Desurvire E. Classical and Quantum Information Theory: An Introduction for the Telecom Scientist

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17.2 Quantum measurements and types 347

Substituting the expansion of |ψ, with complex coordinates x

, over the eigenstate

basis {|λ

}, we obtain for the measurement probability p(λ

) and corresponding post-

measurement state |ψ



:

p(λ

) =ψ|P

|ψ



i=1

λ

|λ

λ







j=1

|λ









i=1



j=1

=|x

(17.57)

|ψ



=



p(λ

)

|ψ

|λ

λ



i=1

|λ



|λ





i=1

≡|λ

.

(17.58)

We will use the above two results in Chapter 18, which concerns projective qubit

measurements.

Let us now analyze some interesting properties of the projective or von-Neumann

measurement.

Since each eigenvalue λ

has an associated measurement probability, p(λ

), we can

deﬁne the average value or mean of the observable measurement as follows:

A≡λ=



m=1

p(λ

) =



m=1

ψ|P

|ψ

=ψ |



m=1

|ψ≡ψ|A|ψ.

(17.59)

The nice conclusion from the above result is that the average value of an observable A is

deﬁned as the bracket ψ|A|ψ. Likewise, we can deﬁne the mean-square and variance

of the observable as follows:

A

≡λ

=



m=1

p(λ

) ≡ψ |A

|ψ, (17.60)

A

≡λ

−λ

=A

−A

(17.61)

(the notation  A corresponds to a real number, not an operator). We have, thus, obtained

the deﬁnitions of the two main statistical parameters characterizing any observable

measurement, namely the mean and the variance of the observed physical quantity. We

call the uncertainty of the observable A the measured standard deviation

√

A

= A.

Similar measurements can be performed with any other observable quantity B, with

uncertainty B. A fundamental result in quantum mechanics, known as the Heisenberg

uncertainty principle, states that given two Hermitian operators A, B the product of their

348 Quantum measurements

uncertainties measured in the system state |ψ  must satisfy:

AB ≥

|ψ|[ A, B]|ψ|, (17.62)

where [A, B] = AB − BA is the commutator of A, B (see demonstration in Appendix

O). In the particular case (called conjugate observables) where [ A, B] =±i , and where

|ψ has a modulus of unity, we have

AB ≥

, (17.63)

which represents the more well-known form of the uncertainty principle. The key con-

clusion of this principle could be stated as follows: two independent measurements of

noncommuting observables cannot both reach arbitrary accuracy. The product of the

corresponding uncertainties has a nonzero lower bound given by the right-hand side of

Eq. (17.62), which we may call ε. Thus, a relatively accurate measurement of A, e.g.,

A = 1/N where N may be any large number, implies that the accuracy in the measure-

ment of B cannot be better than N ε.Thus,ifN →∞,wehaveA → 0, corresponding

to absolute accuracy in the observable A, while we have B →∞, corresponding to

an absolute “indetermination” of the observable B. It would be incorrect, however, to

conclude that in any case, a ﬁrst measurement of A actually inﬂuences that of B, and the

reverse. The two measurements are assumed to be independent, both in terms of time

and order sequence. Only in the case where the two observables commute can the two

measurements reach arbitrary accuracy.

POVM measurements

A third type of measurement is referred to as positive-operator-valued-measure,or

POVM. Deﬁne a ﬁnite set of Hermitian operators {E

} with any E

being a posi-

tive operator (meaning ψ|E

|ψ≥0 for any normalized state |ψ), and altogether

satisfying the completeness relation



= I. (17.64)

The operators E

that satisfy the above conditions are then said to be POVM operators.

As we have seen in the previous section, any Hermitian operator A can be represented

by a diagonal matrix with nonnegative coefﬁcients, hence, there exists a diagonal oper-

ator B =

√

A, such that B

= A, and which is also Hermitian (B

is real, so B

is).

Therefore, for each E

we can associate a Hermitian operator M

√

that satisﬁes

= E

, and consequently the two above POVM conditions. According to the

general deﬁnition, the set {M

} is, thus, a truly eligible set of measurement operators,

even if their number, with respect to the number n of pure states, is not speciﬁed. The

projection or von-Neumann measurements, with the set {M

=|λ

λ

|}being deﬁned

by the n eigenstates of an observable, represent a speciﬁc case of POVM set.

I shall now illustrate an important application of POVM, which no other measurement

type can provide. Consider a simple example in the basis V ={|0, |1}. Assume the

17.2 Quantum measurements and types 349

Table 17.1 Probabilities associated with different POVM measurement possibilities when the input

state is either |ψ=|0 (left) or |ψ=|+(right).

Input state |ψ=|0 Input state |ψ =|+

p(1) =0|E

|0=0 p(1) =+|E

|+ =

p(2) =0|E

|0=

p(2) =+|E

|+ = 0

p(3) =0|(I − E

− E

)|0=1 −

p(3) =−|(I − E

− E

)|− = 1 −

three POVM operators:

= u|11|

= u

|0−|1

√

0|−1|

√

≡ u|−−|

= I − E

− E

(17.65)

with u =

√

2/(1 +

√

2). By summation of the above three equations, we immediately

observe that the completeness relation is satisﬁed. It is also easily veriﬁed that the three

operators E

, E

are positive (it is left as an exercise to prove that in the general

case, the condition u ≤ 2/3 must be satisﬁed). We have, thus, deﬁned a valid POVM set

of three measurement operators from the two-element basis V .

The following will show how the above POVM measurement can be applied. Assume

that the system state |ψ has been prepared in such a way that it is certain to be either

|ψ=|0 or |ψ=|+, to the exclusion of any other possibility. It is also assumed

that we have this key information about the two system-state possibilities, prior to any

measurement, but we don’t know, a priori, which ones the system chooses to be in. The

probabilities associated with our three possible measurements associated to each case

are summarized in Table 17.1.

First, the table indicates that our measurement fails to convey any information in two

cases:

If we use E

when the input is |ψ=|0;

If we use E

when the input is |ψ=|+.

This is because, in each case, the measurement operators project the input state on its

orthogonal counterpart, namely |1 or |−, respectively, resulting in the measurement

probabilities for E

of p(E

) =0|E

|0=0or p(E

) =+|E

|+ = 0, and showing

that these measurement have no possible outcomes.

Second, the table indicates that there is a ﬁnite probability u/2 that we obtain a

measurement

Of E

,ifweuseE

when the input is |ψ=|+;

Of E

,ifweuseE

when the input is |ψ=|0.

Then comes a nice subtlety in the interpretation of these two measurement possibilities.

Indeed, if we happen to measure E

the system cannot be in state |ψ =|0, since E

350 Quantum measurements

a projector on |1. In this case, we reach the absolute conclusion that the system must

have been in state |ψ=|+. Likewise, if we happen to measure E

, the system cannot

be in state |ψ=|+, since E

is a projector on |−. In this case, we reach the absolute

conclusion that the system must have been in state |ψ=|0. However, we obtain this

absolute information only with probability u/2, meaning that there is a ﬁnite probability

1 − u/2 that the two measurements fail to convey any information.

If we were to use the measurement E

, Table 17.1 shows that the measurement works

in all cases (|ψ=|0 or |+), but with a probability 1 −u/2. By deﬁnition, E

is a

projector on all states that are neither |1 or |−. Thus, the positive outcome of any E

measurement only tells us that the system state is neither |1or |−, but we already know

this for a fact, which represents no information! Therefore, there is no point in using E

as a means to measure the system state.

Let me, then, clarify what is meant by “positive outcome” and “failure” of any of the

above measurements, using a ﬁgurative analogy with a physical measurement. Compare

the system state |ψ to a light source that randomly emits in two possible color tones,

either A or B (standing for |0 or |+, respectively), these tones being invisible to the

naked eye. Our measurement consists of determining which color tone is emitted by the

source by observing it through a set of “magic ﬁlters,” called 1, 2, 3, (for E

, E

Such ﬁlters have the following strange properties: ﬁlter 1 does not react to tone A,

while it makes tone B visible to the eye with probability p (or u/2); ﬁlter 2 does not

react to tone B, while it makes tone A visible to the eye with same probability p; and

ﬁlter 3 does not react to any tone other than A, B (but this is not useful here) but

makes tone A, B visible with probability 1 − p (or 1 −u/2). Basically, if we choose

either ﬁlter 1 or ﬁlter 2, we have a chance p of seeing something, and 1 − p of seeing

nothing!

Figure 17.1 shows what we can see through the magic ﬁlters, according to the twelve

possible cases, namely, determined by the two source-tone possibilities A, B and our

three possible magic-ﬁlter choices 1, 2, 3. A bright spot indicates that we observe some-

thing, corresponding to a “positive” measurement. The absence of a spot, or dark image

indicates a “negative” or “failed” measurement. The ﬁgure shows that the combina-

tions (A, 1) and (B, 2) are certain to fail, while the other combinations ( A, 2) (B, 1)

or ( A, 3) (B, 3) have a ﬁnite chance, p or 1 − p, respectively, of succeeding. The suc-

cessful measurements in the two cases ( A, 2) or (B, 1), as marked with a cross (×),

correspond to absolute certainty that the source tone is A or B, respectively. The other

two successful measurements (A, 3) (B, 3), like all failed measurements, do not convey

any information, as mentioned earlier. The success of our measurement and the absolute

conclusion therein, is, thus, dependent on our ﬁlter choice (1 or 2), which is essentially

a matter of guesswork.

From any academic standpoint, it remains debatable whether the above ﬁctitious

“physical measurement” through a set of “magic ﬁlters” may, in some way, help clarify

the essence of a true quantum POVM measurement. If it has any merit, however, it

helps to stress the point that quantum measurements are based on the observer’s choice

of a measurement operator (the magic ﬁlter). The measurement or observation of an

outcome may either succeed or fail (seeing or not seeing a spot). In the case of success,

17.3 Quantum measurements on joint states 351

1 −

Figure 17.1 Fictitious analogy of POVM measurement of tones A, B and a magic-ﬁlter set 1, 2, 3

to characterize a source. Given the tones A, B and choices 1, 2, 3, a positive observation of the

tone is characterized by the bright spot on the images shown, with various probabilities

0, 1, p, 1 − p. The two images marked with a cross (×) correspond to absolutely certain

measurements that the source tone is A or B.

the observer may or may not be able to derive an absolute conclusion as to what state

was observed (the source tone), depending on his or her choice.

Leaving ﬁction behind, we can conclude that POVM measurements make it pos-

sibletogetabsolutely certain information about the input state |ψ , despite the fact

that it is, a priori, unknown, just as the right measurement operators are unknown.

We have noted that while the corresponding measurements may sometimes fail, the

conclusion is never wrong when they succeed. This remarkable property is referred to

as unambiguous quantum state discrimination, of UQSD. Remarkably, while UQSD

has no classical explanation, or no counterpart in classical physics, it can be imple-

mented in actual physical measurements, for instance, in quantum cryptography (see

Chapter 25).

17.3 Quantum measurements on joint states

In Chapter 16, we introduced the concept of tensor states, which correspond to systems

being described by more than one quantum state, e.g., |ψ

 and |ψ

. Here, we shall

call such systems composite systems. As we have seen, the corresponding state can be

written in the tensor form |ψ=|ψ

⊗|ψ

,or|ψ =|ψ

|ψ

, or, for short |ψ=

|ψ

,ψ

, and even, sometimes, |ψ=|ψ

, e.g., the notations |ψ =|i ⊗|j≡

|i|j ≡|i, j≡|ij are all equivalent. These tensor states are also referred to as joint

states. Regardless of the notation conciseness for joint states, however, it is important to

352 Quantum measurements

keep in mind the underlying tensor product concept, which is also associated with that

of tensor operators. Thus, the tensor operator U = A ⊗ B acts on the joint state |ψ=

|ψ

⊗|ψ

 according to the rules U |ψ =A|ψ

⊗B|ψ

 and ψ|U

=ψ

⊗

ψ

, hence, the imperative need of keeping the single quantum state notation in the

right order sequence inside the ket |·, and choose the appropriate notation for the tensor

products when operators are being used.

We are now interested in deﬁning a generic measurement operator for joint states. The

following may look complicated at ﬁrst but, as we shall see, the result makes complete

sense in view of what we have learnt from the previous section. Assume ﬁrst a single-state

space V , as deﬁned by the orthonormal basis V

={|x

} (for simplicity we identify

here the “space” and one corresponding “basis” with the same letter V ). Assume next

that {M

} represents a valid set of measurement operators for V (meaning that M

are positive and verify the completeness relation). For the set {M

} there is a space

M of orthonormal states {|m}, corresponding to each of the different possibilities of

post-measurement states resulting from the action of M

. I shall refer to M as an ancilla

space.Let|

m be a ﬁxed state of M, to be used as an arbitrary reference, which we

might index |

m=|0. We can now work with the joint states |ψ|0=|ψ, 0 from the

extended space V ⊗ M of joint states |ψ|m=|ψ, m. Next, deﬁne the tensor operator

U whose action on |ψ, 0 results in:

U |ψ, 0=



|ψ, m. (17.66)

We then have for any two states |ψ, |ψ



:

ψ



, 0|U

U |ψ, 0=



ψ



, k|M

|ψ, m



ψ



|ψδ



ψ



|ψ

=ψ





|ψ≡ψ



|ψ.

(17.67)

The above result shows that the transformation U preserves the inner product of all

states |ψ, 0 as a unitary operator for the subspace V ⊗{|0}. Actually, we can make

U a unitary operator for the entire space V ⊗ M, which applies to any state |ψ, m,

Indeed, deﬁne U



as the sum of two projectors on the two orthogonal spaces V ⊗{|0} and V ⊗{|m = 0}

for any state |ψ∈V , as follows: U



= U |ψ, 0ψ, 0|+



m=0

|ψ, mψ, m|. It is clear that since U

is unitary, U



is unitary. We also have U



|ψ, 0=U



|ψ, 0 and, for k = 0, U



|ψ, k=U|ψ, 0ψ, k|+



m=0

|ψ, mψ, m|ψ, k=U δ



m=0

|ψ, mδ

≡|ψ, k.

17.3 Quantum measurements on joint states 353

according to the alternative deﬁnition:

|ψ, k=



⊗|km|)|ψ, k



|ψ⊗(|km|k)



|ψ⊗(|kδ

)



|ψ, m⊗(|kδ

(17.68)

As an exercise, it is possible to check that ψ



, k



|ψ, k≡ψ



|ψδ



which shows

that the new deﬁnition of U (i.e., U

)inEq.(17.68) is indeed unitary over V ⊗ M.Having

found the above unitary operator U that conserves the inner product of all joint states

|ψ, k in V ⊗ M, we can now introduce the joint-state measurement (JSM) operator E

= U

U, (17.69)

with

= I

⊗|mm|, (17.70)

where I

is the identity operator on V and |mm|the projector over |m∈M.

Next, we

shall calculate the bracket function p(m) =ψ, k|E

|ψ, k, which, using Eq. (17.66)

with |0=|k,is:

p(m) =ψ, k|U

U |ψ, k



ψ, p|M

⊗|mm|)



|ψ, q



ψ|M

|ψp|mm|q



ψ|M

|ψδ

≡ψ |M

|ψ.

(17.71)

We immediately recognize from the result that p(m)isthemeasurement probability

associated with the operator M

on the state |ψ∈V , as deﬁned in Eq. (17.46).After

Eq. (17.50), and writing the JSM operator E

= (U

U , we obtain the joint

post-measurement state as the result of the transformation:

|ψ



, k



=



p(m)

U |ψ, k



ψ|M

|ψ

|ψ, m.

(17.72)

The full deﬁnition of the JSM operator actually being E

= ( P

U = U

U ≡ U

where we used the projector and Hermitian properties of P

354 Quantum measurements

This ﬁnal result shows that after the measurement, the state |ψof space V is transformed

into

|ψ



=



ψ|M

|ψ

|ψ, (17.73)

while the ancilla state |k is transformed into |m. In the rest of this book, we will not

have to be concerned about the ancilla space M. Such a space is just used here as a

mathematical tool to allow the construction of a quantum measurement operator for joint

states. Bearing this in mind, we can now expand the conclusion to measurements of joint

states of any dimension from V

⊗n

⊗ M

⊗n

, by means of the n-ancilla space, M

⊗n

, i.e.,

|ψ

,ψ

,...,ψ

, k

,...,k

=|ψ

⊗|ψ

⊗···⊗|ψ

⊗|k

⊗k

⊗···⊗|k

.

.(17.74)

We, thus, know that it is possible to deﬁne a measurement operator E

that can act on

the subspace pair |ψ

, |k

 and collapse it into the pair M

|ψ

, |m, such that

|ψ

,ψ

,...,ψ

, k

,...,k













|ψ

⊗|ψ

⊗···⊗



ψ

|ψ



|ψ

⊗···|ψ



⊗|k

⊗k

⊗···⊗|k

i−1

⊗|m⊗|k

i+1

⊗···|k













(17.75)

We may just forget now about the ancilla bits and consider simply that

|ψ

,ψ

,...,ψ

≡|ψ

⊗|ψ

⊗···⊗



ψ

|ψ



|ψ

⊗···⊗|ψ

.

(17.76)

To provide an illustration, assume that M

=|mm|. Letting |ψ

=



(i)

|n,we

obtain

|ψ

,...,ψ

=|ψ

⊗|ψ

⊗···⊗

(i)

|m⊗···⊗|ψ



(i)

|ψ

,...,ψ

i−1

, m,ψ

i+1

,...,ψ



≡|ψ

,...,ψ

i−1

, m,ψ

i+1

,...,ψ

,

(17.77)

as obtained within an unobservable phase factor e

iδ

= x

(i)

/|x

(i)

Finally, consider for simplicity two joint states in 2-state space V ⊗ V , namely |φ=

|φ

,φ

 and |ψ=|ψ

,ψ

, and their superposition |χ with complex coefﬁcients α, β

(satisfying |α|

+|β|

= 1):

|χ=α|φ+β|ψ=α|φ

,φ

+β|ψ

,ψ

. (17.78)

Owing to the linearity and distributiveness of the measurement operators, we have

|χ=E

(α|φ+β|ψ) = αE

|φ

,φ

+β E

|ψ

,ψ

=α|1,φ

+β|1,ψ



|χ=E

(α|φ+β|ψ) = αE

|φ

,φ

+β E

|ψ

,ψ

=α|φ

, 2+β|ψ

, 2

(17.79)

17.4 Exercises 355

and

|χ=E

|χ=α|1, 2+β|1, 2≡(α + β)|1, 2, (17.80)

which actually illustrates the great simplicity of joint-state measurements and their

properties. These can now be applied to the case of single and multiple qubit systems,

which are described in Chapter 18.

17.4 Exercises

17.1 (B): Calculate the moduli and scalar products of the 2-qubits

|q=i|00+|01−|11,



=|00+|01−i|10+|11.

17.2 (B): Show that |qq| is the projector operator over the qubit |q.

17.3 (B): Determine the eigenstates and diagonal representations of the Pauli matrix

Y and Hadamard matrix H .

17.4 (T): Show that the transition operator T deﬁned by its matrix element

=x|y

in different orthogonal bases V

={|x} and W

={|y} satisﬁes the property

T = TT

= I.

Then show that T = T

in the general case.

17.5 (M): Show that two successive measurements from different operators M

and

(in this order) are equivalent to a single measurement of operator deﬁnition

= L

17.6 (T): Show that the three operators {E

, E

} deﬁned by

= u|11|,

= v(|0−|1)(0|−1|),

= I − E

− E

form a complete POVM set over the space V ={|0, |1} if the constants u,v

satisfy the two conditions

0 < u ≤

,v=

18 Qubit measurements, superdense

coding, and quantum teleportation

This chapter is concerned with the measure of information contained in qubits. This can

be done only through quantum measurement, an operation that has no counterpart in the

classical domain. I shall ﬁrst describe in detail the case of single qubit measurements,

which shows under which measurement conditions “classical” bits can be retrieved. Next,

we consider the measurements of higher-order or n-qubits. Particular attention is given

to the Einstein–Podolsky–Rosen (EPR) or Bell states, which, unlike other joint tensor

states, are shown being entangled. The various single-qubit measurement outcomes from

the EPR–Bell states illustrate an effect of causality in the information concerning the

other qubit. We then focus on the technique of Bell measurement, which makes it possible

to know which Bell state is being measured, yielding two classical bits as the outcome.

The property of EPR–Bell state entanglement is exploited in the principle of quantum

superdense coding, which makes it possible to transmit classical bits at twice the classical

rate, namely through the generation and measurement of a single qubit. Another key

application concerns quantum teleportation. It consists of the transmission of quantum

states over arbitrary distances, by means of a common EPR–Bell state resource shared

by the two channel ends. While quantum teleportation of a qubit is instantaneous, owing

to the effect of quantum-state collapse, it is shown that its completion does require

the communication of two classical bits, which is itself limited by the speed of light.

We then brieﬂy consider the possibility of denser teleportation schemes, taking, for

example, the case of two qubits. Quantum entanglement is also shown to be applicable

to the teleportation quantum gates. This opens the perspective of distributed quantum

computing: the possibility to manipulate qubits from a distance, or, in a futuristic view,

to share the resources of remote quantum networks.

18.1 Measuring single qubits

This section describes the effect of quantum measurements on qubits. We shall ﬁrst

recall some key results obtained in previous chapters, and apply the principles to the

case of single qubits and then joint qubit states, including the intriguing case of entan-

gled qubits.

In Chapter 17, we learnt that given a space of quantum states |ψ deﬁned

by an orthonormal base V

={|x

}, there exist different possibilities of quantum-

measurement operator sets, called {M

}, corresponding to a number n of possible