W-Theory: A Local Realist Interpretation of Quantum Mechanics

26 min readMay 26, 2025

Yes, you read the title right. This w-theory is a local realist interpretation. Unlike something like pilot wave theory, it does not conflict with special relativity as it is local and doesn’t even introduce anything fundamentally new, only reinterprets the pre-existing mathematics. As a local realist interpretation, it allows you to assign values to all observables of all particles throughout the system, and gives clear solutions to all of the paradoxes in quantum mechanics.

Note that I am not even positing here that this is the correct way to think about quantum mechanics. I just think this is an interesting exploration to think about. It is not original to me but largely comes from a paper which you can read here. The authors did not even give it a name. I feel like I need to call it something, so for the sake of the article I am referring to it as w-theory, even though it’s more of an interpretation than a theory as it doesn’t actually add or remove any postulates from quantum mechanics.

Some parts of it are also m

The Uncertainty Principle

The uncertainty principle disallows you from making absolute predictions into the future. Let’s focus on qubits and spin-½ for simplicity. They have three observables you can measure called X, Y, and Z, and their values are either measured to be +1 or -1.

For an electron, its spin value follows such rules. You can measure it with a device called the Stern-Gerlach apparatus. If you measure one of the observables, such as its Z value, you can physically rotate the measuring device 90 degrees to measure it on another axis, such as its X value.

When you do this, you find that if it has a previously known value on a particular axis, then measuring it on a different axis gives you a random value. And, in fact, how random the value is depends upon the angle of rotation. Below is a simple demonstration of this where the not-dotted line represents a starting known value of +1 and the dotted line represents a starting known value of -1.

As it is rotated, the expectation value of what will be measured in the subsequent measurement changes. This expectation value is a weighted average where the +1 and -1 are multiplied by their likelihoods of occurring and summed. When the expectation value is 0, they have equal likelihoods, 50%/50%, of occurring. When it is +1, there is a 100% chance of measuring +1. So on and so forth.

This fact seems to prevent us from knowing the values of all the observables simultaneously, because measuring one of them, such as Z, and then rotating the measuring device 90 degrees to measure X, seems to give us a random value, and then if we rotate it back to measure Z, we get another random value. What gives?

Bell’s Theorem

One intuitive explanation for quantum randomness is that measurement disturbs the system. For instance, measuring property X scrambles Z, and vice versa, so repeated measurements yield unpredictable results. However, this view is usually refuted with Bell’s theorem and the Greenberger–Horne–Zeilinger (GHZ) experiment.

In the GHZ setup, three entangled particles are measured. Each can be tested for one of two properties: X or Y, with possible results of +1 or -1. There are four experimental runs, each measuring different combinations:

X₁, Y₂, Y₃ → X₁Y₂Y₃ = -1
Y₁, X₂, Y₃ → Y₁X₂Y₃ = -1
Y₁, Y₂, X₃ → Y₁Y₂X₃ = -1
X₁, X₂, X₃ → X₁X₂X₃ = +1

If all these properties existed simultaneously, multiplying the first three equations would give: (X₁Y₂Y₃)(Y₁X₂Y₃)(Y₁Y₂X₃) = -1, and cancelling repeated terms gives us X₁X₂X₃ = -1.

Here’s the kicker: if you conduct the fourth experiment in the real world, it gives you X₁X₂X₃ = +1, a contradiction to our assumption that all the values pre-exist simultaneously.

You might argue that the values do not need to exist simultaneously because different sets of measurements disturb the system in different ways, and thus each run of the experiment is genuinely a different experiment.

If this is true, then each particle’s outcome would need to depend on which measurements are performed on other particles. This is because the local measurement on a single particle does not provide sufficient information to distinguish between different combinations of measurements. If the second particle‘s X observable is measured, that is not sufficient information to distinguish it between the Y₁X₂Y₃ case and the X₁X₂X₃ case.

Since there are three separate particles, you could imagine spatially distributing them and performing the measurements with three different measuring devices. To explain the results by measurement disturbance, changes in the settings of your measuring device would need to disturb all of them faster-than-light, violating special relativity.

Hence, many physicists have just come to argue that hidden variable theories are impossible. Either they would not violate the speed of light limit, but could not reproduce the predictions of quantum mechanics (a local hidden variable theory), or they could reproduce the predictions of quantum mechanics but would also violate the speed of light limit (a nonlocal hidden variable theory). Neither solution seems to be desirable, and so many physicists instead come to believe there just are no hidden variables.

The position and momentum of a particle do not commute, meaning, they are subject to the uncertainty principle and thus only one of them can be known at a time. Many interpret this to mean the particle only genuinely possesses one of these properties at a time.

The Arrow of Time

If you watch a video of particles bouncing in a box and then are asked whether or not the video is playing in the reverse or not, you cannot tell, because the fundamental laws that govern their motion work the same in both directions of time. This makes it appear as if the arrow of time is not actually fundamental.

Many people will bring up entropy as a solution to this. Imagine particles all start in one corner. As time moves forward they will spread out evenly and almost never return to that corner. That behavior tells you the forward direction in time.

Here is the catch. The laws of physics allow a return to the corner just as much as a spread out arrangement. Every individual configuration, or microstate, has the same probability. The sense of improbability only appears when you group microstates into macrostates. For example, put every corner configuration into macrostate X and every uniform configuration into macrostate Y. Macrostate Y contains far more microstates than X. Entropy really is just a measure how many microstates belong to the macrostate that a system is within. When the system shifts from X to Y we say entropy has increased.

But those groupings are arbitrary. Suppose you purchase a new deck of cards and you find it to be in sequential order, and you call that ordering macrostate X and all other orderings macrostate Y. If you shuffle the cards, it then raises entropy as it is far more likely that the shuffle will place the cards into macrostate Y. Yet, each deck order is equally likely. On another planet, the factory might ship decks in a configuration that you would consider shuffled but they would not, and those orders would be low entropy. Whether entropy rises or falls would depend entirely on how you defined your macrostates, making it seem subjective.

The way out of this is the past hypothesis. We tie our macrostates to an objective feature of the universe’s beginning. Near the Big Bang, matter was extremely compacted into a small space. As space expands there are vastly more ways for particles to be spread out than to remain compact. By defining low entropy macrostates as highly compacted arrangements and high entropy macrostates as widely spread out arrangements, we obtain a universal arrow of time rooted in the universe’s compacted origin.

A Hidden Assumption

Bell’s theorem and the PBR theorem both implicitly assume an arrow of time in their construction of causality. Causes always precede effects. Any local hidden variable theory, then, only needs to condition on the causes, and that would provide sufficient constraints to then predict the future effects with absolute certainty.

Microscopic physics remains fundamentally indifferent to the arrow of time unless we introduce the past hypothesis, yet the past hypothesis appeals to the Big Bang, which is a feature of macroscopic cosmology and general relativity. These are separate theories entirely, and you cannot, as far as we know, derive anything about general relativity from the postulates of quantum mechanics. If we stick to just quantum mechanics, there is no arrow of time. That is a macroscopic feature of reality, not a microscopic feature.

Both Bell’s theorem and the PBR theorem make the assumption that causality absolutely has a temporal direction. Even though the theory itself does not give you a definite arrow of time, whatever reference point you do choose at t=0, all causes must precede effects once you evolve the system forwards in time from its initial conditions, and this should be sufficient in any hidden variable theory to predict all future states of the system.

One of the assumptions going into the derivation of Bell’s theorem is ‘no future input-dependence,’ i.e., ‘no model parameter associated with time t can be dependent upon model inputs associated with times greater than t.’
— Emily Adlam, “Two Roads to Retrocausality”

This is not to say that to escape Bell’s theorem or the PBR theorem we need to replace regular old causality with retrocausality. To replace causality only going forwards in time with causality going backwards in time is to just replace one arrow-dependent notion of causality with another. What we can instead consider is a notion of causality that is perfectly time-symmetrical, meaning it can neither be said to be regular old causality or retrocausality.

Global Determinism and “All-at-once” Causality

In regular old causality, all causes lie in the past. You begin a past state, such as t=-1, and then evolve it to the present state, t=0. We can express this notion of causality in the form [t=-1]→[t=0], where the past conditions are always sufficient conditions — sufficient causes — to predict the present conditions with absolute certainty.

Retrocausality would posit the opposite, that you have to begin with the final state and evolve it backwards to the present state you are interested in, and this is a sufficient condition to predict the state of the system at the present moment in time. We could express this notion of causality instead using the form [t=1]→[t=0], where the future state can be seen as sufficient cause to determine the present state of a system with certainty.

Both of these suggest an arrow-dependent notion of causation. To get to an arrow-independent notion of causation, we can imagine that neither conditioning on the future state nor the past state is a sufficient constraint to the determine the present properties of the system.

Instead, you would need to condition on both the initial state at time t=-1 and the final state at time t=1 and evolve them towards each other, computing the evolution of the system from both ends to where they meet at some intermediate point: [t=-1]→[t=0]←[t=1]. This intermediate point would have its properties absolutely determined by both its past and future state simultaneously, providing sufficient conditions to predict it with certainty.

You should not think of this as if particles are actually sending messages back and forth in time. It is more that if you trace how a system evolves forwards in time, in traditional causality, this provides sufficient information to predict its value at any future point. In an arrow-independent notion of causality, you are tracing the evolution of the system from both ends simultaneously until they meet at a central point, and those together provide sufficient constraints to predict the value of the system there with absolute certainty.

This is sometimes referred as global determinism, or sometimes an “all-at-once” notion of causality. If Laplace’s demon knew the initial and final state of the universe, then the evolution of the universe would become absolutely deterministic, because any time particles interact, he could evolve (mathematically) the initial state of that particle at the beginning of the universe forwards in time and the final state of that particle at the end of the universe backwards in time, then find where the two meet, and that would tell him with certainty what the outcome of the interaction should be.

This, of course, only makes sense if you presume that the future state of the universe really does, in some way, pre-exist. It demands a sort of Block universe picture of the universe where all spacetime events are already sitting there in the block. If this is the case, then such a notion of causality would place constraints on how those events could possibly be organized, because the moment you fill out the initial and final states, the rest of the intermediate states would be absolutely determined and you would have no leeway in how you fill them out.

To give an intuitive picture of what the notion of global determinism entails, I suggest the following analogy. In a globally deterministic world, the laws of nature prescribe a unique course of history in the same way as a properly set up game of sudoku has a unique solution; the laws of nature govern the whole course of history at once, rather than moment-by-moment, in the same way as the rules of the game of sudoku apply to the whole grid at once, rather than dictating the entries column by column.
— Emily Adlam, “Quantum Mechanics and Global Determinism”

A globally deterministic model would not describe the universe such that everything can be predicted just by picking an arbitrary starting point and evolving it towards some direction in time. Rather, the universe’s evolution would have to follow a set of rules and constraints that are globally defined and time-independent. A system may evolve into a particular state that could not be explained from a cause from its past, or even from some nonlocal “spooky action at a distance,” but because it must evolve into that particular state to satisfy certain constraints in order to be consistent with its future and past evolution taken together.

The W-Equation

If the universe were indeed globally deterministic, then it would appear random to us when only considering a forwards evolution through time from the initial state. This would be because the properties of systems, their hidden variables, are determined both by the simultaneous local evolution of the system from an initial state forwards in time as well as its local evolution backwards through time from a future state. If we only condition on one, it would not give of sufficient information to actually predict the future state of the system.

Indeed, if a completed sudoku square were to be revealed column by column to an observer who could not see the whole square at once, each column would appear to be related to the previous one in a probabilistic way (there would be obvious patterns of dependence which could be described by probabilistic rules, but there would not usually be enough information available to determine the next column exactly), and I suggest that similarly, the apparently probabilistic nature of quantum theory is a consequence of our inability to see the whole picture.
— Emily Adlam, “Quantum Mechanics and Global Determinism”

Indeed, this seems to be how quantum mechanics works. If we take the system and evolve it forwards in time from an initial starting point, we run into ambiguities regarding the outcomes of certain interactions due to the uncertainty principle. We thus end up having to consider all possible outcomes using the limited information we have, giving us a probabilistic description rather than a complete description.

If quantum mechanics is globally deterministic, then we should be able to build a complete picture of the system retrospectively. After the experiment is already completed, we should be able to take the final state of the system and evolve it backwards in time to where it meets an intermediate point, and take a past state of the system and evolve it forwards in time to where it meets an intermediate point, and both constraints taken together should reveal the values of the observables at the intermediate points.

Are there any pre-existing equations in quantum mechanics that condition on both the future and past state simultaneously? There is, it’s known as the weak value equation, related to the two-state vector formalism.

https://en.wikipedia.org/wiki/Weak_value

Weak values were constructed as a way to try to learn something about the intermediate states of the system. You see, if the value at an intermediate point is random, then fluctuations in that value should have rippling impacts throughout the system until also altering the final state. If you fix the final state as well, that is to say, you decide to only consider cases where both the initial and final state are very specific values (meaning, you throw out any data where the outcome isn’t the fixed one you decided upon), then that should put more constraints on the system such that the intermediate states would not fluctuate as much.

Interestingly, when you do this, what comes out of it is a single, deterministic value for each observable. The uncertainty principle tells us that we can only know a single one of the X, Y, and Z at a time, but with this equation you could compute all three simultaneously through retrospection.

Now, this could just be a mathematical trick and the values we are computing have no physical meaning. Yet, what the authors of the paper point out is that these values do indeed seem to be rather physically meaningful. They do actually seem to reflect something about the intermediate states of the system, and they evolve locally through time.

The Einstein-Podolsky-Rosen Paradox Resolution

If these values are indeed physically meaningful, then they should resolve the EPR paradox. We should be able to prove that the particles become correlated when they locally interact and, in that moment, “choose” the particular outcome that will be later revealed by measurement. This value would be decided upon locally and long before you make the measurement, thus eliminating the “spooky action at a distance” associated with entanglement.

Here is a simple algorithm for a quantum computer that reconstructs an EPR-style state where the qubits are guaranteed to have opposite values.

Supposedly, the qubits don’t decide what their values are up until we measure it, so we could in principle separate the qubits with vast distance and measuring one suddenly determines the other nonlocally, as the old story goes.

If we compute the weak values, we have to condition on both the initial and final state. We can then compute for all X, Y, and Z observables for both qubits. Below shows us computing the values for the observables twice because there are two possible outputs to this program: |01⟩ and |10⟩. I wrote the observables at each step in the form (X,Y,Z) where the left-hand side is the most-significant qubit.

|00⟩ → |01⟩
t=0: (0,0,+1);(+1,+i,+1)
t=1: (0,0,+1);(+1,-i,+1)
t=2: (0,0,+1);(0,0,+1)
t=3: (0,0,+1);(0,0,-1)
|00⟩ → |10⟩
t=0: (0,0,+1);(-1,-i,+1)
t=1: (0,0,+1);(+1,+i,-1)
t=2: (0,0,-1);(0,0,-1)
t=3: (0,0,-1); (0,0,+1)

One thing you might notice is that the values are not simply +1 or -1 but can be any complex number where the values of the real or the imaginary component can range from ±tan(π/(2θ)). The angular nature of it is due to the fact that the values are relative to the measurement basis. What counts as the Z axis is ultimately arbitrary as you can rotate the measuring device and make the Z axis something else.

Another thing you will notice is that the qubits chose the correlated value at t=2, when we applied the CX operator to entangle them, even prior to when we applied the X operator at t=3. You can see in the first case at t=2 that they chose |00⟩ because +1 is the same as |0⟩ on the Z basis and both of their Z values are +1. In the second case they chose |11⟩ at t=2. Hence, when we applied the X operator to the least significant qubit, in the first case it changed to |01⟩, and in the second case when we apply the operator it changes to |10⟩.

Hence, the EPR paradox is not actually spooky action at a distance as they chose their values ahead of time.

The W-Vector

Notice that, in the EPR case, the initial conditions are different as one has a +i at the beginning and the other has a -i at the beginning, and these are the values of observables we did not measure as we can only know the initial state of one of the observables at a time. These differences produce differences in the final output.

If you are just using single qubits placed in a superposition of states, then you can always explain differences in the final output on differences in the initial conditions of the observables you did not measure.

Single qubits can be represented in Dirac notation as shown below.

|w⟩ = a|x⟩ + b|y⟩ + c|z⟩

The column vectors are defined as…

|x⟩ = [1; 0; 0]
|y⟩ = [0; 1; 0]
|z⟩ = [0; 0; 1]

Note that these don’t represent superpositions of states like you have with |ψ⟩. They represent the concrete values of the observables computed from the weak value equation. If, for example, c=+1, then that means if you had measured it you would measure |0⟩ and if c=-1 then if you measured it you will have measured |1⟩ as that factor represents the value of Z.

You can evolve the w-vector similarly to how you would evolve the state vector by just matrix-multiplying an operator by the vector. You can convert any unitary operator into an operator that acts directly on the w-vector with the equation below. The values j,k on the left-hand side are for the rows and columns for the operator, and when they used on the right-hand side they are indexing into a list of observables, in this case [X, Y, Z].

Let’s take a simple example of the program below. The qubit is placed into a superposition of states with equal probability of giving you a |0⟩ and a |1⟩.

Now, let’s compute the initial weak values prior to the qubit being placed into a superposition of states and not bother computing the weak values for it after. We will do it twice to condition on |0⟩ and |1⟩ to see what is going on in both cases.

|0⟩ → |0⟩
t=0: (+1,+i,+1)
|0⟩ → |1⟩
t=0: (-1,-i,+1)

Now, why didn’t we compute the weak values after the qubit is placed into a superposition of states? Why did we just compute the initial conditions? The reason is that we can convert the H operator (Hadamard gate) using that R(U) function into a gate that we can apply directly to the w-vector. Below I demonstrate this in Octave.

octave:7> RH
RH =

        0 +      0i        0 +      0i   1.0000 +      0i
        0 - 0.0000i  -1.0000 +      0i        0 +      0i
   1.0000 +      0i        0 - 0.0000i        0 +      0i

octave:8> w0
w0 =

   1 + 0i
   0 + 1i
   1 + 0i

octave:9> w1
w1 =

  -1 + 0i
   0 - 1i
   1 + 0i

octave:10> RH * w0
ans =

   1 + 0i
   0 - 1i
   1 + 0i

octave:11> RH * w1
ans =

   1 + 0i
   0 + 1i
  -1 + 0i

As you can see, in the first case we get an output of (+1,-i,+1) which means that if we measured it in this state we would measure |0⟩. In the second case, we get (+1,+i,-1) which means if we measured it in this state we would measure |1⟩. In both cases, they began such with the initial conditions of |0⟩, but because of differences on their other obervables we did not measure, when we applied the H operator (Hadamard gate) then it changes their Z values differently.

This all seems so simple, but sadly it all is not so simple. When it comes to single qubits, or spin-½ particles, all you need to know is the initial conditions and you can apply operators to the qubits to evolve it without issue. When it comes to multipartite interactions, such as with the CX operator or the SWAP gate, it becomes more complicated.

The W-Field

The w-vector, as we have shown before, is a three-vector called |w⟩. The w-field is instead an all-permeating vector field with a vector |W⟩ assigned to every point in space and time. Although, at most points it is a null vector where |W⟩=∅. The only places where it actually has elements are the spacetime coordinates where interactions take place.

If you have a system with two qubits, you can express their individual observables using things like X⊗I or I⊗Y where you combine the product with the identity matrix. In the first example, that would represent the X observable for the most significant qubit. For the second example, that would represent the Y observable for the least significant qubit. We can call these unipartite observables.

For qubits considered separately, combinations of observables like X⊗Y or Z⊗Z do not make much sense. We can call these multipartite observables. These only make sense at the locations where particles meet and interact. Hence, the values for multipartite observables are always to be found in |W⟩, fixed to the spacetime coordinates of the w-field where the two moving w-vectors intersect.

A block universe depiction where upwards movement represents movement through time and side-to-side and back-to-front movement represents movement through space. You can imagine particles (w-vectors) traveling along pre-existing deterministic trajectories. The point where the two intersect and thus where the two particles interact has specific values assigned there in the w-field for the multipartite observables.

Computing the values of |W⟩, the multipartite observables at that intersection, is done using the same w-equation we used for computing the values for the unipartite observables. This, again, requires knowing both an initial and future state of the system. You trace the evolution of the system from the initial state into the future, and the future state into the past, until they meet at those coordinates where they interact, and then you compute all possible permutations of the multipartite observables and assign that value to |W⟩ at that location.

When the two particles do meet up at that location, their unipartite values in their respective w-vectors can be combined with the multipartite values in the w-field at that instantaneous moment to produce a complete vector that has values for all permutations of the Pauli group. That would give you a vector of sixteen elements. Note that the value of the observable I⊗I is always equal to 1.

Once you have this big vector, you can then use the same exact R(U) function to convert a standard unitary operator, like the CX operator or SWAP gate, into an operator that can be applied to this big vector. You then apply it to the big vector and take note of how it updates the unipartite values, and throw all other information out.

You see, the multipartite values for the w-field only exist in that very precise moment in space and time where the particles interact. Since it has a set coordinate in time as well as space, the moment it comes into being, it ceases to exist, as time will have progressed beyond its spacetime coordinates. It’s not something that exists beyond the very brief moment that the interaction takes place, and therefore it doesn’t need to have its values updated.

All you care about when multiplying this big vector by an operator built with R(U) is how it alters the unipartite observables. Those are the updates it applies to respective w-vectors of the two particles as a result of the interaction. All other information is irrelevant and is never needed again after the interaction takes place.

It is not possible to simply start with the initial state of all the w-vectors and evolve them forwards in time. You need to also know the values distributed across the w-field. The randomness in quantum mechanics, if we were to take this viewpoint seriously, would arise from the fact that the values of the w-field are determined by both the initial and future state of the universe taken together.

You cannot predict them ahead of time because it requires knowing the final state of a system, but that’s exactly what we’re trying to predict. You cannot measure them ahead of time because they don’t even exist ahead of time but they only exist on-the-fly in the brief moment of the interaction, and aren’t even a tangible thing you could observe in the first place as they are simply vector values in a field.

Hence, if you are beginning with just the initial values, the best you can do is a probabilistic prediction into the future, hence why we have to describe things in phase space and superpositions of states. It’s not that particles literally exist in multiple places at once, but that we just cannot predict what their values would be ahead of time. If you do know the final state of the system, however, you could use this to retrospectively compute its local and deterministic evolution.

As long as you know the initial state of the w-vectors and all of the values of the w-field, you can compute its evolution forwards in time from the initial state, and it evolves locally and deterministically.

Local and Continuous Evolution

The evolution of the values of the observables are continuous. You can use the same Schrodinger equation to compute the time-evolution operator and then use the R(U) function to convert that into an operator to be applied to the w-vector for the qubits.

You might be worried that, since it still relies on multipartite operators when particles interact, that maybe something nonlocal could show up. However, this isn’t the case. The vector |W⟩ only has values defined for the multipartite operators relevant to the particles interacting at those spacetime coordinates. It never contains multipartite operators for any other particles.

The evolution of the values of the w-vectors, even during interactions with other particles where values in the w-field become relevant, is absolutely continuous. The values of the w-vectors never jump suddenly. In the paper, the authors point out that if you apply the SWAP gate continuously over and over again, the values of the observables do not suddenly swing back and forth discontinuously, but oscillate around each other like a sinusoidal wave. It impacts other observables as well, like the X observable. Below, you can see how the X observables are oscillating around a central point as we continuously apply the SWAP gate multiple times in succession.

This is a recreation of what the authors did in the original paper.

Hence, you could map the change of the values of the particles to differential equations like you could in classical mechanics. Whenever they interact, only the local values of the w-vectors and of the w-field at that particular point are relevant in the interaction. Only the values of the w-vectors actually change, and they change in a way that is completely continuous.

The authors point out that the SWAP gate’s simple evolution can be used as the basis for universal computation as any set of gates can be decomposed into SWAP gates and unipartite operators, and so if you created a simplified model that predicted the behavior of the weak values through the SWAP gates then you might be able to do away with having to actually evolve the state vectors from both ends altogether, and it could be replaced with a much simpler rule. Although, they do not actually develop such a model in their paper.

The Frauchiger-Renner Paradox Resolution

Perhaps the most bizarre paradox in quantum mechanics is the Frauchiger-Renner paradox.

In the experiment, Alice measures the Coin in a superposition of states and based on the measurement result prepares a Note which is also either in a superposition of states if she measures a |1⟩, or she just prepares the Note with |0⟩ if she measures a 0. She sends it to Bob who then reads the Note. At the end of the experiment. Charlie measures the whole lab of Alice and the Coin in the Bell basis, and Danny measures the whole lab of Bob and the Note in the Bell basis.

We can represent Alice, the Coin, the Note, and Bob as qubits in the order BNAC. Charlie would be measuring XXAC and Danny would be measuring BNXX.

If we compute the |ψ|² for Charlie, we get…

0000: 0.6667
1100: 0.1667
1101: 0.1667

If Charlie measures XX01, then the only possibility is 1101 for BN, which Charlie did not touch or alter in any way. Therefore, Charlie can conclude that Bob received the Note of |1⟩, which is only possible if Alice had seen the Coin in a state of |1⟩.

If we compute the |ψ|² for Danny, we get…

0000: 0.1667
0011: 0.6667
0100: 0.1667

If Danny measures 01XX, then the only possibility is 0100 for AC, which Danny did not touch or alter in any way. Therefore, Danny can conclude that Alice observed the Coin in the |0⟩ state and therefore must have prepared a |0⟩ to send to Bob, which is what Bob also would report to have observed.

It gets weird if the then compute the probability distribution for Charlie and Danny’s measurements taken together.

0000: 0.75
0001: 0.0833
0100: 0.0833
0101: 0.0833

Notice that there is a small probability of 0101. That means both Charlie and Danny would measure a 01, and thus Charlie would conclude Alice and Bob both saw |1⟩, whereas Danny would conclude Alice and Bob both saw |0⟩. A contradiction. A paradox.

The most naive solution actually deepens the paradox. Maria Violaris has argued that the solution is to simply recognize that it’s unreasonable for Charlie and Danny, who can derive conclusions about one part of the system from their measurement, to then use that conclusion in a chain of reasoning to derive conclusions about the other part, because they are on incompatible bases and thus subject to the uncertainty principle.

This, however, merely deepens the paradox. If it is subject to the uncertainty principle, then, even after they know the outcome for part of the system, they have to treat the other part as uncertain. If it is uncertain, it is not perfectly statistically correlated. If it is not perfectly statistically correlated, it means it’s possible that Alice would believe that she has sent a |0⟩ to Bob, yet Bob would have received a |1⟩, which would contradict Alice’s own account of what is going on.

This is clear if you look the final probability distribution. If Charlie measures 01XX then actually both 0100 and 0101 are possible If Danny measures XX01 then both 0001 and 0101 are possible. Hence, Violaris is right that it is subject to the uncertainty principle which prevents you from deriving conclusions about the full system, but the key to the paradox is that it is this very uncertainty that makes it possible that, if Charlie and Danny compare measurements, you can have a case where it appears as if Alice viewed |0⟩, believes she viewed |0⟩, prepares the Note in the |0⟩ state, sends it to Bob, and somehow Bob receives a |1⟩.

If we compute the weak values for this strange case where the final outcome is measured to be 0101, what we find is rather interesting.

As the values can be ±tan(π/(2θ)), I am interpreting +tan(π/(2θ)) to be +1 and -tan(π/(2θ)) to be -1.

At t=2 Alice measures the Coin and finds it is in the |0⟩ state and chooses to send a |0⟩ to Bob. But notice what happens at t=3 when Alice applies the CH operator. It should, according to traditional quantum logic, not have any effect on the Note and it should remain in the |0⟩ state, but it gets altered by the interaction and changes to the |1⟩ state! Bob then reads the Note and finds it is a |1⟩, which, according to traditional quantum logic, would lead him to believe Alice must have necessarily measured the Coin to be a |0⟩.

What we find when we analyze weak values is that these multipartite operators really do change their behavior based on future conditions. It is the very fact that in the future these qubits will end up in certain states that will show up on the detector that leads to the multipartite CH operator to not behave as Alice would typically expect.

Alice observed a |0⟩ on the Coin and then applies the CH operator to prepare the Note, and the CH operator does nothing if control is |0⟩, so she believes she prepared |0⟩ to send to Bob but is mistaken. Bob will actually receive a |1⟩ because the CH operator is behaving unexpectedly and alters the values of the Note in a way she didn’t anticipate.

This is the only case where it does this. I went through the tables for all the other cases and the results were exactly as you’d expect. If Alice perceives the Coin to be a |0⟩ then she will prepare the Note as |0⟩ and send it to Bob, Bob will receive a |0⟩ and this is consistent with what Charlie and Danny will conclude. If Alice perceives the Coin to be a |1⟩ she will prepare the Note in a superposition of states and send that to Bob, Bob will receive it and perceive either |0⟩ or |1⟩, and these two cases would be consistent between what Charlie and Danny would conclude.

Only in this one strange case that occurs 1/12th of the time does the CH operator not behave as Alice would expect. The future measurements of Charlie and Danny interfere with the behavior of her CH operator! That is because the value of the w-field at the spacetime coordinates of the CH operator has dependence upon the future measurement conditions.

You see, how the paradox is usually constructed, when Alice sends the Note to Bob, she mathematical formulation in the original paper by Frauchiger and Renner does not include Alice actually verifying that what she wrote on the paper is what she actually intended to write. It’s represented just with a CH operator where she trusts the CH operator did what she wanted when she sends the Note off to Bob.

If we actually add an additional step so that Alice does indeed look at the Note to verify the CH operator wrote it correctly (using a CX operator) then we find the possibility of 0101 actually disappears.