Quantum Decoherence

Part 4 of "An Introduction to Quantum Reality"

Hopefully by this stage you will have read the previous pages on Quantum Mechanics: An Introduction, The Quantum Casino, and Quantum Entanglement. On those pages several weird and puzzling questions were raised. In this page we start to see some answers.

Perhaps the first principle to understand is the idea that we cannot separate an object being measured (observed) from the apparatus performing the measurement. This is clearly shown in the quantum world where you cannot divorce the property you are trying to measure from the type of observation you make: the property is dependent on the measurement.

It's actually quite like a rainbow. When a person looks at a rainbow he sees it starting in a certain position and ending in a certain position. However, when a second person - who is standing in another place - looks at the rainbow he will see it starting and ending in a completely different spot. So the two people are effectively seeing different rainbows, with different starting and ending positions. That's why you will never find a pot of gold at the end of a rainbow!

How is this possible? It's possible because you have to consider the rainbow and the observer as a single system. Actually, this is true of all measurements and observations: you really cannot separate the object being measured from the device performing the measurement or observation - you have to consider them as a single system. For example, when you take the temperature of an object using a thermometer, you have to remove a very small sample of heat from the object. The measuring device has altered the object - the two entities are not separate. The object and the device performing the measurement are bound together as a single system.

This idea of the observed object and the observer being bound together as a single system will be shown to be the key to providing the mechanism for the apparent "collapse of the wavefunction". Because, in reality, a quantum particle is rarely completely isolated from its environment. Rather, the particle and the environment are bound together as one system.

How the environment eliminates interference effects

In the page on The Quantum Casino we have seen that when a measurement of an observable is performed, the quantum state appears to "jump" to a particular eigenstate (with the observable taking the associated eigenvalue). This apparent jumping puzzled physicists for many years because it was not understood how and why the usually linear time-evolution of the Schrödinger equation should suddenly decide to make a sudden jump.

Also, as a quantum state can be viewed as a superposition of many other states, the question can also be asked as to why we never see these other states in macroscopic objects. For example, why is Schrödinger's cat never seen as being both alive and dead at the same time?

However, in the double-slit experiment we do see the other states of the superposition, as they provide constructive and destructive interference effects (see Quantum Mechanics: An Introduction). Why do these so-called interference states appear in the double-slit experiment but apparently vanish in macroscopic objects?

Let's remind ourselves of how a quantum state can be expressed as a linear combination of components of other eigenstates (this was considered in the page on The Quantum Casino):

Now here is the absolutely key point: every component eigenstate has an associated phase (this was considered back in The Quantum Casino). It is this phase which gives the wavefunction its "wavelike" character (in complex space, remember). In order for the components to combine together correctly to produce a superposition state, they must be in the same phase (must be coherent). This is what happens in the double-slit experiment: interference components possessing the same phase combine to produce the interference effects.

As explained in the "rainbow" example at the top of this page, we cannot separate an observed object from the observer: we have to treat the resultant combined system as one system. What happens in the real world is that a particle is not perfectly isolated: a particle inevitably interacts with the environment. These interactions have the effect of the particle "being observed" by the environment - the "environment" might very well be a man-made measuring device, for example. (For a technical discussion about the measured system and the measuring apparatus acting as one system, see here. For more on the idea of environmental interactions producing a measurement, see the page on Quantum Reality).

What happens to a quantum particle in the real world is that each of its component states gets entangled (separately) with different aspects of its environment. As seen in the page on Quantum Entanglement, when particles become entangled you have to consider them as one single, entangled state (you use the tensor product to calculate the resultant state). So each component of our quantum particle forms separate entangled states. The phases of these states will be altered. This destroys the coherent phase relationships between the components. The components are said to decohere.

If a particle interacts with just a single photon, for example, then the two particles will enter an entangled state and that will be enough to trigger the onset of decoherence (for example a single photon entering the double-slit experiment will be enough to destroy the interference pattern). However, for all interference effects to disappear, the particle must have a macroscopic (rather than a microscopic) effect by forming entanglements with billions of particles in, say, a Geiger counter. This is described in the book Quantum Enigma: "Whenever any property of a microscopic object affects a macroscopic object, that property is 'observed' and becomes a physical reality" (this idea of "decoherence=observation" is considered in greater detail in the next page on Quantum Reality). In that case, if there are no longer any interference terms then to all intents and purposes the particle is now in a single, quantum state - one of the component eigenstates:

(In the page on Quantum Entanglement it was shown how the dimensionality of the Hilbert state space increases rapidly with each entanglement, thus further reducing the chance of coherent interference effects - see here)

Note that the interference components do not actually disappear - because they are out of phase we just don't notice them at the macroscopic level. In fact, they just get dissipated out into the wider environment. I always imagine them as little ripples in the ocean - we only ever notice the big (macroscopic) waves in the ocean. The little ripples get entangled with other little ripples until it is impossible to tell from which big wave each little ripple came.

Imagine you throw a rock in the sea off the coast of the United Kingdom. After the initial big splash, the ripples dissipate and apparently disappear. But of course, they haven't really disappeared. The ripples have decreased in size, and they have mixed and interfered with other waves, but they have not disappeared. Two weeks later, on the rocky shore of Tierra del Fuego off the Argentinian coast, one of the small waves washing to shore is maybe an imperceptible fraction of one micron higher because of that rock you threw.

So the ripples (interference terms) do not actually disappear. They dissipate into the environment and become effectively undetectable. And it's certainly not possible to associate the microscopic change in the height of the wave in Tierra del Fuego with the rock you threw - there have been so many interactions with other waves along the way. In this sense, the process of decoherence is irreversible - and that's a key feature of decoherence: we can't reverse the process (to regenerate the initial interference components) - they're gone for good. And even the "little ripple" echoes of the interference effects have become imperceptible due to interactions with the environment. Then, for all intents and purposes, the interference effects (ripples) have completely disappeared.

At last we seem to have found the mechanism behind the disappearance of the interference effects, the truth behind the mysterious "collapse of the wavefunction".

Decoherence, then, is not a sudden "jumping" effect. Rather, the interference terms disappear due to the progressive influence of billions of particles (and associated entanglements) as a particle passes through our measuring apparatus. So there is a progressive filtering (of the interference terms) and amplification (of the eventual measurement - Bohr referred to an "irreversible act of amplification"). As Brian Greene explains in his book The Fabric of the Cosmos: "Decoherence forces much of the weirdness of quantum physics to 'leak' from large objects since, bit by bit, the quantum weirdness is carried away by the innumerable impinging particles from the environment."

However, the decoherence process fooled physicists for many years because it is such an efficient process - decoherence happens so fast (in the region of 10^-27 seconds!) - giving a false impression of a discontinuous, instantaneous quantum "jump". However, recent experiments have managed to delay decoherence by decoupling quantum particles from their environment. If decoherence is delayed then the superposition states become evident. As an example, an electric current has been made to flow in opposite directions at the same time by using a superconducting ring (see this Physics World article which considers the effect of decoherence on Schrödinger's cat, and this New Scientist article).

The reason we never see Schrödinger's cat both dead and alive at the same time is because decoherence takes place within the box long before we open it. This is due to the device housing the radioactive nucleus and the poison. This is what forms the macroscopic environment immediately surrounding the radioactive nucleus.

So why does the electron in the double-slit experiment still show interference effects? Why does it not decohere? The answer is because it is not a macroscopic object, it is an isolated microscopic object. While decoherence happens extraordinarily fast for macroscopic objects, for an electron the decoherence time (the so-called coefficient fluctuation time) is about 10⁷ seconds, or about a year - plenty of time to perform the double-slit experiment and see interference effects.

For a clear explanation of decoherence, I can recommend Chapter 7 of Brian Greene's book The Fabric of the Cosmos.

Decoherence and Entropy

There is a parallel here with thermodynamic behaviour, considering the statistical movement of particles under heat. The second law of thermodynamics says that the amount of entropy in a closed system (the amount of disorder, basically) will always increase (see the Arrow of Time page for full discussion about entropy). For example, if we have a sample of gas in a closed container in a corner of a room, when we open the container the gas will spread throughout the room. Eventually we will reach a state when all the molecules of the gas are completely randomly orientated throughout the room. This state is called thermal equilibrium. It is the state of maximum entropy (disorder).

The dissipation of the interference components into the environment during decoherence behaves in a similar way. The environment can be considered a heat bath into which the interference terms spread and become completely disordered. At that point, the process is said to be thermodynamically irreversible. Interference is gone for good.

The "collapse of the wavefunction" is like a snooker break-off shot. Imagine each ball represents an interference term of the quantum state. Before the shot, (before we make a quantum observation), we see low entropy - everything is nicely ordered. All the interference terms are coherent, and capable of producing interference patterns.

After the shot, the system of balls represents a system with greatly-increased entropy (disorder). This is what happens when we make a quantum observation: interference terms dissipate into the "heat bath" environment, and all coherence is lost in the confused mess. The situation is now one of thermal irreversibility: it is extremely unlikely the original ordered situation could re-form itself. We therefore only see the collapse of the wavefunction operating the forward time direction (for the same reason we don't see broken eggs mending themselves).

See the Arrow of Time page for a full discussion about entropy and the thermodynamic arrow of time.

Decoherence and the Many-Worlds Interpretation

The so-called "quantum measurement problem" has baffled physicists ever since quantum mechanics was first discovered: What constitutes a measurement? What random process selects the observed value from the possible values in the superposition? What happens to the other terms in the superposition?

In 1957, Hugh Everett proposed the many-worlds interpretation (MWI) of quantum mechanics in an attempt to provide an answer to the quantum measurement problem. The MWI suggests that when we make a measurement, the universe itself splits into different parallel universes, each universe containing one possible outcome of the observation. For example, in the case of Schrödinger's cat, when we open the box the universe splits into two: the cat is alive in one universe, and dead in the other.

I believe far too much is read into the so-called "collapse of the wavefunction". Physicists go to quite fantastic lengths to explain the phenomenon including postulating parallel universes! But while we can explain the process by an existing physical principle (increasing entropy and the thermodynamic arrow of time - discussed immediately above), I do not believe we should not resort to introducing unnecesarily fantastical solutions (see Is Big Physics peddling science pornography?, also, according to H. Dieter Zeh in his arXiv paper Quantum Theory and Time Asymmetry quant-ph/0307013: "The existence of many unobserved world components - postulated in this interpretation - is usually regarded as an unnecessary and extravagant complication.") Also, the direction of the thermodynamic arrow of time is defined by the very special initial conditions of the universe which provides a very natural solution to the question of why entropy increases in the forward time direction, but what is the cause of the time asymmetry in the Many Worlds interpretation, i.e., why do universes "split" only in the forward time direction?

To my mind, the Many Worlds interpretation seems very much a product of the fifties. Recent results in quantum decoherence have given us new insights into the quantum measurement problem, and there is no longer any need to propose parallel universes to explain the process. As was explained previously on this page, interference terms get dissipated out into the wider environment and become effectively undetectable. We do not need to propose that they teleport into a parallel universe - the terms remain firmly in this universe.

As was explained in the main text, there is now experimental support for this decoherence viewpoint. If particles can be isolated from the environment we can manage to view multiple interference superposition terms as a physical reality in this universe. For example, the electric current being made to flow in opposite directions (see this Physics World article, which also explains Schrödinger's cat in terms of decoherence), or the research at NIST which has created an atom in two places at the same time (see this excerpt from Jim Al-Khalili's Quantum here, or see this press release from NIST). If the interference terms had really escaped to a parallel universe then we should never be able to observe them both as physical reality in this universe.

Decoherence in an Ensemble of Particles

So decoherence solves the mystery of apparent wavefunction collapse, and also explains why we do not see superposition states in macroscopic objects, but it does not explain which particular eigenstate is selected during the "measurement" process. As explained in the page on The Quantum Casino, the selection of a particular eigenstate is governed by a purely probabilistic process, so in order to analyse this probabilistic behaviour we should consider a number of quantum particles in a similar state (called an ensemble), and then we can use a useful statistical tool called a density matrix.

Let us consider a ensemble of particles in a box. The whole box can then be treated as a single quantum system. When we extract a particle from the box and measure it we find it to be either "blue" or "green", say. Before measurement, this system can then be in one of two states (see this paper by John Boccio):

• A pure state - each of the particles is in the same state with the same state vector. For example, let's suppose all the particles in our are in the same superposition state before measuring, an equal superposition of blue and green.
• A mixed state - the particles are all in different classical states, i.e., the particles are all either blue or green: no particles are in superposition states. This is just a classical mix of blue and green particles.

When we extract particles from the box one-by-one and measure each particle to determine if it is "blue" or "green" we find that, in both the case of the pure state and the mixed state, 50% of the particles measure as "blue" and 50% of the particles measure as "green". So, after measurement, the two quantum systems appear to be identical. However, the states of the two systems before measurement were clearly different: the pure state could be described by a single state vector (all of the particles were in the same superposition state). The mixed state, on the other hand, could not be described by a single state vector (because the particles were not all in the same state: some were green, some were blue). The statistical properties of both systems before measurement, however, could be described by a density matrix. So for an ensemble system such as this the density matrix is a better representation of the state of the system than the state vector.

So how do we calculate the density matrix? The density matrix is defined as the weighted sum of the tensor products over all the different states (see the page on The Quantum Casino for a description of the tensor product, also see here for more details):

Where p and q refer to the relative probability of each state. For the example of particles in a box, p would represent the number of particles in state , and q would represent the number of particles in state .

Let's imagine we have a number of qubits in a box (these can take the value or , see the page on Quantum Entanglement). Let's say all the qubits are in the following superposition state:

In other words, the ensemble system is in a pure state, with all of the particles in an identical quantum superposition of states and . As we are dealing with a single, pure state, the construction of the density matrix is particularly simple: we have a single probability p, which is equal to 1.0 (certainty), while q (and all the other probabilities) are equal to zero. The density matrix then simplifies to:

This state can be written as a column ("ket") vector (see back to the page on The Quantum Casino for a discussion of bra-ket notation). Note the imaginary component (the expansion coefficients are in general complex numbers):

In order to generate the density matrix we need to use the Hermitian conjugate (or adjoint) of this column vector (the transpose of the complex conjugate of ). So in this case the adjoint is the following row ("bra") vector:

So, in this case, we can calculate the density matrix defined as the single tensor product:

What does this density matrix tell us about the statistical properties of our pure state ensemble quantum system? For a start, the diagonal elements tell us the probabilities of finding the particle in the or eigenstate. For example, the 0.36 component informs us that there will be a 36% probability of the particle being found in the state after measurement. Of course, that leaves a 64% chance that the particle will be found in the state (the 0.64 component).

The way the density matrix is calculated, the diagonal elements can never have imaginary components (this is similar to the way the eigenvalues are always real - see back to the page on The Quantum Casino). However, the off-diagonal terms can have imaginary components (as shown in the above example). These imaginary components have a associated phase (complex numbers can be written in polar form). It is the phase differences of these off-diagonal elements which produces interference (for more details, see this extract from the book Quantum Mechanics Demystified here). The off-diagonal elements are characteristic of a pure state. A mixed state is a classical statistical mixture and therefore has no off-diagonal terms and no interference.

So how do the off-diagonal elements (and related interference effects) vanish during decoherence?

The off-diagonal (imaginary) terms have a completely unknown relative phase factor which must be averaged over during any calculation since it is different for each separate measurement (each particle in the ensemble). As the phase of these terms is not correlated (not coherent) the sums cancel out to zero. The matrix becomes diagonalised (all off-diagonal terms become zero). Interference effects vanish. The quantum state of the ensemble system is then apparently "forced" into one of the diagonal eigenstates (the overall state of the system becomes a mixture state) with the probability of a particular eigenstate selection predicted by the value of the corresponding diagonal element of the density matrix.

Consider the following density matrix for a pure state ensemble in which the off-diagonal terms have a phase factor of :

I have written a JavaScript program to show how the decoherence process averages over the off-diagonal terms of the density matrix shown above. Click the Start Decoherence button below to start the sequence. The process averages over the ensemble of particles, each iteration adding a new particle with a randomly-selected value of . As each new particle is added and averaged, the off-diagonal interference terms reduce, eventually vanishing to zero:

	1	0.7071 +0.7071 i
	0.7071 -0.7071 i	1

(If you find one of the numbers fails to decrease to zero then it means there has been a rupture in the very fabric of spacetime and the universe will explode in 30 seconds).

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