???
• A silver atom consists of a nucleus and 47 electrons
• 46 out of 47 electrons form a “cloud’’ without net angular momentum
• Magnetic moment m of the atom ] electron spin S of the 47th electron.
• Interaction energy of the magnetic moment with the magnetic field is – m . B
• The z-component of the force experience by the atom is
That’s just
great!
@#%!@!!
z
y
It’s really
simple!
If the atom is in state a|up> + b|down>
then it will take the
• upper path with probability |a|2
• the lower path probability |b|2
z
z
y
y
If the King’s men measure in the z-direction, and I
measure in the same direction, then I will get the exact
same result and they will set me free!
But what if they had measured
the x- or y-direction?
If the King’s men measure the y-direction and I
measure the z-direction, then both paths are
equally likely, since the measurements are
mutually unbiased; so they might kill me...
z
y
x
y
Actually, this has an interesting application. If I send a
stream of silver atoms to Tweedledum and encode up or
down in the x-, y- or z-direction, and Tweedledum
measures in one of these three directions at random, then
we can get a perfectly secure key exchange!
up = 1 and down = 0
Alice:
z=0
x=1
y=0
z=1
z=0
Twdl:
x
x
y
x
z
ok
ok
ok
You simply cannot reliably copy an unknown quantum state
because quantum mechanics is linear.
But what if
the evil
Queen Eve
just copies
the state of
the atoms?
We do not have time for this!
Obviously, you cannot solve the King’s
problem with a single silver atom!
Yes, but what
if we take two
silver atoms...
Tweedledeee,
Tweedledum!
We can describe the state of a silver atom by a vector in
C2, the state of two silver atoms by a vector in C4, and
the dimension doubles with each silver atom.
If we have a system in state v and another in state w,
then the combined system is in state
For example,
A linear combination of such tensor products spans the
space. Not all states are such products; if they are not,
then they are called entangled states.
Fine with me, as
long as we don’t
do any
experiments
z
x
y
If I use two silver atoms, then I can prepare them in
an entangled state, for example a superposition of 2
spin up and 2 spin down.
This is simply the state
I feel extremely
uncomfortable,
and its not just all
the talk about
quantum
mechanics.
It is curious that
for a = x,y, and z, isn’t it?
• We prepare the two silver atoms in the state
• The King’s men measure the state of one
atom, so the state will collapse to
•We need to device a measurement of both atoms such that the
result specifies a function f from bases to measurement values
• If the King reveals the basis a, then f(a) should give the value b
which has been measured by the King’s men.
So suppose we have a function
f:{x,y,z}->{0,1}
then we can define an entangled state by
These state will define our measurements
of the states.
Did you notice that we average over the
three bases?
• Do sets of n+1 mutually unbiased bases exist in dimension n?
• Connection with affine 2-designs
• State tomography
• Quantum cryptography
• The mean King’s problem can be generalized
• We have a solution whenever an affine plane exists
• Open problem:
•Are mutually unbiased bases related to finite geometries?
• Solutions
of quantum problems can shed new
light on classical problems
• Interesting fundamental problems
• Quantum algorithms CPSC 640