Review of Basics and Elementary introduction to quantum postulates

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Review of Basics and Elementary introduction to
quantum postulates
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Requirements On Mathematics Apparatus

• Physical states
• Mathematic entities
• Interference phenomena
• Nondeterministic predictions
• Model the effects of measurement
• Distinction between evolution and measurement

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Whats Quantum Mechanics

• A mathematical framework
• Description of the world known
• Rather simple rules
• but counterintuitive
• applications

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Introduction to Linear Algebra

• Quantum mechanics
• The basis for quantum computing and quantum
  information
• Why Linear Algebra?
• Prerequisities
• What is Linear Algebra concerning?
• Vector spaces
• Linear operations

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Basic linear algebra useful in QM

• Complex numbers
• Vector space
• Linear operators
• Inner products
• Unitary operators
• Tensor products

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Dirac-notation Bra and Ket

• For the sake of simplification
• ket stands for a vector in Hilbert
• bra stands for the adjoint of
• Named after the word bracket

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Hilbert Space Fundamentals

• Inner product space linear space equipped with
  inner product
• Hilbert Space (finite dimensional) can be
  considered as inner product space of a quantum
  system
• Orthogonality
• Norm
• Unit vector parallel to v?

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Hilbert Space (Contd)

• Orthonormal basis
• a basis set where
• Can be found from an arbitrary basis set by
  Gram-Schmidt Orthogonalization

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Inner Products
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Inner Products

• Inner Product is a function combining two vectors
  
• It yields a complex number
• It obeys the following rules

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Unitary Operator

• An operator U is unitary, if
• Preserves Inner product

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Tensor Product

• Larger vector space formed from two smaller ones
• Combining elements from each in all possible ways
• Preserves both linearity and scalar
  multiplication

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Qubit on Bloch Sphere
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Mathematically, what is a qubit ? (1)

• We can form linear combinations of states
• A qubit state is a unit vector in a two
  dimensional complex vector space

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Qubits Cont'd

• We may rewrite as
• From a single measurement one obtains only a
  single bit of information about the state of the
  qubit
• There is "hidden" quantum information and this
  information grows exponentially

We can ignore eia as it has no observable effect
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Any pair of linearly independent vectors can be a
basis!
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Measurements of the same qubit in various bases
1/?2
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Bloch Sphere
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Measurements
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AXIOMS OF QUANTUM MECHANICS
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Postulates in QM

• Why are postulates important?
• they provide the connections between the
  physical, real, world and the quantum mechanics
  mathematics used to model these systems
• - Isaak L.
  Chuang

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Physical Systems - Quantum Mechanics Connections
Postulate 1 Isolated physical system ? ? Hilbert Space
Postulate 2 Evolution of a physical system ? ? Unitary transformation
Postulate 3 Measurements of a physical system ? ? Measurement operators
Postulate 4 Composite physical system ? ? Tensor product of components
entanglement
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Summary on Postulates
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Postulate 3 in rough form
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From last slide
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Manin was first
compare
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Postulate 4
You can apply the constant to each
Distributive properties
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Entanglement
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Entanglement
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Some convenctions implicit in postulate 4
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Entangled state as opposed to separable states
We assume the opposite
Leads to contradiction, so we cannot decompose as
this
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• Composed quantum systems results of Postulate 4

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Composite quantum system
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This was used before CV was invented. You can
verify it by multiplying matrices
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The Measurement Problem
Can we deduce postulate 3 from 1 and 2?
Joke. Do not try it. Slides are from MIT.
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Quantum Computing Mathematics and Postulates
Sources
Presented by Chensheng Qiu Supervised by Dplm.
Ing. Gherman Examiner Prof. Wunderlich
Anuj Dawar , Michael Nielsen

• Advanced topic seminar SS02
• Innovative Computer architecture and concepts
• Examiner Prof. Wunderlich

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• Covered in 2007, 2011