Quantum Entanglement
Part 3 of "An Introduction to Quantum Reality"
It is possible for a particle to interact with another particle in such a way that the quantum states of the two particles form a single entangled state. The definition of an entangled state is that it is not entirely independent of other states: its state is dependent on another state in some way. Because of this dependency it is a mistake to consider either state in isolation from the other. Rather we should combine the states and treat the result as a single, entangled state.
For example, a light beam is a composed of a stream of photons. The direction of light's electric field is its direction of polarization. The polarization direction of a photon can be at any particular angle, for example "vertical" or "horizontal". It is possible to generate a pair of entangled photons if, for example, a laser is shone at a crystal. In that case a single photon can split to become two photons. Each photon produced in this way will always have a polarization orthogonal to the other photon (see this Physics World article or this Laser Focus World article). For example, if one photon has vertical polarization then the other photon must have horizontal polarization (this is due to the law of the conservation of angular momentum: angular momentum of the system before the split must equal the angular momentum of the system after the split).
So if two people each receive one of the entangled photons and performs a measurement, they will find that the other person's photon has orthogonal polarization. There is an apparent connection between the particles, no matter how far apart they are taken (see this interview with Anton Zeilinger about "spooky action").
"I cannot seriously believe in quantum theory because it cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance." - Albert Einstein
The EPR Paradox
Einstein was never happy with the implications of quantum theory and at the legendary 1927 Solvay conference (and throughout the late 1920s) he proposed several thought experiments which he believed revealed flaws in the theory. However, all of these objections were successfully refuted by Niels Bohr, and a humbled Einstein went home to lick his wounds. However, in 1935 Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) described a thought experiment commonly referred-to as the EPR paradox, the implications of which shook quantum theory to its core.
Einstein's great objection to quantum theory came from its denial of physical reality before observation (quantum theory says that only after we measure a property value of a particle does that property gain physical reality - before we measure it we must consider it to be in a superposition state). We often quote Einstein's rejection of quantum indeterminacy: "God does not play dice", but his less-quoted objection to quantum theory's denial of physical reality reveals his more serious concern: "I like to think the moon is there even if I am not looking at it."
In order to dispute quantum theory on this basis, the EPR paper contained - for the first time in the history of science - an operational definition of physical reality:
"If, without in any way disturbing a system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity."
- Definition of physical reality from the EPR paper.
Let's say the same thing in other words: If a physical property of an object can be known without it being observed, then that property could not have been created by observation. If it wasn't created by observation, it must have existed as a physical reality before its observation.
Einstein believed that quantum entanglement could be used to reveal a flaw in quantum theory, because he thought that quantum entanglement could determine physical reality before observation - at odds with the principles of quantum theory. Consider two entangled photons, one of which is sent to observer Alice, and the other is sent to observer Bob (see the diagram below). The two observers could be a considerable distance apart. Now remember from the discussion at the top of this page that two entangled photons must have orthogonal polarizations. So when Alice measures the polarization of her photon and finds it to be, say, vertically polarized we instantly know that Bob's photon will have horizontal polarization - even though Bob has not yet measured it!
But quantum theory says that before Bob measures his photon it can have no defined value for its polarization property - it is in a superposition state. Only when Bob measures it does its value become physically real. How then can we know the result Bob will obtain before Bob measures it? The solution according to quantum theory is that it is the measurement of Alice which collapses the wavefunction of both Alice's and Bob's photons. It is the observation of the polarization of one of the photons as being, say, vertical that instantaneously collapses both photons, resulting in Bob's photon having horizontal polarization.
However, Einstein realised that such instantaneous communication of polarization value between the two photons was forbidden by his own theory of Special Relativity (nothing travels faster than light). Hence, Einstein believed that the model of quantum mechanics was incomplete: it did not describe the physical reality of Bob's photon before observation.
Einstein believed the correct way out of this paradox was to assume that Bob's photon (and all particles) possessed some sort of fixed properties which were hidden from our view (generally referred-to as hidden variables). No faster-than-light communication is then required: the particle properties were set when the particles were created. Crucially, though, this would mean the particles possessing more information than quantum theory said they should have. If particles had these hidden variables then quantum theory was wrong.
An associate of Bohr tells that "this onslaught came down upon us like a bolt from the blue. Its effect on Bohr was remarkable ... as soon as Bohr heard my report of Einstein's argument, everything else was abandoned."
Bell's Inequality
In 1964, John Bell devised an ingenious test for the existence of hidden variables. Bell's theorem (which is commonly called Bell's Inequality) has been called "the most profound discovery of science" (see here).
Bell showed that for a group of objects with fixed properties A, B and C, the number of objects which have property A but not property B plus the number of objects which have property B but not property C is greater than or equal to the number of objects which have property A but not property C.
This can be written more compactly as:
Number(A, not B) + Number(B, not C) >= Number(A, not C)
An easy-to-understand version of this inequality is provided by David M. Harrison of the University of Toronto (see here). Let's consider our collection of objects with fixed properties to be a collection of people. And let their fixed properties be the following:
- A: Sex ("Male" or "Female")
- B: Height (over 5' 6" ("Tall") or under 5' 6" ("Short" - don't be offended!))
- C: Eye colour ("Blue" or "Green")
Then, no matter which group of people you are dealing with, you are always able to issue the following statement (inequality): "The number of short males plus the number of tall people, male and female, with green eyes will always be greater than or equal to the number of males with green eyes. I absolutely guarantee that for any collection of people this will turn out to be true."
That's always true. Isn't that amazing? That's a bit of quantum mechanics you can try out at your next party!
It's relatively simple to prove this. Note that every person can be classified into one of the following eight groups:
Referring to this diagram, Bell's Inequality is saying that:
(Group 1 + Group 2) + (Group 4 + Group 8) >= (Group 2 + Group 4)
Which, if you study it, is clearly always going to be true.
So Bell's Inequality will always hold true for normal, everyday objects with fixed properties. But now let's analyse the situation for quantum particles. Let's consider the polarization (more commonly called "spin") of a photon as the property to be measured. We now find we have a limitation imposed by quantum mechanics: the Heisenberg Uncertainty Principle says we cannot obtain the correct value for the spin in two different directions, for example, we cannot know the spin of a particle in both the 90° direction ("up") and the 45° direction at the same time. So how can we perform a test of Bell's Inequality on a particle? This is where quantum entanglement appears to come to our rescue, suggesting we can find the values of two properties if we have two entangled particles. Let's treat "spin up" as our Property A, and "spin 45°" as our Property B. If we measure one entangled particle for Property A then it would appear we can test the other still-unmeasured second particle for Property B (note: this would imply we have somehow "beaten" the Uncertainty Principle).
Referring back to our original statement of Bell's Inequality:
Number(A, not B) + Number(B, not C) >= Number(A, not C)
We will have to divide our quantum particles into three groups (called ensembles), and extract entangled pairs of particles from each group:
Because we cannot measure all three of the properties of each particle (i.e., we cannot measure A, B, and C for each particle - we can only measure the values of two of the properties), we cannot conclusively say if the inequality is broken or not. The best we can do is run the test on an ensemble of many thousands of particles and consider the statistics of the results. The first published experiment was by Clauser, Horne, Shimony and Holt in 1969 using photon pairs (with the different properties corresponding to polarization angles of 0, 45°, 22.5° and 67.5° - see here) and it was found that the statistics strongly suggested that the inequality was, indeed, violated.
What does this mean? Well, taking our example given earlier of a collection of people, considering the right-hand side of that inequality, in order for the inequality to be violated a person would have to be more likely to have green eyes if the other (entangled) person had been found to be a male. In other words, the property measurement was dependent on the type of observation performed on the other entangled particle: there is an instantaneous connectedness between the particles. What happens to one particle can instantaneously affect the other (as a result, it shows that we failed to "beat" the Uncertainty Principle - we cannot get a true measurement of two property values: when we measure one property, we "poison" the other reading - see here).
Summary
To sum up, Einstein believed the following:
- There was no "spooky action-at-a-distance" which apparently violated Special Relativity (i.e., there was locality).
- Objects had definite reality, properties independent of observation.
However, the violation of Bell's Inequality reveals that the reverse is true:
- There is a strange connection between particles which instantaneously informs the undisturbed particle of the type of measurement just carried out on its partner (however Special Relativity is not violated because no information can be transmitted using this method).
The implications of this result will be considered further in the later page on Quantum Reality.
The Leggett Inequality
So Einstein believed in a theory in which particles had pre-existing hidden properties independent of measurement ("realism"), and there was no "spooky" action at a distance (i.e., there was "locality"). But the violation of Bell's Inequality showed there was, indeed, some form of spooky action-at-a-distance (i.e., there was non-locality). However, it did not go so far as to prohibit the possibility of some form of pre-existing hidden variables in particles (realism).
However, in 2003 Anthony Leggett devised an new inequality which - if experimentally violated in the same way as Bell's Inequality - would invalidate the possibility of hidden variables, showing that particles could not possess "realism" in the conventional sense. So Einstein's twin principles of locality and realism would both be shown to be violated.
Leggett's Inequality is based on the fact that if we have two numbers - A and B - which can both take the values of +1 or -1 then the following is always true:
(We could, for example, consider the +1 and -1 values as representing photon polarization either up (+1) or down (-1) for two particles, A and B).
It is easy to see that this is true just by setting A and B to either +1 or -1 and examining the four possible situations:
The inequality based on this principle was recently tested experimentally and, indeed, found to be violated (see arXiv paper quant-ph/0704.2529v1 for details of the experiment and a description of Leggett's Inequality, also see this Physics World article or this New Scientist article). So it would appear that Einstein's twin pillars of realism and locality both come crumbling down.
(However, in the page on Quantum Reality we will see that it might be possible to consider the quantum superposition state as "reality before observation". In that case, what Leggett's Inequality reveals is that "reality before observation" - our A and B photons above - cannot be a clear-cut +1 or -1 form (for example, photon polarization up or down). Instead, any reality would have to be a superposition mix of the two values: +1 and -1. This eliminates most conventional views of realism. However, we already knew from experimental results such as interference effects in the double-slit experiment that particles behave as though they are in a strange quantum superposition state before observation, so I'm not sure this tells us anything new. Any "reality before observation" would have to be a rather peculiar "multi-valued" reality, and Leggett's Inequality would not appear to eliminate pre-existing reality of this form (for more on multi-valued reality see the section "Beyond Copenhagen" on the Quantum Reality page).
Qubits
The secret for analysing entangled systems is that you can no longer talk of the wavefunction for just a single localised particle, you have to to talk of one single wavefunction for the entire system.
It's quite useful at this point to introduce another useful piece of common notation. In the
previous page on The Quantum Casino we were
introduced to the common "bra-ket" notation, with a state being denoted by
. If we have two states,
and
, then the joint state is the
tensor product of the two states (see back to the explanation of the tensor product in
The Quantum Casino). The resultant, single state is then:
For an example, consider a particle which, after measurement, can be in one of two states:
or
(a particle such as this is known
as a Qubit and is used in the exciting new field of
Quantum Computing).
The orthonormal basis of the Hilbert state space
should therefore have two vectors representing the
and
states (a two-dimensional Hilbert space is denoted by H2 -
see back to the page on
The Quantum Casino for an explanation of Hilbert spaces and
the orthonormal basis):
Before measurement, and "wavefunction collapse", the qubit's state could be any mix (superposition)
of these two states. But after measurement, the qubit's state is always found to be in state
or
(the superpositions apparently disappear).
What happens if our qubit is a photon and has an entangled partner photon (as explained earlier)? You might think each photon has its own state, and hence each photon would have its own separate Hilbert state space:
That's wrong.
Remember that the states of two entangled photons are dependent on each other. For our
entangled qubits, let's say that if the first qubit is in state
then the second qubit is constrained to be in state
as well, and if the first qubit is in state
then the second entangled qubit is constrained to be in state
as well.
Now instead of having two separate state spaces for the two qubits we have one entangled
state space, and instead of one wavefunction for each qubit we now have one wavefunction for the
entire system (composed of two qubits).
In order to construct the entangled space, we first see that each of the two vector spaces is spanned by similar sets of two basis vectors:
The resultant single, joint state space can be constructed by taking the tensor product of the two sets of basis vectors of the two original state spaces (the tensor product of two vectors is a matrix - see back to the discussion of bra-ket notation in The Quantum Casino). This is achieved by performing a matrix multiplication of the two sets of basis vectors:
We can see that the original two-dimensional state spaces combine to produce a single four-dimensional state space:
So the resultant set of four basis vectors for the new entangled space is:
Note that with just one entanglement the Hilbert space has rapidly increased from two dimensions to four dimensions. This rapid increase in the dimensions of the Hilbert space due to entanglement will be shown to be key to what really happens during "wavefunction collapse". This will be explained in the next page on Quantum Decoherence.
For more on qubits and this method, see this PowerPoint presentation.
Separable and Entangled Systems
After receiving the comments of AKP below, I decided it would be useful to clarify the distinction between separable and entangled systems.
Let's consider eye colour (which can be either blue or green) of two non-related people. The two people could be treated as a single system, in which case the resultant combined system (I see Wikipedia call it a composite system here) could have four possible states for eye colour (essentially, each person is like a qubit, and we are dealing with a two-qubit system):
It is possible to decompose the system because there was no dependence between the eye colour of person 1 and the eye colour of person 2: they were completely independent systems. However, if we now consider the situation of identical twins we find there is now a dependence: the eye colour of person 1 depends on the eye colour of person 2 - they are identical.
It is now not possible to decompose this system into two separate systems (people) because of that dependence between the two twins. For example, you are no longer free to select blue eyes for person 1 and green eyes for person 2 - that combined state is forbidden. For this reason, this combined system with the two twins is non-separable: it is an entangled state. So this is the reason why an entangled two-qubit system cannot be decomposed into two independent one-qubit systems.
For a mathematical description of this, see the PowerPoint presentation, especially pages 16-20 on "Two Qubits". On page 20,
"Two Qubits, Entangled", you could think of the 
entangled state as representing
the situation when both identical twins have blue eyes, and the 
entangled state as representing the situation when they both have green eyes.
Comments
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maybe you could expand the heisenberg uncertainty comments into a paragraph. i think its important to talk about the difference between two conjugate measurements on one particle and these measurements being performed on separate particles from the same ensemble. and being a statement about standard deviations in measurements on two different groups of particles, the principle is not telling us how one measurement will "disturb" the results of a second conjugate measurement.
regarding particle spin measurements, if i have an ensemble of x-spin-up particles and i measure y-spin and z-spin each on a half of the ensemble, dont i get 1/2, 1/2 in each case? isnt this perfectly precise? what is lower bounded here is the product of the std devs of these two probability distributions. again being measurements on different particles.
i really liked the clarity of the bell inequality discussion. it would be cool to see a picture like the green eyes, etc. with the quantum tests inserted.
jmahoneyatphysicsdotucdavisdotedu - john, 29th March 2007
Regarding your comments about particle spin, I'm really referring to what might be called a "generalized Uncertainty Principle" - not the same as the complementary pair idea (see http://en.wikipedia.org/wiki/Spin-1/2 ). This just states that you can't measure all spin values simultaneously: taking a measurement along one axis destroys the information in the other axes. - Andrew Thomas, 30th March 2007
"then the entangled state is the tensor product of the two states (see back to the explanation of the tensor product in The Wavefunction). The resultant, single entangled state is then..."
Perhaps you mean "the composite state" or similar instead of "entangled state"? Keep up the good work! - AKP, 16th June 2007
But you're right - I shouldn't have described the entangled state space as resulting from the tensor product of the two state spaces. I've changed the wording to "joint state space" instead of "entangled state space". Thanks. - Andrew Thomas, 18th June 2007
Number(A, not B) + Number(B, not C) >= Number(A, not C)
with C:Eye colour ("Blue" or "Green") but "not C" is "not Blue" rather than "Green" as you have it later. Blue and green are not the only eye colours.
B: Height (over 5' 6" ("Tall") or under 5' 6" ("Short")) is OK since it covers everyone
A: Sex ("Male" or "Female") could also be dubious. - John Middlemas, 22nd July 2007
So if property A is "true" we're actually saying it's a male, and if property A is "false" then it's a female. And when I say "The number of objects which have property A (i.e., property A is true) but not property B" I mean the number of short males.
If instead we consider the "The number of objects which do NOT have property A but DO have property B" it would be the number of tall women. I hope that's clarified what I meant in the text (or maybe not!). - Andrew Thomas, 30th September 2007
This is the MOST LUCID site I have found on QM intro principles, great work !
One thing I don't understand is the association of the 3 Bell properties ( e.g. A,B,C) with the photon polarization ( 0,45, and 90 ), particularly 45 degrees, as both articles which talk about photon entanglement generation reference vertical/horizontal polarization pairs that are 90 degrees related. How do we get the 45 degree polarized photons, and how are they related to the entanglement state ? - Dan, Philadelphia Pa. USA, 19th February 2008
Also, I'm wondering if the Second Law of Thermodynamics can be explained in a similar way with quantum methods, especially taking entanglement into account? - DJ Fadereu, 21st March 2008
First of all - thanks for such a wonderful set of articles on QM for the layman.
I had a question on entanglement: is it possible to determine experimentally whether a particle is part of an entangled pair or not, if you have no idea of the history of that particle?
Thanks,
Gopi - Gopi, 5th May 2008
In quantum mechanical terms, it means that the value we get when we take a measurement can depend on the type of measurement we perform, and can even depend on the value we obtain when we measure its entangled twin ... many thousands of miles away! - Andrew Thomas, 15th May 2008
A few ideas... By presupposing that blind chance established the universe, scientism denies that there is rationality behind nature. Reason cannot possibly comprehend a non-rational universe. To assume that you have already reached the limits of knowledge is either 'weak conceit' or 'ill-applied moderation'. (Francis Bacon) The universe is neither eternal nor infinite, yet it is real and good. This is a necessary presupposition of science. - Addamstaft, 9th June 2008
Suppose you had a pair of socks ('entangled particles') and sent one off on a spaceship to Andromeda. If the Andromedan found it was a right sock, why should it imply "spukhafte Fernwirkung" when we (obviously, I would have thought) immediately find the earthbound sock to be left? - John Marks, Gisborne, Newzealand, 3rd July 2008
Imagine we shine a light beam through a beam splitter so that half goes to one planet, and the other half of the beam goes to another planet many light years away. Now we turn down the intensity of the light so that only one photon is emitted at a time and passes through the beam splitter. If we have two experimenters - one on each planet - and they try to detect the photon then only one experimenter will detect the photon, i.e., the photon has taken just one path to one of the planets. This is as you would expect. But now imagine if the experimenters do not try to detect the photon but instead both experimeters reflect their received light beams onto a third planet (i.e., both beams go to the same third planet). Then it would be possible to create an interference pattern on that third planet like in the double-slit experiment - even though we are only dealing with a single photon. In other words, the photon has apparently travelled both paths to both planets! That's the only way an interference pattern could be produced.
This shows that before we measure a particle we should consider it to be in a quantum superposition of all possible states - both left sock and right sock at the same time! - Andrew Thomas, 3rd July 2008
Thanks for your amazingly speedy reply. I appreciate and understand what you're saying, in particular the weirdness of the double-slit experiment with a single photon producing interference patterns.
Instead of socks, let a pair of 'gewirrt' (entangled) photons, such as might be created from the annihilation of an electron, be the two travellers, one staying here (reflected round a bathtub, say) and the other going off to Andromeda. As I understand it, each photon will have an opposite spin, let us call them 'right' and 'left', though we would not know which has which.
Then Schroedinger's phenomenon (of entanglement) seems to say that, if I measure the earth-bound photon to have a left-hand spin, the Andromedan must find his to have a right-hand spin. And, analogous to the socks, there would be nothing mysterious about this and no need to invoke Einstein's 'spukhafte Fernwirkung'.
Certainly, as beautifully described by Feynmann and others, the double-slit experiment encapsulates the mystery of wave-particle duality. But there is nothing there about spukhafte Fernwirkung and I don't understand why this problem is raised. In the example you give of the single photon interfering with itself, you basically describe the double-slit experiment. What puzzles me is the invocation of Einstein and Schroedinger's spukhafte Fernwirkung: it seems to me unnecessary but I assume I haven't understood the problem posed by Einstein et al. and Schroedinger back in 1935. How is the problem that Aspect tested different in principle from the double-slit experiment? Why is there the concern over faster than light communication when the problem is conceptually no different from a pair of socks? - even though they may be quantum socks! - John Marks, 3rd July 2008
But when we measure one entangled particle it has to take a determined (fixed) value. Which implies that the other particle instantaneously also has to take the opposite value. So Einstein suggested there had to be instantaneous communication between the two. - Andrew Thomas, 3rd July 2008
Thanks again. Yes, I understand that, although Schroedinger's cat is neither alive nor dead until we observe it, it is undoubtedly alive or dead when we do. Similarly, our photons are neither 'right' nor 'left' but we do know they come in such pairs. Before observation, there is a sense in which each is both right and left (Schroedinger superposition) or, at the very least, undetermined. But measurement of one of a pair determines the other by logic not by supraluminal communication nor "entanglement".
Please send me off to a lower class if you think I'm being tiresome or stupid, but I do not see the "paradox" in the EPR paradox. - John Marks, Gisborne, Newzealand, 4th July 2008
You say: "But measurement of one of a pair determines the other by logic not by supraluminal communication nor "entanglement". Yes, indeed, by logic you know that if you find your particle "spin up" then you know the other particle will be found "spin down". If an observer on the other planet now measures his particle he will, indeed, find it "spin down". Now, if those particles had those fixed characteristics BEFORE anyone performed a measurement then that would be the least surprising experiment anyone has ever performed - as you suggest. But quantum theory says those particles DON'T have fixed characterisics - the characteristics are randomly chosen at the time of measurement. This is what Einstein found so hard to believe about the Copenhagen interpretation of quantum mechanics. Because if the Copenhagen interpretation was right and the characteristic is randomly chosen at the time of measurement, then when you measure your particle and find it "spin up" then this would require instantaneous communication to the other particle saying "You're no longer in an undetermined state, with a value to be selected at random - you now have to be spin down". - Andrew Thomas, 4th July 2008
I now see why Einstein couldn't swallow Bohr's assertion - "the Copenhagen interpretation". But he needn't have been worried because he beats Bohr at his own game. Bohr's quantum theory asserts the indeterminacy of the spin before observation. Such a thesis is in the same category as epicycles, ether and phlogiston: things no-one observes nor, in the case of Bohr's indeterminate quantum state, can it be observed because it is defined as the state before observation.
Observation, i.e. empirical experiment, demonstrates the "conservation of spin" in photon pairs. You can't in principle know the spins beforehand but, although randomly assigned, the assignment takes place in pairs. This pairwise assignment at a pair's creation may be A spin-up and B spin-down or, randomly, vice-versa. Experiment shows this to be the case. - John Marks, Gisborne, Newzealand, 4th July 2008
I realize that Andrew is trying to say that experiments such as double slit conclusively indicate that particles "must" be in superposition, because supposedly there is no other way to explain the interference pattern. But I do not agree with the logic of this assertion.
Simply stated, because the notion of superposition (an unobserved proposition) may provide an explanation on a piece of paper, what's to say that there is not some other unrealized intrinsic property to matter light which manifests as the double slit effect? - Stefan Hostetler, 8th August 2008
I put some mathematical formulae on the site for people who want to dig deeper. This is an inclusive site for everyone - I'm not just aiming it at f%$K Wits! You're right, I want to "Combine the minds of a world". Nice line.
Heck, I've had some great comments on this site recently! - Andrew Thomas, 11th October 2008
I will inquire once agian on something soon enough, but first i shall give you some praise (as others have been) on such an awesome site! i have read some other things on it (even did some research on a lower level of this dastardly stuff) and on those a lot of the time its put out in a lot more complicated manor which takes a little longer to get my head wrapped around it, but here its well laid out and in a mixed level between avr. joes and smarty pants. which is awesome!
Ok, now for my point to raise. i think Einstien was on a correct path. Becuase these experiments havent proven that the protons don't gain their spin at the begining of the experiment and retain that same value throghout? i know that this interfers with particle uncertainty but the whole theory is based off of the theory that particles change thier actions based on observation. Doesnt this mean that if someone finds a way to prove that one observation isn't what chooses how a particle will act that the whole quantrum physics theory will crumble? i also realise this is extremely hard to figure out a way to do this, but the whole reason the theory stands is becuase apparently if we observe a particle it changes the action that it was going to perform. - Austin Cunningham, California, 19th October 2008
if bob and alice's photons are no entirely independent of each other, then why wouldnt measuring alice's photon be exactly the same as measuring bob's photon. Therefor by measuring alice's photon, we are actually measuring bob's, hence the photon changes from its superpositioned state, to a defined value.
Probably a silly question but thanks anyway, this site is really helpful and easy to follow along - corey, 5th November 2008