Hello, so today I wanted to finish this short chapter about how to measure entanglement

and then I want to go on to describe the measurement process.

Now with regard to measuring entanglement, the standard setting is we just have two subsystems,

A and B, that you can think of as being very far apart, but they have been in contact with

each other so there has been some interaction going on and you want to find out is the state

entangled, the joint state of A plus B, is it entangled, and if yes, how much entanglement

is there in this state.

So this would be A and that would be B and we are considering the density matrix of the

joint state that we want to find out ways to quantify entanglement.

Now the most important concept there is this notion of local operations and classical communication

because these are all the operations that you would think should not increase entanglement.

If you apply local, the unitary operations, either onto A or B or you do measurements

on A or B or you even reveal those results of measurements from A to B, that should not

allow you to increase entanglement.

So then the hope would be to use this to provide an ordering of states because if one state

can be produced out of another state by using local operations and classical communication,

then you would say apparently this state doesn't have more entanglement.

The catch is that sometimes there are states that cannot be compared, where you can't go

from one to the other and also you can't go back and so you don't know which one is more

entangled and there it helps to expand this notion and actually to consider many copies

of one state.

So you still have A and B as two subsystems, but then you don't take only one possibly

entangled state, but many copies of this entangled state have been prepared and then you want

to allow operations such as acting on both of these qubits, measuring them and again

allowing for communication and you want to find out whether this multi-copy state can

be transferred into multiple copies of another state.

Then you would take the ratio of the number of copies to find out which state is more

entangled.

So that has actually been the basis of definitions of entanglement measures and the question

is always you want to compare states with regard to their entanglement and what would

you compare against?

Probably you would want to compare against maximally entangled states, bell states.

And so for example one question you can ask is given some arbitrary state how many bell

states would I need to produce this arbitrary state?

Or else you could say given some arbitrary state if I apply very smart operations on

that state or more precisely on many copies of that state how many bell states could I

get out of it?

And these are just the two possibilities that people really have considered.

The first one is found entangled and closed and it's denoted by EC of rho where rho is

our arbitrary state of the total system A plus B. The idea is very simple in principle.

You say let me take one of those maximally entangled bell states for example phi plus

which you remember was the state up up plus down down and then I take many copies of that

state and I hope to be able to convert this via local operations and classical communications

into many copies of my target state rho.

And the smallest R for which this is possible then tells me something about what is the

cost of producing this target state.

And again last time I already told you that you would take the limit n tending to infinity

and that this operation may work only up to some little error that then however should

tend to zero and goes to infinity.

So this is a reasonable definition.