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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 ???

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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

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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?

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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...

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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.

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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!

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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

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z x y

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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

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It is curious that for a = x,y, and z, isn’t it?

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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.

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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?

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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?

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Solutions of quantum problems can shed new light on classical problems Interesting fundamental problems Quantum algorithms CPSC 640

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