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

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18.1 Measuring single qubits 357

measurements indexed by m (m = 1...n). To recall, the two requirements for any such

set are (a) that all operator products M

should be positive (which result in the

unitarity of M

), and (b) that they satisfy the completeness relation



= I .

The two key results are, ﬁrst, that the probability of measuring m is given by the bracket:

p(m) =ψ|M

|ψ, (18.1)

and, second that the measurement projects the input state |ψinto the post-measurement

state |ψ



 deﬁned as

|ψ



=



p(m)

|ψ=



ψ|M

|ψ

|ψ. (18.2)

Out of many possibilities for any quantum measurement, we can use the measurement in

the state basis, where M

=|x

x

| is the projector on the pure state |x

.Giventhe

state |ψ=



, it is clear from the deﬁnition in Eq. (18.1) with M

=|x

x

that the probability that the measurement ﬁnds |ψ in the state |x

 is p(m) =|x

.As

we have seen in Chapter 17, the post-measurement state is

|ψ



=

. (18.3)

Let the input state be a qubit |q=α|0+β|1, as deﬁned from the orthonormal base

V ={|0, |1}, and where (to recall) α, β are complex numbers satisfying |α|

+|β|

1. It is clear that α, β represent the complex probability amplitudes of the qubit state, and

that |α|

or |β|

represent the probabilities of the qubit being found in either pure state

|0 or |1 of the base V . According to Eq. (18.3), the corresponding post-measurement

states are











|α|

|0=e

iδ

|0

|β|

|1=e

iδ

|1,

(18.4)

where

are unobservable phase factors, which can be omitted.

Since the qubit is a superposition of states |q=α|0+β|1, the action of any

measurement in the state-base V , as deﬁned by the operators M

=|00|or M

=|11|,

is, thus, to project, or collapse it into one of the two states |0 or |1.UsingM

,the

probability that |q collapses into |0 is p(0) =|α|

.UsingM

, the probability that

|q collapses into |1 is p(1) =|β|

= 1 −|α|

. We can interpret the result of such

a measurement as reducing the qubit information into the classical bit values 0 or

1, respectively. But unlike the classical case, there is no absolute certainty that the

resulting information is, indeed, 0 or 1, when using measurement operators M

, M

, since

either outcome is associated with some ﬁnite probabilities. For illustration, consider the

following possibilities:

(a) α = 0,β = 1, yielding p(0) = 0, p(1) = 1;

(b) α = β = 1/

√

2, yielding p(0) = p(1) = 1/2;

√

3,β= 2/

√

3, yielding p(0) =1/3, p(1) = 2/3;

(d) α = 0.9,β= 0.1, yielding p(0) =0.99, p(1) = 0.01.

358 Qubit measurements, coding, and teleportation

It is clear that in case (a), there is absolute certainty that the qubit measurement will

result in the bit information 1. In case (b), the outcome has equal chances of being

0 or 1. In case (c) the odds on measuring 1 are twice those of measuring 0. In case

(d), the odds are 99% for 0 and 1% for 1. It must be emphasized here that the above

ﬁgures concern the measurement outcomes obtained by using the corresponding opera-

tors M

, M

. Here comes a subtlety: in case (c), for instance, we know that we have a

1/3 chance of measuring 0 when using M

. This means that there is a 2/3 chance that

this measurement fails. Likewise, there is a 1/3 chance that the measurement of 1 fails

when using M

. But what does “measurement failure” mean? Basically, that whenever

we choose to use a given measurement projector M

, the input state is not absolutely

certain of collapsing into the corresponding state |x

. It will only happen “successfully”

with probability |x

. If this collapse does not happen (with probability 1 −|x

), then

nothing is measured, and the measurement is, thus, failed. Also, the quantum state col-

lapses into nothingness, which here means qubit annihilation and irreversible loss of its

information. We can mathematically deﬁne such an annihilated, informationless state as



=∅.

To clarify the picture, or to alleviate some possible unease with the notions of “suc-

cessful” and “failed” qubit measurements, consider the following (ﬁctitious) physical

analogy, which we have already used in Chapter 17 for POVM measurements.

Compare the qubit |q with a source that has two possible tones, A or B (standing for

the states |0 or |1, respectively), and which are not visible to the eye. Our measurement

consists of determining the color tones by observing this source through a set of two

“magic ﬁlters,” called 0, 1(forM

, M

). As we have seen earlier, such ﬁlters have

strange properties. Indeed, ﬁlter 0 does not react to tone B, while it makes tone A

visible to the eye, but only with probability p (or |α|

). Filter 1 does not react to tone

A, while it makes tone B visible to the eye, but only with probability q = 1 − p (or

|β|

). Even stranger here (than the description in Chapter 17) is that the two ﬁlters

A, B are of complementary nature, since their probabilities of positive reaction are now

complementary (q = 1 − p). Figure 18.1 shows the eye images obtained in the eight

possible cases, according to the two pure-tone possibilities A or B (but not both) and

their mixture A, B, and the two ﬁlter options 0, 1. The observation of a bright spot

indicates a positive, or successful, measurement (seeing something). The absence of a

spot indicates a negative, or failed, measurement (not seeing anything).

Considering ﬁrst the case of pure tones (A or B), the ﬁgure indicates that there are

actually two “successful” measurement outcomes, as characterized by the presence of

the bright spot with nonzero probability. These successes correspond to the two input or

ﬁlter cases ( A, 0), (B, 1), which positively identify the tones to be A or B with absolute

certainty ( p = 1forA,orq = 1 − p = 1forB). In the two other cases, ( A, 1), (B, 0),

we never see any spot, or the measurement is always “failed.” But here comes a subtlety:

if we know prior to the measurement that the color tones are pure, the failed measurement

is also a positive indication. Indeed, a failure to measure A means an absolute certainty

that the input is B, and the reverse. The certainty is the same regardless of our ﬁlter choice

(0 or 1). In this case, the classical bit measurement from the qubit is 100% accurate. The

18.1 Measuring single qubits 359

1 −

0 1

Figure 18.1 Fictitious analogy of qubit measurement of tones A, B, or their mixture, and a

magic-ﬁlter set 0, 1 to characterize a source. Given the tones A, B and choices 1, 2, a positive

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

probabilities 1, 0, p, 1 − p.

information gained is a 0 or 1 classical bit, in 100% correspondence with the qubit pure

state. But to achieve this performance without any concern for the measurement option

, M

), it costs the tax of one bit of advance information (i.e., “pure input state?” =

YES, NO). In the case of the tone mixture (Fig. 18.1, right) we observe only two

measurement “successes,” with probability p when using the ﬁlter 0, and probability

1 − p when using the ﬁlter 1. The information obtained in these two cases is a classical

bit of information 0 or 1, as we have projected the qubit onto the corresponding pure

states. But, as the ﬁgure indicates, there is a ﬁnite probability 1 − p or p that using

either ﬁlter 0 or 1 results in a measurement failure, and in these cases we do not get any

classical bit information. There is one exception to this, however: if we know in advance

that the qubit is in the superposition α = β = 1/

√

2 (case (b)), there are equal chances

of success or failure, regardless of the measurement operator (or tone ﬁlter). There is no

point in making any measurement, since we know for a fact that a successful outcome

will give either a 0 or a 1 classical bit, with equal likelihood. We may just pick a bit

value at random. The tax to pay for this result is one bit of advance information (i.e.,

“case (b)?” = YES, NO).

The above description, despite its questionable usefulness from a purist or academic

standpoint, made it possible to “visualize” somewhat and get a better feel of the nature of

quantum measurements, when applied to single qubits. It is noteworthy, however, that the

ﬁctitious “magic-ﬁlter” experiment can actually be implemented in the physical world,

as described in Chapter 25 in the part concerning quantum cryptography. The lesson

learnt is that any “successful” qubit base-state measurement results in its collapse into

one of the pure states, and the reduction of its intrinsic information into a classical 0 or

1 bit.

In Chapter 17, we have also described another type of measurement referred to as

projective or von Neumann measurement. In this approach, we can deﬁne the Hermitian

observable A according to its spectral decomposition over the eigenstates |λ

, |λ



360 Qubit measurements, coding, and teleportation

forming an orthonormal base V ={|λ

, |λ

}:

A =



m=1



m=1

|λ

λ

≡ λ

|λ

λ

|+λ

|λ

λ

(18.5)

where λ

,λ

are the corresponding eigenvalues, and P

=|mm| the measurement

operators. Next, we recall the two key results obtained in Chapter 17, while applying

them to the case of qubits. First, given a qubit |q=x

|λ

+x

|λ

 as expressed

in the eigenstate base V , the probability of measuring the eigenvalue λ

is given

by p(λ

) =|x

. With the usual qubit notations |q=α|λ

+β|λ

, we, thus,

obtain:

p(λ

) =|α|

, p(λ

) =|β|

. (18.6)

Second, the post-measurement state |q



 is



=|λ

. (18.7)

We may conclude from the two above results that the projective measurement of qubits is

essentially similar to the previously described base measurement in the pure states, except

that the measurements project the qubit onto the eigenstates of A. Such a conclusion

is true, but it fails to convey an important property. Indeed, if we know prior to the

measurement that the qubit must be in one of the eigenstates, then we are in the same

situation as in the previous pure-state case. No matter whether we use P

=|λ

λ

| or

=|λ

λ

| for the qubit measurement, we always retrieve a classical 0 or 1 bit. To

recall, if we use P

when the qubit is in state |λ

 or P

when the qubit is in state |λ

,

the measurements are certain to “succeed,” yielding the classical 0 or 1 bits, and the

collapsed qubit state remains invariant. But if we use the wrong operator (e.g., P

when

the qubit is |λ

), the measurement “fails,” and the qubit state is annihilated. However,

such a failure indicates that the qubit must have been in the other orthogonal state, and,

therefore, we can conclude with certainty the value of the corresponding classical bit.

The only penalty in this event is that we have lost the qubit, and it cannot be reused for

another quantum computation, for instance as an input to a subsequent quantum gate

within a quantum circuit. The key conclusion is that measurements can be destructive

(even when informative) and, therefore, it is safer to proceed to quantum measurements

only at the end of a purposeful computational chain. If intermediate measurements are

required, one must ensure that these are always nondestructive, so that the collapsed

qubit may be reused in the rest of the circuit computation.

In quantum-gate circuits (Chapter 16), any quantum measurement is conventionally

represented by an “ammeter” symbol, as illustrated in Fig. 18.2. The two output lines

shown at the right correspond to the collapsed or post-measurement state (|λ

) and

to the measurement value λ

. In the case where the measurement fails, we may write



=∅for the input state annihilation.

18.2 Measuring

-qubits 361

=´

Figure 18.2 Quantum-circuit representation of measurement apparatus to measure qubits with

operator M

, yielding measurement value λ

18.2 Measuring

-qubits

In this section, I describe quantum measurements of higher-order or n-qubits, which

have fundamental implications in both QIT and quantum computing (QC). This is a

turning point, where we shall begin to grasp the strange beauty (and the oddity!) of the

quantum properties of qubits.

As we have seen in Chapter 17, any base-state measurement E

on an n-state |ψ 

from V

⊗n

results in the collapsing effect

|ψ=E

|ψ

,...,ψ



=|ψ

,...,ψ

i−1

, m,ψ

,...,ψ

.

(18.8)

In the case of an n-qubit |q=|x

, x

,...,x

, with x

= 0, 1, we have:

|q=E

, x

,...,x



=|x

, x

,...,x

i−1

, m, x

i+1

,...,x

.

(18.9)

With this measurement tool in hand, we can now perform measurements on any qubit

superpositions or mixtures in the subspace V

⊗2

(for instance), such as:

|q=u|x

, x

+v|x

, x

, (18.10)

with u,vbeing complex amplitudes satisfying |u|

+|v|

= 1, so that |rhas a modulus

of one. For each qubit i in the subspace V , there are two possible measurement operators,

namely E

and E

. To illustrate the action of these operators, consider the following

example (using the compact notation |x

, x

≡|x

):

|q=

√

(|00+|01+|10). (18.11)

If we want to measure a classical bit 0 in the ﬁrst qubit position (i = 1), we use E

and

obtain

|q=

√

|00+E

|01+E

|10)

√

(|00+|01+∅)

√

(|00+|01).

(18.12)

362 Qubit measurements, coding, and teleportation

Thus, the probability of measuring 0 in the ﬁrst qubit is:

i=1

(0) =q|E

|q=

(00|+01|)(|00+|01) ≡

. (18.13)

It is now a straightforward exercise to evaluate mentally the other probabilities:

i=1

(1) =

, p

i=2

(0) =

, p

i=2

(1) =

. (18.14)

Next, we shall look at a peculiar family of some 2-qubits, which are of utmost interest for

qubit measurements and for the “quantum manipulation” of information. In Chapter 15,

we identiﬁed four state superpositions, referred to as Einstein–Podolsky–Rosen (EPR)

or Bell states (or pairs), and deﬁned:











|β

≡

√

(|00+|11)

|β

≡

√

(|01+|10)

|β

≡

√

(|00−|11)

|β

≡

√

(|01−|10).

(18.15)

These EPR–Bell states may seem innocuous at ﬁrst glance, but they actually reserve

a certain number of interesting properties, and as we shall see further down, really

“stunning” applications.

As a ﬁrst property, there exists no tensor product of pure states or of qubits that is

capable of generating any of the EPR–Bell states. This is left as a straightforward exercise

to check. Because the EPR–Bell states cannot be decomposed as a qubit tensor product,

they are called entangled states. A faster way to realize the nature of entanglement is to

observe that none of the states in Eq. (18.15) can be factorized into a pure-state product

of the form |i|j =|0|0, |0|1, |1|0, |1|1. Thus, it is not possible to tell what the

states of the ﬁrst and the second qubits are.

While the EPR–Bell states are not tensor products of qubit states, they can, however, be

generated by means of quantum gates. The circuit shown in Fig. 18.3, which comprises

a Hadamard gate and a CNOT gate (Chapter 16) represents one possibility for the

generation of EPR–Bell states. As we saw in that earlier chapter, the Hadamard gate

transforms the input qubit |0 into the mixed state |+ = (|0+|1)/

√

2, and the input

qubit |1 into the mixed state |− = (|0−|1)/

√

2. Thus, for the quantum gate in

Fig. 18.3, the tensor input |0⊗|y results in

(H |x) ⊗|x ⊕ y=(|x = 0+|x = 1) ⊗|x ⊕ y/

√

= (|0⊗|0 ⊕ y+|1⊗|1 ⊕ y)/

√

≡ (|0, y+|1, 1 ⊕ y)/

√

(18.16)

Hence, |0⊗|0→(|0, 0+|1, 1)/

√

2 ≡|β

 and |0⊗|1→(|0, 1+|1, 0)/

√

2 ≡|β

. The two other results for the inputs |1⊗|y are also readily veriﬁed.

18.2 Measuring

-qubits 363

Figure 18.3 Quantum circuit for the generation of EPR–Bell states |β

as deﬁned in Eq. (18.15).

As a second property, the EPR–Bell states form an orthonormal basis for any 2-qubit

|kl and any superposition thereof. Indeed, it is straightforward to see from Eq. (18.15)

that











|00=

√

(|β

+|β

)

|01=

√

(|β

+|β

)

|10=

√

(|β

−|β

)

|11=

√

(|β

−|β

),

(18.17)

hence, any 2-qubit superposition of the above pure |ij states accepts a unique decom-

position over the EPR–Bell states.

Third, we notice from the deﬁnitions in Eq. (18.5) that the EPR–Bell states represent a

mix of 2-qubits |ijand |klsatisfying i = 0, k = 1, and j, l = 0, 1. For the measurement

probabilities, this leads to the following property (as is easily mentally checked):

i=1

(0) = p

i=1

(1) = p

i=2

(0) = p

i=2

(1) =

. (18.18)

This result shows that any measurement E

of the EPR–Bell states has a 50% chance

of successfully projecting into any of the pure states. The 16 possible measurements

resulting in the corresponding state collapses (phase factors

−

π/2

being omitted) are

summarized in Table 18.1. As expected, we observe from the table that the EPR–Bell

states collapse into any of the pure states |00, |01, |10, |11. We see that using a

positive measurement E

necessarily collapses the EPR–Bell pair into the pure state

|m⊗|x, and a positive measurement E

necessarily collapses the EPR–Bell pair into

the pure state |x⊗|m, where x = 0, 1. This has the consequence that regardless of the

four possible EPR–Bell pairs, a single positive measurement yields absolute information

on one of the two qubits in the collapsed state.

Because the ﬁrst measurement gives only one qubit information, a second measure-

ment would be necessary to get the missing information concerning the other qubit. But

we do not need to make a second measurement if we know in advance the initial EPR–Bell

pair. Indeed, Table 18.1 illustrates that the advance knowledge of |β

, |β



364 Qubit measurements, coding, and teleportation

Table 18.1 State collapses resulting from single-qubit measurements E

on the EPR–Bell states |β

, |β

.

|β

|β



|00|01|00|01

|11|10|11|10

|00|10|00|10

|11|01|11|01

()

1001

Alice

Bob

Measures

Collapse

Measures

Figure 18.4 Two successive measurements by Alice then Bob in an entangled quantum state, here

|β

.

and the choice of a single-measurement operator E

, E

absolutely deﬁnes

the collapsed state and, hence, the value of the two entangled qubits. For instance, if

we use |β

, the single measurement E

yields the 2-qubit |01=|0⊗|1 and the

absolute information therein. Thus, given the knowledge of the system EPR–Bell pair,

any positive measurement of one of the two qubits conditions the information knowledge

of the other qubit. Because the information of one qubit measurement conditions that

of the second qubit measurement, the EPR–Bell states are said to be entangled.InQIT

jargon, EPR–Bell states are also referred to as ebits, for “entanglement bits” (one ebit =

one single entangled pair).

To provide a visual illustration of this, imagine two people A and B, who are tradi-

tionally named Alice and Bob, and who live far apart. However, both have access to a

quantum system, which is in the entangled EPR–Bell state |β

, as represented by a

cloud in Fig. 18.4. The state remains in |β

 until Alice unilaterally takes the initiative

to measure the ﬁrst qubit, through the operator E

, for instance. This measurement

immediately results in the EPR–Bell pair collapsing into the pure state |10. Bob’s mea-

surement of the second qubit can only yield |0 as a positive outcome. Bob and Alice

can exchange roles. For instance, Bob may ﬁrst measure the second qubit through E

resulting in the system collapsing into the state |01, and leaving |0 to Alice as the only

positive measurement for the ﬁrst qubit. In each case, the measurement of the ﬁrst qubit

conditions the measurement of the second, because of the effect of state collapse that

occurs in between.

18.3 Bell state measurement 365

Table 18.2 Outcome of Bell measurement with EPR–Bell states as input (normalization

factor being overlooked).

Input state |xy Pre-measurement state Cbit output m, m



|β

=|00+|11|00 00

|β

=|00+|11|01 01

|β

=|00−|11|10 10

|β

=|01−|10|11 11

Figure 18.5 Principle of Bell measurement or measurement in Bell states.

From any physical perception of our “classical world,” the effect of state entanglement

seems rather nonintuitive. We may illustrate it through a ﬁgurative experiment. Assume

that we place two coins in two separate boxes. In each box, the position (or states) of

the coins can be either heads or tails, but we have no awareness of which one. In the

classical world, opening a box tells us only that the coin is heads or tails, but we cannot

know of the other coin until we open the other box. In the quantum world, the coin states

may be entangled. This means that opening only one box conditions the state of the coin

in the other box. The point here is not about the advance knowledge of the entanglement

possibilities, as represented by the four possible EPR–Bell states. Rather, it is the matter

that a single measurement in one physical system determines “for absolute certain”

the state of the other physical system. As described later in this chapter, the properties

of EPR–Bell state entanglement are the key to applications called quantum superdense

coding and quantum teleportation, both of which having no counterpart in the classical

world.

18.3 Bell state measurement

In this section, I describe a 2-qubit measurement scheme, referred to as measurement in

the Bell basis,orBell state measurement. The principle is illustrated in Fig. 18.5.The

circuit includes a Hadamard gate and a CNOT gate, followed by two qubit measurements.

366 Qubit measurements, coding, and teleportation

Bob

Alice

Figure 18.6 Circuit used for superdense coding.

The measurements, which are made in the base {|00, |01, |10, |11}, output the clas-

sical bits m, m



. What happens if we input to this gate circuit any of the EPR–Bell states

|β



? It is left as a simple exercise to check that the pre-measurement states are |mm





and the resulting measurement cbits (short for “classical bits”) are m, m



,asshownin

Table 18.2. This means that the apparatus is able to positively single out or identify any of

the four EPR–Bell states, hence, the name “measurement in the Bell basis” or “Bell state

measurement.” Such a denomination may sound confusing, since the measurement is

actually performed in the {|00, |01, |10, |11}basis. The point is that while the circuit

represents a true EPR–Bell state measurement apparatus, it can be used for other types

of application, as will be shown next when describing superdense coding and quantum

teleportation.

18.4 Superdense coding

In this section, I describe a ﬁrst application of quantum entanglement, which enables

quantum superdense coding. The key concept behind this appellation is the possibility

for Alice to send to Bob two classical bits m, m



, which she has in her possession, by

means of communicating only a single qubit. The generic circuit for superdense coding

is represented in Fig. 18.6. Alice and Bob must share an EPR–Bell state, for instance

|β

. Alice controls a gate U , which she can change at will into I, Z, X , and iY.Bob

has the control of a Bell-state measurement apparatus, as shown by the shaded subsystem

in the ﬁgure. It can be checked as an easy exercise that the choice of U transforms |β



into any EPR–Bell state |β

, |β

, according to

I |β

=|β

; X |β

=|β



Z|β

=|β

; iY|β

=|β

.

(18.19)

Thus, by switching the U gate into I , Z , X, and iY, Alice can send Bob the classical

bit pairs 00, 01, 10, and 11, respectively, at an information rate of two bits per qubit,

which represents twice the classical information rate, hence the name superdense coding.

Note that from any realistic telecom viewpoint, twofold improvements in information-

rate capacities do not deserve such superlatives as “super” or “dense,” or, worst of

all, “superdense,” notwithstanding the conceptual importance of the potential capacity

expansion of the “old-fashioned” classical bit.