Desurvire E. Classical and Quantum Information Theory: An Introduction for the Telecom Scientist
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17.1 Dirac notation 337
and
U
x
|x
=(|xx|)|x
=|xx|x
=δ
xx
|x. (17.16)
These two results confirm that U
x
=|xx| is the unique projector on the pure state |x.
It is clear that for any state |q of unity length (such as a qubit), the operator |qq| is
the projector over the state |q, which is left as an easy exercise to demonstrate.
Change of basis
Let W
n
={|y} be a new orthonormal basis. Consistently, we can expand |q over W
n
,
using a new system of coordinates y defined over the states |y. How can we relate the
new coordinates y to the old coordinates x? Here again, Dirac notations turn out to be
quite handy. First, let us insert the completeness relation of W
n
into the V
n
expansion
according:
|q=
x∈w
n
x|x=
x∈V
n
x
y∈V
n
|yy|
|x
=
y∈V
n
3
x∈V
n
xy|x
4
|y≡
y∈V
n
y|y.
(17.17)
The result in Eq. (17.17), thus, yields the definition of the new y coordinates with respect
to the old x coordinates:
y =
x∈V
n
y|xx. (17.18)
Unitary transformations
As seen in Chapter 16, a unitary transformation is characterized by an operator U
satisfying the property U
+
U = I , where U
+
is the Hermitian conjugate of U . Define
|q
=U|q as the output state resulting from the transformation of input state |q.
Consistently with previous definition of the “bra,” we can equivalently write q
|=
q|U
+
as the transformation of q| into q
|. Developing the inner product q
|q
and
using the unitary condition U
+
U = I yields
q
|q
=(q
|U
+
)(U|q)
=q|U
+
U |q=q|I |q≡q|q,
(17.19)
which shows that unitary transformations conserve the state modulus. Furthermore,
unitary transformations also conserve the inner product. This is readily verified given
|a
=U|a and |b
=U|b, which yields a
|b
=a|U
+
U |b≡a|b.
Operator matrix elements
Given the basis V
n
={|x}, the action of operator U on the qubit or state |q, which
yields |q
=U|q, can be characterized by a change of coordinates from x to x
,as
338 Quantum measurements
follows:
|q
=U|q=U
3
x∈V
n
x|x
4
=
x∈V
n
xU|x
=
x∈V
n
x
3
x
∈V
n
|x
x
|
4
U |x=
x∈V
n
x
x
∈V
n
|x
x
|U|x
=
x
∈V
n
3
x∈V
n
x
|U|xx
4
|x
≡
x
∈V
n
x
|x
(17.20)
with
x
=
x∈V
n
x
|U|xx. (17.21)
In Eq. (17.21), the complex coefficients x
|U|x represents the matrix elements of the
operator U, as expressed in the basis V
n
.
Assume next a different orthonormal basis W
n
={|y}. What are the matrix elements
of U in this new basis? To answer this question, we effect the following development:
y
|U|y=y
|
3
x
∈V
n
|x
x
|
4
U
3
x∈V
n
|xx|
4
|y
=
x
∈V
n
x∈V
n
y
|x
x
|U|xx|y≡
x∈V
n
x
∈V
n
T
y
x
U
x
x
T
xy
(17.22)
with
T
yx
= T
xy
= T
+
yx
=y|x, (17.23)
where T is called the transition operator. The result in Eq. (17.22) shows that the matrix
U in basis W
n
(call it
˜
U) can be calculated from the matrix U in basis V
n
according to
the transformation:
˜
U = TUT
+
, (17.24)
which is referred to as similarity transformation. It is easily established that T
+
T =
TT
+
= I in any basis (noticing that T
yx
= T
+
yx
does not imply T = T
+
, which is left as
an exercise to show).
Eigenvalues and eigenstates
Given an operator A,anystate|v such that A|v=λ|v, with λ being a complex
number, is called an eigenstate of A. The corresponding number λ is called the
eigenvalue of |v. The eigenvalues are the solutions of the characteristic equation
|A −λI |=0, where |Q| (also sometimes noted det |Q|)isthedeterminant of the
matrix Q.
2
It is customary to label the eigenstates after their eigenvalues, thus, the
eigenstate |λ
i
implicitly satisfies A|λ
i
=λ
i
|λ
i
. Two or more eigenstates may have
2
For a 2 × 2 matrix Q =
ac
bd
, the determinant is |Q|=ad − bc. For a general definition see, for
instance, http://en.wikipedia.org/wiki/Determinant.
17.1 Dirac notation 339
the same eigenvalue, in which case the eigenvalue is said to be degenerate (in this
case the corresponding eigenstates may be labeled as |λ
(1)
i
, |λ
(2)
i
···). We also have
λ
i
|A
+
=
¯
λ
i
λ
i
|, thus, λ
i
|A
+
A|λ
i
=|λ
i
|
2
λ
i
|λ
i
.IfA is Hermitian (A = A
+
), we
have λ
i
|A
+
A|λ
i
=λ
i
|AA|λ
i
=λ
2
i
λ
i
|λ
i
, which shows that λ
2
i
=|λ
i
|
2
, or that the
eigenvalues of a Hermitian operator are real. It can be shown that any linear operator
has at least one eigenvalue and eigenvector (the characteristic equation having at least
one root). The complete set of eigenstates defines the operator’s eigenspace. A key prop-
erty of the eigenstates |λ
i
of any operator A is that they form an orthonormal basis, i.e.,
λ
i
|λ
j
=δ
ij
. Therefore, the completeness relation applies to the set of eigenstates:
i
|λ
i
λ
i
|=I. (17.25)
Using the completeness relation, we can write the matrix elements of A in the form:
x
|A|x=x
|A
3
i
|λ
i
λ
i
|
4
|x=
i
x
|A|λ
i
λ
i
|x
=
i
x
|λ
i
|λ
i
λ
i
|x≡x
|
i
λ
i
|λ
i
λ
i
|
|x,
(17.26)
which gives
A =
i
λ
i
|λ
i
λ
i
|. (17.27)
The definition in Eq. (17.27), which decomposes A into a sum of eigenstate projectors
|λ
i
λ
i
|, is called the diagonal representation,orspectral decomposition of A.For
instance, the diagonal-operator representation of the Pauli matrix Z is:
Z =
10
0 −1
=|00|−|11|. (17.28)
It is left as an easy exercise to determine the eigenstates and diagonal representations
of the Pauli matrix Y and the Hadamard matrix H . More generally, in the diagonal
representation of A, the matrix elements λ
i
|A|λ
j
can be developed as follows:
λ
i
|A|λ
j
=λ
i
|
k
λ
k
|λ
k
λ
k
|
|λ
j
=
k
λ
k
λ
i
|λ
k
λ
k
|λ
j
=
k
λ
k
δ
ik
δ
kj
≡ λ
i
δ
ij
,
(17.29)
which shows that all nondiagonal elements of the matrix are identically zero, while the
diagonal elements are equal to the eigenvalues λ
i
, i.e.,
A =
λ
1
0 ··· 0
0 λ
2
···
.
.
.
.
.
.
.
.
.
.
.
.
0
0 ··· 0 λ
n
. (17.30)
340 Quantum measurements
An operator matrix that can be transformed into the diagonal form shown in Eq. (17.30)
is said to de diagonalizable. For this, its n × n matrix must have n eigenstates. Clearly, if
the matrix has fewer than n eigenstates, it cannot be diagonalized, as there remain some
nonzero off-diagonal elements in the matrix. An operator is said to be normal if it satisfies
AA
+
= A
+
A. A remarkable property: any normal operator is diagonalizable, and any
diagonalizable operator is normal.
3
Consequently, since they are normal, Hermitian
operators are diagonalizable.
Matrix trace
The trace of a matrix A, noted tr( A) is the sum of all its diagonal elements A
ii
, according
to
tr(A) =
n
i=1
A
ii
. (17.31)
It is easily established that for any two operators A, B and complex number α,wehave
the properties:
tr(α A) = α tr(A)
tr(A + B) = tr( A) + tr(B)
tr(AB) = tr(BA).
(17.32)
In the diagonal representation (Eq. (17.30)), it is clear that the trace of a matrix is the
sum of its eigenvalues. But the trace of a matrix does not depend on the choice of the
3
This can be shown through the following. Since A is a linear operator, it must have at least one eigenvalue,
say λ.DefineP =|λλ| as the projector on the corresponding eigenspace |λ (where AP = λ P), and
Q = I − P the projector on the orthogonal subspace (hence P + Q = I and PQ = QP = 0). We also
have P
2
= P, Q
2
= Q,andP, Q are both Hermitian (P
+
= P, hence Q
+
= Q). We develop A according
to the following:
A = (P + Q) A = (P + Q) A(P + Q) = PAP+ QAP + PAQ + QAQ
= PλP + λQP + PAQ+ QAQ = λP + PAQ+ QAQ.
Next we can show that PAQ = 0ifA is normal. Indeed, we have AA
+
|λ=A
+
A|λ=λ A
+
|λ, which
shows that |λ
=A
+
|λ is also an eigenstate of A and belongs to its eigenspace. Therefore, QA
+
= 0,
hence, QA
+
P = (PAQ)
+
= 0 = PAQ. We, thus, obtain A = λ P + QAQ. Finally, we can show that
QAQ is normal. We have
(QAQ)(QA
+
Q) = QAQQA
+
Q
= QAQA
+
Q = QA(I − P) A
+
Q
= QA(I − P) A
+
Q = QAA
+
Q − QAPA
+
Q
= QA
+
AQ − λQPA
+
Q = QA
+
AQ
= QA
+
(P + Q)AQ = Q( A
+
P)AQ + QA
+
QAQ
= QA
+
QQAQ = (QA
+
Q)(QAQ),
which proves that QAQ is normal. We have found that A is decomposed as the sum A = λ P + A
, with P
being diagonal in the eigenspace |λ,andA
= QAQ being normal. Since A
is a linear operator, it must
have at least one eigenvalue, say λ
, thus, it can be decomposed into A
= λ
P
+ A
,whereP
is diagonal
in the eigenspace |λ
,and A
is normal. By induction, we conclude that there exists an orthonormal basis
{|λ, |λ
...} of the same dimension as A for which the matrix A is diagonal.
17.1 Dirac notation 341
basis representation. Indeed, using the basis transformation in Eq. (17.24), and the third
property in Eq. (17.32), we obtain:
tr(
˜
U) = tr(TUT
+
) = tr(T
+
TU) ≡ tr(U). (17.33)
Density operator or matrix
Let V
n
={|x
i
} = {|x
1
, |x
2
,...,|x
n
} an orthonormal basis for the space of quantum
states |ψ, i.e., satisfying x
i
|x
j
=δ
ij
. In this basis, the states have a unique decompo-
sition, which takes the form
|ψ=x
1
|x
1
+x
2
|x
2
+···+x
n
|x
n
=
n
i=1
x
i
|x
i
, (17.34)
where x
i
(i = 1,...,n) represents the complex coordinates. If the modulus or length of
|ψ is unity, we have
ψ|ψ=
n
i=1
n
j=1
¯
x
i
x
j
x
i
|x
j
=
n
i=1
|x
i
|
2
= 1. (17.35)
As we have seen in Chapter 16, for qubits, the number p
i
=|x
i
|
2
represents the probabil-
ity of finding (or measuring) the state |ψ in the basis state |x
i
. Hence, the coordinates
x
i
represent complex amplitude probabilities. We can now define the density operator
or matrix associated with the state |ψ as:
ρ =|x
1
|
2
|x
1
x
1
|+|x
2
|
2
|x
2
x
2
|+···+|x
n
|
2
|x
n
x
n
|
=
n
i=1
|x
i
|
2
|x
i
x
i
|.
(17.36)
The density matrix is diagonal, since its elements ρ
ij
are given by
ρ
ij
=x
i
|ρ|x
j
=x
i
|
3
n
k=1
|x
k
|
2
|x
k
x
k
|
4
|x
j
=
n
k=1
|x
k
|
2
x
i
|x
k
x
k
|x
j
=
n
k=1
|x
k
|
2
δ
ik
δ
kj
≡|x
i
|
2
δ
ij
.
(17.37)
Hence, the density matrix operator takes the diagonal matrix representation:
ρ =
|x
1
|
2
0 ··· 0
0 |x
2
|
2
···
.
.
.
.
.
.
.
.
.
.
.
.
0
0 ··· 0 |x
n
|
2
=
p
1
0 ··· 0
0 p
2
···
.
.
.
.
.
.
.
.
.
.
.
.
0
0 ··· 0 p
n
. (17.38)
342 Quantum measurements
It is immediately noted that the trace of the density matrix is unity, since
tr(ρ) =
n
i=1
|x
i
|
2
=
n
i=1
p
i
= 1. (17.39)
We now have the tools to make a short hint at quantum information theory.Thismay
also constitute a nice reward for having gone through the lengthy description of Dirac
notation!
Let us introduce a new operator, called U log U , where U is assumed to be diagonal
with nonnegative coefficients. To calculate the matrix coefficients of U log U ,wemust
first define log U . Assume, then, a linear operator V , which satisfies U = exp(V ), which
defines V = log U . Formally, the exponential operator is determined by the infinite
series:
U = exp(V ) =
∞
n
V
n
n!
. (17.40)
Since U is diagonal, any of the powers V
n
must be diagonal. The diagonal coefficients of
U are, thus, given by U
ii
= exp(V
ii
)orV
ii
= log(U
ii
). The matrix W = UV = U log U
is also diagonal. It is clear that its coefficients are given by W
ii
= U
ii
V
ii
= U
ii
log(U
ii
).
This result shows that the matrix W = U log U is analytically defined for any diagonal
matrix U with coefficients U
ii
≥ 0.
4
We conclude that the density matrix U = ρ,for
which the coefficients are nonnegative, is an eligible candidate for the operator U log U .
We,thus,have(ρ log ρ)
ii
= ρ
ii
log ρ
ii
and the matrix definition
ρ log ρ =
|x
1
|
2
log |x
1
|
2
0 ··· 0
0 |x
2
|
2
log |x
2
|
2
···
.
.
.
.
.
.
.
.
.
.
.
.
0
0 ··· 0 |x
n
|
2
log |x
n
|
2
=
p
1
log p
1
0 ··· 0
0 p
2
log p
2
···
.
.
.
.
.
.
.
.
.
.
.
.
0
0 ··· 0 p
n
log p
n
.
(17.41)
Finally, we find that the trace of ρ log ρ is given by the expression:
tr(ρ log ρ) =
n
i=1
|x
i
|
2
log |x
i
|
2
=
n
i=1
p
i
log p
i
. (17.42)
Based on our background of Shannon’s information theory (Chapter 4), we can heuris-
tically define an “entropy” H for the quantum state described by the density matrix ρ in
the form
H =−
n
i=1
p
i
log p
i
, (17.43)
4
By application of the property lim
x→0
(x log x) = 0, which defines the function x log x analytically for any
real x ≥ 0.
17.2 Quantum measurements and types 343
which, from Eq. (17.42), yields:
H =−tr(ρ log ρ). (17.44)
The above result quite elegantly connects the concept of Shannon’s entropy to the corre-
sponding “notion” in quantum information theory, which is based on the density-matrix
operator. The definition H =−tr(ρ log ρ)isreferredtoasvon Neumann’s entropy,asI
shall describe in Chapter 21. Anticipating a key result, the entropy corresponding to a
qubit state |q=α|0+β|1, with p =|α|
2
= 1 −|β|
2
,isgivenby
H =−|α|
2
log |α|
2
−|β|
2
log |β|
2
=−p log p − (1 − p)log(1− p) ≡ f ( p).
(17.45)
In the result in Eq. (17.45), we recognize the Shannon entropy of a two-event source
X
2
={0, 1}, corresponding to the two possible states of a classical information bit. As
we have seen, however, the qubit is a superposition of both information states, which
we referred to as |0, |1, with corresponding probabilities |α|
2
, |β|
2
.Wenowhavethe
required conceptual tools to analyze the notion of quantum measurement.
17.2 Quantum measurements and types
In this section, I introduce and analyze the concepts associated with different types
of quantum measurement. A general definition for quantum measurement operators
will first be introduced. This definition will then be applied to measurements in the
orthonormal basis,totheprojective (or von Neumann) measurements and to the so-
called POVM measurements.
Through Dirac notation we have made a formal description of the quantum states and
their various transformation properties through the action of linear operators. Such a
description did not require any quantum-mechanics background, because in Chapters 15
and 16 we obtained a solid view of the world of qubits, as described by 2D complex
vectors, and their operator transformations, as described by unitary rotations on the
Bloch sphere. The simpler world of qubits, thus, offers a convenient introduction to
the greater view of quantum states |ψ and their linear operator transformations |ψ
=
A|ψ. In the same spirit of conceptual simplification, we can view a quantum system as
being a physical system with which one can associate a quantum state |ψ, as expressed
onto some pure-state basis V
n
={|x}, and a set of linear operators {A}.Wearenow
interested in learning about what can be physically measured in such a system, and how
it may possibly be measured.
The following will show that there are different approaches for performing physical
measurements in quantum systems. Assume, first, that for a quantum system with n
pure states, there exist a certain number n of possible measurements, which we index
by m (m = 1,...,n). We then introduce the most general definition of a measurement
operator, M
m
, the collection of which forms the finite operator set {M
m
}. Calling p(m)
344 Quantum measurements
the probability of measuring m, we have by postulate:
p(m) =ψ|M
+
m
M
m
|ψ. (17.46)
Since the probabilities must sum to one, we also have
n
m=1
p(m) =
n
m=1
ψ|M
+
m
M
m
|ψ
=ψ |
3
n
m=1
M
+
m
M
n
4
|ψ=1,
(17.47)
which implies the “completeness” relation:
m
M
+
m
M
m
= I. (17.48)
As a second postulate, the measurement M
m
causes the system state |ψ to be trans-
formed into the post-measurement state |ψ
=γ M
m
|ψ, where γ is a complex number.
The probability q that the system is in the post-measurement state |ψ
is unity. We,
thus, obtain
q =ψ
|ψ
=ψ |¯γ M
+
m
M
m
γ |ψ
=|γ |
2
ψ|M
+
m
M
m
|ψ≡|γ |
2
p(m) = 1,
(17.49)
which shows that |γ |=1/
p(m), hence γ = e
iδ
|γ |=e
iδ
/
p(m). Within an “unob-
servable” phase term e
iδ
, which we shall overlook here, we can set γ = 1/
p(m), and
the post-measurement state is now completely defined as:
|ψ
=
1
p(m)
M
m
|ψ=
1
ψ|M
+
m
M
m
|ψ
M
m
|ψ. (17.50)
Using the expansion in Eq. (17.34) of the input state |ψ, we obtain the expansion of the
post-measurement state |ψ
according to
|ψ
=
1
ψ|M
+
m
M
m
|ψ
M
m
|ψ
=
1
ψ|M
+
m
M
m
|ψ
M
m
n
i=1
x
i
|x
i
=
n
i=1
x
i
ψ|M
+
m
M
m
|ψ
M
m
|x
i
.
(17.51)
We have, thus, obtained a general expression for the post-measurement state |ψ
,
given the quantum measurement operator M
m
. What could be the result of two
successive measurements from different operators M
m
and L
l
? The answer is that
17.2 Quantum measurements and types 345
the two successive measurements (in this order) are equivalent to a single mea-
surement of operator definition K
lm
= L
l
M
m
, which is left as an exercise to
demonstrate.
An important consequence of the above general definition for the quantum-
measurement operator M
m
is that two states |ψ and e
iδ
|ψ, which differ by the phase
factor e
iδ
, have the same measurement properties and outcome. Hence such a phase
factor is said to be “unobservable.” It should not be concluded, however, that in quan-
tum mechanics, phase is definitely unobservable, with no corresponding measurement
operator. Indeed, more recent (1989) and less known work has shown that Hermitian
phase-measurement operators and phase eigenstates can truly be defined.
5
Here, we
shall overlook this academic note, and abide by the above general definition of quantum-
measurement operators, for which phase (and multiplying factors e
iδ
in quantum states)
is indeed “unobservable.”
We shall consider next three possibilities for the measurement operator M
m
.
Quantum measurements in the orthonormal basis
We look for a quantum measurement operator M
m
that projects the input state |ψ
into the pure state |x
m
from the orthonormal basis V ={|x} = {|x
1
, |x
2
,...,|x
n
},
namely, having the action |ψ
=M
m
|ψ=µ|x
n
, where µ is a complex number. As we
know, the projector over |x
m
is defined as P
m
=|x
m
x
m
|, with P
+
m
P
m
= P
m
.Thesetof
Hermitian operators {P
m
=|x
m
x
m
|} satisfies the completeness relation, hence M
m
=
|x
m
x
m
| is a valid measurement operator. We first derive the bracket ψ|M
+
m
M
m
|ψ
according to
ψ|M
+
m
M
m
|ψ=ψ|P
m
|ψ
=
3
n
i=1
¯
x
i
x
i
|
4
|x
m
x
m
|
n
j=1
|x
j
x
j
=
n
i=1
n
j=1
¯
x
i
x
j
δ
im
δ
jm
≡|x
m
|
2
,
(17.52)
which shows that the probability of measuring m (or equivalently, the probability that
|ψis in the state |x
m
)is p(m) =|x
m
|
2
. Then, with substitution of M
m
=|x
m
x
m
|into
5
See, for background reference: E. Desurvire, Erbium-Doped Fiber Amplifiers, Device and System Devel-
opments (New York: J. Wiley & Sons, 2002), pp. 97–111. As an advanced topic for quantum-mechanics
specialists, attention is drawn to phase states |δ
m
, which have eigenvalues δ
m
, and which are defined for
electromagnetic fields by
|δ
m
=
1
√
s + 1
s
k=0
e
ikδ
m
|k,
where |k is a photon-number state, and the series applies in the limit s →∞. For full description, see:
S. M. Barnett and D. T. Pegg, On the Hermitian phase operator. J. Modern Optics, 36 (1989), 7–19 and D. T.
Pegg and S. M. Barnett, Phase properties of the quantized in single-mode electromagnetic field. Phys. Rev.
A, 39 (1989), 1665.
346 Quantum measurements
Eqs. (17.51) and (17.52), we obtain
|ψ
=
1
ψ|M
+
m
M
m
|ψ
M
m
|ψ
=
1
|x
m
|
2
|x
m
x
m
|
n
i=1
x
i
|x
i
=
n
i=1
x
i
|x
m
|
|x
m
x
m
|x
i
=
n
i=1
x
i
|x
m
|
|x
m
δ
mi
≡
x
m
|x
m
|
|x
m
.
(17.53)
The result in Eq. (17.53) indicates that the post-measurement state is indeed of the form
µ|x
m
, but with µ = x
m
/|x
m
| being a complex number of unity modulus, or µ = e
iδ
m
being an “unobservable” phase term that can be overlooked.
We will use the above results in Chapter 18, which concerns qubit measurements in
the orthonormal basis.
Projective or von-Neumann measurements
Another type of quantum measurement whose operator satisfies the above two postulates
is referred to as projective or von-Neumann measurement. As we shall see, it is not
formally different from a measurement in an orthonormal basis, except that we shall
now take for this basis the one formed by the eigenstates {|λ
m
} of a given Hermitian
operator A. Define the corresponding eigenvalues as {λ
m
}. Since A is Hermitian, we
know from the previous section that the eigenvalues are real numbers. For this reason,
the operator A could be associated with “real” physical quantities to characterize the
system, hence the operator A is referred to as an observable.
Given the set of projectors P
m
=|λ
m
λ
m
|, the spectral decomposition of A is given
in Eq. (17.27), which I reproduce for convenience:
A =
n
m=1
λ
m
P
m
=
n
m=1
λ
m
|λ
m
λ
m
|. (17.54)
As we have previously established, the projector P
m
=|λ
m
λ
m
| is a valid measurement
operator and, hence, the probability of measuring the eigenvalue λ
m
from a system in
the state |ψ, as decomposed in the eigenstates basis {|λ
m
} is given by
p(λ
m
) =ψ |P
m
|ψ, (17.55)
and we have for the post-measurement state:
|ψ
=
1
p(λ
m
)
P
m
|ψ
=
1
√
ψ|P
m
|ψ
P
m
|ψ.
(17.56)