Postulates of Quantum Mechanics

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.

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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).

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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

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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??
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A great breakthrough
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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

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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
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Postulate 1 State Space
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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
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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)

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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.

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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

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An example of a state space
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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
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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.

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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.

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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.

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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.

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Postulate 2 Evolution
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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
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Example of evolution
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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

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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.

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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
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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

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

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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
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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.

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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
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Postulate 3 Quantum Measurement
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Computational Basis a reminder
Observe that it is not required to be
orthonormal, just linearly independent
We recalculate to a new basis
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Example of measurement in different bases
1/?2
The second with probability zero
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• 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
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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
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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
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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.

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Observables
40
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
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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.

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Density Operators
43
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.

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M Hermitian observable
45
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
Hermitian observable
Derived in next slide
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How to survive reading papers

1. In all these calculations we have a ket part
   and a bra part
2. Remember how to create Bra part from Ket Part.
3. Remember about dual vectors
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.

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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
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Qubit example calculate the density matrix
Conjugate and change kets to bras
Density matrix
Density matrix is a generalization of state
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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
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Bloch Sphere
51
Very General Formula

• probability p(m) ??Pm??.

Any type of measurement operator
insert Operator between Braket rule
Pm?? state (column vector)
??Pm?? number probability
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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
53
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.
54
General Measurement
Review
To prove it it is sufficient to substitute the
old base and calculate, as shown
55
Illustration of the formalisms used. You can
calculate measurements from there
q
State Vector
Density State
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Postulate 3, rough form
This is calculate as in 2 slides earlier
57
Ensamble Quantum Computers and Mixed States
Particle qubit
Very many identical Molecules Ensamble quantum
computer
Molecule set of qubits single quantum computer
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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!

59
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
60
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
61
Trace of Matrix
62
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
  

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The trace operation
64
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
65
Measurement of a density state for circuit with
entanglement
66
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
67
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.
68
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
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REMINDER Ensemble point of view
Probability of outcome k being in state ?j
Probability of being in state ?j
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Postulate 3 Quantum Measurement
71
Postulate 3 Quantum Measurement
Now we can formulate precisely the Postulate 3
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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
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How the state vector changes in measurement?
M0 ?gt
0gt (lt0 ?0 0gt lt0 ?1 1gt 0gt ?0
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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
75
Projective Measurements Average Values and
Standard Deviations
Observable
Can write
scalar
Average value of a measurement
scalar
Standard deviation of a measurement
scalar
76
Phase
Global phase
Relative phase can be physically distinquished
Relative phase depends on the basis
77
Postulate 4 Composite Systems
78
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?.

79
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?

80
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
81
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.

82
Postulate 4 Composite Systems
83
Summary on Postulates
Hilbert Space
Evolution
Measurement
Tensor Product
84
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 ?????.

85
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

86
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.

87
Additional Slides
88
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
89
General Measurements in compound spaces
90
Uncertainty Principle
91
Positive Operator-Valued Measurements (POVM)
92

• This lecture was taught in 2005, 2007, 2011
• There is more about trace and partial trace in
  future lectures and examples.