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Review
Quantum vs. Classical Probabilistic Evolution
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r Let us illustrate, on an example, a principal difference
between a quantum evolution and a classical probabilistic
evolution.
r If a qubit system develops under the evolution
r then, after one step of evolution, we observe both |0⟩ and |1⟩with the probability 1/2, but after two steps we get
r and therefore any observation gives |0⟩ with probability 1.
PROBABILISTIC vs.
QUANTUM SYSTEM
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r On the other hand, in case of the classical probabilistic
evolution
r we have after one step of evolution both 0 and 1 with the
same probability 1/2 but after two steps we have again
r and therefore, after two steps of evolution, we have again
both values 0 and 1 with the same probability 1/2.
In the quantum case, during the second evolution step, amplitudes at |1⟩ cancel each other and we have so-called destructive interference. At the same time, amplitudes at |0⟩ amplify each other and we have so-called constructive interference.
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Outline for this lecture
BASIC CONCEPTS and RESULTS
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HILBERT SPACE BASICS
r The Hilbert space is a mathematical framework suitable
for describing concepts, principles, processes and laws of
the theory of quantum world called (for historical
reasons) quantum mechanics, in general; and quantum
information processing and communication (QIPC) in
particular.
r In this chapter those basics of Hilbert space theory are
introduced and illustrated that play an important role in
QIPC.
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QUANTUM SYSTEM
= HILBERT SPACEr
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HILBERT SPACE BASICS
r This allows us to introduce on H a metric topology
and such concepts as continuity.
r All n-dimensional Hilbert spaces are isomorphic.
Their vectors of norm 1 are called pure quantum
states.
r Their physical counterparts are n-level quantum
systems.
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MORE ABOUT RELATIONS
BETWEEN QUANTUM SYSTEMS AND
HILBERT SPACES
r Basic assumption: With every quantum systems S there is
associated a Hilbert space HS, whose dimension depends
on the nature of the degree of freedom being considered for
the system.
r Example: If only the spin orientation of an electron (a spin-
1/2 particle) is considered, then the corresponding Hilbert space is a two dimensional Hilbert space H2.
r However, if the position of an electron is of concern, which
can be in any point of some space, then the corresponding
Hilbert space is usually taken to be continuous and is therefore infinite-dimensional.
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BRA-KET NOTATION
r P. Dirac introduced a very handy notation, the so called
bra-ket notation, to deal with amplitudes, quantum states and linear mappings f : H→ C.
r If ψ, φ ∈ H, then:
⟨φ | ψ⟩ − a number: the scalar product between ψ and φ (or
the amplitude of going from ψ to φ)
|ψ⟩ − a ket-vector: a column vector – an equivalent to ψ
⟨φ | − a bra-vector: a row vector - the conjugate transpose of |φ⟩, i.e., linear functional on H such that ⟨φ | (| ψ⟩) = ⟨φ | ψ⟩. 10
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r If φ = (φ1,… , φn) and ψ = (ψ1,… , ψn) then
is a ket vector
is a bra vector
with inner product, i.e., scalar product:
and outer product:
r The meaning of the outer product | φ ⟩⟨ ψ | is that of the mapping that maps any state |γ⟩ into the state | φ ⟩⟨ ψ |(|γ⟩) = | φ⟩(⟨ψ| γ⟩) = ⟨ψ| γ⟩ | φ⟩.
r It is often said that physical counterparts of vectors of n-dimensional Hilbert spaces are n-level quantum systems.
Example
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GENERAL DEFINITION
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DUAL HILBERT SPACE
r The set of dual vectors of a Hilbert space H forms
so the called Dual Hilbert space H*.
r A dual vector ⟨φ | to a vector | φ⟩ is often denoted as
⟨φ |†.
r If ⋃ {|β�⟩}�� forms an orthogonal basis of a Hilbert
space H, then ⋃ ⟨β� }�� forms an orthogonal basis
of H*.
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DUAL HILBERT SPACE
r If H is a Hilbert space, then the set of linear
operators on H forms again a Hilbert space,
denoted usually L(H) and all inner products of an
orthogonal basis of H forms an orthogonal basis of
L(H).
r As a consequence if ⋃ {|β�⟩}�� forms an orthogonal
basis on H, then every linear operator O on H can
be expressed in the form
O = ∑ � ∑ tn,m |β�⟩ ⟨βm |
for some constants
tn,m = ⟨βn|| O |β�⟩. 14
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ORTHOGONALITY of STATES
r Two vectors ⟨φ | and ⟨ψ | are called orthogonal vectors if
⟨φ | ψ⟩ = 0.
r Physically fully distinguishable are only orthogonal vectors
(states).
r By a basis B of Hn we will understand any set of n vectors
|β1⟩, |β2⟩, … , |β�⟩ in Hn of norm 1 which are mutually
orthogonal.
r Given a basis B, any vector | ψ⟩ from Hn can be uniquely
expressed in the form
| ψ⟩ = ∑ α�|β�⟩.��
r A set S of vectors is called orthonormal if all vectors of Shave norm 1 and are mutually orthogonal.
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ORTHOGONALITY of STATES
r Definition: A subspace G of a Hilbert space H is a
subset of H closed under addition and scalar
multiplication.
r An important property of Hilbert spaces is their
decomposability into mutually orthogonal
subspaces.
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Decomposition
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OPERATORS of HILBERT SPACES
r A linear operator on a Hilbert space H is a linear mapping
A : H→ H.
r An application of a linear operator A to a vector | φ⟩ is
denoted A| φ⟩.
r A is also a linear operator of the dual Hilbert space H*, mapping each linear functional ⟨φ | of the dual space to the linear functional, denoted by ⟨φ |A.
r A linear operator A is called positive or semi-definite, notation A ≥ 0, if ⟨φ | A | φ⟩ ≥ 0 for every | φ⟩ ∈ H.
r The norm ||A|| of a linear operator A is defined as
||A|| = sup||φ||=1
||A| φ⟩||.
r A linear operator is called bounded if its norm is finite. 18
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REPRESENTATIONS
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B.
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TRACE of OPERATORS
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SELF-ADJOINT OPERATORS
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SELF-ADJOINT OPERATORS —
HERMITIAN MATRICES
:
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SPECTRAL REPRESENTATION of
SELF-ADJOINT OPERATORS
we define for a
def
def
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SPECTRAL REPRESENTATION of
UNITARY OPERATORS
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Therefore,
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QUANTUM TIME EVOLUTION
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QUANTUM TIME EVOLUTION
,
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ANOTHER VIEW OF QUANTUM
DYNAMICS
,
,
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r A quantum state is observed (measured) with respect to an
observable — a decomposition of a given Hilbert space into
orthogonal subspaces (such that each vector can be
uniquely represented as a sum of vectors of these
subspaces).
r There are two outcomes of a projection measurement of a
state |φ⟩:
1. Classical information into which subspace projection of |φ⟩ was made.
2. A new quantum state |φ′⟩ into which the state |φ⟩ collapses.
r The subspace into which projection is made is chosen
randomly and the corresponding probability is uniquely
determined by the amplitudes at the representation of |φ⟩ at
the basis states of the subspace.
QUANTUM (PROJECTION)
MEASUREMENTS
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PROJECTION MEASUREMENT -
ANOTHER VIEW
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Example
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PROBABILITIES
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ANOTHER VIEW of MEASUREMENT
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ANOTHER VIEW of MEASUREMENT
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r Quantum mechanics is a probabilistic & mathematical theory for
describing the physical world.
r However, probability involved is not probability of some dynamic
variables having a particular value in some state.
r Rather, it represents the probability of finding a particular value
of a dynamical variable if that dynamical variable is measured.
r Quantum mechanics says nothing about values of dynamical
variables when the system is not subjected to any measurement.
IMPORTANT OBSERVATION
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EXPECTATION VALUE
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Outline for next lecture
Quantum Circuits
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Questions?