Reversing the order of an inner product complex-
conjugates it:
(hψ|φi)
∗
= hφ|ψi (6)
2.2 Matrices
Matrices are sums of outer products |·ih·|. For ex-
ample, the matrix (M
i, j
)
i, j
can be thought of as
the weighted sum of “base-matrices” B
i, j
≡ |b
i
ihb
j
|,
whose components are all 0 except for a 1 in the i-th
row and j-th column. The outer product extends lin-
early to non-base kets in the following manor:
|ψihφ| = (
∑
i
ψ
i
|b
i
i)
∑
j
φ
∗
j
hb
j
|
=
∑
i, j
ψ
i
φ
∗
j
|b
i
ihb
j
|
(7)
This is analogous to the conventional multiplication:
ψ
1
.
.
.
ψ
n
(φ
∗
1
··· φ
∗
n
)=
ψ
1
φ
∗
1
··· ψ
1
φ
∗
n
.
.
.
.
.
.
.
.
.
ψ
n
φ
∗
1
··· ψ
n
φ
∗
n
(8)
We will also make use of the tensor product. Its ap-
plication to kets, bras and outer products is linear:
(|ai+ |bi) ⊗|ci = |ai⊗|ci+ |bi⊗|ci
(ha|+ hb|) ⊗hc| = ha|⊗hc|+ hb|⊗hc|
(|aihb|+ |cihd|) ⊗|eihf | =
(|ai⊗|ei)(hb|⊗hf |) + (|ci⊗|ei)(hd|⊗hf |)
(9)
For convenience we omit “⊗” where no confusion
arises, e.g., |ai ⊗ |bi = |ai|bi. When applied to
Hilbert spaces, the tensor product creates the com-
posed Hilbert space H = H
1
⊗... ⊗H
n
whose base
kets are simply induced by the tensor product of its
subspaces’ base kets:
base(H
1
⊗... ⊗H
n
) =
(
n
O
i=1
|bi
i
: |bi
i
∈ base(H
i
), 1 ≤ i ≤ n
)
(10)
Whereas the order of composed kets |ai|bi|ci usu-
ally suffices to identify which subket lives in which
subspace, we make this explicit by giving subkets
the same subscript as the corresponding subspace.
Thus, the order no longer matters, as in the follow-
ing inner product of composed kets:
(ha|
1
hb|
2
hc|
3
)(|ei
3
|di
1
|f i
2
) = ha|dihb|f ihc|ei (11)
Definition 1. Self-adjoint Matrix
A matrix M is self-adjoint iff M
i, j
= M
∗
j,i
for all i, j.
Consequently, all diagonal elements are real-valued,
and M = M
†
is its own transpose conjugate.
Definition 2. Density Matrix
A self-adjoint matrix M is a density matrix iff
it is positive semi-definite, i.e., hφ|M|φi ≥ 0 for
all |φi ∈ C
n
, and it has unit trace, i.e., Tr(M) =
∑
|bi∈B
hb|M|bi = 1.
The term “density matrix” is synonymous with
“density operator”. Any density matrix ρ can
be decomposed arbitrarily as ρ =
∑
i
p
i
|s
i
ihs
i
|, the
weighted sum of sub-matrices |s
i
ihs
i
| with p
i
∈ R
>0
and hs
i
|s
i
i= 1. The p
i
need not sum to 1. In fact the
decomposition where the p
i
sum to 1 and the |s
i
i are
mutually orthogonal is unique and is called the eigen
decomposition. Consequently B
eig
= {|s
i
i}
i
consti-
tutes an orthonormal base, ρ’s so-called eigen base.
Density operators are used in quantum theory to de-
scribe the state of some system. If the system’s state
ρ is certain we call it a pure state and write ρ = |sihs|
for some unit ket |si. Systems whose state is uncer-
tain are described by a mixed state ρ =
∑
i
p
i
|s
i
ihs
i
|
which represents an ensemble of substates or pure
states {(p
i
,s
i
)}
i
where the system is in substate s
i
with probability p
i
. Hence, the term “density” as in
probability density.
It is possible to normalize a density matrix
without committing to any particular decomposi-
tion. Only the trace function is required, because
norm(ρ) = ρ/Tr(ρ). Definition 2 mentions what the
trace function does. However, notice that the same
result is produced for any orthonormal base B , in-
cluding ρ’s eigen base B
eig
= {|e
i
i}
i
. Even though
we do not know the content of B
eig
, we know that it