Weak measurements break our understanding of the universe
And philosophers are not talking about it enough
Since Heisenberg introduced his uncertainty principle in 1927, it has been misunderstood. The principle that says that a object’s position and momentum cannot be known with arbitrary precision at the same time. This is sometimes explained with the example that, if probing a particle with a beam of light, the higher frequency light will give a better position because of the shorter wavelength. Because it is more energetic, however, it will modify the momentum of the particle. In the same way, a lower frequency beam will give a worse position thanks to the longer wavelength but give a better idea of the momentum. This idea really has little to do with Heisenberg’s principle because it makes it sound like a practical problem. It, in fact, has nothing to do with the accuracy of measurements, and everything to do with the inherent randomness in them.
The conventional, and incorrect, explanation of Heisenberg’s is like this: imagine measuring a grain of pollen in a petri dish, moving randomly by Brownian motion, by using a sticky film to catch it. We can get an exact position at that point but no momentum. Conversely, if we hit the pollen with some kind of tiny probe (or laser) and send it skittering away, we can use that to measure the momentum but the position will be long gone. These imply, however, that the pollen’s position and momentum could be measured simultaneously if we were a bit more clever.
Despite the often lofty explanations invoking quantum interpretations of Heisenberg’s, the truth is prosaic. A particle has a probability distribution of postitions and of momenta described by the wavefunction. When we measure these quantities, we are measuring random variables out of that distribution. Heisenberg’s principle is simply the spread of those variables that we would see if we were to make repeated measurements. The distribution of position and momentum cannot both be arbitrarily narrow.
The difference (not the only one) between the pollen and an electron is that the pollen has a diffusive probability distribution, meaning that, as time goes on, the probability that it is a particular distance from its starting point increases. With the wavefunction, the probabilities follow a wave pattern instead, so that you have probabilities that go from high to low and high again in a pattern.
This minimal interpretation of Heisenberg leaves open the question as to whether a particle does indeed have a definite position and momentum. It is simply a law about the probabilities of measurements. It also doesn’t actually say that we can’t measure the position and momentum simultaneously, but we know from the formal equations of quantum mechanics that there is no way to do it, i.e., there is no observable that would measure both on the same particle.
This also applies to ordinary randomly moving objects. You can think of this as watching our bit of pollen under a microscope where we have a strobe light and measure it at two times. We use those two position measurements to determine the momentum but we cannot determine the momentum instantaneously at either time. So we have two positions and an average momentum. If we try to reduce the time between strobes, our measurements will become more and more precise, but, since the pollen is moving randomly, we can never get down to a time where the momentum is not an average at two random times.
If we, nevertheless, let the time interval go to zero, we would still be forced to assign the momentum the first or second position we measured. With continuous motion, this doesn’t matter, they become they same position, but with random motion it does. They do not become the same position. Therefore, you get two different results depending on which position you use. (In fact, with Brownian motion, for an infinitesimal amount of time, the distance the pollen travels will be proportional to the square root of that time, which means that any instantaneous rate of change of position (velocity) is not well defined at all.)
This is the essence of non-commutative variables in all random processes, including quantum physics. You cannot “commute” the position and momentum because it matters if you get the position first and then the momentum (equivalent to using the first position) or get the momentum first and then the position (equivalent to using the second one).
The same principle applies to other measurements as well. These are the “non-commuting” variables of quantum theory such as time and energy, different axes of spin, and so on.
In this article, I want to talk about a way of measuring particles that doesn’t run afoul of Heisenberg’s uncertainty principle and what it is telling us about the strange world of quantum physics. This method has astounding implications for philosophy but most philosophers aren’t even aware of it. This is a shame because it shows that matter can violate what we would consider to be self-evident principles of reality.
I’ll describe how quantum paradoxes arise, the quantum measurement problem with some helpful analogies, and jump into the analysis of a particular paradox (Hardy’s). Then I’ll talk about the implications to philosophy.
Many paradoxes exist in quantum mechanics and they tend to be resolved by violating one of three principles.
These are the principles of
Counterfactual definiteness
Locality
No conspiracy
The first means that if a quantum object is measured to have a particular state, it would have had that same state if not measured. That is, if I measure an electron to have a certain property, I can assume it had that property at an earlier time even though I didn’t measure it at that time.
The word “would” is important here because it implies a hypothetical or, in philosophical terms, a counterfactual reality. Any time you say that something “would” have been the same regardless of whether you observe it or not, you are implying counterfactual definiteness.
Locality means that quantum objects do not communicate faster than light.
No conspiracy is the oft forgotten third condition which means that quantum objects do not “conspire” to be correlated with one another as if they “knew” how they would be measured. In other words, we are free to make whatever measurements we want independently of other scientists.
Every quantum interpretation theory violates one of these. The so-called Copenhagen or wavefunction collapse interpretation violates #1. Bohmian mechanics violates #2. Superdeterminism violates #3. (Many Worlds Interpretation violates #1 but also violates factual definiteness in that whatever we observe is not definite but only an example of one world.)
All of these interpretations are intended to solve the quantum measurement problem. This problem, what it means or what happens when a quantum particle is measured, however, only applies to disruptive measurements, meaning measurements that are strong enough to banish the wave-like nature of the particle. All of these measurements, therefore, violate Heisenberg’s principle.
Recent studies of what are called weak measurements, introduced by Aharonov, Albert, Vaidman in 1988 as minimally disruptive measurements that do not violate Heisenberg’s uncertainty principle, show that self-evident axioms may be false when supposedly contradictory states are realized within a single world.
With weak measurements we can measure multiple aspects of a particle, such as both its position and momentum, at the same time, but at low resolution. Weak measurements are a kind of tradeoff. You learn less about what you are measuring than you would with a strong one but you also disturb less. You might think this wouldn’t have much philosophical value, but outcomes of weak measurement studies violate some of our most basic beliefs.
You can think of this as having a corridor with openings at either end. The corridor has a motion detector but all it can say is if someone is in the corridor or not and possibly how many people. A person enters the corridor and the detector registers that a presence is there. We know about how long the corridor is and we can measure how long the person is in there, so we can come up with a rough idea of how fast they moved from one end to the other. This is just an average of course. Therefore, we know their position vaguely (inside the corridor) and their momentum vaguely (based on the two times and the known length). This is what a weak measurement is like. The only disruption to the person is the motion detector light.
This is clearly different from, for example, measuring their position by dropping them through a trapdoor which would arrest their progress.
These two measurement techniques matter when we look at a paradox called Hardy’s which people tend to resolve by invoking various philosophical arguments requiring quantum interpretations. Weak measurements get us around that need, allowing us to come up with a minimal resolution that leaves the interpretation of quantum physics open.
The following is a detailed description of Hardy’s paradox and how weak measurements resolves it. If you want to skip the physics nitty gritty, you can go to the following section.
Here is a description of Hardy’s paradox by Lundeen and Steinberg: “Hardy’s paradox is a contradiction between classical reasoning and the outcome of standard measurements on an electron E and positron P in a pair of Mach-Zehnder interferometers.”
The basic idea is that you start with an electron and a positron. These two particles, should they encounter one another during the experiment, will annihilate each other.
Suppose we fire these particles at two separate 50:50 beam splitters BSP1 and BSE1. Provided we don’t observe the particles, they will each go through their respective beam splitter to enter a Schroedinger’s cat state where they have each, for lack of better words, gone left and right at the same time.
We have set up our experiment so that if the positron goes right and the electron goes left, they will encounter each other and annihilate with 100% probability.
Now, we provide some mirrors (we will need four, one for each possible path of the two particles) to bounce the electron and positron such that, no matter what path they take, they will be directed at another pair of beam splitters, BSP2 and BSE2. These beam splitters direct the particles at a set of detectors called “ports”. BSP2 directs the positron to a “bright” port and a “dark” port and likewise BSE2 directs the electron to its own pair of ports.
Now exploiting the wavelike nature of electrons and positrons, we set up our experiment so that the electron will interfere with its own wave such that it cannot arrive at the dark port, only the bright one, unless there is some object interfering. We do likewise for the positron. We expect, therefore, that as long as nothing interferes with the electron, only the bright port will ever “click”. If the dark port clicks, it means the positron must have interfered with the electron or vice versa.
Because of their wavelike nature, the positron can interfere with the electron without encountering it, but we infer from this that the electron and positron both went right. By the same token, if the positron dark port clicks we infer they both went left.
The whole setup suggests that we cannot have both dark detectors “click” at the same time because that would indicate that the electron and positron encountered one another and, by all logic, they should have annihilated and not made it to the detectors. Only one or the other dark detector may click. Not both.
Yet, we find that they do both click sometimes. This leads to a paradox. How can they both make it to the dark ports, having interfered with one another, and not be destroyed?
If we use another detector to determine the path the electron and positron follow, that detector itself would interfere with them and cause them to be detected at the dark ports. The paradox disappears.
So, in the case where there is a paradox, it is only because we have invoked counterfactual reasoning. We have not measured the electron and positron yet we imagine a possible world where we did and infer there is a paradox, but we are trying to compare two books for two different worlds as if they are one. Yet, in the book of this world, there is no proposition for our measuring them and hence we cannot logically infer that they are actually there in any real sense.
This reasoning is sound of course because it says that the problem is in believing that a counterfactual reality is factual.
When we calculate the result of making weak measurements of the particles, however, another reality appears that defies conventional reasoning. Quantum mechanical mathematics, in fact, tell us that, when we get detections at both dark ports, weak measurements will indicate the following:
Number of electrons that went left is one.
Number of positrons that went right is one.
Number of electrons that went right is zero.
Number of positrons that went left is zero.
We expect that if the electron goes left and the positron goes right they will encounter one another. But how do they avoid annihilation? Well, it has to do with pairs of particles. The electron-positron pair can have a different state than the individual particles, and we have to look at all those pair possibilities separately.
Number of pairs that converged (electron left, positron right) is zero.
Number of pairs where both went right is 1.
Number where both went left is 1.
It seems we have one more pair of particles than we put into the machine. We aren’t making matter out of nothing and violating the conservation of energy, however, because there is one more pair possibility.
The number of pairs where the particles diverged (electron right, positron left) is negative 1.
In other words, nature manufactures something out of nothing to resolve the paradox. We still have a total of one pair of particles, but we can have pairs doing two different things at the same time by having a negative number of pairs doing something else.
Since the number of pairs that converged is zero, that explains why they didn’t annihilate. The two particles don’t actually encounter each other. Instead, they individually pass through the same location without ever both being there at the same time.
This defies two axioms about the world: (1) that if you have a number of objects, that number cannot be negative and (2) that if you have two individual objects in a location then you also have a pair of objects in that location. Neither of these turns out to be a property of the quantum world.
Modus ponens and all the rest of logic remains perfectly intact, but that which we consider to be self-evident must fall to quantum mechanics in the same way, for example, the parallel axiom of Euclid falls to Einstein’s general relativity.
This experiment was done in 2009 (by two independent teams with different setups) and the measured results were close to what was predicted theoretically.
This isn’t evidence for or against the existence of possible worlds, but it does show that quantum mechanical paradoxes can be explained without invoking counterfactual (or factual) worlds at all provided we admit non-intuitive facts.
Weak measurements can also be applied to entangled particles and show again that nature can resolve paradoxes in unique, counter-intuitive ways. In particular, since entanglement, like Hardy’s paradox, involves both particles as individuals and as pairs, and we know that individual particles can have different states than pairs, this provides a unique resolution to paradoxical correlations between entangled particle states.
At the heart of entanglement is the EPR paradox. One feature of the paradox is that, if Alice measures particle A, she can infer with 100% probability what Bob will measure for particle B. That means that one consequence of measuring an entangled particle is that it instantly confers a definite factual reality on the other while removing the entanglement between the two.
From a philosophical perspective, if we want to understand what propositions are true about this world, it matters whether that reality is conferred by measurement or pre-exists. The problem here is that Alice only infers the state of the other particle. She has not observed it. That means we can create three possible interpretations of what has happened when Alice makes her measurements: (1) Alice disturbs Bob’s particle nonlocally, meaning at a distance, (2) Alice infers Bob’s particle state with 100% certainty but it has no reality, or (3) Alice discovers the pre-existing reality of Bob’s particle without disturbing it.
All these possibilities, however, assume that Alice is making strong measurements of her particle, but what if her measurements are weak?
Weak measurements of entangled particles can extract some information about their states while leaving them still (more weakly) entangled. This has a lot of practical value for quantum computers which rely on entanglement to compute, but also has philosophical value in understanding what is going on when particles are seemingly in contact with one another at a distance.
For example, Alice can measure her particle and determine if its spin in the z-direction is up or down (the only two quantum possibilities). If she measures her particle to be z-up, then she can infer with 100% certainty that Bob will measure z-down on his particle.
If Bob instead chooses to measure x instead of z, then he will get a completely independent measurement from Alice but be able to infer Alice’s x spin in the same way she can infer his z-spin.
If, on the other hand, we simply wish to measure whether Alice’s x spin and Bob’s z spin are the opposite of each other, we can do so. This leads to a paradox, however, if we measure them to be opposite, and then Alice and Bob measure their particles z and x spin respectively. Quantum theory says these three measurements are actually independent of one another because they are all measuring different things. Therefore, if we find that Alice’s x and Bob’s z spin are opposite of each other, but Alice’s z spin is up and Bob’s x spin is up, then what do we say about reality? How can individual spins be the same (Alice infers Bob’s z spin is down and Alice infers Bob’s x spin is down) but also the opposite?
Counterfactual reasoning resolves this by saying that, since Bob never measured his z spin and Alice never measured her x spin, then we cannot infer a counterfactual world where they did. Such a world does not exist.
Weak measurements, however, resolve this paradox too in its characteristically bizarre way. It says that their spins are 50% both down, 100% opposite, and -50% both up. We are cautioned to interpret these not as negative probabilities but negative mass or energy. Basically, we have introduced new sets of particles with negative energy in order to offset the requirements of quantum probabilities.
Vaidman called weak measurements a “generalization of the usual concept of the (strong-measurement) element of reality”, meaning that what we think of as “real” is actually only “strong measurement” real. It is the abridged book of this world, not maximal. Weak measurements give the unabridged version that tells us everything we can know about this one world and still be logically consistent.
The general conclusion from weak measurements is that it is possible to have negative numbers of particles and this helps to resolve paradoxes by allowing particles to be in multiple places at once without violating the total count of particles. Moreover, it is possible to have individual objects both present in a location without having the pair present in that location.
This has important implications to the philosophy of mathematics because it means that axioms that support the concept of counting such as Peano’s five axioms are insufficient to describe our universe. Essentially, the natural numbers cannot count all sets of objects because some have a count that is negative.
Moreover, it means that collections of objects can have different properties than the aggregate of the individual members.
Both of these apparently self-evident properties of the universe are only classical approximations.
It seems to me that these have vast philosophical implications that only physicists have really looked at despite 35 years of research.