Quantum Coherence – now Nature hides stuff from us.
The Heisenberg Uncertainty Principle puts a limit on our ability to measure microscopic properties of matter. This means that when a system is probed we can miss things that are actually there. Using a probe is the only way we can “see” what is going on. In a pitch black room you must turn on the lights. When the lights are out we assume things in the room have not moved. However, is the situation the same when we “turn on the lights” at the microscopic level? If we cannot see something, then we can incorrectly conclude nothing is there while in fact it just might not be detectable.
This is not the usual way one thinks about measurement. Rather, quantum mechanics is believed to give a complete description of the microscopic and misses nothing. The Heisenberg Uncertainty Principle is a fundamental property of waves which tells us that canonically conjugate observables cannot be simultaneously measured with the same apparatus because they are mutually exclusive. Position and momentum simultaneously exist in Nature but quantum mechanics cannot formulate them simultaneously. This is why EPR believed quantum mechanics to be incomplete.
Because of restrictions due to Heisenberg, Quantum mechanics should be treated as a theory of measurement of the microscopic. When measurement misses properties, experiments may not make sense. In order to account for the quantum correlation that leads to the violation of Bell’s Inequalities entanglement is assumed to persist between separated particles. How? Non-locality is well established in physics yet no one understands how it works. How can two particles in different light cones instantaneously “know” the state of its partner? What is the mechanism for this connectivity? How is the connectivity maintained? What are quantum channels? No one knows.
Nature hides things from us in coherent quantum states. The Heisenberg Uncertainty Principle restricts us, not Nature. So if there is information in a system that we cannot measure then we cannot know we have missed it. For example a lot of the mass in the universe is missing—dark matter, and another, if spin has more than one axis of quantization, can we know? Could these properties be stored in undetectable modes? We cannot know because we can only count events: we cannot count undetected events.
In this blog I want to show the importance of quantum coherence with the simplest of quantum system, two levels. From a two level system many other properties can be built: like higher angular momentum states and it gives insight into more complicated systems.
Quantum coherence is a unique property of the microscopic. It usually decoheres at the macroscopic level because quantum states are composed of a statistically large number of particles: again stressing these decohered properties cannot be measured but nonetheless are present. They are just as real as properties that can be measured as confirmed in situations where quantum coherence does make it to the macroscopic: think of lasers, superfluids and superconductors. As far as Nature is concerned, there is no delineation between what we can and cannot measure. In fact quantum coherence carries information about properties which can be as large as what we actually measure. This is what I want to show.
Take a spin density operator and ignore quantum coherence for now, (click equations to enlarge)
given in terms of the Pauli spin vector operator and a polarization vector, P, also called the Bloch vector. Assume a pure state, so that |P|=1. Equation (1.1) tells us there is one axis of spin quantization so choose it in the Z direction, as usual, which leads to,
The eigenstates are the usual ones,
and the expectation values of a spin being deflected up or down is found to be unity,
This simply tells us that the probability of measuring spin up or down is 100%. The eigenvalue equation is satisfied,
Now consider a two level system that has quantum coherence. Let us define such a state operator as,
Note that the two Pauli spin components do not commute, [σZ,σX]=iσY , so immediately from Heisenberg we know that we cannot measure the diagonal and off-diagonal terms simultaneously.
Take a Hermitian operator of trace 1, A. This observable must mix the plus and minus states so assume it has the following representation,
with coefficients that obey -1 ≤ ai ≤ +1, i=1,2. Call Eq.(1.7) the Z representation. This is not the superposition principle of quantum mechanics whence the quantum state is considered to result from a statistical ensemble of particles. In contrast, Eq.(1.7) describes one particle which cannot be in more than one state at any instant. Therefore Eq.(1.7) describes the four states that a single particle can occupy, either a diagonal state or an off-diagonal state,
where a2 =d2cosξ2 ; c2 =d2sinξ2. The four possible outcomes of A are:
1. Two exact (dispersion free) probabilities of being in the plus or minus states,
AZ±± = 1/2(1±a1) (1.9)
2. Two pure coherences which are phases,
AZ±(-/+) = 1/2d2exp(±iξ2) (1.10)
We have already concluded quantum coherence cannot be detected. Hence half the states will be missed in an experiment in this representation. Only half the clicks that are there can be counted.
Nonetheless the properties contained in the quantum coherences are real and we can get to them by simply transforming from the Z representation to the X representation (i.e. do another experiment) using
The operator is not changed by using different basis states, only its representations as seen by comparing the two,
The quantum coherence in the Z representation can be recovered in the X representation and vice versa. Another way of saying this is that Nature hides half the correlation from us in one experiment which can only be recovered by doing a second experiment.
This example shows probability and coherence are complementary which underpins the Heisenberg Uncertainty Principle. The information stored in the quantum coherence is real and present even though blind to us. Complementary experiments are necessary but might not be technologically possible or give results that are inconclusive. For example if a spin had two axes of quantization, like the example here, Eq.(1.6), then we can never know because only one axis can be detected in an experiment. Doing another experiment would not be conclusive because the X spin looks identical to the Z spin, so the conclusion is the spin just reoriented.
The Heisenberg Uncertainty Principle does more than restrict what we can measure, it also result in us missing properties that might actually be present.