De\ufb01nition 3.1. The tensor product of two matrices A of dimensions m \u00d7 n<\/p>\n\n\n\n
and B of any dimensions is given by<\/p>\n\n\n\n
A \u2297 B =<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
A<\/p>\n\n\n\n
11<\/p>\n\n\n\n
B A<\/p>\n\n\n\n
12<\/p>\n\n\n\n
B . . . A<\/p>\n\n\n\n
1n<\/p>\n\n\n\n
B<\/p>\n\n\n\n
A<\/p>\n\n\n\n
21<\/p>\n\n\n\n
B A<\/p>\n\n\n\n
22<\/p>\n\n\n\n
B . . . A<\/p>\n\n\n\n
2n<\/p>\n\n\n\n
B<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
.<\/p>\n\n\n\n
A<\/p>\n\n\n\n
m1<\/p>\n\n\n\n
B A<\/p>\n\n\n\n
m2<\/p>\n\n\n\n
B . . . A<\/p>\n\n\n\n
mn<\/p>\n\n\n\n
B<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
. (3.31)<\/p>\n\n\n\n
Thus we have<\/p>\n\n\n\n
|00i =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
; |01i =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
;<\/p>\n\n\n\n
|10i =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
; |11i =<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
0<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
.<\/p>\n\n\n\n
(3.32)<\/p>\n\n\n\n
Thus we have the natural basis for the 4-dimensional vector space from those<\/p>\n\n\n\n
of two 2-dimensional spaces.<\/p>\n\n\n\n
Example 3.5.1. Direct products: to express \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
on a 2-qubit state as a matrix,<\/p>\n\n\n\n
we take the direct product of two \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
s acting on each single qubit state:<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
\u2297 \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
=<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
\u2297<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
1 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
=<\/p>\n\n\n\n
\uf8ee<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8ef<\/p>\n\n\n\n
\uf8f0<\/p>\n\n\n\n
0 0 0 1<\/p>\n\n\n\n
0 0 1 0<\/p>\n\n\n\n
0 1 0 0<\/p>\n\n\n\n
1 0 0 0<\/p>\n\n\n\n
\uf8f9<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fa<\/p>\n\n\n\n
\uf8fb<\/p>\n\n\n\n
.<\/p>\n\n\n\n
We can generalize to n qubits: the natural basis of the n-qubit Hilbert<\/p>\n\n\n\n
space H<\/p>\n\n\n\n
\u2297n<\/p>\n\n\n\n
consists of 2<\/p>\n\n\n\n
n<\/p>\n\n\n\n
orthogonal vectors<\/p>\n\n\n\n
{|0i, |1i, |2i, …, |2<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\u2212 1i}. (3.33)<\/p>\n\n\n\n
The interpretation as an n-bit register is straightforward when the labels are<\/p>\n\n\n\n
written in binary. For example, the 8<\/p>\n\n\n\n
th<\/p>\n\n\n\n
basis vector for a 4-qubit Hilbert space<\/p>\n\n\n\n
will be<\/p>\n\n\n\n
|7i = |0111i = |0i \u2297 |1i \u2297 |1i \u2297 |1i.<\/p>\n\n\n\n
The algebra of multi-qubit states generalizes in a natural manner from<\/p>\n\n\n\n
that of single qubits.<\/p>\n\n\n\n
<\/p>\n\n\n\n
Problems 59<\/p>\n\n\n\n
Box 3.8: Algebra of Tensor Product States<\/p>\n\n\n\n
Consider two distinct physical systems A and B with Hilbert spaces H<\/p>\n\n\n\n
A<\/p>\n\n\n\n
of dimensions 2<\/p>\n\n\n\n
n<\/p>\n\n\n\n
and H<\/p>\n\n\n\n
B<\/p>\n\n\n\n
of dimensions 2<\/p>\n\n\n\n
m<\/p>\n\n\n\n
. Let the basis vectors of these two<\/p>\n\n\n\n
spaces be denoted {|i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i}, i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
= 0, 1, …2<\/p>\n\n\n\n
n<\/p>\n\n\n\n
\u22121, and {|\u00b5<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i}, \u00b5<\/p>\n\n\n\n
B<\/p>\n\n\n\n
= 0, 1, …2<\/p>\n\n\n\n
m<\/p>\n\n\n\n
\u22121.<\/p>\n\n\n\n
If I pick a state |\u03c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i from A and a state |\u03c8<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i from B, I can form a state in<\/p>\n\n\n\n
the tensor product Hilbert space H<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
= H<\/p>\n\n\n\n
A<\/p>\n\n\n\n
\u2297 H<\/p>\n\n\n\n
B<\/p>\n\n\n\n
as<\/p>\n\n\n\n
|\u03a6<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
i = |\u03c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i|\u03c8<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
\u2022 Probability amplitude hi<\/p>\n\n\n\n
A<\/p>\n\n\n\n
, \u00b5<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|\u03a6i = hi<\/p>\n\n\n\n
A<\/p>\n\n\n\n
|\u03c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
ih\u00b5<\/p>\n\n\n\n
B<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2022 Inner product h\u03a6<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|\u03a6<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i = h\u03c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|\u03c6<\/p>\n\n\n\n
A<\/p>\n\n\n\n
2<\/p>\n\n\n\n
ih\u03c8<\/p>\n\n\n\n
B<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|\u03c8<\/p>\n\n\n\n
B<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2022 Basis states for H<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
is the set of product basis vectors {|\u03c5<\/p>\n\n\n\n
i\u00b5<\/p>\n\n\n\n
i =<\/p>\n\n\n\n
|i<\/p>\n\n\n\n
A<\/p>\n\n\n\n
i|\u00b5<\/p>\n\n\n\n
B<\/p>\n\n\n\n
i}<\/p>\n\n\n\n
\u2022 The most general state in H<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
is a linear combination of these basis<\/p>\n\n\n\n
states:<\/p>\n\n\n\n
|\u03a8i<\/p>\n\n\n\n
AB<\/p>\n\n\n\n
=<\/p>\n\n\n\n
X<\/p>\n\n\n\n
i\u00b5<\/p>\n\n\n\n
C<\/p>\n\n\n\n
i\u00b5<\/p>\n\n\n\n
|\u03c5<\/p>\n\n\n\n
i\u00b5<\/p>\n\n\n\n
i.<\/p>\n\n\n\n
\u2022 If two operators<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A and<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B act on each space independently then the<\/p>\n\n\n\n
action on the product space is given by the operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
C =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A \u2297<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
B.<\/p>\n\n\n\n
Summary: The Math and the Physics<\/p>\n\n\n\n
The arena of quantum mechanics is the Hilbert space H, the state vectors<\/p>\n\n\n\n
live here, and transformations of the state vector are operators in H. To be able<\/p>\n\n\n\n
to work e\ufb03ciently with the maths, we summarize the correspondence between<\/p>\n\n\n\n
the mathematical concept and the physical quantities in Table 3.2. Note that<\/p>\n\n\n\n
this is for \u201cpure\u201d states of isolated quantum systems. (We will discuss mixed<\/p>\n\n\n\n
states of systems that are in\ufb02uenced by some environment in a later chapter.)<\/p>\n\n\n\n
Problems<\/p>\n\n\n\n
3.1. Prove that a Hermitian matrix has real eigenvalues and its eigenvectors<\/p>\n\n\n\n
corresponding to distinct eigenvalues are orthogonal to each other.<\/p>\n\n\n\n
3.2. Prove that a unitary matrix has complex eigenvalues of unit magnitude, and<\/p>\n\n\n\n
that its eigenvectors corresponding to distinct eigenvalues are orthogonal.<\/p>\n\n\n\n
<\/p>\n\n\n\n
60 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
TABLE 3.2: Correspondence between the math and the physics of quantum<\/p>\n\n\n\n
mechanics.<\/p>\n\n\n\n
Math Physics<\/p>\n\n\n\n
Normalized vector |\u03c8i \u2208 H pure state<\/p>\n\n\n\n
Hermitian operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A on H physical observable<\/p>\n\n\n\n
Eigenvalues {a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
} of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A set of all possible values obtainable on measur-<\/p>\n\n\n\n
ing the observable A<\/p>\n\n\n\n
Eigenvector |a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
i of<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
A state in which measuring A gives a value a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
Computational basis {|ii},<\/p>\n\n\n\n
i = 0, 1, 2…<\/p>\n\n\n\n
eigenstates of a suitable \ufb01ducial observable<\/p>\n\n\n\n
Inner product hi|\u03c8i probability amplitude for the state |\u03c8i to be in<\/p>\n\n\n\n
the basis state |ii<\/p>\n\n\n\n
Amplitude squared |ha<\/p>\n\n\n\n
i<\/p>\n\n\n\n
|\u03c8i|<\/p>\n\n\n\n
2<\/p>\n\n\n\n
probability of obtaining the value a<\/p>\n\n\n\n
i<\/p>\n\n\n\n
on mea-<\/p>\n\n\n\n
suring A in the state \u03c8<\/p>\n\n\n\n
Matrix element A<\/p>\n\n\n\n
ij<\/p>\n\n\n\n
= hi|A|ji amplitude for producing a transition from |ji<\/p>\n\n\n\n
to |ii by the action of A (No assumption is<\/p>\n\n\n\n
made here about the nature of the operation)<\/p>\n\n\n\n
Diagonal element h\u03c8|A|\u03c8i average value of the observable A in the state<\/p>\n\n\n\n
|\u03c8i<\/p>\n\n\n\n
Unitary operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U possible evolution operator that changes the<\/p>\n\n\n\n
state reversibly<\/p>\n\n\n\n
3.3. Show that if<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H is a Hermitian operator then e<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
H<\/p>\n\n\n\n
is a unitary operator.<\/p>\n\n\n\n
3.4. Given a unitary operator<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U, show that the operator i( +<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U)( \u2212<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
U) is<\/p>\n\n\n\n
Hermitian.<\/p>\n\n\n\n
3.5. For a Hermitian or unitary matrix, show that the sum of diagonal elements<\/p>\n\n\n\n
(the trace) equals the sum of the eigenvalues, and the determinant equals<\/p>\n\n\n\n
the product of the eigenvalues.<\/p>\n\n\n\n
3.6. For each of the following matrices, \ufb01nd if they are unitary or Hermitian or<\/p>\n\n\n\n
neither. Find their eigenvalues and eigenvectors. Find if their eigenvectors<\/p>\n\n\n\n
are orthogonal.<\/p>\n\n\n\n
(a)<\/p>\n\n\n\n
“<\/p>\n\n\n\n
1 i<\/p>\n\n\n\n
i \u22121<\/p>\n\n\n\n
#<\/p>\n\n\n\n
(b)<\/p>\n\n\n\n
“<\/p>\n\n\n\n
0 1<\/p>\n\n\n\n
0 0<\/p>\n\n\n\n
#<\/p>\n\n\n\n
<\/p>\n\n\n\n
Problems 61<\/p>\n\n\n\n
3.7. For the three Pauli matrices \u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
, \u03c3<\/p>\n\n\n\n
y<\/p>\n\n\n\n
, and \u03c3<\/p>\n\n\n\n
z<\/p>\n\n\n\n
,<\/p>\n\n\n\n
(a) Show that \u03c3<\/p>\n\n\n\n
2<\/p>\n\n\n\n
i<\/p>\n\n\n\n
= .<\/p>\n\n\n\n
(b) Show that \u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u2019s are Hermitian as well as unitary.<\/p>\n\n\n\n
(c) Find the commutator [\u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
, \u03c3<\/p>\n\n\n\n
j<\/p>\n\n\n\n
] = \u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
j<\/p>\n\n\n\n
\u2212 \u03c3<\/p>\n\n\n\n
j<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
.<\/p>\n\n\n\n
(d) Find the anti-commutator {\u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
, \u03c3<\/p>\n\n\n\n
j<\/p>\n\n\n\n
} = \u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
j<\/p>\n\n\n\n
+ \u03c3<\/p>\n\n\n\n
j<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
i<\/p>\n\n\n\n
.<\/p>\n\n\n\n
3.8. Show that all the eigenvalues of any projection operator are either 1 or 0.<\/p>\n\n\n\n
3.9. Show that the operator which performs a transformation from the Z basis<\/p>\n\n\n\n
to the X basis has the following matrix representation:<\/p>\n\n\n\n
H =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
1 1<\/p>\n\n\n\n
1 \u22121<\/p>\n\n\n\n
!<\/p>\n\n\n\n
.<\/p>\n\n\n\n
This operator is also known as the Hadamard operator and is very useful in<\/p>\n\n\n\n
quantum computation.<\/p>\n\n\n\n
Verify that this operator is Hermitian. Show that it can be expressed as a<\/p>\n\n\n\n
linear combination of the Pauli matrices.<\/p>\n\n\n\n
3.10. Show that for any two operators A and B,<\/p>\n\n\n\n
AB =<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
[A, B] +<\/p>\n\n\n\n
1<\/p>\n\n\n\n
2<\/p>\n\n\n\n
{A, B}.<\/p>\n\n\n\n
3.11. Given a unit vector \u02c6e = (e<\/p>\n\n\n\n
x<\/p>\n\n\n\n
, e<\/p>\n\n\n\n
y<\/p>\n\n\n\n
, e<\/p>\n\n\n\n
z<\/p>\n\n\n\n
) in an arbitrary direction, we can de\ufb01ne<\/p>\n\n\n\n
the component of spin along \u02c6e by<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
e<\/p>\n\n\n\n
= e<\/p>\n\n\n\n
x<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
+ e<\/p>\n\n\n\n
y<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
y<\/p>\n\n\n\n
+ e<\/p>\n\n\n\n
z<\/p>\n\n\n\n
\u03c3<\/p>\n\n\n\n
z<\/p>\n\n\n\n
.<\/p>\n\n\n\n
(a) Show that \u03c3<\/p>\n\n\n\n
2<\/p>\n\n\n\n
e<\/p>\n\n\n\n
= .<\/p>\n\n\n\n
(b) Find the eigenvalues and eigenvectors of \u03c3<\/p>\n\n\n\n
e<\/p>\n\n\n\n
.<\/p>\n\n\n\n
3.12. De\ufb01ne a \u201cvector matrix\u201d ~\u03c3 =<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
i\u03c3<\/p>\n\n\n\n
x<\/p>\n\n\n\n
+<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
j\u03c3<\/p>\n\n\n\n
y<\/p>\n\n\n\n
+<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
k\u03c3<\/p>\n\n\n\n
z<\/p>\n\n\n\n
. Show that<\/p>\n\n\n\n
(~a. \u00b7 ~\u03c3)(<\/p>\n\n\n\n
~<\/p>\n\n\n\n
b \u00b7 ~\u03c3) = (~a \u00b7<\/p>\n\n\n\n
~<\/p>\n\n\n\n
b) + i(~a \u00d7<\/p>\n\n\n\n
~<\/p>\n\n\n\n
b) \u00b7 ~\u03c3 (3.34)<\/p>\n\n\n\n
for vectors ~a and<\/p>\n\n\n\n
~<\/p>\n\n\n\n
b.<\/p>\n\n\n\n
3.13. Find the expectation value of \u03c3<\/p>\n\n\n\n
e<\/p>\n\n\n\n
in the state |0i. Generalize this result<\/p>\n\n\n\n
to \ufb01nd the expectation value of \u03c3<\/p>\n\n\n\n
e<\/p>\n\n\n\n
in a state |<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
f+i where<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
f is a general<\/p>\n\n\n\n
direction making angle \u03b8 with the \u02c6z axis.<\/p>\n","protected":false},"excerpt":{"rendered":"
De\ufb01nition 3.1. The tensor product of two matrices A of dimensions m \u00d7 n and B of any dimensions is given by A \u2297 B = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 A 11 B A 12 B . . . A 1n B A 21 B A 22 B . . . A 2n […]<\/p>\n","protected":false},"author":1,"featured_media":3974,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[489],"tags":[],"class_list":["post-3999","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-4-quantum-mechanics"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/quantum-computer-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3999","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=3999"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3999\/revisions"}],"predecessor-version":[{"id":4548,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3999\/revisions\/4548"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3974"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3999"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}