let's focus in on

I just say something on the blackboard and then I switch to the blackboard.

But the first thing is just to show you,

so we will be talking now about the very least possible conditions of minor spheres.

And first let me just recall one thing,

so let's look at Gaussian states.

It's very nice.

So Gaussian states are determined by three parameters,

I already said,

so let's start with x,

and then x for a one degree.

So this is b,

and c,

and c,

that brings us some properties that Gaussian states will fill,

and then I want to use the parameters,

is first that it will be a kind of a restricted Heisenberg principle.

If I multiply b, I get the same thing.

If I multiply c, well, it has to be always larger than h bar over 4,

which is a common for a Gaussian state.

I can also define the beauties,

so how pure the Gaussian state is,

that's going to be my beauty as the base of rho squared,

for a pure state rho is the projectile,

so rho squared is itself,

and the state of rho is 1,

so for a pure state the beauty is 1,

for a big state the beauty is 1,

and for a Gaussian state,

this beauty can be written always like one half of h,

and then the end will be minus c squared.

This can never be negative because of Heisenberg's uncertainty principle.

Then another parameter that will be usable,

is the so-called finite contingency length,

so basically this is a length scale that we want to characterize

how off-diagonal the density matrix is to the position basis.

So basically if you take the off-diagonal terms in that way,

so I'll say, okay,

what happens with a Gaussian state if I take the off-diagonal term at a distance,

basically x minus x divided by 2 from the diagonal,

then this can be as an interval written as this nice one,

as a number written here,

and this is minus x squared and this coherent length,

g squared,

and this g squared is given in a very compact and beautiful way,

but it is small.

I think that's going to be a very important parameter

because that's going to be telling me how large,

how expanded my weight is,

how continually expanded my weight is,