Monday, 24 February 2014

Understanding quantum entanglement

As mentioned before, a quantum computer exploits the rules of quantum theory for accomplishing feats that are considered impossible with conventional computers. Two features of quantum mechanics are often involved in achieving such feats: superposition, which was discussed in our last post, and entanglement, which is the topic here.

Quantum entanglement is considered to be one of the counterintuitive aspects of quantum mechanics. However it is not a particularly difficult concept to grasp if we start with the notion of correlation. Roughly speaking, entangled quantum systems are objects whose properties so strongly correlated that using a state to describe all of them as a single unit describes them better than assigning a state to individual parts.

To describe quantum entanglement in more detail, we need to introduce the idea of correlation between 2 or more objects. Suppose we have two coins. Each coin has two possible states: heads (H) or tails (T). Taken together, there are four possible outcomes when both coins are tossed: HH, HT, TH, and TTIn general, a correlation measures the probability we get the same outcome when we measure the same property for two or more systems.

We say that the coins are perfectly correlated if we can determine the state of one coin with certainty when are told about the state of the other coin. One somewhat contrived way to achieve this would be to join a pair of coins like in the diagram below, so that they end of heads or tails together. In this case, if the chance of landing heads or tails is the same, then the state of the two coins is described by 1/2 HH + 1/2 TT.

Two coins attached to each other by a stick so that when they are flipped they also land heads or tails together. We say that the outcomes of the flip is perfectly correlated.

Perfect correlation is usually better understood when two states involved are always opposite each other (in this case we usually say they are anti-correlated), like if I promise that two closed boxes contain a pair of socks and you open one to reveal a left sock, then you know that the other box held a right sock (if I was telling the truth).  The point remains that when objects are perfectly correlated or anti-correlated, information about some property of one can be used to determine the same property for the other. 

Let's consider the perfectly correlated qubits. Suppose we have two electrons, each with spin pointing up, which we write as the state |u),  or down, which we write as state |d). Taken together, we have four states that we can perfectly distinguished by measurement: |u,u), |u,d), |d,u), and |d,d), where |u,d) means the first electron has spin pointing up and the second electron has spin pointing down, and so on.

An example of a perfectly correlated state for this pair of qubits is 1/sqrt(2) |u,u) + 1/sqrt(2) |d,d). This is an example of an entangled quantum state for two qubits. Observe that it is a superposition state and we know from the previous post that this state leads to probabilities of a very different quality than what we get from a pair of random coins. However, that is still a somewhat mathematical way of looking at it.  Is there an experimental way to distinguish between this entangled state, and a correlated mixture of bits?

To find out, recall that for a qubit such as the spin of an electron, up and down can be measured along any direction in 3-dimensional space. Using the coordinates on a globe to label directions, with "up or down" going along the line between north and south pole, we can try to measure the spin along "|+) or |-)" direction on the equator. With 

|+) = 1/sqrt(2) |u) + 1/sqrt(2) |d), 
|-) = 1/sqrt(2) |u) - 1/sqrt(2) |d)

we can write |u) and |d) in terms of |+) and |-). Carrying out the calculation step-by-step, we have

|u,u) = |u)|u) = [ 1/sqrt(2) |+) + 1/sqrt(2) |-) ]  [ 1/sqrt(2) |+) + 1/sqrt(2) |-) ] 
          = 1/2 |+,+) + 1/2 |+,-) + 1/2 |-,+) + 1/2 |-,-)

where roughly speaking, what we did was multiply numbers with numbers and states with states. Similarly, we get

|d,d) = |d)|d) = 1/2 |+,+) - 1/2 |+,-) - 1/2 |-,+) + 1/2 |-,-).

Adding the two expressions, we see that 

|u,u) + |d,d) =|+,+) +  |-,-), 

meaning that if we had measured the spins in "|+) or  |-)" direction, we would still get correlated results. That is, if the first spin is in state |+), the second one will also be found in state |+), and similarly for |-).

If you don't find this to be surprising, remember how the "|u) or |d)" and "|+) or |-)" refer to two properties of an electron spin that obey an uncertainty principle, like position and momentum. Because we have so many different possible directions available, this means that we have an assortment of properties, any one of which can be used to find correlated results on spins measured in the same way. It is often in this sense that we say that a pair of entangled spins are not really two separate quantum systems but just a single one.

Also, if we can keep the entangled electrons sufficiently isolated from the environment, we can separate them as far apart as galaxies before measuring their spins and the outcomes will still be correlated. This means that quantum entanglement is not due to some force or influence being exchanged by the electrons on each other to keep them synchronized, since any such signal must ultimately be limited by the speed of light according to Einstein's relativity.

If each spin is measured along a different direction, there is some randomness to the outcomes so we wouldn't be able to guess accurately the result for one spin just from information of the other spin. This is why we cannot use entangled quantum states to send information faster than light, since any two people using entangled spins for example would have to measure them along the exact same direction, and guaranteeing that requires some for of communication between them. 

In those cases where the spins are measured along different orientations, there is still a way of showing through calculations that in general the correlations are stronger than one can achieve with classical bits of information, through violation of so-called Bell inequalities. 

A Bell inequality establishes some bound on the correlation between two systems if we assume that each system is local, which just means one system cannot suddenly affect the other if they are sufficiently far apart, and has properties that have well-defined values (that exist even if we don't measure them.) It can be shown that quantum states with some degree of entanglement can violate such a bound for some choices of measurement. 

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