Figure 1
When a ballerina does a pirouette she must escape the friction of the ground in order to get the freedom to move. (Figure 1: Photo by Michael Garner, courtesy of English National Ballet.) She does this by restricting her contact with the ground to a point. In a recent paper I and my collaborators Andrew Garner and FQXi's Vlatko Vedral show that quantum theory in a very similar way escapes a fundamental constraint on movement by accepting uncertainty.
Quantum systems are associated with states which encode the statistics of future possible measurements. The collection of such states may be represented as a geometric shape. In the smallest possible quantum systems, single qubits (quantum bits), this shape is a sphere, called the Bloch sphere.
For example, think about a property of a qubit, such as its position: the qubit could be associated with two possible positions, A and B, say, or it can be in a fuzzy superposition where it exists in both of these mutually incompatible states simultaneously, before being observed. If it's in a superposition then although experimenters cannot know with certainty what position they will find it in when they make a measurement, they will have some sense of the probability of getting a certain outcome. The Bloch sphere helps to visualise this odd feature and the probabilistic nature of quantum mechanics. In the example, a vector pointing to the north pole of the sphere could represent position A, while the south pole represents position B. (In a classical system, this would represent the only two options available for a binary digit, or bit, to access). However, a qubit can also be represented by a vector pointing elsewhere on the surface of the sphere, corresponding to the fuzzy in-between states.
Figure 2
The maximal state space conceivable would actually be the cube outside of the sphere, as shown in figure 2. The quantum state space is the sphere, but if there were no uncertainty principle all states in the outer cube could be allowed. In this case certain measurements could all have predictable outcomes at the same time, in violation of the quantum uncertainty principle.
One may ask why quantum theory is restricted to the sphere, and accordingly to having the uncertainty principle.
We came across an intriguing answer when thinking about how the cube state space would handle an interferometer. In an interferometer the particle or photon is firstly placed in a superposition of being in two places and then operations are done on each site. Now when you have two different sites fundamental locality restrictions come into play. In particular, we point out that if a system has 0 probability of being found on site B, then an operation on site B must leave the state of the system invariant. Otherwise we could do action at a distance. Contrary to some popular science depictions, quantum theory does not allow action at a distance. The universe would be almost inconceivably odd and complicated if action at a distance were possible. We would not be able to make a statement about an individual system without taking into account what happens everywhere else in the world.
On the Bloch diagram, state transformations move points around, e.g. by rotating the shape. So, if one accepts that this locality restriction holds, it turns out that operations on site B must leave all points (states) on the lower plane of the cube invariant. It is like the points are stuck by total friction between the shape and the lower plane. As a result the cube has a big disadvantage over the sphere because if the entire square face touching the ground is restricted, then the whole cube gets stuck and no states can change.
But now imagine metamorphosing the cube into a sphere, or indeed something else with only one point on the lower plane, like how the ballerina goes up on one toe. Then the shape, with all the quantum states in it, can move. The quantum sphere has the advantage over the cube that it can rotate even if there is full friction with the lower (and/or upper) plane, just as the ballerina accepts the uncertainty of only having a point in contact with the ground in return for the ability to pirouette.
One may say that uncertainty, rather than being just limiting, liberates quantum states to change.
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Oscar Dahlsten is affiliated with Oxford University
The paper appears in Nature Communications.