The Wave Function is Unnecessary

Quantum Mechanics does not need Wave Functions

Much of the discussion around interpretations of quantum mechanics center around interpreting the physical meaning of the wave function. However, it is possible to do calculations without invoking a wave function at all and get all the right predictions. Indeed, when you avoid using the wave function, you also find that imaginary numbers disappear and you are left with just real numbers. What you are left with is a much clearer picture of what the calculations physically represent.

This does not mean we should stop using wave functions in our calculations. The wave function can be seen as a mathematical simplification that gets rid of physical redundancies in the mathematics. Those physical redundancies still physically exist, but because they aren’t relevant to your final prediction, you can simplify the mathematics by getting rid of them. This, however, leads to confusion if you then try to physically interpret the system as equivalent to its simplified form, because it would appear to be missing a lot of stuff.

Quantum mechanics without the wave function simply requires expanding it to its redundant form. When we do this, its physical meaning becomes entirely unambiguous and we can work solely with real numbers without any imaginary components. This expanded form is mathematically equivalent to the reduced form and thus does not require any additional postulates.

The Expanded Vector Form

For a single qubit, if we are given a wave function associated with it, then its values of the wave function associated with it will be…

• |ψ⟩ = [ α; β ]

A qubit, however, has three observables, not just two. It has the observables X, Y, and Z, that can each be either +1 or -1, as well as a constant observable I that is always equal to +1. The expanded form we can express as…

• |w⟩ = [ 1; x; y; z ]

We can then convert between any wave function and the expanded form using the equations shown below.

Due to the uncertainty principle, we can only know the value of one of these three observables at a time. If we know one of them with absolute certainty, then the rest we will maximally uncertain. We can express certainty using expectation values. Maximal certainty would be equivalent to either a +1 or -1, relating to which value we know for certain. The expectation value of 0 then represents complete uncertainty, as it represents a uniform probability distribution of 50% +1 and 50% -1.

In quantum computing, we always begin an algorithm with qubits in a known state of |ψ⟩ = [ 1; 0 ]. If we use the equations above to expand this out, we find that it is equivalent to |w⟩ = [ 1; 0; 0; 1 ]. This means we know for certain that the Z observable has a value of +1, but are maximally uncertain of the other two observables.

Indeed, in this form, we find that the expanded vector never at any point represents a system being in “multiple states at once.” It always represents a concrete physical property of the system that if we were to measure, the measurement outcome would be absolutely certain and predetermined with no probabilities at all.

For example, we can take the superposition of states…

• |ψ⟩ = [ 1/sqrt(2); 1/sqrt(2)]

This is called the plus state and is often interpreted as the qubit having both a value of 0 and 1 simultaneously. Yet, if we convert it to its expanded form, we find that it is equivalent to…

• |w⟩ = [1; 1; 0; 0 ]

In other words, it is merely a statement that we know its value on the X axis to be +1 but do not know its value on the Z axis. If we were to measure the qubit on the X axis, then the measurement outcome would be absolutely determined to give us +1.

In the expanded form, we find that the wave function is just a way of expressing a subset of the system’s physical properties. Since the uncertainty principle disallows us from knowing all of its properties simultaneously, we represent it just with a subset of those properties that we do know, and, as the particles undergo physical interactions, we keep track of how this influences the subset of information we can say about the system.

The expand vectors also do not have imaginary numbers.

The Expanded Operator Form

If we want to apply an operator to a qubit, we can apply it in the same way we do for the wave function by simply multiplying the operator by the vector. However, we also need to expand the operator as unipartite operators for single qubits are 2x2 yet our expanded vector is 1x4. To allow for a multiplication, we need to expand the operator to 4x4.

The equation below will take any operator of the reduced form U into its expanded equivalent R(U).

When we expand the operators, the imaginary components disappear, and it becomes more physically obvious the effect of the operators. Each row can be thought of as an input eigenvector whereas each column is an output eigenvector. Since any vector can be decomposed into a linear sum of its eigenvectors, the operator matrix can be interpreted as taking each decomposition, passing it into the table, getting the output vector, and then summing them all up.

In the Mach-Zehnder interferometer, the beam splitter is represented using this 2x2 operator.

If we expand the operator using R(U), we find that it expands to…

If we recall that this represents a kind of input-output table, then we can interpret this to mean…

• I = (I)
• X = (X)
• Y = (Z)
• Z = (-Y)

The values in parenthesis are the initial values, whereas the values not in parentheses are the output values. As we can see, the beam splitter operator effectively swaps Y and Z while also flipping the sign of the Y value that will become the new Z value.

One of the most common operators in quantum computing is the Hadamard operator, or the H operator. As a 2x2 operator, it is represented as shown below.

If we expand it, we get…

If we recall that this represents a kind of input-output table, then we can interpret this to mean…

• I = (I)
• X = (Z)
• Y = (-Y)
• Z = (X)

The operator flips the sign of the Y value and swaps the X and Z value.

Now, it should become obvious as to why, when we apply a single H operator, that the measurement outcome is then uniformly random. If we begin with a known value of Z=+1, and then swap the X and Z value, then the known value is now X=+1. If we measure the Z value, then we would be measuring an unknown value, and hence it would be random.

It should also become obvious as to why, when we apply the H operator twice in a row, it cancels itself out and we do not get anything random. We swap the Z and X values and negate the Y, then swap the Z and X values and negate the Y again, which undoes itself.

Weak Measurements

In quantum mechanics, we don’t have to just measure the X, Y, or Z value. We can measure fractions in between those values, and when we do so, they are called weak measurements. We can represent weak measurements using the rotation operators Rx, Ry, and Rz, and in their expanded forms, the imaginary component also disappears.

When working with continuous values, we can convert the expanded form into an expectation value first by using the S(U) value to get the expanded expectation observable…

And then we can use this to compute the expectation value for a given observable using the vectors in the expanded form.

We can then derive the uncertainty principle just by applying rotation operators and looking at how the expectation values change.

Complementarity

A perfect and ideal measuring device would have an input-output sequence that looks like this…

• IX = (IX)
• IY = (IY)
• IZ = (IZ)
• XI = (XI)
• YI = (YI)
• ZI = (IZ)

If you notice, almost everything is unchanged. The only thing that is changed is that ZI = IZ. If we assume the left-hand letter represents the most significant qubit’s observable which is the qubit representing the memory for our measuring device, then this ideal measuring device would simply copy the Z value of another qubit into its own internal memory without perturbing anything else.

However, this is physically impossible. Why? Well, first, it’s not time-reversible. The value of ZI before the measurement, that is to say, the measuring device’s memory state prior to the measurement, is simply overwritten, so the information associated it would be lost.

We could instead change ZI = (IZ) into ZI = (ZZ). This is time-reversible because IZ is not modfied and thus we can compute the previous ZI from the new ZI/IZ. As long as the know that ZI=+1 initially, i.e. we know the initial value of our measuring device’s memory state, then we can figure out what IZ is from how ZI changes.

This, still, violates a different and more subtle law, which is that all operators must be a completely positive trace-preserving map. This is because, as discussed in the previous section, we can compute expectation values, and thus probabilities, for the values we don’t know from the values we know. For this to be mathematically consistent, the operators cannot produce negative probabilities, which is physically meaningless, and the operator described thus far would indeed violate this rule.

We would test this by constructing a Choi matrix as shown below from R(U) and then computing the eigenvalues of the Choi matrix. If any are negative then it’s not a completely positive trace-preserving map and is thus not a physically valid operator.

In reality, the measurement operator we are trying to construct will end up having to look something like…

• II = (II)
• IX = (XX)
• IY = (XY)
• IZ = (IZ)
• XI = (XI)
• XX = (IX)
• XY = (IY)
• XZ = (XZ)
• YI = (YZ)
• YX = (ZY)
• YY = (-ZX)
• YZ = (YI)
• ZI = (ZZ)
• ZX = (-YY)
• ZY = (YX)
• ZZ = (ZI)

What is important here is that IX=(XX), IY=(XY), and IZ=(IZ). This operator, called the CX operator, does not perturb the value we are trying to measure, in this case the Z value of the least-significant qubit, so it eveals what is really there. However, it does perturb the X and Y values, and, even worse, it perturbs them in a way that depends upon the measuring device’s X value.

In order to account for this perturbation, we would need to know the measuring device’s X value. So we would need a new measuring device that measures X values and not Z values like our current one. If we can construct an operator for measuring X values….

• II = (II)
• IX = (IX)
• IY = (XY)
• IZ = (XZ)
• XI = (XI)
• XX = (XX)
• XY = (IY)
• XZ = (IZ)
• YI = (YX)
• YX = (YI)
• YY = (ZZ)
• YZ = (-ZY)
• ZI = (ZX)
• ZX = (ZI)
• ZY = (-YZ)
• ZZ = (YY)

Now, look, IY=(XY) and IZ=(XZ). We are perturbing the Y and Z value of the measuring device we’re trying to measure based on the X value of measuring device measuring the measuring device! We need a stable Z value for the measuring device or else it can’t function as a measuring device, so we would need to introduce a measuring device to measure the measuring device to measure the measuring device’s X value to account for this perturbation, but then this would perturb that measuring device’s Z value.

In other words, we get an infinite regress, and hence, measuring all values of all observables simultaneously is physically impossible. This does not need to be taken as a postulate. It can be demonstrated, as shown here.

Entanglement

Consider that we have two qubits, Q#0 and Q#1. They both begin in a known state of Z=+1. We then swap the X and Z values of Q#0 using the H operator, and so now we know X=+1 for Q#0 but don’t know its Z value. Then, we use the CX operator to flip the Z value for Q#1 based on the Z value of Q#0.

Since we don’t know the Z value of Q#0, we can’t know the Z value of Q#1. Even worse, the CX operator, as shown previously, disturbs the X value of, in this case, Q#0 in a way that depends upon Q#1’s X value, but we don’t know Q#1’s X value either. Hence, we now know nothing about any of the observables of the system at all!

Or… do we? Yes, we don’t know any specific values for the observables, but we do know that Q#1’s Z value was initially +1 before it was perturbed, and if Q#0’s Z value is +1, then it won’t flip Q#1’s Z value and thus it will remain +1, but if Q#0’s Z value is -1, it will flip it, and thus Q#1’s Z value will become -1. Even though we don’t know either of their values, we know that they must be matching, that is to say, they are positively correlated with one another.

We can express this as ZZ=+1. If we know one, we can know the other, because they must be matching.

Indeed, if we look back up at the input-outputs for our CX operator, we see on there…

• II = (II)
• XX = (IX)
• YY = (-ZX)
• ZZ = (ZI)

We knew the Z value of Q#1 prior to the CX operator to be Z=+1 as well as the X value of Q#0 prior to the CX operator to be X=+1. And thus, we can know that II=(+1)(+1)=+1, XX=(+1)(+1)=+1, YY=-(+1)(+1)=-1, and ZZ=(+1)(+1)=+1, that is to say, we can know the X values and Y values must be positively correlated with one another and the Y values are negatively correlated with one another.

As a wave function, we would express this as…

• |ψ⟩ = [ 1/sqrt(2); 0; 0; 1/sqrt(2)]

This seems a bit mysterious as to what it means and people often interpret it as saying that a system is in “multiple states at once.” However, this is clearly not the case if we expand it into the its expanded form, it simply becomes…

• |w⟩ = [ 1; 0; 0; 0; 0; 1; 0; 0; 0; 0; -1; 0; 0; 0; 0; 1 ]

This is mathematically equivalent.

This also explains why we need to express the whole system as a giant vector that grows exponentially. If we simply apply the operator, take the output values on how they alter the X, Y, and Z observables, and plug them back into the qubits, well, in the case shown above, we actually don’t know what the X, Y, and Z observables are after the CX operator is applied, so we would end up plugging all 0s into the qubits which would represent their values all being uniformly random, and could not make any prediction at all.

We need this very large vector that keeps track of not only the isolated values of the qubits (IX, IY, IZ, XI, YI, and ZI) but also the correlatitive values (XX, XY, XZ, YX, YY, YZ, ZX, ZY, and ZZ) or else we would not be able to make any sort of prediction at all.

If you notice, our vector does have a lot of 0s, so it can be simplified. This is ultimately what the wave function is. The expanded form usually is mostly all 0s and is only expressed in terms of real numbers. It can thus be drastically simplified, and the wave function is just that mathematical simplification.

Conclusion

The point is, however, that when we see [ 1/sqrt(2); 0; 0; 1/sqrt(2)], we should not think of it as “the particle is in the states 00 and 11 simultaneously.” We should think of it as “XX=+1, YY=-1, and ZZ=+1,” i.e. it’s a description of a physical property of the system, and if we went to measure those correlations, we are guaranteed to measure those values without any probabilities at all. The measurement outcome would be absolutely deterministic.

It is only not predictable when we measure something that the wave function doesn’t describe. It is sort of like, If I told you the mass of a car and nothing else about it, if you went to measure its mass, you could predict the outcome with complete certainty, but if you went to measure its color, you wouldn’t be able to know it with certainty because it wasn’t part of the description I gave you.

Does that make the description of the car’s mass epistemic? No, it is an ontic description of a property of the car, it is just a subset of its properties, not all of them. Similarly, that is ultimately what the wave function represents. It is a subset of the system’s properties. It describes a property of the system that if you went out and measured, you would measure the value it describes with absolute certainty.

However, due to complementarity and the uncertainty principle, if you measure a different property, then it would be unpredictable with certainty. Although, quantum mechanics allows you at least give probabilistic predictions based on the limited knowledge you do know, given the rule we showed before for computing expectation values from the expanded vector.

Of course, here we only discuss the simple case of spin-1/2 particles, but the principle is generally the same. Each wave function is just a condensed vector representing all the possible permutations of the system containing expectation values relating to the knowledge you possess on the system, and so they can always be expanded in this way.