Physics 580
Handout 16
16 November 2010
Quantum Mechanics I
webusers.physics.illinois.edu/
∼
goldbart/
Homework 12
Prof. P. M. Goldbart
University of Illinois
1) Quantifying entanglement
: Consider a quantum system involving
two parties
, tradi
tionally called Alice and Bob. We call such systems
bipartite
. They may each be,
e.g.
, a
quantum spin (say, one spin1 and one spin5/2), but for now let us leave them arbitrary and
denote orthonormal sets of vectors spanning Alice’s Hilbert space
{
α
⟩}
and Bob’s Hilbert
space
{
β
⟩}
. Then arbitrary states

ψ
A
⟩
of Alice can be written

ψ
A
⟩
=
∑
n
a
α
=1
A
α

α
⟩
; arbitrary
states

ψ
B
⟩
of Bob can be written

ψ
B
⟩
=
∑
n
b
β
=1
B
β

β
⟩
.
The generic states of the composite bipartite system

Ψ
⟩
can be expressed in terms of
the amplitude Ψ
αβ
as

Ψ
⟩
=
±
αβ
Ψ
αβ

α
⟩ ⊗ 
β
⟩
.
The
unentangled
states of the composite bipartite system are the subset for which the ampli
tude Ψ
αβ
factorises: Ψ
αβ
=
A
α
B
β
. The
entangled
states of the composite bipartite system
are the subset for which the amplitude Ψ
αβ
does not factorise.
a) Show that if the amplitude Ψ
αβ
factorises then the state of the composite bipartite
system

Ψ
⟩
factorises.
Let
O
A
and
O
B
be operators, respectively acting solely on Alice’s and Bob’s Hilbert spaces.
(Think,
e.g.
, of the spin and position operators for a particle with spin.)
b) Show that for unentangled states the expectation value
⟨
Ψ
O
A
O
B

Ψ
⟩
of the product
operator
O
A
O
B
factorises into a product of expectation values, one involving each
party. Brieﬂy explain why we say that such states possess no quantum correlations
between the parties.
Suppose we are concerned only with properties of one of the parties, say Alice. Rather than
retain the full information

Ψ
⟩
(or, equivalently, the full density matrix

Ψ
⟩⟨
Ψ

) describing
the composite system, we may
trace out
Bob’s Hilbert space. In this way, we develop a
reduced density matrix
ρ
A
≡
Tr
B

Ψ
⟩⟨
Ψ

,
acting solely on Alice’s Hilbert space. As long as we are only concerned with computing
expectation values of operators on Alice,
i.e.
, operators of the form
O
A
⊗
I
B
, where I
B
is the
identity on Bob’s Hilbert space, this tracing presents no loss of options.
c) Show that if
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