A silver atom consists of a nucleus and 47 electrons 46 out of 47 electrons form a “cloud’’...
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![Page 1: A silver atom consists of a nucleus and 47 electrons 46 out of 47 electrons form a “cloud’’ without net angular momentum Magnetic moment.](https://reader036.fdocuments.us/reader036/viewer/2022062620/551b51d3550346ae7a8b531b/html5/thumbnails/1.jpg)
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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 47th 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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zy
zy
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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zy
yx
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=0 x=1 y=0 z=1 z=0
Twdl: x x y x z
ok ok ok
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 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
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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