Download QUANTUM SUPERPOSITION PRINCIPLE

Lecture 9
Born’s statistical interpretation and Quantum
superposition principle
波恩统计解释及态叠加原理
Outline
Part 1 Born’s statistical interpretation
Part 2 Quantum superposition principle
Part 3 Hilbert Space 希尔伯特空间
Part 4 Application of Hilbert Space
Part 1 Born’s statistical interpretation
• What is this "wave function" and what can it tell you? After all, a
particle, by its nature, is localized at a point, whereas the wave
function is spread out in space (it's a function of x, for any given
time t). How can such an object (which is the wave function) be
said to describe the state of a particle? Born gave the answer.
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• Born's statistical interpretation:
• The intensity of wave function at one position is proportional to the
probability that particle presents there.
• 在某一坐标点的波函数的强度正比于粒子在该点出现的几率。
•
Probability that particle presents in the point r ( x, y, z ) at time t  C ( x, y, z , t )
2
Probability at ( x, y, z , t )  C ( x, y, z , t )   ( x, y, z , t )
2
2
• The absolute square of wave function is the probability that
particle presents there.
•
 ( x, y, z , t ) dx dy dz 代表在 t 时刻点 r ( x, y, z ) 附近的小体积元 dV  dx dy dz 内
2
找到粒子的几率 dW 
•
•
 ( x, y, z, t ) dx dy dz .
2
dW
2
  ( x, y , z , t )
dV
The difference between probability wave and classical wave
• In classical physics:
The spatial probability density (空间几率密度) w 
•
•
•
two wave functions  ( x, y, z , t ) and C ( x, y, z, t ) denote different energy or
intensity, such as sound wave.
In quantum mechanics:
two wave functions describe the same quantum state, because their
relative value of probability distribution at different spatial positions is
equal.
(ri , t )
2
 ( rj , t )
2

C (ri , t )
2
C  ( rj , t )
2
•
For two arbitrary position (ri , rj ),
•
If a wave function is multiplied by a constant, the relative probability of
finding a particle in a certain position is not changed.
• 相差常数因子的波函数表示的在某处找到粒子的相对几率是不变的
•
So the wave function has to be normalized.
 C ( x, y, z, t )

2
dx dy dz  1  C 
1
2
 ( x, y, z, t )
2
dx dy dz

•
The condition which wave function can be normalized is that

( x, y, z, t ) dx dy dz has to be converged (收敛的),
2

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2
i.e.

( x, y, z, t ) dx dy dz  M , where M is a real and limited constant.
2

波函数能归一化的条件：要求波函数是平方可积的.
• The uncertainty of phase of the normalized wave function
• 归一化波函数相位角的不确定性
•
The normalized wave function includes an uncertain phase factor
of modulus being one
• 任何一个归一化的波函数都包含一个相角任意的模为
i
1 的因子 e
,   .
2
i
•
e ( x, y, z, t )  1
• Conclusion and examples for the Part 1
•
The value of  ( x, t )
2
gives the probability to find the particle in point x at time

t if
 dx ( x, t )
2
 1.

• 一维束缚粒子
• For a 1-D bound particles which is confined in a limited space
by some forces or a potential gradient.
lim  ( x, t )  0
2
•
x 
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lim  ( x, t )  0
2
x 
Quite likely to find the particle near A, and relatively unlikely near B
Let’s see a wave function f ( x, t )
z
)]
v
z
f ( z  vt )  f ( z  z  vt  z )  f [( z  z )  v (t  )]
v
f ( z  vt )  f ( z  z  vt  z )  f [( z  z )  v(t 
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• 束缚粒子波函数（一个例子而已，仅代表某种一维势场中的粒子）
2
h 2    x, t 



U
(
x
)

x
,
t

i
h
  x, t 


2
2m  x
t
 ( x, t )  A exp(
 k2 x 2
2
) exp i(k0 x  t ) 
 ( x, t )
 k2 =
1
2
2
2

k
exp(  k2 x 2 )，

， =
1
2 k
 ( x, t )  A* A exp( k2 x 2 )
2
A 
1
2


 exp(
2 2
k
x )dx
1

k

k


k
 k2 x 2
 ( x, t )  4 exp(
) exp i(k0 x  t ) 
2


1
p(x) 
e
 2
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(x  )2
2
2
gaussian
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• 束缚粒子（as shown above）
• 自由粒子(following below)
• For a 1-D free particles which can go everywhere free of any forces
（力） or a potential gradient (势能梯度) all over the whole space.
 ( x, t )  0 .
• That is  ( x, t )  0 whatever x or t is, even if xlim

2
2
p02
E0 
,
2
•
0 
h
,
p0
k0 
2
0

2 p0 p0
 ，p0  h k0 , E0  h 
h
h
E 
 p
 ( x, t )  A exp i (k0 x  t )   A exp i ( 0 x  0 t ) 
h 
 h
• The wave function of a free particle is
•
i

 ( x, t )  A exp  ( p0 x  E0t ) 
h

• The intensity of its wave function is
 ( x, t )  AA*  A  constant
2
•
2
• This wave function is not able to be normalized in the position
representation coordinating systems!!!
•
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 ( x, t ) 
1
i

exp  ( p0 x  E0t ) 
2 h
h

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Part 2 Quantum superposition principle
Superposition is a principle of quantum theory that describes a
challenging concept about the nature and behavior of matter and
forces at the sub-atomic level. The principle of superposition claims
that while we do not know what the state of any object is, it is actually
in all possible states simultaneously, as long as we don't look to
check. It is the measurement itself that causes the object to be limited
to a single possibility.
量子力学中的态叠加原理是在亚原子尺度上的一个难以理解的概
念，但是它反映了微观物质粒子的本性和行为。态叠加原理指出：
只要我们不去测量/查看粒子处在什么状态上,即当我们根本不知
道粒子处在哪一个态上时，粒子其实是同时处在所有的可能态上。
只有测量本身才能使得粒子“被限制”于某一个态上。
• Suppose I measure the position of the particle, and I find C. Question:
Where was the particle just before I made the measurement?
Indefinite Positions
Definite Position
•
• The particle wasn't really anywhere.
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It was the act of measurement that
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forced the particle to "take a stand."
Observations not only disturb what is
to be measured, they also produce it .... We compel the particle to assume a
definite position. This view (the so-called Copenhagen interpretation) is
associated with Bohr and his followers. Among physicists it has always been
the most widely accepted position.
Note, however, that if it is correct there
is something very peculiar about the act of measurement - something that
over half a century of debate has done precious little to illuminate.
注意，如果这个理论是正确的，那么测量的作用一定是非常不寻常的、特
别的，大家争论了半个世纪，也没有很好地解释这种不寻常的作用。
• Suppose I measure the position of the particle, and I find C. Question:
Where will be the particle immediately after?
• Of course in C. How does the orthodox interpretation explain that the
second measurement is bound to give the value C? 第二次测量怎么会与
第一次测量值 C 绑定在一起呢？ Evidently the first measurement radically
alters the wave function, so that it is now sharply peaked about C. The wave
function collapses upon measurement (but soon spreads out again, following
the Schrödinger equation, so the second measurement must be made quickly).
• There are, then, two entirely distinct kinds of physical processes:
"ordinary", in which Ψ evolves under the Schrödinger equation, and
"measurements", in which Ψ suddenly collapses.
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•
The superposition state of Schrödinger's cat
In 1935, Erwin Schrödinger proposed an analogy to show how superposition
would operate in the every day world: the somewhat cruel analogy of
Schrödinger's cat. Here's Schrödinger's (theoretical) experiment: We place a living
cat into a steel chamber, along with a device containing a vial of hydrocyanic acid
氢氰酸. There is, in the chamber, a very small amount of a radioactive
substance. If even a single atom of the substance decays during the test period,
a relay mechanism will trip a hammer, which will, in turn, break the vial and kill
the cat.
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The observer cannot know whether or not an atom of the substance
has decayed, and consequently, cannot know whether the vial has
been broken, the hydrocyanic acid released, and the cat killed. Since
we cannot know, the cat is both dead and alive according to quantum
law, in a superposition of states.
It is only when we break open the box and learn the condition of the
cat that the superposition is lost, and the cat becomes one or the
other (dead or alive). This situation is sometimes called quantum
indeterminacy or the observer's paradox（悖 bèi 论）: the observation
or measurement itself affects an outcome, so that the outcome as
such does not exist unless the measurement is made. (That is, there
is no single outcome unless it is observed.)
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A cat is placed in a box, together with a radioactive atom. If the atom
decays, and the Geiger Counter detects an alpha particle, the hammer
hits a flask of prussic acid (HCN), killing the cat. The paradox lies in
the clever coupling of quantum and classical domains. Before the
observer opens the box, the cat's fate is tied to the wave function of
the atom, which is itself in a superposition of decayed and undecayed
states. Thus, said Schrödinger, the cat must itself be in a
superposition of dead and alive states before the observer opens the
box, ``observes'' the cat, and ``collapses'' its wave function.
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Thomas Young's double-slit experiment
Superposition is well illustrated by Thomas Young's double-slit
experiment, developed in the early nineteenth century to prove that
light consisted of waves. In fact, the noted physicist Richard
Feynman claimed that the essentials of quantum mechanics could be
grasped by an exploration of the implications of Young's experiment.
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Quantum Superposition
The double slit experiment
Which slit
does an
electron
pass
through ?
Interference of electrons
Quantum Superposition
The double slit experiment
Which slit
does an
electron
pass
through ?
No
interference
when you
watch the
electrons
Interference of electrons
1
1

1 
2
2
2
The principle of superposition states that if the world can be in any
configuration, any possible arrangement of particles or fields, and if
the world could also be in another configuration, then the world can
also be in a state which is a superposition of the two, where the
amount of each configuration that is in the superposition is specified
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by a complex number.
叠加原理指出：如果粒子可以处于一种状态 1（无论 1 随空间坐标的分
布如何），如果粒子还可以处于另一种状态  2 （无论  2 随空间坐标的分
布如何）
，那么   C11  C22 一定也是粒子的状态（其中 C1 , C2 是与空间坐
标无关的复常数），即粒子一定也处于状态  上。如果你观察到粒子处
于 1 态，则说明粒子的状态从叠加态  “塌陷”到 1 态上了，“塌陷”
到 1 态的概率幅是 C1 ，“塌陷”到 1 态的概率是 C1*  C1  C1 2 ，是实数。
In Classical Probability theory, every possible event has a non-negative
real number between zero and one associated to it, the probability, which
gives the chance that it happens.
If you want to know the probability of two independent events both
occurring, for example the probability that it will rain and also that the stock
market will go up, you multiply the probability for each event. If there is a
0.3 chance that it will rain and a 0.6 chance that the market will go up, there
is a 0.18 chance that both will happen.
If the market can go up because of two exclusive （相互排斥的）events, and
you want to know the total probability that the market will go up, you add
the probability for the two events. For example, if on the condition that
interest rates are lowered by 2 points or by 3 points the market will go up,
the probability that the market will go up is the sum of the probability of the
interest rate being lowered by 2 points or by 3 points.
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Quantum mechanics （Non-classical Probability theory）has the exact
same rules for multiplying and adding numbers associated with events,
except that the quantities are complex numbers called amplitudes instead
of positive real numbers called probabilities.
The superposition principle says that the way to describe the world is to
assign such a complex number to every possible situation, and that the
way to describe how things change is to treat these numbers
mathematically as if they were probabilities.
Because these numbers can be positive or negative, quantum mechanics
allows the counterintuitive （违反直觉的）phenomenon that sometimes
when there are more ways for a thing to happen, the chance that it
happens goes down. An event with a negative amplitude can cancel with
an event with a positive amplitude.
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Part 3 Hilbert Space 希尔伯特空间
A mathematical description of quantum superposition principle
The orthonormality（正交性） of wave function
Consider a system with k distinguishable states. For example, the electron
in an atom might be either in its ground state or one of k−1 excited states,
each of progressively higher energy.
The (k-1)th excited state  k 1
…
The second excited state  2
The first excited state  1
The ground state  0
The superposition state of the k states above is
k
  C0 0  C1 1  C2 2  L  Ck 1 k 1   C j j
j 0
As a classical system, we might use the state of this system to store
a number between 0 and k−1. The superposition principle says that
if a quantum system is allowed to be any one of number of different
states then it can also be placed in a linear superposition of these
states with complex coefficients. Thus the quantum state of the
k-state system above is described by a sequence of k complex
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numbers c0, . . . , ck−1∈C. The c j is said to be the (complex)
amplitude with which the system is in state j. We will require that the
k
amplitudes are normalized so that  C j  1 .
2
j 0
k
C0  C1  C2  L  Ck 1   C j  1
2
2
2
2
2
j 0
It is natural to write the state of the system as a k dimensional
vector in which
 0 ,1, 2 ,L , k 1 are the basis vectors（基矢）:
k
  C0 0  C1 1  C2 2  L  Ck 1 k 1   C j j
j 0
 C0 


C
1



  C2    C0


M


C 
 k 1 
C1 C2 L
Ck 1 
 x
r
r
r
r
r  
r  xex  yey  zez  r   y    x
z
 
T
y
z
T
The normalization on the complex amplitudes means that the state
of the system is a unit vector（单位矢量） in a k dimensional complex
vector space — called a Hilbert space（希尔伯特空间）.
k
r
r
r
r
r
r
 (r , t )  C0 0 (r , t )  C1 1 (r , t )  C2 2 (r , t )  L  Ck 1 k 1 (r , t )   C j j (r , t )
j 0
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
r
(
r
, t ) 组成一个函数集合，集合中的元素具有封闭性，即它们的任意线性
j
r
组合得到的新函数仍然是这个集合中的元素。所以由于集合  j (r , t ) 中的每
k
r
r
r
一个元素 j (r , t ) 都描述粒子的一个状态，而 (r , t )   C j j (r , t ) 也是集合
j 0
r
中的一个元素，因此 (r , t ) 也是描述粒子状态的函数。

r

数学上，把这样一个函数集合  j (r , t ) 叫做一个线性空间，它是一个函数空

r
r

间，因为每一个函数 j (r , t ) 是这个空间中的一个基矢，k个基矢  j (r , t ) 张
r
成一个k维函数空间，粒子（或物理系统）的任意态函数 (r , t ) 完全可以在
这个k维函数空间中表示出来。但是数学上规定希尔伯特空间的基矢必须是正
交归一的.
k constitutes an orthonormal basis of a Hilbert space：
r
r
0 n  m
 (r , t ) m (r , t ) dV   nm  

1 n  m

*
n
态叠加原理的简练表述为：
物理系统的状态由希尔伯特空间中的矢量描写。
当粒子处于态与态的线性叠加态时，粒子既处在态 1 ，又处
在态 2 。量子力学中的这种态的叠加，导致在叠加态下的观
测值的不确定性。
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Part 4 Application of Hilbert Space
用一维自由粒子的波函数构成一个希尔伯特空间
等同于：用一维动量本征函数构成一个希尔伯特空间
 p0 ( x, t ) 
1
i

exp  ( p0 x  E0t ) 
2 h
h

一维动量本征函数集合：

1

i

¤    p ( x, t ) 
exp  ( px  Et )  : p  
2 h
h



If
is any set, then one can form a Hilbert space of sequences with
index of the set
, defined by p   .
For the reason of simplification,
1

px 

exp  i
¤   p ( x) 
 : p 
 h 
2 h


用一维动量本征函数  p ( x) : p  构成一个希尔伯特空间,要
求满足正交归一条件，即
, p '  p
  ( x)  p ( x) dx    p ' p    0, p '  p

*
p'
必须成立，这是  p ( x) : p  作为希尔伯特空间基矢的前提条
件。
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Speaking of δ- function, there are three definitions of it:
, x  0
  x  
① Definition 1
 0, x  0
② Definition 2




f ( x ) 1



dx

1


x
a 0


f ( x)  x  a dx  f (a)

 ipx
e
 dx  2   p 
③ Definition 3

To prove the third definition of δ- function by Fourier Transform
1
f ( x) 
2
1
g ( p) 
2
1
f ( x) 
2
1

2


 1
  2



g ( p) eipx dp



f ( x) eipx dx


f ( x ') e  ipx ' dx ' eipx dp



f ( x ') dx '  e  i ( x ' x ) p dp

1
If define  (x ' x)=
2

f ( x) 





e
i ( x ' x ) p
dp

f ( x ')  (x ' x)dx '  f ( x)

That is the third definition of δ- function
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 20 of 25
1
 (x ' x)=
2

e
i ( x ' x ) p

1
dp   (x)=
2

 ip x
e
 dp

Now we can prove that
, p '  p
 0, p '  p

*

 p ' ( x)  p ( x) dx    p ' p   

is true by using the definition 3 ofδ- function.

1
*

(
x
)

(
x
)
dx

 p ' p
2 h
1

2

e
i ( p ' p )

x
h

e
i
p'x
px
i
h
h
e
dx

, p '  p
x
d    p ' p   
h
 0, p '  p
Now take
1

px 

exp  i
:
p

¤   p ( x) 


 h 
2 h


as the base vectors of a Hilbert Space which is call momentum space.
The problem 3.6 at the page101 in the text book is as following.
The state function of a particle at time t=0 is
 ( x)  A sin 2 kx  cos kx 

1
2

Please find the average momentum and the average kinetic energy of
the particle.
The equations below may be helpful for the problem solving.
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 21 of 25
⑴
Probability Distributions and Averages
Observed Distribution of Measurements
Let’s see the results of the measurement of the lifetime of electrical
bulbs below.
P(x)
Normal Distribution
1
x x 
N
n
P( xi )  i 
N
xi 1300

xi  400
ni xi 
ni
xi 400

xi  400
,x 
xi 1300

xi 1300
ni
xi 1300
ni
xi  845.2hours, N   ni
N
xi  400
xi 1300

xi 400
xi 1300
P( xi ) xi ,

xi 400
P( xi )  1
-- For the discrete variables
N measurements, xi, with ci repeats, of k possible outcomes.
k
k
ci
ci
1 k
x  x   ci xi , P( xi )   k , x   P( xi ) xi ,  P( xi )  1
N i1
N
i 1
i 1
 ci
i 1
-- For continuous variables
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 22 of 25
P( x) 
c( x)

¡
c( x)dx
, x   p( x) xdx,  p( x)dx  1
¡
¡
⑵ The 1-D eigenfunction of momentum
px
i
1
 p ( x) 
eh
2 h
⑶ Euler formula
eix  eix
eix  eix
sin x 
, cos x 
2i
2
Now let’s go back to the question and study on the wave function
 ( x)  A sin 2 kx  cos kx 
1
2


 eikx  e ikx  2 1  eikx  e ikx  
 A 

  2
2i

2



A
  2  e 2ikx  e 2ikx  eikx  e ikx 
4
2 khx
2 khx
khx
khx
i
i
i
i

A  2 h 
1
1
1
1
1
i0
h
h
h

e 
e

e

e 
e h 
2
4
2 h
2 h
2 h
2 h
2 h


A  2 h
 20 ( x)  2 kh ( x)  2 kh ( x)  kh ( x)   kh ( x)
4
A  2 h
A  2 h
A  2 h
A  2 h
A  2 h

0 ( x) 
 2 kh ( x) 
 2 kh ( x) 
 kh ( x) 
  k h ( x)
2
4
4
4
4
 C00 ( x)  C2 kh 2 kh ( x)  C2 kh 2 kh ( x)  Ckh kh ( x)  C kh   kh ( x)



用 ( x)  A sin 2 kx  cos kx  描述的粒子

1
2

处于 0 ( x) 的几率是 C0  2
2
处于 2 k ( x) 的几率是 C2 k
Created by Tianxin Yang, College
2
A2
，这也是粒子动量为零 p  0 的几率
4
A2
 2
，这是粒子动量为零 p  2k 的几率
16
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 23 of 25
处于 2 k ( x) 的几率是 C2k
处于 k ( x) 的几率是 Ck
2
处于  k ( x) 的几率是 C k
2
 2
A2
，这是粒子动量为零 p  2k 的几率
16
A2
 2
，这是粒子动量为零 p  k 的几率
16
2
A2
，这是粒子动量为零 p  k 的几率。
16
 2
说明该态下的粒子只能有5种动量取值，分别是 0, 2k ,  2k , k ,  k ，对应
的几率分别是 2
A2
A2
A2
A2
A2
, 2
, 2
, 2
, 2
，这些几率总和应该为1，
4
16
16
16
16
因为粒子只能出现在这5个态上，不会有其他可能的状态了。
A2
A2
A2
A2
A2
1
2
 2
 2
 2
 2
  A2  1  A 
4
16
16
16
16

C0  2
2
C2 k
Ck
C k
A2 1
 2

16 8
2
C2 k
2
A2 1

4 2
2
 2
 2
2
A2 1

16 8
A2 1

16 8
A2 1
 2

16 8
p  0,
p 2  0, T 
p  2k ,
p  4k
2
2
p2
0
2
4k 2 2 2k 2
, T

2

p  2k ,
p 2  4k
pk ,
p2  k 2 2 ,
p  k ,
p k
2
2
2
2
2 2
,2 T 
T
,
4k 2 2 2k 2

2

2
k2 2
2
k2 2
T
2
1  2
1

sin
kx

cos
kx

2
 h 
 C00 ( x)  C2 kh 2 kh ( x)  C2 kh 2 kh ( x)  Ckh kh ( x)  C kh  kh ( x)
 ( x) 
=
1
1
1
1
1
0 ( x)  2 kh ( x)  2 kh ( x) 
 kh ( x) 
  kh ( x)
2
8
8
8
8
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 24 of 25
1
1
1
1
1
p   0   2 k   ( 2 k )   ( k )   (  k )  0
2
8
8
8
8
1
1
1
1
1
5
p 2   0   4k 2 2   4k 2 2   k 2 2   k 2 2  k 2 2
2
8
8
8
8
4
1
1 2 k 2 2 1 2 k 2 2 1 k 2 2 1 k 2 2 5k 2 2
T  0  
 
 
 

2
8 
8 
8 2 8 2
8
Dirac’s Notations
In quantum mechanics it is customary to use the Dirac’s ket notation
to write vectors. As we shall see later, this is a particularly useful
notation in the context of quantum computation. In the ket notation,
the above state is written as:
j k
   C j j ‡ˆ ˆ †ˆ  ( x, y, z, t )   C j j ( x, y, z, t )
j 0
j
Here 0  1 0   0 , 1   0 1   0  and k  1   0 0   1 . The
T
T
T
Dirac notation has the advantage that the it labels the basis vectors
explicitly. This is very convenient because the notation expresses
both that the state of the qubit is a vector, and that it is data (0 or 1)
to be processed. The { 0 , 1 , …, k  1 } basis is called the standard
or computational basis.
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 25 of 25