CSE 599d Quantum Computing Problem Set 3
Author: Dave Bacon (
Department of Computer Science & Engineering, University of Washington
)
Due:
February 17, 2006
For this problem set recall that the Pauli
X
,
Y
, and
Z
are
X
=
0 1
1 0
,
Y
=
0

i
i
0
,
and
Z
=
1
0
0

1
.
(1)
Exercise 1:
Tsirel’son’s Inequality
Suppose that
A
,
A
,
B
,
B
are operators on some Hilbert space
H
which satisfy
A
2
=
A
2
=
B
2
=
B
2
=
I
and [
A, B
] = [
A, B
] = [
A , B
] = [
A , B
] = 0 (where the commutator is [
M, N
] =
MN

NM
.)
(a) Define
C
=
AB
+
AB
+
A B

A B
. Show that
C
2
= 4
I

[
A, A
][
B, B
].
(b) The
sup norm
of an operator
M
is defined as

M

sup
= sup

ψ
=0

M

ψ


ψ

(2)
where
 · 
is the standard norm on our Hilbert space. Prove that

M
+
N

sup
≤ 
M

sup
+

N

sup
(3)
and

MN

sup
≤ 
M

sup

N

sup
(4)
(c) Use these properties of the sup norm to show that

C

sup
≤
2
√
2
(5)
This is Tsirel’son’s (or Cirel’son’s) inequality. Suppose we are working on a Hilbert space of two qubits. If we take
A
=
A
1
⊗
I
,
A
=
A
2
⊗
I
,
B
=
I
⊗
B
1
, and
B
=
I
⊗
B
2
, then this expression is

A
1
⊗
B
1
+
A
1
⊗
B
1
+
A
2
⊗
B
1

A
2
⊗
B
2

sup
≤
2
√
2
(6)
Recall that from class we saw that for local hidden variable theories satisfy the CHSH inequality:

C
 ≤
2.
So
Tsirel’son’s inequality bounds the “amount” of violation that quantum states can have over the CHSH inequality. In
fact quantum theory can saturate this bound.
Exercise 2:
A Quantum Error Detecting Code
In this problem we will examine a quantum error detecting code on four qubits.
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 Fall '08
 Staff
 Computer Science, Hilbert space, Pauli, Wolfgang Pauli, equations Si ψ

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