1
 alive  dead
cat 
2
Presented by:
Erik Cox, Shannon Hintzman,
Mike Miller, Jacquie Otto,
Adam Serdar, Lacie Zimmerman

What’s to come…
-Brief history and background of quantum
mechanics and quantum computation
-Linear Algebra required to understand
quantum mechanics
-Dirac Bra-ket Notation
-Modeling quantum mechanics and
applying it to quantum computation
History of Quantum
Mechanics
Classical (Newtonian) Physics
Sufficiently describes everyday things and
events.
Breaks down for very small sizes (quantum
mechanics) and very high speeds (theory of
relativity).
Why do we need Quantum
Mechanics?
In short, quantum mechanics describes
behaviors that classical (Newtonian) physics
cannot. Some behaviors include:
- Discreteness of energy
- The wave-particle duality of light and matter
- Quantum tunneling
- The Heisenberg uncertainty principle
- Spin of a particle
Spin of a Particle
- Discovered in 1922 by Otto Stern and
Walther Gerlach
- Experiment indicated that atomic particles
possess intrinsic angular momentum, called
spin, that can only have certain discrete values.
The Quantum Computer
Idea developed by Richard
Feynman in 1982.
Concept:
Create a computer that uses the effects of
quantum mechanics to its advantage.
Classical
Quantum
Computer vs. Computer
Information
Information
- Bit, exists in two
states, 0 or 1
- Qubit, exists in two
states, 0 or 1, and
superposition of both
Why are quantum computers
important?
Recently, Peter Shor developed
an algorithm to factor large
numbers on a quantum computer.
Since factoring is key to current
encryption, quantum computers
would be able to quickly break
current cryptography techniques.
In the beginning, there
was Linear Algebra…
- Complex inner product spaces
- Linear Operators
- Unitary Operators
- Projections
- Tensor Products
Complex inner product spaces
An inner product space is a complex
vector space V, together with a map
f : V x V → F where F is the ground field
C. We write <x, y> instead of f(x, y) and
require that the following axioms be
satisfied:
x V , x, x  0, and x, x  0 iff x  0
(Positive Definiteness)
a  F , x, y, z V , z, ax  y  a z, x  z, y
(Conjugate Bilinearity)
x, y V , x, y  y, x *
(Conjugate Symmetry)
* denotes complex conjugate
Complex Conjugate:

Let z  C
z  x  iy
where i   1
z*  x  iy
Example of Complex Inner Product Space:
V  C  z1 , z2 , z3 ,..., zn  z j  C , j  1..n
Let v, w  V
n
v, w  v *1 w1  v *2 w2  ...  v *n wn
Linear Operators
Let V and W be vector spaces over C
, then
Aˆ : V  W is a linear operator if c  C ,
x, y  V. The following properties exist:
 
 ˆ 
ˆ
ˆ
A x  y   A x   A y  (Additivity)


ˆ
ˆ
Acx   cA x 
(Homogeneity)
Example:
d / dx f x  gx  d / dx f x  d / dxg x
d / dxcf x  cd / dx f x
Unitary Operators
Properties:
1
U U
t
t
UU  U U  I
t
Norm Preserving…
Inner Product Preserving…
t denotes adjoint
a11a21a31...an1 
a a a ...a 
12 22 32
n1 

A
...



a1n a2 n a3n ...ann 
Adjoints
Matrix
Representation
a *11 a *12 a *13 ...a *1n 
a * a * a * ...a * 
21
22
23
1n 
t

A 
...



a *n1 a *n 2 a *n3 ...a *nn 
Definition of
Adjoint:
 
t 
v Aw  A v w
 
Suppose U : V  V , v , w  V
 
t  
Uv Uw  U Uv w
 
 
 Iv w  v w

Uv 
 
Uv Uv 
(Inner Product
Preserving)
 
 (Norm
v v  v
Preserving)
• In quantum mechanics we use orthogonal
projections.
• Definition: Let V be an inner product space
over F. Let M be a subspace of V. Given an
element y  V then the orthogonal projection of
y onto M is the vector Py  M which satisfies
y  Py  v
where v is orthogonal to every element m M.
A projection operator P on V satisfies
t
2
PP P
We say P is the projection onto its range, i.e.,
onto the subspace
W  v V : Pv  v
In quantum
mechanics tensor
products are used
with :
• Vectors
• Vector Spaces
• Operators
• N-Fold tensor
products.
m
n
W

C

V

C

If
and
, there is a natural mapping
T : W  V  C mn defined by
T x1 ,..., xm ,  y1 ,..., yn   x1  y1 ,..., yn ,..., xm  y1 ,..., yn 
 x1 y1 ,..., x1 yn ,..., xm y1 ,..., xm yn 
We use notation w  v to symbolize T(w, v) and
call w  v the tensor product of w and v.
• W  V means the vector space consisting
of all finite formal sums:
a w v
ij
i
j
where wi W
and v j  V
If A, B are operators on W and V we define AB
on WV by
A  B aij wi  v j    aij Awi  Bv j
4 Properties of Tensor Products
1.
a(w  v) = (aw)  v = w  (av) for all a in C;
2.
(x + y)  v = x  v + y  v;
3.
w  (x + y) = w  x + w  y;
4.
 w  x | y  z  =  w | y   x | v .
Note: | is the notation used for inner products in quantum mechanics.
Property #1:
a(w  v) = (aw)  v = w  (av) for all a in C
Example in C
 :
2
a(w  v)  a(w1v1, w1v2 , w2v1, w2v2 )
 (aw1v1 , aw1v2 , aw2v1 , aw2v2 )
 :
Example in C
2
(aw)  v  a(w1 , w2 )  v
 (aw1 , aw2 )  v
 (aw1v1, aw1v2 , aw2v1, aw2v2 )
 :
Example in C
2
w  (av)  w  a(v1 , v2 )
 w (av1 , av2 )
 (aw1v1 , aw1v2 , aw2v1 , aw2v2 )
Property #2:
(x + y)  v = x  v + y  v
2
Example in C
 :
( x  y)  v  (( x1 , x2 )  ( y1 , y2 ))  v
 ((( x1 , x2 )v1 ), (( x1 , x2 )v2 ))
 ((( y1 , y2 )v1 ), (( y1 , y2 )v2 ))
 ( x1v1 , x2v1 , x1v2 , x2v2 )
 ( y1v1 , y2v1 , y1v2 , y2v2 )
 :
Example in C
2
x  v  y  v  ( x1v1 , x1v2 , x2v1 , x2v2 )
 ( y1v1 , y1v2 , y2v1 , y2v2 )
Property #3:
w  (x + y) = w  x + w  y
2
Example in C
 :
w  ( x  y)  w  (( x1, x2 )  ( y1 , y2 ))
 (( w1 ( x1, x2 )), (w2 ( x1 , x2 )))
 (( w1 ( y1, y2 )), (w2 ( y1, y2 )))
 (w1 x1 , w1 x2 , w2 x1 , w2 x2 )
 (w1 y1 , w1 y2 , w2 y1 , w2 y2 )
Example in C
 :
2
w  x  w  y  (w1 x1 , w1 x2 , w2 x1 , w2 x2 )
 (w1 y1 , w1 y2 , w2 y1 , w2 y2 )
Property #4:
wx|yz=w|yx|z
2
 :
Example in C
 w x | y  z 
 (w1x1, w1x2 , w2 x1, w2 x2 ) | ( y1z1, y1z2 , y2 z1, y2 z2 ) 
 (w1 x1 ) * ( y1 z1 )  (w1 x2 ) * ( y1 z2 )
 (w2 x1 ) * ( y2 z1 )  (w2 x2 ) * ( y2 z2 )
Example in
C
2
:
 w | y  x | z 
 (w1, w2 ) | ( y1, y2 )  ( x1, x2 ) | ( z1, z2 ) 
 ((w1 ) * ( y1 )  (w2 ) * ( y2 ))
(( x1 ) * ( z1 )  ( x2 ) * ( z2 ))
 (( w1 ) * ( y1 )( x1 ) * ( z1 ))  (( w1 ) * ( x2 ) * ( z2 )( y2 ))
 (( w2 ) * ( y2 )( x1 ) * ( z1 ))  (( w2 ) * ( x2 ) * ( z2 )( y2 ))
 (w1 x1 ) * ( y1 z1 )  (w1 x2 ) * ( y1 z2 )
 (w2 x1 ) * ( y2 z1 )  (w2 x2 ) * ( y2 z2 )
Dirac Bra-Ket Notation
Notation
Inner Products
Outer Products
Completeness Equation
Outer Product Representation of
Operators
Bra-Ket Notation Involves
Vector Xn can be
represented two
ways
Ket
Bra
|n>
<n| = |n>t
v 
 
 w
x 
 
y
z 
 
v w x y z 
*
m
*
* *
* *
is the complex conjugate of m
Inner Products
An Inner Product is a Bra multiplied by a Ket
<x| |y> can be simplified to <x|y>
<x|y> =
v
*
w* x* y * z *

l 
 
 m
 n  = lv*  mw*  nx*  oy*  pz*
 
o 
 p
 
Outer Products
An Outer Product is a Ket multiplied by a Bra
l 
 lv * lw*
 
 * *
 m
 mv mw
n  *

*
*
*
*
*
*
|y><x| =
=
nv
nw
  v w x y z

o 
 ov* ow*
 * *
 p
 
 pv pw

By Definition

x
y
v
lx *
mx*
nx*
ox*
px*
 y v x
ly *
my*
ny *
oy*
py*
lz * 

*
mz 
*
nz 
oz * 
*
pz 
Completeness Equation
is used to create a identity operator
represented by vector products.
Let |i>, i = 1, 2, ..., n, be a basis for V
and v is a vector in V
| i  i || v    i | v | i | v 
So Effectively
| i  i | I
Proof for the Completeness
Equation
Using Linear Algebra, the basis of a vectors space
can be represented series of vectors with a one in
each successive position and zeros in every other
(aka {1, 0, 0, ... }, {0, 1, 0, ...}, {0, 0, 1, ...}, ...)
So |i><i| will create a matrix with a one in each
successive position along the diagonal.
1

0
0

 ...

0 0 ...  0
 
0 0 ...  0
0 0 ...  0
 
... ... ...  ...
0 0 ...

1 0 ...
0 0 ...

... ... ...
0

0
0

 ...

0 0 ...

0 0 ...
0 1 ...

... ... ...
etc.
Completeness Cont.
Thus
| i  i |
1

0
0

 ...

0 0 ...
0 0 0


0 0 ...
0 1 0
+ 

0 0 ...
0 0 0


 ... ... ...
... ... ...

 1 0 0 ...


 0 1 0 ...
 0 0 1 ...


 ... ... ... ...


=
...

...
+

...

...
= I
0

0
0

 ...

0 0 ...

0 0 ...
+ ... =

0 1 ...

... ... ...
One application of the Dirac
notation is to represent
Operators in terms of inner and
outer products.

n 1
 i A j i j
i , j 0
and
Aij  i A j
• If A is an operator, we can represent A by applying the
completeness equation twice this gives the following
equation:

n 1
 i A j i j
i , j 0
• This shows that any operator has an outer product
representation and that the entries of the associated
matrix for the basis |i are:
Aij  i A j
Projections
• Projection is a type of operator
• Application of inner and outer
products
Linear Algebra View
  
We can represent v  x  y
graphically:


v
y

x

u
Using the rule of dot products we
 
know x  y  0
 
Given that c  0 we can say x  cu
Linear Algebra View (Cont.)
Using these facts we can solve


for x and y

 
v  cu  y
 
  
v  u  (cu  y )  u
  
   
(cu  y )  u  cu  u  y  u
 
Again using the rule of dot products u  y  0
 
2
We get v  u  c u
Linear Algebra View (Cont.)
 
u v
So c   2
u
Plugging this
 back
 into the original
equation x  cu
 
Gives us:

u

v
 
x    2 u
 u 


 

     u v 
yvx v 2 u
 u 


Linear Algebra View (Cont.)


If u is a unit vector u  1

  
x  (u  v )u
    
y  v  (u  v )u
Projections in Quantum
Mechanics
Given that W  V and
v
v V
y
W
x
This graph is a representation of
Given Pv  x and
xW
v  x y
Projections in QM (cont.)
1 , 2 ,... k being the full basis of W
We can regard the full basis of
as being
V
{| 1 , | 2 ,... | k , | k  1 ,... | n }

For some c j  C
On Basis v  c1 | 1  c2 | 2  ...  ck | k 
 ck 1 | k  1  ...  cn | n 
Projections in QM (cont.)
Taking the inner products gives
1 v  c1 1 1  c2 1 2  ...  cn 1 n
Therefore
generally
n
c1  1 v and more
cj  j v
n
So
v j v j  j v j 
j 1
j 1
n

j  k 1
jv j
Projections in QM (cont.)
n
Now set x   j v j W
y
j 1
n

j v j
j  k 1
So Pv   j v j    j j  v
k
k
j 1
k
j 1
P j j
j 1
Computational Basis
V  C
V
n
2
 V  V  ...  V  V
2
2
2
Basis for V  C
2
0
0  (1,0)
, 1
2n

1  (0,1)
Computational Basis (cont.)
A basis for C will have basis
vectors:
2
00...0 , 00...01 , 00...10 ,..., 11...1
Called a computational
n
basis V
Notation: 00...01  0 0 ... 0 1
Quantum States
Thinking in terms of directions
z1, z2 : z1, z2  C 2


model quantum states by directions
in a vector space
z2
1
1,0
0,1
0
z1
Associated with an isolated quantum system is an
inner product space
V  C
n
called the “state
space” of the system. The system at any given
time is described by a “state”, which is a unit
vector in V.
2
• Simplest state space - V  C
 or Qubit
If | 0 and | 1 form a basis for V ,
then an arbitrary qubit state has the form
| x  a | 0  b | 1 , where a and b in C

2
2
|
a
|

|
b
|
 1.
have
• Qubit state differs from a bit because
“superpositions” of an arbitrary qubit state
are possible.
The evolution of an isolated quantum system is
described by a unitary operator on its state space.
The state |  (t1 ) is related to the state |  (t2 ) by a
unitary operator U t t i.e., |  (t2 )  U t ,t |  (t1 ) .
1, 2
1 2
Quantum measurements are described by a
finite set, {Pm}, of projections acting on the
state space of the system being measured.
•
If the state of the system is |   immediately
before the measurement, then the probability that
the result m occurs is given by p(m)   | Pm |   .
• If the result m occurs, then the state of the
system immediately after the measurement is
Pm |  
Pm |  

1/ 2
 | Pm |  
p(m)
The state space of a composite quantum system is
the tensor product of the state of its components.
If the systems numbered 1 through n are prepared
in states |  (ti ), i = 1,…, n, then the joint state of
the composite total system is |  1      |  n .
Product vs. Entangled
States
n


Product State – a state in V is called a
product state if it has the form:
       
Entangled State – if   is a linear
combination of i ' s that can’t be written as a
product state
Example of an
Entangled State


The 2-qubit in the state    00  11 / 2
Suppose:
|00 + |11 = |a  |b for some |a and |b. Taking inner
products with |00, |11, and |01 and applying the state space
property of tensor products (states |i, i=1, …, n, then the joint
state of the composite total system is |1  · · ·  |n) gives
0|a 0|b = 1, 1|a 1|b = 1, and 0|a 1|b = 0, respectively.
Since neither 0|a nor 1|b is 0, this gives a contradiction
Tying it all
Together
With an example of a 2-qubit
Example of a 2-qubit
• A qubit is a 2-dimensional quantum system
(say a photon) and a 2-qubit is a composite of
two qubits
• 2-qubits “live” in the vector space C  C
2
2
Suppose that  is an example of a 2 component system
with  being a linear combination of basic qubits with
amplitude being the coefficients:
   a0 00  a1 01  a2 10  a3 11
In which
a0  a1  a2  a3  1
2
2
2
2
Measuring the
st
1
qubit
•When we measure the first qubit in the composite system,
the measuring apparatus interacts with the 1st qubit and
leaves the 2nd qubit undisturbed (postulate 4), similarly
when we measure the 2nd qubit the measuring device
leaves the 1st qubit undisturbed
•Thus, we apply the measurement P0 , P1,in which
P0  0 0  I
P1  1 1  I
Leading to the probabilities and
post measurement states…
Using postulate 3 the probability that 0 occurs is given by
p1 0   P0    a0 00   a1 01  a0  a1
2
If the result 0 occurs, then the state of the system immediately
after the measurement is given by

0
1

P0 
p1 0

a0 00  a1 01
a0  a1
2
2
2
Similarly we obtain the result 1 on the
1st qubit with probability…
p1 1   P1   a2  a3
2
2
Resulting in the post-measurement
state…

1
1

P1 
p1 1

a2 10  a3 11
a2  a3
2
2
In the same way for the second qubit…
p2 0  a0
2
 a2 ,
2
p2 1  a1  a3 ,
2
2

0
2

a0 00  a2 10
a0

1
2

2
 a2
2
a1 01  a3 11
a1  a3
2
2
Consider the entangled
2-qubit
   00  11  / 2
   00  11  / 2
We consider
 with amplitudes
1
1
a0 
, a1  0, a2  0, a3 
2
2
After applying Quantum
Measurement Techniques
The probabilities for each state for each qubit are all 1/2
1
p1 0   p1 1  p2 0   p2 1 
2
the post measurement states are
v10  v20  00 ,
v11  v12  11
(A Perfectly Correlated Measurement)
Conclusion
• Brief History of Quantum Mechanics
• Tools Of Linear Algebra
– Complex Inner Product Spaces
– Linear and Unitary Operators
– Projections
– Tensor Products
Conclusion Cont.
• Dirac Bra-Ket Notation
– Inner and Outer Products
– Completeness Equation
– Outer Product Representations
– Projections
– Computational Bases
Conclusion Cont. (again)
• Mathematical Model of Quantum Mech.
– Quantum States
– Postulates of Quantum Mechanics
– Product vs. Entangled States
Where do we go from here?
• Quantum Circuits
• Superdense Coding and Teleportation
Bibliography
http://en.wikipedia.org/wiki/Inner_product_space
http://vergil.chemistry.gatech.edu/notes/quantrev/node14.html
http://en2.wikipedia.org/wiki/Linear_operator
http://vergil.chemistry.gatech.edu/notes/quantrev/node17.html
http://www.doc.ic.ad.uk/~nd/surprise_97/journal/vol4/spb3/
http://www-theory.chem.washington.edu/~trstedl/quantum/quantum.html
Gudder, S. (2003-March). Quantum Computation. American Mathmatical
Monthly. 110, no. 3,181-188.
Special Thanks to:
Dr. Steve Deckelman
Dr. Alan Scott