Download The Weird World of Quantum Information

The Weird World of Quantum
Information
Marianna Safronova
Department of Physics and Astronomy
What do we need to build a computer?
Memory
Initialization: ability to prepare one certain state
repeatedly on demand, for example put all to zero at the
start.
Ability to perform (universal) logical operations.
No or very small error rate (that can be fixed).
Ability to efficiently read out the result.
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Outline
Quantum Information: fundamental principles
(and how it is different from the classical one).
Bits & Qubits
Spin, atoms, and periodic table
Quantum weirdness: entanglement, superposition & measurement
Classical and quantum Logic gates
Cryptography & quantum information
Real world: what do we need to build a quantum computer?
Why quantum information?
Information is physical!
Any processing of information
is always performed by physical means
Bits of information obey laws of classical physics.
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Why quantum information?
Information is physical!
Any processing of information
is always performed by physical means
Bits of information obey laws of classical physics.
Why Quantum Computers?
Computer technology is
making devices smaller
and smaller…
…reaching a point where classical
physics is no longer a suitable model for
the laws of physics.
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Bits & Qubits
Fundamental building
blocks of classical
computers:
BITS
STATE:
Definitely
0 or 1
Fundamental building
blocks of quantum
computers:
Quantum bits
or
QUBITS
Basis states: 0 and 1
Superposition:
ψ =α 0 +β 1
A very brief introduction into quantum mechanics
Problem: indeterminacy of the quantum mechanics. Even if you know
everything that theory (i.e. quantum mechanics ) has to tell you about the
particle (i.e. wave function), you can not predict with certainty where this
particle is going to be found by the experiment.
Quantum mechanics provides statistical information about possible results.
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One of the biggest difference between
classical and quantum physics: superposition
If your quantum system (particle) has three possible
states,
ψ 1 , ψ 2 , and ψ 3
it may be in superposition of these three states
ψ = a1 ψ 1 + a2 ψ 2 + a3 ψ 3
If you make a measure the wave function will collapse to “eigenstate”
ψ 1 , ψ 2 , and ψ 3
The probability to “catch” particle in state 1 is
The probability to “catch” particle in state 2 is
The probability to “catch” particle in state 3 is
a1
a2
a3
2
.
2
.
2
.
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Bits & Qubits:
Superposition
primary differences
ψ =α 0 +β 1
Example: two spin states of spin ½ particle
Example: spin and measurements
In 1922, O. Stern and W. Gerlach conducted experiment to measure
the magnetic dipole moments of atoms. The results of these
experiments could not be explained by classical mechanics. First, let's
discuss why would atom poses a magnetic moment.
Even in Bohr's model of the hydrogen atom, an electron, which is a
charged particle, occupies a circular orbit, rotating with orbital angular
momentum L. A moving charge is equivalent to electric current, so an
electron moving in a closed orbit forms a current loop and this,
therefore, creates a magnetic dipole. The corresponding magnetic
dipole moment is given by:
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If the atom with a magnetic moment
a net force F,
is placed in a magnetic field B, it will experience
Stern suggested to measure the magnetic moments of atoms by deflecting atomic beam by
inhomogeneous magnetic field. In the experimental setup, the only force on the atoms
is in z direction and
The direction of magnetic moment in the beam is random, so every value of
in the
range
is expected. As a result, the deposit on the collecting
plate is expected to be spread continuously over a symmetrical region about the point of no
displacement.
Electronic configurations of atoms in Stern-Gerlach experiments:
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Conclusion: elementary particles carry intrinsic angular momentum S in addition to L.
Spin of elementary particles has nothing to do with rotation, does not depend on
coordinates
and
, and is purely a quantum mechanical phenomena.
S = ( Sx , S y , Sz )
Spin
, therefore
and there are two eigenstates
We will call them spin up
and spin down
.
Taking these eigenstates to be basis vectors, we can express any spin state of a
particle with spin
as:
ψ =α 0 +β 1
m=1/2
m=-1/2
Two states deflected
differently in magnetic
field.
In atoms, such states
have different energy
levels in magnetic field.
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Modern version of Stern-Gerlach experiment
Measuring expectation values of
Sx , S y , Sz
S = ( Sx , S y , Sz )
Note on spin quantum numbers: S and M (or Mz)
To fully described spin quantum number, one of the direction (z) is picked
Simulation:
https://phet.colorado.edu/en/simulation/legacy/stern-gerlach
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Electron spin and periodic table
Single electron states are labeled by quantum numbers: n, l, ml, s, ms
Rule:
In an atoms, all electrons have to differ in at least one quantum number
Electrons are fermions and have to be in different states – remember that
this leads to electron degeneracy pressure in white dwarfs.
n is principal quantum numbers, 1, 2, 3, 4, …
l is orbital angular momentum quantum numbers 0< l < n-1
ml is corresponding magnetic quantum number - l ≤ l ≤ l
m
s is spin s=1/2
ms corresponding
magnetic quantum
number - s ≤ ms ≤ s, so ms=-1/2, +1/2
H: electron is in n=1, l=0 state (1s)
https://phet.colorado.edu/en/simulation/build-an-atom
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