# Postulates of quantum mechanics

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```							Postulates of
Quantum
Mechanics
SOURCES
Angela Antoniu, David Fortin,
Artur Ekert, Michael Frank,
Kevin Irwig , Anuj Dawar ,
Michael Nielsen
Jacob Biamonte and students
Short review

Linear Operators
• V,W: Vector spaces.

• A linear operator A from V to W is a linear function A:VW. An
operator on V is an operator from V to itself.

• Given bases for V and W, we can represent linear operators as
matrices.

• An operator A on V is Hermitian iff it is self-adjoint (A=A†).
• Its diagonal elements are real.
Eigenvalues & Eigenvectors
• v is called an eigenvector of linear operator A iff A just
multiplies v by a scalar x, i.e. Av=xv
– “eigen” (German) = “characteristic”.

• x, the eigenvalue corresponding to eigenvector v, is just
the scalar that A multiplies v by.

• the eigenvalue x is degenerate if it is shared by 2
eigenvectors that are not scalar multiples of each other.
– (Two different eigenvectors have the same eigenvalue)

• Any Hermitian operator has all real-valued eigenvectors,
which are orthogonal (for distinct eigenvalues).
Exam Problems
• Find eigenvalues and eigenvectors of operators.
• Calculate solutions for quantum arrays.
• Prove that rows and columns are orthonormal.
• Prove probability preservation
• Prove unitarity of matrices.
• Postulates of Quantum Mechanics. Examples and
interpretations.
• Properties of unitary operators
Unitary Transformations
• A matrix (or linear operator) U is unitary iff its inverse
equals its adjoint: U1 = U†

• Some properties of unitary transformations (UT):
–   Invertible, bijective, one-to-one.
–   The set of row vectors is orthonormal.
–   The set of column vectors is orthonormal.
–   Unitary transformation preserves vector length:
|U| = | |
• Therefore also preserves total probability over all states:
   ( si )
2               2

i
– UT corresponds to a change of basis, from one orthonormal basis
to another.
– Or, a generalized rotation of in Hilbert space

Who an when invented all this stuff??
A great
breakthrough
Postulates of Quantum Mechanics
Lecture objectives
• Why are postulates important?
– … they provide the connections between the physical,
real, world and the quantum mechanics mathematics
used to model these systems

• Lecture Objectives
–   Description of connections
–   Introduce the postulates
–   Learn how to use them
–   …and when to use them
Physical Systems -
Quantum Mechanics
Connections
Isolated physical      
Postulate 1                                 Hilbert Space
system            

Evolution of a physical          Unitary
Postulate 2
system               transformation

Measurements of a          Measurement
Postulate 3
physical system            operators

Composite physical          Tensor product
Postulate 4
system                  of components
Postulate 1:
State Space
Systems and Subsystems
• Intuitively speaking, a physical system consists of a
region of spacetime & all the entities (e.g. particles &
fields) contained within it.
– The universe (over all time) is a physical system
– Transistors, computers, people: also physical systems.

• One physical system A is a subsystem of another
system B (write AB) iff A is completely contained
within B.
B
A
• Later, we may try to make these definitions more
formal & precise.
Closed vs. Open Systems
• A subsystem is closed to the extent that no particles,
information, energy, or entropy enter or leave the
system.
– The universe is (presumably) a closed system.
– Subsystems of the universe may be almost closed

• Often in physics we consider statements about closed
systems.
– These statements may often be perfectly true only in a
perfectly closed system.
– However, they will often also be approximately true in any
nearly closed system (in a well-defined way)
Concrete vs. Abstract Systems
• Usually, when reasoning about or interacting with a system, an entity
(e.g. a physicist) has in mind a description of the system.

• A description that contains every property of the system is an exact
or concrete description.
– That system (to the entity) is a concrete system.

• Other descriptions are abstract descriptions.
– The system (as considered by that entity) is an abstract system, to some
degree.

• We nearly always deal with abstract systems!
– Based on the descriptions that are available to us.
States & State Spaces
• A possible state S of an abstract system A (described by a
description D) is any concrete system C that is consistent
with D.
– I.e., it is possible that the system in question could be completely
described by the description of C.

• The state space of A is the set of all possible states of A.

• Most of the class, the concepts we’ve discussed can be
applied to either classical or quantum physics
– Now, let’s get to the uniquely quantum stuff…
An example of a state space
Schroedinger’s Cat and
Explanation of Qubits
Postulate 1 in a
simple way: An
isolated physical
system is described
by a unit vector (state
vector) in a Hilbert
space (state space)
Cat is isolated in the
box
Distinguishability of States
• Classical and quantum mechanics differ regarding the
distinguishability of states.

• In classical mechanics, there is no issue:
– Any two states s, t are either the same (s = t), or different (s  t),
and that’s all there is to it.

• In quantum mechanics (i.e. in reality):
– There are pairs of states s  t that are mathematically distinct, but
not 100% physically distinguishable.
– Such states cannot be reliably distinguished by any number of
measurements, no matter how precise.
• But you can know the real state (with high probability), if you prepared the
system to be in a certain state.
Postulate 1: State Space
– Postulate 1 defines “the setting” in which Quantum Mechanics
takes place.
– This setting is the Hilbert space.
– The Hilbert Space is an inner product space which satisfies the
condition of completeness (recall math lecture few weeks ago).

• Postulate1: Any isolated physical space is associated
with a complex vector space with inner product called
the State Space of the system.
– The system is completely described by a state vector, a unit
vector, pertaining to the state space.
– The state space describes all possible states the system can be
in.
– Postulate 1 does NOT tell us either what the state space is or
what the state vector is.
Distinguishability of States, more
precisely
• Two state vectors s and t are (perfectly) distinguishable or orthogonal
(write st)                                                        t
iff s†t = 0. (Their inner product is zero.)
     s

• State vectors s and t are perfectly indistinguishable or identical
(write s=t)
iff s†t = 1. (Their inner product is one.)

• Otherwise, s and t are both non-orthogonal, and non-identical. Not
perfectly distinguishable.

• We say, “the amplitude of state s, given state t, is s†t”.
– Note: amplitudes are complex numbers.
State Vectors & Hilbert Space
• Let S be any maximal set of distinguishable possible
states s, t, … of an abstract system A.

• Identify the elements of S with unit-length, mutually-
orthogonal (basis) vectors in an abstract complex vector
space H.
– The “Hilbert space”

• Postulate 1: The possible states  of A
can be identified with the unit                        t
vectors of H.                                             s
Postulate 2:
Evolution
Postulate 2: Evolution
• Evolution of an isolated system can be expressed as:
v(t 2 )  U(t 1 , t 2 ) v(t 1 )
where t1, t2 are moments in time and U(t1, t2) is a unitary
operator.
– U may vary with time. Hence, the corresponding segment of time is
explicitly specified:
U(t1, t2)
– the process is in a sense Markovian (history doesn’t matter) and
reversible, since
UU v  v
†

Unitary operations preserve inner product
Example of evolution
Time Evolution
• Recall the Postulate: (Closed) systems evolve (change state) over
time via unitary transformations.
t2 = Ut1t2 t1

• Note that since U is linear, a small-factor change in amplitude of a
particular state at t1 leads to a correspondingly small change in the
amplitude of the corresponding state at t2.

– Chaos (sensitivity to initial conditions) requires an ensemble of initial states that
are different enough to be distinguishable (in the sense we defined)

• Indistinguishable initial states never beget distinguishable outcome
Wavefunctions
• Given any set S of system states (mutually
distinguishable, or not),

• A quantum state vector can also be translated to a
wavefunction  : S  C, giving, for each state sS, the
amplitude (s) of that state.

–  is called a wavefunction because its time evolution obeys an
equation (Schrödinger’s equation) which has the form of a wave
equation when S ranges over a space of positional states.
Schrödinger’s Wave Equation for particles
We have a system with states given by (x,t) where:
– t is a global time coordinate, and
– x describes N/3 particles (p1,…,pN/3) with masses (m1,…,mN/3)
in a 3-D Euclidean space,
– where each pi is located at coordinates (x3i, x3i+1, x3i+2), and
– where particles interact with potential energy function V(x,t),
• the wavefunction (x,t) obeys the following (2nd-order,
linear, partial) differential equation:

Planck
Constant

 N 1 1  2                     

                        ( x, t )   V ( x, t )  i   ( x, t )
2  j 0 m j / 3 x j

2            
                   t
Features of the wave equation
• Particles’ momentum state p is encoded implicitly
by the particle’s wavelength : p=h/

• The energy of any state is given by the frequency
 of rotation of the wavefunction in the complex
plane: E=h.

• By simulating this simple equation, one can
observe basic quantum phenomena such as:
– Interference fringes
– Tunneling of wave packets through potential barriers
Heisenberg and Schroedinger
views of Postulate 2
This is Heisenberg picture

This is Schroedinger
picture

..in this class we are interested in Heisenberg’s view…..
The Solution to the Schrödinger Equation
• The Schrödinger Equation governs the transformation of
an initial input state 0to a final output state t .
• It is a prescription for what we want to do to the
computer.
 t ˆ         
 t   T expi  H  d   0  U t   0
ˆ
 0           

•   H  
ˆ
is a time-dependent Hermitian matrix of size 2n
called the Hamiltonian
• U t  is a matrix of size 2n called the evolution matrix,
ˆ

• Vectors of complex numbers of length 2n
• Tτ is the time-ordering operator
The Schrödinger Equation
• n is the number of quantum bits (qubits) in the quantum computer

• The function exp is the traditional exponential function, but some
care must be taken here because the argument is a matrix.

 xn 
expx    
We discussed this
 n!      power series

• The evolution matrix U t  is the program for the quantum
ˆ
computer. Applying this program to the input state produces the
output state t     ,which gives us a solution to the problem.

 t ˆ         
 t   T expi  H  d   0  U t   0
Formula
from                                               ˆ
 0           
last slide
The Hamiltonian Matrix in Schroedinger Equation

• The Hamiltonian is a matrix that tells us how the quantum
computer reacts to the application of signals.

• In other words, it describes how the qubits behave under the
influence of a machine language consisting of varying some
controllable parameters (like electric or magnetic fields).

• Usually, the form of the matrix H needs to be either derived
by a physicist or obtained via direct measurement of the
properties of the computer.

 t ˆ         
 t   T expi  H  d   0  U t   0
ˆ
 0           
The Evolution Matrix in the Schrodinger Equation
• While the Hamiltonian describes how the quantum computer
responds to the machine language, the evolution matrix describes
the effect that this has on the state of the quantum computer.

• While knowing the Hamiltonian allows us to calculate the
evolution matrix in a pretty straightforward way, the reverse is not
true.

• If we know the program, by which is meant the evolution
matrix, it is not an easy problem to determine the machine
language sequence that produces that program.

• This is the quantum computer science version of the compiler
problem. Or “logic synthesis” problem.
Hamiltonian matrix          Evolution matrix

 t ˆ         
 t   T expi  H  d   0  U t   0
ˆ
 0           
Postulate 3:
Quantum
Measurement
Computational Basis – a reminder

Observe that it is not     We recalculate to a new
required to be orthonormal,            basis
just linearly independent
Example of measurement
in different bases

1/2

The second with
probability zero
• You can check from definition that inner product
of |0> and |1> is zero.
• Similarly the inner product of vectors from the
second basis is zero.
• But we can take vectors like |0> and 1/2(|0>-
|1>) as a basis also, although measurement will
perhaps suffer.

Good
base         Not a base
A simplified Bloch Sphere to illustrate
the bases and measurements

You cannot add more vectors that would be orthogonal together with blue or red
vectors
Probability and Measurement
• A yes/no measurement is an interaction designed to
determine whether a given system is in a certain state s.

• The amplitude of state s, given the actual state t of the
system determines the probability of getting a “yes”
from the measurement.

• Important: For a system prepared in state t, any
measurement that asks “is it in state s?” will return “yes”
with probability Pr[s|t] = |s†t|2

– After the measurement, the state is changed, in a way we will
define later.
Inner product
between states
A Simple Example of distinguishable, non-
distinguishable states and measurements
• Suppose abstract system S has a set of only 4
distinguishable possible states, which we’ll call
s0, s1, s2, and s3, with corresponding ket vectors
|s0, |s1, |s2, and |s3.
• Another possible state is then the vector
1       i               1 2 
s0      s3
2        2                   
 0 
• Which is equal to the column matrix:      0 
      
• If measured to see if it is in state s0,  i 2 
      
we have a 50% chance of getting a “yes”.
Observables
Observables
• Hermitian operator A on V is called an
observable if there is an orthonormal (all unit-
length, and mutually orthogonal) subset of its
eigenvectors that forms a basis of V.
There can be
measurements that
are not observables

Observe that
the
eigenvectors
must be
orthonormal
Observables
• Postulate 3:
– Every measurable physical property of a system is
described by a corresponding operator A.
– Measurement outcomes correspond to eigenvalues.

• Postulate 3a:
– The probability of an outcome is given by the
squared absolute amplitude of the corresponding
eigenvector(s), given the state.
Density
Operators
Density Operators
• For a given state |, the probabilities of all the
basis states si are determined by an Hermitian
operator or matrix  (the density matrix):
 c1 c1  cn c1 
*       *

               
  [  i , j ]     [c j ci ]      
*

c1 cn  cn cn 
*       *
               
• The diagonal elements i,i are the probabilities of
the basis states.
– The off-diagonal elements are “coherences”.
• The density matrix describes the state exactly.
M=
Hermitian
observable
Towards QM Postulate 3 on
measurement and general formulas
A measurement is described by an Hermitian
eigenvalue
operator (observable)
Hermitian observable
M =  m Pm
m

– Pm is the projector onto the eigenspace of M with
eigenvalue m                              Pm|
– After the measurement the state will be p(m) with
probability p(m) = |Pm|.
– e.g. measurement of a qubit in the computational basis
• measuring | = |0 + |1 gives:
• |0 with probability |00| = |0||2 = ||2   Derived in
• |1 with probability |11| = |1||2 = ||2   next slide
1. In all these calculations we have a “ket part” and a “bra
part”
2. Remember how to create Bra part from Ket Part.
4. Remember about inner products, how to calculate them
5. Remember that there are many rules to operate on these
vectors so you do not have to do all calculations on
Heisenberg matrices.
6. If you are confused with notations, always convert to
Heisenberg matrices.
An example how Measurement Operators act
on the state space of a quantum system
Measurement operators act on the state space of a quantum system
Initial state:
 0
Hadamard   Operate on the state space with an operator that preservers unitary evolution:

0 1         1 1
  H op 0                                         = |0+|1
2        2 1
 
Define a collection of measurement operators for our state space:

M1  1 1                   M0  0 0                                     Half
probability of
Act on the state space of our system with measurement operators:                      measuring 0
 1 0  1 1 1
0 0 
1
1 1 
 0 0  2 1  2
                                             = p0
2                 
Half
 0 0  1 1 1
1
1 1 
probability of
1 1             0 1  2 1  2
                                                   measuring 1
2                 
But now it is a part of formal calculus
Qubit example: calculate the density matrix

Suppose   0 with probability 1.
1          1 0
 0 0 .
Then   0 0    1 0  

0             
Suppose   1 with probability 1.
0                 0 0 
Then   1 1    0 1          0 1  .
 1   
    
0 i 1
Suppose                               with probability 1.
2
 0  i 1  0  i 1  1 1           1 1 i 
Then                       i  1 i   i 1  .
    2        2      2         2
     
Conjugate and change
kets to bras
Density matrix is a         Density
generalization of state
where    pj  j  j is the density matrix.                                  matrix
j
Measurement of a state vector using
projective measurement
Operate on the state space with an operator that preservers unitary evolution:

0 1       1 1
 0                         H op 0                  
2         2 1
 

Define observables:

0 1                    0  i                    1    0
x  
 1 0
             y                     z  
 0  1

i  0                           
                           

Act on the state space of our system with observables
(The average value of measurement outcome after lots of measurements):
Vector is on
1 0                   1 0  1 1
axis X
 0  1   2 1 1   0  1  2 1  0
1
                                 
                                 

0 1                  0 1  1 1
 1 0    2 1 1   1 0   2 1  1
1
                                             This type of measurement represents
                                             the limit as the number of
measurements goes to infinity
0  i                0  i  1 1
 i 0    2 1 1   i 0   2 1  0
1
                               
                               

Here 3 may be enough, in general you need
four
Bloch Sphere





Very General Formula
• probability p(m) = |Pm|.

Any type of
measurement
operator

“insert Operator
between Braket” rule

Pm| = state (column vector)

|Pm| = number = probability
More examples and generalizations: Duals and
Inner Products are used in measurements
<|

This is inner product not
tensor product!

(        )                                      Remember this is a
number

We prove from general properties of operators
Review: Duals as Row Vectors

To do bra from ket you need transpose and conjugate to
make a row vector of conjugates.
Do this always if you are in trouble to understand some
formula involving operators, density matrices, bras and
kets.
Review:     General
Measurement

To prove it it is sufficient
to substitute the old base
and calculate, as shown
Illustration of the formalisms used. You
can calculate measurements from there
               i       
  cos       0  e sin                 1
2                         2
 e i      0 
1 0 
z                                              Z    
 0

i 
 0  1
         0                                              e 
                       
0 1                                                 cos   sin  
x  
 1 0
                                       Y ( )  
 sin   cos  
                                                                       

0  i                                              cos   i sin  
y                                            X ( )  
  i sin   cos  
i 0                                               


     
1
  0   1                                      *              * e t 
      * t
  e                   * 

State Vector                                                      

Density State
Postulate 3, rough form

This is calculate as in 2
slides earlier
Ensamble Quantum
Computers and
Mixed States
Particle = qubit
Very many identical
Molecule = set of     Molecules =
qubits = single   Ensamble quantum
quantum computer        computer
Mixed States
• Suppose one only knows of a system that it is in one of a statistical
ensemble of state vectors vi (“pure” states), each with density
matrix i and probability Pi.

• This is called a mixed state.

• This ensemble is completely described, for all physical purposes,
by the expectation value (weighted average) of density matrices:

   Pi i
– Note: even if there were uncountably many state vectors vi, the state remains fully
described by <n2 complex numbers, where n is the number of basis states!
Ensemble of quantum states
• Quantum states can be expressed as a density
matrix:                      Sum of these probabilities is one

   pi  i  i                                    For the
i                                          ensemble

• A system with n quantum states has n entries across
the diagonal of the density matrix. The nth entry of
the diagonal corresponds to the probability of the
system being measured in the nth quantum state.

• The off diagonal correlations are zeroed out by
decoherence.                                                       
U U       *T
Unitary operations on a density matrix
• Unitary operations on a density matrix are
Old density
expressed as:                                           matrix

New density
matrix
    piU i i U   UU 
i
• In other words the diagonal of density matrix is left as
weights corresponding to the current states projection
onto the computational basis after acted on by the
unitary operator U
– Much like an inner product.

   pi  i  i
i                              U   U *T
Trace of
Matrix
Trace of Matrix
• Trace of a matrix (sum of the diagonal elements):

tr ( A)   Aii
i

•   Unitary operators are trace preserving.

• The trace of a pure state is 1, all information about the system is
known.

• Operators Commute under the action of the trace:

tr( XY )  tr (YX )
U   U *T
The trace operation
tr A    j Ajj
0 1                            1 0
Examples: X       tr  X   0;         I 
0 1
tr I   2.
 1 0                              
Cyclicity property: tr AB  =tr BA  .

tr AB    j AB  jj   jk Ajk Bkj   jk Bkj Ajk  k BA kk  tr BA 
tr ( A)   Aii
Partial Trace of a Matrix                                   i

U   U *T
• Partial Trace trB (  AB ) (defined by linearity)
• If you want to know about the nth state in a system, you
can trace over the other states.
Partial trace

trB( a1 a2  b1 b2 )  a1 a2 tr( b1 b2 )

Density
matrix    trace
Measurement of a
density state for
circuit with
entanglement
Measurement of a density state for
circuit with entanglement
  
1       0 0 0
                              H
0       0 0 0
Initial state:                 00 00  
0      0 0 0
             
0       0 0 0
             

Operate on the state space with an operator that preservers unitary evolution (H gate first bit):

 '  H1  I  00 00 H1  I   U1U                                       
1
1   0 0 1
         
Now act on system with CNOT gate:                                                           1 0   0 0 0
 '  CNOT12 U 1 U 1 CNOT12   U 2,1 U 2,1    
2 0    0 0 0
         
1   0 0 1
         
We still define collections of measurement operators to act on the state space of our system:

M 0  00 00               M 1  01 01              M 2  10 10                  M 3  11 11
Measurement of a density state
The probability that a result m occurs is given by the equation:

0
trB( a1 a2  b1 b2 )  a1 a2 tr( b1 b2 )   0 0 0 1      0 0 tr1P
    
                             
k

M m    tr 
0 0 0 1

p(m)  tr M m M m   tr

        0
0    0 0 0 1  0
0 0 0 2  0   0 0 0      2
                             
0      0 0 1 1      0 0 1
                             

0000
0000
M3                   ½ tr    0000     =1/2
1001

recall                          Probability of outcome k             tr Pk 
For most of our purposes we can just use
state vectors.
REMINDER: Ensemble point of view

Imagine that a quantum system is in the state  j with
probability pj .           Probability of outcome k being in state j

We do a measurement described by projectors Pk .

k

Probability of outcome k   Pr k | state  j pj     
   j Pk  j pj              Probability
Formula linking          k                           of being in

trace and density
k

  pj tr  j  j Pk          state j

matrix
REMINDER: Ensemble point of view
Probability of outcome k being in state j

k

Probability of outcome k   Pr k | state  j pj        
   j Pk  j pj
k                                    Probability

               
of being in
  pj tr  j  j Pk                    state j
k

Probability of outcome k  tr P 
k

where    pj  j  j is the density matrix.
j

 completely determines all measurement statistics.
Postulate 3:
Quantum
Measurement
Postulate 3:
Quantum Measurement

Now we can formulate
precisely the Postulate 3
Now we use this notation for
an Example of Qubit
Measurement

0                      0   We show here two
methods to derive
10
[0* 1*]        10
= [0* 1*]
it. Check for
00   00   1
10   1   homeworks and
exam
00
How the state vector changes in
measurement?

M0 |> =

= |0> (<0| 0 |0> + <0| 1 |1> = |0> 0
What happens to a system after a
Measurement?
• After a system or subsystem is measured from outside, its state
appears to collapse to exactly match the measured outcome

– the amplitudes of all states perfectly distinguishable from states consistent with
that outcome drop to zero

– states consistent with measured outcome can be considered “renormalized” so
their probabilities sum to 1

Only orthogonal state can be
distinguished in a measurement
Projective Measurements:
Average Values and Standard
Deviations
Observable:

Can write:                              scalar

Average value of a measurement:          scalar

Standard deviation of a measurement:

scalar
Phase
Global phase

Relative phase can be physically
distinquished

Relative phase depends on the basis
Postulate 4:
Composite
Systems
Compound Systems
• Let C=AB be a system composed of two
separate subsystems A, B each with vector
spaces A, B with bases |ai, |bj.

• The state space of C is a vector space
C=AB given by the tensor product
of spaces A and B, with basis states
labeled as |aibj.
Composition example
The state space of a composite physical system is
the tensor product of the state spaces of the
components
– n qubits represented by a 2n-dimensional Hilbert space
– composite state is | = |1  |2 . . . |n
– e.g. 2 qubits:
|1 = 1|0 + 1|1
|2 = 2|0 + 2|1
| = |1  |2 = 12|00 + 12|01 + 12|10 + 12|11
– entanglement
2 qubits are entangled if |  |1  |2 for any |1, |2
e.g. | = |00 + |11
Entanglement
• If the state of compound system C can be expressed as a
tensor product of states of two independent subsystems A
and B,
c = ab,                 Entanglement results from
axioms/postulates. It exists only
for quantum states

• then, we say that A and B are not entangled, and they
have individual states.
– E.g. |00+|01+|10+|11=(|0+|1)(|0+|1)

• Otherwise, A and B are entangled (basically correlated);
their states are not independent.
– E.g. |00+|11
Size of Compound State Spaces
• Note that a system composed of many separate
subsystems has a very large state space.

• Say it is composed of N subsystems, each with k basis
states:

– The compound system has kN basis states!
– There are states of the compound system having nonzero
amplitude in all these kN basis states!

– In such states, all the distinguishable basis states are
(simultaneously) possible outcomes (each with some
corresponding probability)
– Illustrates the “many worlds” nature of quantum mechanics.
Postulate 4:
Composite Systems
Summary on Postulates
Hilbert Space

Evolution

Measurement

Tensor Product
Key Points to Remember:
• An abstractly-specified system may have many
possible states; only some are distinguishable.
• A quantum state/vector/wavefunction  assigns
a complex-valued amplitude (si) to each
distinguishable state si (out of some basis set)
• The probability of state si is |(si)|2, the square
of (si)’s length in the complex plane.
• States evolve over time via unitary (invertible,
length-preserving) transformations.
• Statistical mixtures of states are represented by
weighted sums of density matrices =||.
Key points to
remember
• The Schrödinger Equation
• The Hamiltonian
• The Evolution Matrix
• How complicated is a single Quantum Bit?
• Measurement
• Measurement operators
• Measurement of a state vector using projective
measurement
• Density Matrix and the Trace
• Ensembles of quantum states, basic definitions and
importance
• Measurement of a density state
Bibliography & acknowledgements
• Michael A. Nielsen and Isaac L. Chuang, Quantum
Computation and Quantum Information, Cambridge
University Press, Cambridge, UK, 2002
• V. Bulitko, On quantum Computing and AI, Notes
for a graduate class, University of Alberta, 2002
• R. Mann,M.Mosca, Introduction to Quantum
Computation, Lecture series, Univ. Waterloo, 2000
http://cacr.math.uwaterloo.ca/~mmosca/quantumcou
rsef00.htm D. Fotin, Introduction to “Quantum
Computing Summer School”, University of Alberta,
2002.
Slides
Distinguishability
Recall that M
is
measurement
operator

We represent |2
in new base

Thus we have contradiction, states can be
On the other hand   distinguished unless they are orthogonal
General Measurements
in compound spaces
Uncertainty
Principle
Positive Operator-Valued
Measurements (POVM)
• This lecture was taught in 2005, 2007, 2011
• There is more about trace and partial trace in
future lectures and examples.

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