A silver atom consists of a nucleus and 47 electrons 46 out of 47 electrons form a “cloud’’ without net angular momentum Magnetic moment  of the atom ] electron spin S of the 47 th electron. Interaction energy of the magnetic moment with the magnetic field is – . B The z-component of the force experience by the atom is ???

z y It’s really simple! That’s just great! 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 y z 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?

z y y x 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...

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=0x=1y=0z=1z=0 Twdl:xxyxz okokok But what if the evil Queen Eve just copies the state of the atoms? You simply cannot reliably copy an unknown quantum state because quantum mechanics is linear.

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 C 2, the state of two silver atoms by a vector in C 4, 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. I feel extremely uncomfortable, and its not just all the talk about quantum mechanics. This is simply the state

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