Quantum
Chapter 1
1.
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Computation is a physical process
R. Feynman noted that quantum systems appear to be exponentially hard to simulate
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with classical computers
Church–Turing thesis（A.M. Turing was a British mathematician）
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Turing machine consists of a tape divided into the squares, a scanner, and a dial
Two central ideas︰state vectors（describe the instantaneous state of the system）&
measurement（describe how information is extracted from the system）
Three important properties﹕interference, superposition, entanglement
Interference﹕Two–Slit Experiment
 Two–slit experiment（Fig. 1）
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3.
Quantum Physics & Church–Turing
Introduction︰Computation & Physics
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2.
Computation
Probability amplitude
Quantum Bits
 A quantum bit（“qubit”）is a two state quantum system, such as the electron in the
two lowest energy levels of a Hydrogen atom（Fig. 2）
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Bra∕Ket Notation：|•＞ is called a “ket”,＜•| is a “bra”, ＜•|•＞ is a “braket”
Geometric view point（Fig. 3）
A paradox：a qubit seems to contain an infinite amount of information, since it is
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in a superposition state
Solution：information can extracted by measurement. If measurement of
|ψ＞＝α|0＞＋β|1＞ results in |0＞, then the state |ψ＞ changes to |0＞ and if
the state is measured again w.r.t the same basis will return |0＞ with probability 1.
Conclusion：a qubit contains exactly the same amount of information as a
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classical bit
Hidden information
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4.
Multiple Qubits
 Individual state spaces of n particles classical combine through the cartesian
product. Quantum states, however, combine through the tensor product.
 Consider two qubits case. The state space of two qubits , each with basis﹛|0＞，
|1＞﹜, has basis ﹛|0＞⊙|0＞，|0＞⊙|1＞，|1＞⊙|0＞，|1＞⊙|1＞﹜ or
﹛|00＞，|01＞，|10＞，|11＞﹜
 Entangled state：for instance, the state |00＞＋|11＞ cannot be decomposed into
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separate states for each of the two qubits, i.e. we cannot find α, β, γ, δ such that
（α|0＞＋β|1＞）⊙（γ|0＞＋δ|1＞）＝|00＞＋|11＞
Measurement：any state of a two qubits system can be expressed as
α|00＞＋β|01＞＋γ|10＞＋δ|11＞. When the first qubit is measured w.r.t the basis
﹛|0＞，|1＞﹜,the probability that the result is |1＞﹜ is |α|＋|β|. To get the state
of system after measurement, we must renormalize so that the total probability
is 1.
Measurement gives another way of thinking about entangled particles. Particles
are not entangled if the measurement of one has no effect on the other.
Bell state
Einstein–Podolsky–Rosen（EPR）paradox（Fig. 4）：
An EPR pair （two maximally edtangled particles）is sent one each to Alice and
Bob. Alice and Bob can be arbitrarily far apart. Suppose Alice measures her
particle and observes state |0＞. This means that the combined state will now be
|00＞ and if now Bob measures his particle he will also observe |0＞.It appears that
this would enable Alice and Bob to communicate faster than the speed of light.
Figure 4： Gedanken experiment
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Solution：the values can be explained equally well by Bob’s measuring first and
causing a change in the state of Alice’s particle, as the other way around. This
symmetry shows that Alice and Bob cannot , in fact, use their EPR pair to
communicate faster than the speed of light.
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