Title: Postulates%20of%20Quantum%20Mechanics
1- Postulates of Quantum Mechanics
- SOURCES
- Angela Antoniu, David Fortin, Artur Ekert,
Michael Frank, Kevin Irwig , Anuj Dawar , Michael
Nielsen - Jacob Biamonte and students
2Linear Operators
Short review
- V,W Vector spaces.
- A linear operator A from V to W is a linear
function AV?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 (AA). - Its diagonal elements are real.
3Eigenvalues Eigenvectors
- v is called an eigenvector of linear operator A
iff A just multiplies v by a scalar x, i.e. Avxv
- 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).
4Exam 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
5Unitary 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 - 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??
6A great breakthrough
7Postulates of Quantum MechanicsLecture 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
8Physical Systems - Quantum Mechanics Connections
Hilbert Space
? ?
Isolated physical system
Postulate 1
Unitary transformation
? ?
Evolution of a physical system
Postulate 2
Measurement operators
? ?
Measurements of a physical system
Postulate 3
Tensor product of components
? ?
Composite physical system
Postulate 4
9Postulate 1 State Space
10Systems 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. - Later, we may try to make these definitions more
formal precise.
B
A
11Closed 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)
12Concrete 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.
13States 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 weve discussed
can be applied to either classical or quantum
physics - Now, lets get to the uniquely quantum stuff
14An example of a state space
15Schroedingers 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
16Distinguishability 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 thats 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.
17Postulate 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.
18Distinguishability of States, more precisely
- Two state vectors s and t are (perfectly)
distinguishable or orthogonal (write s?t) iff
st 0. (Their inner product is zero.) - State vectors s and t are perfectly
indistinguishable or identical - (write st) iff st 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 st. - Note amplitudes are complex numbers.
19State 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 Acan be
identified with the unitvectors of H.
20Postulate 2 Evolution
21Postulate 2 Evolution
- Evolution of an isolated system can be expressed
as - 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
doesnt matter) and reversible, since -
Unitary operations preserve inner product
22Example of evolution
23Time 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
24Wavefunctions
- 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 (Schrödingers
equation) which has the form of a wave equation
when S ranges over a space of positional states.
25Schrödingers 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,
x3i1, x3i2), 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
26Features of the wave equation
- Particles momentum state p is encoded implicitly
by the particles wavelength ? ph/? - The energy of any state is given by the frequency
? of rotation of the wavefunction in the complex
plane Eh?. - By simulating this simple equation, one can
observe basic quantum phenomena such as - Interference fringes
- Tunneling of wave packets through potential
barriers
27Heisenberg and Schroedinger views of Postulate 2
This is Heisenberg picture
This is Schroedinger picture
..in this class we are interested in Heisenbergs
view..
28The Solution to the Schrödinger Equation
- The Schrödinger Equation governs the
transformation of an initial input state to
a final output state . - It is a prescription for what we want to do to
the computer. - is a time-dependent Hermitian matrix of
size 2n called the Hamiltonian - is a matrix of size 2n called the
evolution matrix, - Vectors of complex numbers of length 2n
- Tt is the time-ordering operator
29The 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. - The evolution matrix is the program for
the quantum computer. Applying this program to
the input state produces the output state
,which gives us a solution to the
problem.
We discussed this power series already
Formula from last slide
30The 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.
31The 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.
Evolution matrix
Hamiltonian matrix
32Postulate 3 Quantum Measurement
33Computational Basis a reminder
Observe that it is not required to be
orthonormal, just linearly independent
We recalculate to a new basis
34Example of measurement in different bases
1/?2
The second with probability zero
35- You can check from definition that inner product
of 0gt and 1gt is zero. - Similarly the inner product of vectors from the
second basis is zero. - But we can take vectors like 0gt and
1/?2(0gt-1gt) as a basis also, although
measurement will perhaps suffer.
Good base
Not a base
36A simplified Bloch Sphere to illustrate the bases
and measurements
You cannot add more vectors that would be
orthogonal together with blue or red vectors
37Probability 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 Prst st2 - After the measurement, the state is changed, in a
way we will define later.
Inner product between states
38A Simple Example of distinguishable,
non-distinguishable states and measurements
- Suppose abstract system S has a set of only 4
distinguishable possible states, which well call
s0, s1, s2, and s3, with corresponding ket
vectors s0?, s1?, s2?, and s3?. - Another possible state is then the vector
- 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.
39Observables
- 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
40Observables
- 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.
41Density Operators
- For a given state ??, the probabilities of all
the basis states si are determined by an
Hermitian operator or matrix ? (the density
matrix) - 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.
42Towards QM Postulate 3 on measurement and general
formulas
- A measurement is described by an Hermitian
operator (observable) - M m Pm
- Pm is the projector onto the eigenspace of M with
eigenvalue m - After the measurement the state will be
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
eigenvalue
? m
Derived in next slide
43An 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
Operate on the state space with an operator that
preservers unitary evolution
Hadamard
?0??1?
Define a collection of measurement operators for
our state space
Half probability of measuring 0
Act on the state space of our system with
measurement operators
p0
Half probability of measuring 1
But now it is a part of formal calculus
44Qubit example calculate the density matrix
Conjugate and change kets to bras
Density matrix
Density matrix is a generalization of state
45Measurement of a state vector using projective
measurement
Operate on the state space with an operator that
preservers unitary evolution
Define observables
Vector is on axis X
Act on the state space of our system with
observables (The average value of measurement
outcome after lots of measurements)
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
46Bloch Sphere
47Very General Formula
Any type of measurement operator
insert Operator between Braket rule
Pm?? state (column vector)
??Pm?? number probability
48More examples and generalizations Duals and
Inner Products are used in measurements
lt?
This is inner product not tensor product!
(
)
Remember this is a number
We prove from general properties of operators
49Review 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.
50General Measurement
Review
To prove it it is sufficient to substitute the
old base and calculate, as shown
51Illustration of the formalisms used. You can
calculate measurements from there
q
State Vector
Density State
52Postulate 3, rough form
This is calculate as in 2 slides earlier
53Mixed 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 - Note even if there were uncountably many state
vectors vi, the state remains fully described by
ltn2 complex numbers, where n is the number of
basis states!
54Ensemble of quantum states
- Quantum states can be expressed as a density
matrix - 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.
Sum of these probabilities is one
For the ensemble
55Unitary operations on a density matrix
- Unitary operations on a density matrix are
expressed as - 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.
Old density matrix
New density matrix
adjoint
56Trace of Matrix
- Trace of a matrix (sum of the diagonal elements)
- 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
-
57The trace operation
58Partial Trace of a Matrix
- Partial Trace (defined by
linearity) - If you want to know about the nth state in a
system, you can trace over the other states.
Partial trace
Density matrix
trace
59Measurement of a density state for circuit with
entanglement
H
Initial state
Operate on the state space with an operator that
preservers unitary evolution (H gate first bit)
Now act on system with CNOT gate
We still define collections of measurement
operators to act on the state space of our system
60Measurement of a density state
The probability that a result m occurs is given
by the equation
0000 0000 0000 1001
M3
½ tr 1/2
recall
For most of our purposes we can just use state
vectors.
61REMINDER Ensemble point of view
Probability of outcome k being in state ?j
Probability of being in state ?j
Formula linking trace and density matrix
62REMINDER Ensemble point of view
Probability of outcome k being in state ?j
Probability of being in state ?j
63Postulate 3 Quantum Measurement
Now we can formulate precisely the Postulate 3
64Now we use this notation for an Example of Qubit
Measurement
?0 ?1
?0 ?1
We show here two methods to derive it. Check for
homeworks and exam
1 0 0 0
1 0 0 0
?0 ?1
1 0 0 0
?0 ?1
65How the state vector changes in measurement?
M0 ?gt
0gt (lt0 ?0 0gt lt0 ?1 1gt 0gt ?0
66What 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
67Projective Measurements Average Values and
Standard Deviations
Observable
Can write
scalar
Average value of a measurement
scalar
Standard deviation of a measurement
scalar
68Phase
Global phase
Relative phase can be physically distinquished
Relative phase depends on the basis
69Postulate 4 Composite Systems
70Compound Systems
- Let CAB 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 CA?B
given by the tensor product of spaces A and B,
with basis states labeled as aibj?.
71Composition 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? ?10? ?11??2? ?20? ?21???
?1? ? ?2? ?1?200? ?1?201? ?1?210?
?1?211? - entanglement
- 2 qubits are entangled if ?? ? ?1? ? ?2? for
any ?1?, ?2? - e.g. ?? ?00? ?11?
72Entanglement
- 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 results from axioms/postulates. It
exists only for quantum states
73Size 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.
74Postulate 4 Composite Systems
75Summary on Postulates
Hilbert Space
Evolution
Measurement
Tensor Product
76Key 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 ?????.
77Key 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
78Bibliography 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/quantumcours
ef00.htm D. Fotin, Introduction to Quantum
Computing Summer School, University of Alberta,
2002. -
79Additional Slides
80Distinguishability
Recall that M is measurement operator
We represent ?2? in new base
Thus we have contradiction, states can be
distinguished unless they are orthogonal
On the other hand
81General Measurements in compound spaces
82Uncertainty Principle
83Positive Operator-Valued Measurements (POVM)
84- This lecture was taught in 2005, 2007
- There is more about trace and partial trace in
future lectures and examples.