion across many repeated preparations and measurements of the same state, not a simultaneous measurement of a single particle along two axes.

Fixing θ at 90° and switching φ from 0° to 180° gives the following two states:

In both formulas, the coefficients in front of |0⟩ and |1⟩ have the same magnitude, so measuring along the Z-axis gives 50:50 for both. Yet these two states are precisely the basis states for an X-axis measurement—so measuring along the X-axis yields + with certainty for the first and − with certainty for the second.

In short, probabilities in a single measurement basis cannot distinguish all quantum states. Changing the relative phase can alter measurement probabilities in other bases, and because amplitudes of equal magnitude can add constructively or cancel out depending on their phase, this difference is central to understanding quantum interference as well.

Why a sphere?

This connects back to the Hilbert space discussed earlier: a single qubit is represented by a state vector in a 2D complex Hilbert space.

Two complex numbers contain four real components in total. Normalization removes one degree of freedom, and setting aside the unobservable global phase leaves just two independent real parameters—which, when expressed as θ and φ, map to a point on the surface of a sphere.

The height z on the sphere is cosθ, corresponding to "(probability of 0) − (probability of 1)" along the Z-axis. Since the two probabilities sum to 1, the probability of 0 is (1 + cosθ)/2, which simplifies via the half-angle identity to cos²(θ/2) seen earlier. This is why θ/2 appears in the state formula; the angle measured in the coefficient plane in the Hilbert space note and θ on the Bloch sphere are not the same angle.

This sphere is a mathematical map of quantum states rather than a sphere in physical space, and there is one more distinction to keep in mind:

• Surface of the sphere: pure states specified by a single state vector.
• Interior of the sphere: mixed states formed by probabilistically mixing different pure states.
• Exact center: the maximally mixed state I/2, an equal mixture of two orthogonal basis states.

For instance, preparing |0⟩ or |1⟩ with 50% probability each sits at the center of the sphere. That differs from the pure equatorial state |+⟩: along the Z-axis both give 50:50, but along the X-axis the mixed state at the center still gives 50:50 whereas |+⟩ yields + 100% of the time. A quantum superposition and a classical probabilistic mixture must therefore be distinguished.

The sphere including its interior is sometimes called the Bloch ball. The interactive controls above move across pure states on the surface; a surface point appearing inside the circular boundary on a 2D projection should not be confused with an interior mixed state.

Why does Palmer's Invariant Set Theory use this sphere?

Now let's look at the part most directly connected to the main text. In standard quantum mechanics, every point on the surface of the Bloch sphere is a mathematically permitted pure qubit state. Palmer, however, hypothesizes an additional geometric constraint on a more fundamental state space.

Palmer's model connects state representations on the Bloch sphere to finite-length bit strings, discretized angles, and a fractal invariant set. The key idea here is that even if a state can be written down mathematically in standard quantum mechanics, it might not be realizable within the fundamental state space he assumes.

This forces us to re-examine counterfactual questions like "What if we had kept the particle's hidden state the same and only changed the measurement setting?"—because we must first ask whether the resulting combination still lies within the model's invariant set.

This does not mean the Bloch sphere itself is a fractal; the sphere is the standard representation of quantum states, while Palmer's discretization and invariant set structure are a separate hypothesis proposed on top of it. Whether actual nature obeys such a constraint has not been experimentally established.

Connecting the concepts so far: Hilbert space is the mathematical space representing quantum states, basis vectors are the reference frame used to express them, and the Bloch sphere is a tool that maps single-qubit states onto a 3D figure. Palmer's research asks what gap might exist between our mathematical representation of states and the fundamental states the actual world can take.