Download Matter Waves and Uncertainty Principle

Quantum Mechanics for Quantum
Information & Computation
Anu Venugopalan
Guru Gobind Singh Indraprastha Univeristy
Delhi
_______________________________________________
INTERNATIONAL PROGRAM ON QUANTUM INFORMATION
(IPQI-2010)
Institute of Physics (IOP), Bhubaneswar
January 2010
IPQI-2010-Anu Venugopalan
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Real computers are physical systems
Computer technology in the last fifty years- dramatic
miniaturization
Faster and smaller –
- the memory capacity of a chip approximately doubles
every 18 months – clock speeds and transistor density
are rising exponentially...what is their ultimate
fate????
IPQI-2010-Anu Venugopalan
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Moore’s law [www.intel.com]
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The future of computer technology
If Moore’s law is extrapolated, by the year 2020 the basic
memory component of the chip would be of the size of an
atom – what will be space, time and energy considerations at
these scales (heat dissipation…)?
At such scales, the laws of quantum physics would come into
play - the laws of quantum physics are very different from
the laws of classical physics - everything would change!
[“There’s plenty of room at the bottom”
Richard P. Feynman (1969)
Feynman explored the idea of data bits the size of a single
atom, and discussed the possibility of building devices an
atom or a molecule at a time (bottom-up approach) nanotechnology]
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Quantum Mechanics
_______________________________
• At the turn of the last century, there were several
experimental observations which could not be explained by
the established laws of classical physics and called for a
radically different way of thinking
• This led to the development of Quantum Mechanics which
is today regarded as the fundamental theory of Nature
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Some key events/observations that led to the
development of quantum mechanics…
___________________________________
• Black body radiation spectrum (Planck, 1901)
• Photoelectric effect (Einstein, 1905)
• Model of the atom (Rutherford, 1911)
• Quantum Theory of Spectra (Bohr, 1913)
• Scattering of photons off electrons (Compton, 1922)
• Exclusion Principle (Pauli, 1922)
• Matter Waves (de Broglie 1925)
• Experimental test of matter waves (Davisson and Germer,
1927)
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Quantum Mechanics
___________________________________


Matter and radiation have a dual nature – of both wave
and particle
The matter wave associated with a particle has a de
Broglie wavelength given by
h

p

The wave corresponding to a quantum system is
described by a wave function or state vector

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Quantum Mechanics
___________________________________
Quantum Mechanics is the most accurate and
complete description of the physical world
– It also forms a basis for the understanding of
quantum information
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Quantum Mechanics
_______________________________________________________
Quantum Mechanics
Nature……..
–
most
successful
working
theory
of
The price to be paid for this powerful tool is that some of the
predictions that Quantum Mechanics makes are highly
counterintuitive and compel us to reshape our classical (‘common
sense’) notions.........
Schrödinger Equation
d
i |    H |  
dt
Linear
Deterministic
Unitary evolution
Linear superposition principle
Some conceptual problems in QM: quantum
measurement, entanglements, nonlocality
___________________________________
Quantum Measurement

d
ˆ : a , |   |    c i |  i 
i |    H |  
A
i
i
dt
Basic postulates of quantum measurement
Measurement on |   yields eigenvalue a i with probability | ci | 2
Measurement culminates in a collapse or reduction of |  
to one of the eigenstates, {|  i }
‘non unitary’ process….
Some conceptual problems in QM: quantum
measurement, entanglements, nonlocality
_________________________________________
Macroscopic Superpositions
i
d
|   H | 
dt
linear superposition principle
Schrödinger's Cat
|  atom cat  | | alive | | dead 
Such states are almost never seen for classical
(‘macro’) objects in our familiar physical
world….but the ‘macro’ is finally made up of the
‘micro’…so, where is the boundary??
Conceptual problems of QM: quantum
measurement, entanglements, nonlocality
___________________________________
Quantum entanglements – a uniquely quantum
mechanical phenomenon associated with composite
systems
|  AB   c1 | 1  |  1   c2 | 2  |  2  |  A  |  B 
|  EPR  
A
1
2
{| A  | B  | A  | B }
B
The Qubit
______________________________________
‘Bit’ : fundamental concept of classical computation & info. - 0 or 1
‘Qubit’ : fundamental concept of quantum computation & info
 0  1
|  |2  |  |2  1
1
Normalization
0
- can
be thought of
mathematical objects having
some specific properties
Physical implementations - Photons,
electron, spin, nuclear spin
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Quantum Mechanics
&
Linear Algebra
___________________________________
Linear Algebra: The study of vector spaces and of linear
operations on those vector spaces.
Basic objects of Linear algebra
Vector spaces
Cn
The space of ‘n-tuples’ of complex
numbers, (z1, z2, z3,………zn)
Elements of vector spaces
IPQI-2010-Anu Venugopalan
vectors
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Quantum mechanics & Linear Algebra
___________________________________
Vector :

V
column matrix
The
standard
quantum
mechanical
representation for a vector in a vector
space :

: ‘Ket’
 zz1 
. 2 
. 
 zn 
Dirac notation
The state of a closed quantum system is described by
such a ‘state vector’ described on a ‘state space’
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Quantum mechanics & Linear Algebra
_____________________________________________
Associated to any quantum system is a complex vector
space known as state space.
The state of a closed quantum system is a unit vector in
state space.
A qubit, has a two-dimensional state space C2.
 
 0  1  
 
Most physical systems often
have finite dimensional state
spaces
IPQI-2010-Anu Venugopalan
   0 0  1 1  2 2  ...  d 1 d  1
 0 
 
 1 
  2 
 
 : 
d 1 
‘Qudit’
Cd
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Linear Algebra & vector spaces
___________________________________
•Vector space V, closed under scalar multiplication & addition
•Spanning set: A set of vectors in V : v1 , v3 , v3 ....... vn
such that any vector v in the space V can be expressed
as a linear combination:
v   ai vi
i
Example: For a Qubit: Vector Space
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C2
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Linear Algebra & vector spaces
___________________________________
Example: For a Qubit: Vector Space
C2
 0
1 
v1   ; v2   
1 
 0
v1 and v1
span the Vector space C2
 a1 
v      ai vi  a1 v1  a2 v2
 a2  i
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Linear Algebra & vector spaces
___________________________________
A particular vector space could have many spanning sets.
Example: For C2
1
1 
w1   ; w2   
1
  1
w1 and w2 also span the Vector space C2
 a1  a1  a2
a1  a2
v    
w1 
w2
2
2
 a2 
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Linear Algebra & vector spaces
___________________________________
v1 , v3 , v3 ....... vn are
A set of non zero vectors,
linearly dependent if there exists a set of complex numbers
a1 , a2 ......an with ai  0 for at least one value of i such that
a1 v1  a2 v2  ....................an vn  0
A set of nonzero vectors is linearly independent if they
are not linearly dependent in the above sense
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Linear Algebra & vector spaces
___________________________________
• Any two sets of linearly independent vectors that span
a vector space V have the same number of elements
•A linearly independent spanning set is called a basis set
•The number of elements in the basis set is equal to the
dimension of the vector space V
•For a qubit, V : C 2 ;
IPQI-2010-Anu Venugopalan
0 and 1 are the
computational basis states
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Linear operators & Matrices
________________________________
1 
 0  Computational Basis for a Qubit
0   ; 1   
0
1 
A linear operator between vector spaces V and W is
defines as any function Â
Â:
V
W, which is linear in its inputs
Aˆ  ai vi   ai Aˆ vi
i
i
Î: Identity operator
Ô: Zero Operator
Once the action of a linear operator  on a basis is
specified, the action of  is completely determined
on all inputs
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Linear operators & Matrices
__________________________________
Linear operators and Matrix representations are
equivalent
Examples: Four extremely useful matrices that operate on
elements in C 2
1 0

 0  I  
0 1
IPQI-2010-Anu Venugopalan
0 1

 x  
1 0
0  i

 y  
i 0 
1 0 

 z  
 0  1
The Pauli Matrices
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Linear operators and matrices - some
properties
____________________________________
 v , w 
vw
Inner product - A vector space equipped with an inner product is
called an inner product space- e.g. “Hilbert Space”
 v , w   v , w 
*
Norm:
v 
IPQI-2010-Anu Venugopalan
 v , v  0
v v  1 for a unit vecto r
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Linear operators and matrices - some
properties
____________________________________
Norm:
v 
v v  1 for a unit vecto r
Normalized form for any non-zero vector:
v
v
A set of vectors with index i is orthonormal if each
vector is a unit vector and distinct vectors are orthogonal
i j   ij
IPQI-2010-Anu Venugopalan
The Gram-Schmidt
orthonormalization procedure
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Linear operators and matrices - some
properties
____________________________________
Outer Product
w v :
w v
v :
vector in inner product space V
w :
vector in inner product space W
A linear operator from V to W
w v v'  w v v'  v v' w
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completeness
relation
i
i I
i
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Linear operators and matrices - some
properties
____________________________________
Eigenvalues and eigenvectors
Diagonal Representation
Aˆ v  v v
ˆ
A
 i i i
i
c( )  det Aˆ  Iˆ  0
i 
example
diagonal representation for z
IPQI-2010-Anu Venugopalan
An orthonormal set of
eigenvectors for  with
corresponding eigenvalues i
1 0 
  0 0  1 1
 z  
 0  1
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The Postulates of Quantum Mechanics
____________________________________
Quantum mechanics is a mathematical framework for the
development of physical theories. The postulates of
quantum mechanics connect the physical world to the
mathematical formalism
Postulate 1: Associated with any isolated physical
system is a complex vector space with inner product,
known as the state space of the system. The system is
completely described by its state vector, which is a
unit vector in the system’s state space
A qubit, has a two-dimensional state space: C2.
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The Postulates of Quantum Mechanics
____________________________________
Evolution - How does the state,
change with time?

, of a quantum system
Postulate 2: The evolution of a closed quantum system
is described by a Unitary transformation
 ' U 
A matrix/operator U is said to be Unitary if
U U  I
Unitary operators preserve normalization /inner products
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The Postulates of Quantum Mechanics - Unitary
operators/Matrices
_________________________________________
a b 
A

c
d


Hermitian conjugation; taking the adjoint
 
A†  A*
T
a * c * 
 *

*
b d 
A is said to be unitary if AA†  A†A  I
We usually write unitary matrices as U.
Example:
0 1  0 1   1 0 
XX  

I




 1 0   1 0  0 1 
†
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Linear operators & Matrices –
operations on a Qubit (examples)
___________________________________
The Pauli Matrices- Unitary operators on qubits - Gates
0 1 ˆ ˆ
  X ; X 0  1 ; Xˆ 1  0
 x  
1 0
NOT Gate
0  i ˆ ˆ
  Y ; Y 0  i 1 ; Yˆ 1  i 0
 y  
i 0 
1 0  ˆ
Phase flip
  Z ; Ẑ 0  0 ; Ẑ 1   1
 z  
Gate
 0  1
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Unitary operators & Matrices- examples
___________________________________
Unitary operators acting on qubits
The Quantum Hadamard Gate
1 1 1 
ˆ


H
2 1  1
1
1
ˆ
ˆ
 0  1 ; H 1   0  1 
H 0 
2
2
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The Postulates of Quantum Mechanics
Quantum Measurement
____________________________________
• The outcome of the measurement cannot be determined
with certainty but only probabilistically
• Soon after the measurement, the state of the system
changes (collapses) to an eigenstate of the operator
corresponding to measured observable
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The Postulates of Quantum Mechanics
Quantum Measurement
____________________________________
Postulate 3:. Unlike classical systems, when we measure a
quantum system, our action ends up disturbing the system
and changing its state. The act of quantum measurements
are described by a collection of measurement operators
which act on the state space of the system being measure
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Measuring a qubit
_____________________________________
  0   1
Quantum mechanics DOES NOT allow us to determine  and  .
We can, however, read out limited information about  and  .
If we measure in the computational basis, i.e.,
2
P (0)   ;
P (1)  
0 and 1
2
Measurement unavoidably disturbs the system, leaving it in
a state 0 or 1 determined by the outcome.
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More general measurements
____________________________________
Observable A (to be measured) corresponds to operator Â
 has a set of eigenvectors with corresponding eigenvalues
Aˆ : ai , | i 
To measure  on the system whose state vector is
one expresses |   in terms of the eigenvectors
|    ci | i 
| 
More general measurements
____________________________________
|    ci | i 
1.The measurement on state
the eigenvalues,
ai
|   yields only one of
with probability
| ci |
2
2.The measurement culminates with the state
collapsing to one of the eigenstates,
The process is non unitary
| i 
Quantum
Classical
transition in a quantum
measurement
The collapse of the wavefunction following measurement
Several interpretations of quantum mechanics seek to
explain this transition and a resolution to this apparent
nonunitary collapse in a quantum measurement.
The quantum measurement paradox/foundations
of quantum mechanics