Download 2005-q-0035-Postulates-of-quantum-mechanics

Postulates of
Quantum
Mechanics
SOURCES
Angela Antoniu, David Fortin,
Artur Ekert, Michael Frank,
Kevin Irwig , Anuj Dawar ,
Michael Nielsen
Jacob Biamonte and students
Gates on Multi-Qubit State, a reminder
Example of Complex quantum
system of 3 qubits – other
realization of Toffoli, composed of
2-qubit gates
•All gates are at most 2-qubit
•Only CNOT as 2-qubit gates
•It has 6 not 5 interaction gates
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 selfadjoint (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.
• x is degenerate if it is shared by 2 eigenvectors
that are not scalar multiples of each other.
• 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.
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
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
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.
B
• One physical system A is a subsystem of
A
another system B (write AB) iff A is
completely contained within B.
• 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.
Revised Postulate 1
Distinguishability of States, more
precisely
t
• Two state vectors s and t are (perfectly)
s

distinguishable or orthogonal (write st)
iff s†t = 0. (Their inner product is zero.)
• 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
vectors of H.
t

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.
– When s is another state vector, and the real state is t,
then (s) is just s†t.
–  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
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

2  j 0 m j / 3 x j
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 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  Uˆ t   0
 0

•
Ĥ  
is a time-dependent Hermitian matrix of size 2n
called the Hamiltonian
• Û t  is a matrix of size 2n called the evolution matrix,
• 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    
n 0  n! 


• The evolution matrix Û t is the program for the quantum
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  Uˆ t   0
 0

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 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  Uˆ t   0
 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.
 t

 t   T exp i  Hˆ  d   0  Uˆ t   0
 0

Postulate 3:
Quantum
Measurement
Computational Basis – a reminder
Observe that it is not
required to be orthonormal,
just linearly independent
We recalculate to a new
basis
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.
A Simple Example of distinguishable, nondistinguishable 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
s0 
s3
2
2
1 2 


 0 
 0 


  i 2 
• Which is equal to the column matrix:
• If measured to see if it is in state s0,
we have a 50% chance of getting a “yes”.
Observables
• Hermitian operator A on V is called an
observable if there is an orthonormal (all unitlength, 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
• 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.
Towards QM Postulate 3 on
measurement and general formulas
A measurement is described by an Hermitian
eigenvalue
operator (observable)
M=
m
P
m
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
• |1 with probability |11| = |1||2 = ||2
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
Duals as Row Vectors
To do bra from ket you need
transpose and conjugate to make
a row vector of conjugates.
General
Measurement
To prove it it is sufficient
to substitute the old base
and calculate, as shown
Illustration of some formalisms used, you can
calculate measurements from there
  cos
1 0 

 z  
0

1



2
i
0  e sin
0
0 1

 x  
1
0



0  i

 y  
i
0



1
  0   1
State Vector


2
1
 e i
Z    
 0
0 

i 
e 
 cos   sin  

Y ( )  




sin

cos



 cos   i sin  

X ( )  





i
sin

cos



  *
      * t
  e
Density State
 *e t 

* 
 
Postulate 3, rough form
This is calculate as in
previous slide
The Measurement Problem
Can we deduce postulate 3
from 1 and 2?
Joke. Do not try it. Slides are from MIT.
More examples 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
Operate on the state space with an operator that preservers unitary evolution:
  H op 0 
0 1
2
1 1
 

2 1
Define a collection of measurement operators for our state space:
M1  1 1
M0  0 0
Act on the state space of our system with measurement operators:
 1 0  1 1 1
1

  
1 1 
0 0 
2
 0 0  2 1 2
 0 0  1 1 1
1

  
1 1 
1 1 
2
 0 1  2 1 2
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!
Measurement of a state vector using
projective measurement
Operate on the state space with an operator that preservers unitary evolution:
 0
  H op 0 
0 1
2

1 1
 
2 1
Define observables:
0 1

 x  
1 0
0  i

0 
 y  
i
1
0

 z  
 0  1
Act on the state space of our system with observables
(The average value of measurement outcome after lots of measurements):
1 0  1 1
1 0 
1
  
 
   0
1 1  
 
2
 0  1
 0  1 2 1
0 1  1 1
0 1
1
  
 
   1
1 1  
 
2
1 0
 1 0  2 1
0  i  1 1
0  i
1
  
 
   0
1 1  
 
2
i 0 
 i 0  2 1
This type of measurement represents
the limit as the number of
measurements goes to infinity
Here 3 may be enough, in general you need
four
The Density Matrix and the Trace
Ensembles of quantum states, basic definitions and importance(1)
• Quantum states can be expressed as a density
matrix:
   pi i i
i
• 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
The Density Matrix and the Trace
Ensembles of quantum states, basic definitions and importance (2)
• Unitary operations on a density matrix are
expressed as:
New density
matrix
Old density
matrix
    piU i i U   UU 
i
• In other words the diagonal 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.
U   U *T
The Density Matrix and the Trace
Ensembles of quantum states, basic definitions and importance
• Trace of a matrix (sum of the diagonal elements):
tr( A)   A
• 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:
ii
i
tr ( XY )  tr (YX )
AB
tr
B
(

) (defined by linearity)
• Partial Trace
U   U *T
• If you want to know about the nth state in a system, you
can trace over the other states.
trB( a1 a2  b1 b2 )  a1 a2 tr( b1 b2 )
Measurement of a density state
Initial state:
1

0
     00 00  
0

0

0
0
0
0
0

0
0

0 
0
0
0
0
H
Operate on the state space with an operator that preservers unitary evolution (H gate first bit):
 '  H1  I  00
Now act on system with CNOT gate:
00 H1  I 

 U1U

1
 '  CNOT12 U 1 U 1 CNOT12   U 2,1 U 2,1
1

1 0
 
2 0

1

0
0
0
0
0
0
0
0
We still define collections of measurement operators to act on the state space of our system:
M 0  00 00
M1  01 01
M 2  10 10
M 3  11 11
1

0
0

1 
REMINDER: Ensemble point of view
Imagine that a quantum system is in the state  j with
Probability of outcome k being in state j
probability pj .
We do a measurement described by projectors Pk .


Probability of outcome k   Pr k | state  j pj
k
   j Pk  j pj
k

  pj tr  j  j Pk
k

Probability
of being in
state j
Probability of outcome k  tr  Pk 
where    pj  j  j is the density matrix.
j
 completely determines all measurement statistics.
Measurement of a density state
The probability that a result m occurs is given by the equation:
0

0

p(m)=


p(m)  tr M m M


tr
M


tr
m
m
0

0

trB( a1 a2  b1 b2 )  a1 a2 tr( b1 b2 )


0 0 0 1
 
0 0 0 1  0
0 0 0 2  0
 
0 0 1   1
0 0 tr1P

0 0 0 1



0 0 0
2

0 0 1  
k
M3
recall
Probability of outcome k  tr
For most of our purposes we can just use
state vectors.
 Pk 
Postulate 3:
Quantum Measurement
Now we can formulate
precisely the Postulate 3
Now we use this notation for
an Example of Qubit
Measurement
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
• This “collapse” seems nonunitary (& nonlocal)
– However, this behavior is now explicable as the expected
consensus phenomenon that would be experienced even by entities
within a closed, perfectly unitarily-evolving world (Everett,
Zurek).
Distinguishability
Recall that M
is
measurement
operator
On the other hand
Thus we have contradiction, states can be
distinguished unless they are orthogonal
Projective Measurements:
Average Values and Standard
Deviations
Observable:
Can write:
Average value of a measurement:
Standard deviation of a measurement:
Irrelevance of
“global phase”
Phase
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,
• 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
Entanglement
Entanglement
Some convenctions implicit in postulate 4
Quantum Entanglement
We assume that we
can factorize as
tensor product of |a>
and |b>
Leads to
contradiction
Superdense
Coding
Multiple-Qubit Systems
Postulate 4
Example to calculate state of a composite system from
previous state of it (problem possible for final exam)
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.
Additional
Slides
General Measurements
in compound spaces
Uncertainty
Principle
Positive Operator-Valued
Measurements (POVM)