If there's one thing that makes Quantum Mechanics so quantum is Entanglement. Quantum Entanglement is at the heart of Quantum Mechanics. Here we will dive into this quantum world and untie it's nuts and bolt. I expect you to have some basic quantum mathematical background before marching forward. If you're ready, let's crack it up !
What is QM?
Before we plunge into the world of Entanglement, i would like to give a short overview about what QM is really about and how should you think about it. I discussed this in my previous blog too. QM is a very important mathematical tool that provides a description of the physical properties of nature at the scale of atoms and sub atomic particles. Having being said that you mustn't think that it is only limited to microscopic domain. It applies to macroscopic domain as well, it's just that the manifestation of QM is greater at the smaller scale. QM is a linear theory and the very exciting feature of this theory is that the experimental outcomes are described in terms of probabilities. Now let's build up ourselves into Entanglement.
Composite system
Combining system and forming bigger system is what we mostly do in QM. You could naively think this as combining two single particle to form a two particle system. Study of composite system provides us a better understanding about how intricate system can be derived from simpler ones.
Let's take Alice's and Bob's system. Alice and Bob have become near universal when we discuss about Entanglement. Now, we want to combine Alice's and Bob's system into a single composite system. Let's say Alice's system is described by a space of states Sa and Bob's system by Sb. Let's say Alice's system is a coin with two basis states H and T. This coin is a quantum coin which can be in a superposition of two states
αH |H> + αT |T>
Let's say Bob's system is a die. Bob's space of states would be six dimensional with the basis
|1> , |2> , |3> , |4> , |5> , |6>
Now we can combine the two systems into a single composite system with space of states Sab. It can be done by forming tensor product of Sa and Sb,
Sab = Sa ⊗ Sb
Such that the basis vectors would be
T1 T2 T3 T4 T5 T6
H1 H2 H3 H4 H5 H6
Now there are 12 basis for the combined system. Each basis represents a single state. For eg : H1 represents a state in which Alice's system shows H and Bob's system shows 1. And of course we can form superposition of these basis
αH1|H1> + αT2|T2>
The idea of Spin
When you hear the word spin, you think of an object rotating about an axis. The object has angular momenta and the direction of angular momenta is given by the right hand rule. And if the object is charged, it will produce a magnetic field. This is our classical intuition about spin.
But in QM, the idea of spin is quite subtle. I am not going to dive in too much into spin. I just want you to know that, particles like electrons have an intrinsic(meaning within it) angular momentum that we call spin. And of course spin is a vector meaning it has x, y and z components. The spin is either up or down. In QM, it can be in a superposition of being up and down
α|u> + β|d>
And of course we can form composite spin system
α1β1|uu> + α1β2 |ud> + ......
For more information on spin, you can refer to the famous Stern Gerlach Experiment.
Product state
The simplest type of state for the composite system is called a product state. A product state is the result of completely independent preparation of each system.
Let's say Alice prepares her spin in state
αu|u> + αd|d>
Bob prepares his spin in the state
βu|u> + βd|d>
We assume each state is normalized
αu*αu + αd*αd = 1
βu*βu + βd*βd = 1
One important feature of product state is each system has it's own normalization equation.
The product state describing combined system is
|Product state> = {αu|u> + αd|d>} ⊗ { βu|u> + βd|d>}
Where first factor represents Alice's state and the second represents Bob's. The main feature of product state is that each sub system behaves independently of other.
The emergence of Entangled state
The most general vector in the composite space of state is
ψuu|uu> + ψud|ud> + ψdu|du> + ψdd|dd>
The symbol ψ is just the complex coefficient like α and β. But unlike product state, entangled state has only one normalization condition,
ψuu*ψuu + ψud*ψud + ψdu*ψdu + ψdd*ψdd =1
Also, the space of state is richer than product space. Something new is going on, this new thing is called Entanglement.
An example of maximally entangled state is the singlet state,
|Singlet> = 1/√2 ( |ud> - |du> )
The triplets are also maximally entangled state
1/√2 ( |ud> + |du> )
1/√2 ( |uu> + |dd> )
1/√2 ( |uu> - |dd> )
The two fascinating thing about maximally entangled states are
- An entangled state is a complete description of the combined system. No more can be known about it.
- In a maximally entangled state, nothing is known about the individual sub system.
And that is the mystery of Entanglement, that you can know everything about the combined system yet know nothing about the individual sub system. For a person who grew up with classical intuition, this must feel absurd.
Classical correlation
Let's add a new member to the AB group, Sam (AB group meaning Alice Bob group). Sam has two coins a penny and a cent. Sam shuffles the two coins and gives one coin each to both Alice and Bob. What coin will Alice get is completely random. Then Alice goes to one end of universe and Bob goes to Nepal. They have synchronized their clocks and have agreed to look at their clock at the same instant. As soon as Alice reaches the end of universe and looks at her coin, she immediately knows exactly what coin will Bob see, even before Bob looks at his coin. What? Is this amazing? Have they done faster than light communication? Well no. Alice may know what coin will Bob see but she has no way to tell him. Until Bob looks at his coin, he doesn't know what coin he will get.
Let's do this experiment many times. Sam pains a σ = +1 on each penny and
σ = -1 on each cent. He then shuffles them and the same Alice and Bob going to different place and looking at their coins experiment is done. Then following facts will emerge,
- On average both Alice and Bob will get as many pennies as cent. Calling values of Alice's observation σA and Bob's σB
<σA> =0
<σB> =0
- For each trial, if Alice observed σ = +1 then Bob will observe σ = -1 and vice versa. In other words, the product σAσB always equals 1
< σAσB > = -1
Here, the product of average is not equal to the average of product
<σA> <σB> ≠ < σAσB >
The quantity < σAσB > -<σA> <σB> is called the statistical correlation between Alice's and Bob's observation. When the statistical correlation is non zero, we say that observations are correlated.
Suppose, you have a probability distribution P(a,b) for two variables a and b. Of the variables are correlated completely, then the probability will factorize.
P(ab) = P(a)P(b)
Where P(a) and P(b) are the individual probability for a and b.
In an entangled state the probability won't factorize. Entangled pairs are described in terms of correlation. Entanglement is conditional. The outcome of second depends on the outcome of first.
Spin Polarization principle
It states that for any state of a single spin, there is some direction for which the spin is +1. This means that the expectation value of the components must satisfy,
<σx>² + <σy>² + <σz>² = 1
Which tells us that not all the expectation values can be zero. This fact continues to hold for all product states. But it doesn't hold for the entangled state. For the singlet state, all the three expectation values are zero. This means the experimental outcome is equally likely to be +1 or -1. In other word, the outcome is completely random. Even though we know the exact state vector |singlet>, we know nothing at all the outcome of any component of either spin. You can study this further using the concept of density matrix.
Einstein and Entanglement
I know most of you were waiting for this part, the never ending Einstein and QM debate. In 1935 Einstein along with Nathan Rosen and Boris Podolski proposed a thought experiment in which they argued that the description provided by QM was incomplete. This thought experiment involved a pair particle separated by a distance. If the pair's entangled, a measurement on one particle would instantaneously affect the outcome of second particle. This violated Einstein's notion of locality and causality. He termed it as 'Spooky action at a distance'. The outcome of measurement on first particle is always random (as we learned above that the expectation values of all three component added to 0). Einstein argued that this probabilistic nature arose due to the incomplete information about the particle. There were some additional terms that were not taken into consideration. This idea famously is the hidden variable theory. But later an Irish physicist John Bell in 1964 published a paper that disproved this theory. The experimental version of Bell's theorem was performed by John Clauser in early 70's.
But what I really think is that Einstein understood QM better than anyone else. He always thought it as a correct but an incomplete theory. And what Einstein was really dissatisfied with QM was that observer seem to play a fundamental role. Observers are made up of the same things everything else is made of, so why should there be different laws for them.
Intuitive experiment
QM is not the most intuitive subject but let's have a mental visualization of what we are talking about( I happen to say that visualization is not the most elegant thing to do in QM but it's just to develop an idea of what is going on). We will now do the Quantum version of experiment described in the Classical correlation. Let's say Rafael has an entangled pair of photon. He randomly puts each photon in a box and gives it to Ashesh and Alisha. They both are blind folded at this time so they have no idea which one of the pair they have with them .Then Alisha remains in Earth while Ashesh goes to the Andromeda. At the same instant of time (taking account relativity) they both look in the box. Right at the moment Alisha opens the box and finds her photon spinning up, instantaneously the photon Ashesh has, snaps into spinning down state. Similar scenario if Ashesh observed his particle first. What's happened here is that observation on one particle had immediate and instantaneous effect on the other one even if they were separated by million miles. What? That is the miracle of entanglement which makes it one of the most interesting subject to study.
What Entanglement really is
Once the particles are entangled, they are described by a single wave function and the entangled pairs are described in terms of correlation. A single wavefunction governing two particle combined system (in our case) in which you can know everything about the combined system yet know nothing about the individual sub system is entanglement.
If you measure the spin in same axis you will find a strong correlation (if one is up another is down) but if you measure spin in different axis you won't find that correlation ( if one is up in X axis another is either up or down with 50% probability in Y and Z axis). And if you again measure the spin in the same axis, you again get this correlation ( one is up and another is down or vice versa). That is the beauty of entanglement and this is the thing that is very difficult to understand.
Entanglement doesn't violate special relativity because the experimental outcome is completely random. In entangled state, up down or down up is a single state. So, if Alice measures spin up the wave function spikes into the up down state which concludes that Bob would get down. If Alice measures up then Bob will always measure down and vice versa.
How is entangled pair created?
If two particles are created at the same instant from the same source, for example conversion of a single unpolarized photon into two polarized photons using non linear crystal. These two photons are strongly entangled with respect to each other. If you take two electrons in the ground state of He atom, they would be perfectly entangled. The entanglement happens by interaction or because of common origin. If you randomly put two electrons together, they won't be entangled necessarily. It's not difficult to create the entangled state but what's difficult is to preserve it for long time. You have to take into account several other things else any environmental influence will create new entanglement called decoherence, destroying initial setup.
Scope of Entanglement
Among the best known application of entanglement are superdense coding and quantum teleportation. Most researchers believe that entanglement is necessary to realize quantum computing. Entanglement is used in some protocols of Quantum Cryptography.
We still don't understand Entanglement completely. We see these effects in experiments but we don't know how that happens. It's the fact that' No matter how you set it up but the fact that you can set it up'. So this subject of demystifying entanglement requires more creative and enthusiastic minds and I hope that is going to be one of the guy reading this blog.
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