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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

2
Linear 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.

3
Eigenvalues 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).

4
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

5
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
  • 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??
6
A great breakthrough
7
Postulates 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

8
Physical 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
9
Postulate 1 State Space
10
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.
  • Later, we may try to make these definitions more
    formal precise.

B
A
11
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)

12
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.

13
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 weve discussed
    can be applied to either classical or quantum
    physics
  • Now, lets get to the uniquely quantum stuff

14
An example of a state space
15
Schroedingers 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
16
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 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.

17
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.

18
Distinguishability 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.

19
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 Acan be
    identified with the unitvectors of H.

20
Postulate 2 Evolution
21
Postulate 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
22
Example of evolution
23
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

24
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 (Schrödingers
    equation) which has the form of a wave equation
    when S ranges over a space of positional states.

25
Schrödingers 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
26
Features 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

27
Heisenberg and Schroedinger views of Postulate 2
This is Heisenberg picture
This is Schroedinger picture
..in this class we are interested in Heisenbergs
view..
28
The 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

29
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.
  • 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
30
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.

31
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.

Evolution matrix
Hamiltonian matrix
32
Postulate 3 Quantum Measurement
33
Computational Basis a reminder
Observe that it is not required to be
orthonormal, just linearly independent
We recalculate to a new basis
34
Example 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
36
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
37
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 Prst st2
  • After the measurement, the state is changed, in a
    way we will define later.

Inner product between states
38
A 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.

39
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
40
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.

41
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)
  • 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.

42
Towards 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
43
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
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
44
Qubit example calculate the density matrix
Conjugate and change kets to bras
Density matrix
Density matrix is a generalization of state
45
Measurement 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
46
Bloch Sphere
47
Very General Formula
  • probability p(m) ??Pm??.

Any type of measurement operator
insert Operator between Braket rule
Pm?? state (column vector)
??Pm?? number probability
48
More 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
49
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.
50
General Measurement
Review
To prove it it is sufficient to substitute the
old base and calculate, as shown
51
Illustration of the formalisms used. You can
calculate measurements from there
q
State Vector
Density State
52
Postulate 3, rough form
This is calculate as in 2 slides earlier
53
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
  • 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!

54
Ensemble 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
55
Unitary 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
56
Trace 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

57
The trace operation
58
Partial 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
59
Measurement 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
60
Measurement 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.
61
REMINDER Ensemble point of view
Probability of outcome k being in state ?j
Probability of being in state ?j
Formula linking trace and density matrix
62
REMINDER Ensemble point of view
Probability of outcome k being in state ?j
Probability of being in state ?j
63
Postulate 3 Quantum Measurement
Now we can formulate precisely the Postulate 3
64
Now 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
65
How the state vector changes in measurement?
M0 ?gt
0gt (lt0 ?0 0gt lt0 ?1 1gt 0gt ?0
66
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
67
Projective Measurements Average Values and
Standard Deviations
Observable
Can write
scalar
Average value of a measurement
scalar
Standard deviation of a measurement
scalar
68
Phase
Global phase
Relative phase can be physically distinquished
Relative phase depends on the basis
69
Postulate 4 Composite Systems
70
Compound 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?.

71
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? ?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?

72
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 results from axioms/postulates. It
exists only for quantum states
73
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.

74
Postulate 4 Composite Systems
75
Summary on Postulates
Hilbert Space
Evolution
Measurement
Tensor Product
76
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 ?????.

77
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

78
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/quantumcours
    ef00.htm D. Fotin, Introduction to Quantum
    Computing Summer School, University of Alberta,
    2002.

79
Additional Slides
80
Distinguishability
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
81
General Measurements in compound spaces
82
Uncertainty Principle
83
Positive 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.
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