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

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16.5 Tensor products 327

tum algorithms simply emulate classical ones, with qubits instead of bits! (Chapter 19

and 20 will prove far differently).

16.5 Tensor products

In previous sections, we made use of the notion of tensor product for qubits, to describe

quantum gates with more than one qubit input. This section will make it possible to

clarify such a notion, which is fundamental to the understanding of quantum computing

with multiple states. In the description, we shall proceed gradually, from elementary to

advanced.

Given two qubits |a, |b we called the tensor product the 2-qubit |a|b.Fromnow

on, we shall write the result with the new notation |a|b≡|a⊗|b, with the sign ⊗

standing for the tensor-product operation. Given the computational base V ={|0, |1},

which deﬁned a 2D qubit space, we can form the 4D extended base deﬁning a 4D 2-qubit

space V ⊗ V according to:

V ⊗ V ={|0, |1} ⊗ {|0, |1}

={|0|0, |0|1, |1|0, |1|1}

≡{|0⊗|0, |0⊗|1, |1⊗|0, |1⊗|1}.

(16.45)

In the extended base, |a⊗|b is represented by four complex coordinates

, u

) to be determined. Assuming that |a=a

|0+a

|1 and |b=b

|0+

|1, we obtain

|a⊗|b=(a

|0+a

|1) ⊗ (b

|0+b

|1)

= a

|0⊗|0+a

|0⊗|1+a

|1⊗|0+a

|1⊗|1,

(16.46)

hence (u

, u

) ≡ (a

, a

). In the above, we have implicitly made

use of the linearity and distribution-over-addition properties of the tensor product:



λ(|x⊗|y) = λ|x⊗|y=|x⊗λ|y

|x⊗(|y+|z) =|x⊗|y+|x⊗|z.

(16.47)

The notion of tensor product, along with the same properties, applies to any two nD

and mD spaces deﬁned by computational basis V ={|α

, |α

,...,|α

} and W =

{|β

, |β

,...,|β

}. Hence, given

|a=



i=1

|α

, |b=



j=1

|β

, (16.48)

we have

|a⊗|b=



i=1



j=1

|α

⊗|β

. (16.49)

A qubit |a that is tensored with itself n times will be noted |a

⊗n

, namely,

|a

⊗n

=|a⊗|a⊗,...,⊗|a

n times

. (16.50)

328 Quantum bits and quantum gates

The tensor state |a⊗|b, often noted |a, b for simpliﬁcation, is also referred to as a

joint state. Such a joint state corresponds to the description of two quantum systems,

themselves being in the states |a and |b, respectively. Thus, |a⊗|a corresponds to

the description of two systems being in the same state |a. However, such a possibility

should not lead one to the conclusion that given a quantum system in state |a,itis

possible to duplicate this state into a second quantum system, so as to obtain the joint

state |a⊗|a. It simply cannot be achieved. This is a result of the noncloning theorem,

which is described in Section 16.6.

Next, I shall introduce the notion of tensor product for linear operators (which so

far have been referred to as matrices operating on the qubit vector space). Assume an

operator A deﬁned on the |a qubit space and an operator B deﬁned on the |b qubit

space. The operator tensor product A ⊗ B applies to the tensor states |a⊗|b and is

deﬁned according to the following distribution rule:

A ⊗ B(|a⊗|b) = A|a⊗B|b. (16.51)

An operator A that is tensored with itself n times will be noted A

⊗n

, namely,

⊗n

= A ⊗ A⊗,...,⊗A

n times

. (16.52)

The operator tensor product A ⊗ B also satisﬁes the useful properties according to which

conjugation, transposition, and Hermitian conjugation

are distributive. Namely:







(A ⊗ B)

∗

= A

∗

⊗ B

∗

(A ⊗ B)

= A

⊗ B

(A ⊗ B)

= A

⊗ B

(16.53)

These three properties stem from the deﬁnition of the operator tensor product in

Eq. (16.51), in which the operations of transposition and complex or Hermitian conju-

gation are clearly distributive.

How can we derive the matrix of the tensor operator A ⊗ B? The rule, which is

referred to as the Kronecker product, is quite simple. Assume that A is represented by a

n × m matrix (n lines, m columns), with coefﬁcients A

(i = 1,...,n, j = 1,...,m).

We have, by deﬁnition

A ⊗ B =







B ··· A

. ···

B ··· A







. (16.54)

Thus, in the above reduced form, A ⊗ B is an n × m matrix with coefﬁcients A

B.IfB

is represented by a p × q matrix ( p lines, q columns), then A ⊗ B is clearly an np × mq

matrix. The Kronecker-product rule also applies to single-column matrices, or vectors,

To recall, for any operator A of matrix coefﬁcients A

, the conjugate operator A

∗

has for coefﬁcients

∗

(complex conjugate of A

), the transposed operator A

has for coefﬁcients A

, and the Hermitian-

conjugate (or adjoint) operator A

has for coefﬁcients A

∗

. Also, the transposed and Hermitian conjugate

of the product AB are (AB)

= B

and (AB)

= B

, respectively.

16.5 Tensor products 329

which enables one to calculate the tensor product |a⊗|b. Let us next examine some

illustrative examples.

First, consider the case of the qubit tensor product |a⊗|bin bases V ={|α

, |α

}

and W ={|β

, |β

}. The corresponding matrices are single column, with A

and B

= b

. We obtain:

|a⊗|b=



|b











































, (16.55)

which is the expected result.

Second, consider the tensor product X ⊗ Y of the two Pauli matrices X, Y . We obtain:

X ⊗ Y =



0 × Y 1 ×Y

1 × Y 0 ×Y





0 Y

Y 0









000−i

00i0

0 −i0 0

i000







(16.56)

It is left as an easy exercise to verify the property X ⊗ Y (|a⊗|b) = X|a⊗Y |b,

which, as we have seen, applies to any pairs of operators A, B and qubits |a, |b.

An interesting case of n-tensored operator is provided by H

⊗n

, where H is the

Hadamard gate. Assume the extended computational base V

={|0, |1}

={|a}, with

|asymbolizing any of the n-qubits base element generated by the n-tensor product |a=

⊗|v

⊗···⊗|v

 where v

= 0orv

= 1(i = 1,...,n). As a general property,

it can be shown that

⊗n

|a=

√



(−1)

a∗b

|b, (16.57)

where |b=|w

⊗|w

⊗···|w

 is any base element of V

, and a

∗

b is a scalar

deﬁned as:

∗

b = v

+ v

+···+v



i=1

. (16.58)

With the tensor-product tools developed in this section, it is proposed as a closing

exercise to establish the property in Eq. (16.57) for the case n = 2, then by induction

for the general case.

330 Quantum bits and quantum gates

16.6 Noncloning theorem

Given a quantum system A,inanystate|ψ, and a second quantum system B,inany

pure state |s, is it possible to duplicate the ﬁrst state into the second? Such a “cloning”

operation would correspond to the transformation:

|ψ⊗|s→U(|ψ⊗|s) =|ψ⊗|ψ≡|ψ, ψ, (16.59)

where U is a unitary tensor operator. Assume that such an operator U exists and applies

to any state |ψ of A.Let|φ be another state of A such that |φ =|ψ.Wemustalso

be able to duplicate it into B according to:

U (|φ⊗|s) =|φ⊗|φ≡|φ,φ. (16.60)

By linearity of the transformation, we must also have for any state mixture |χ =

λ|ψ+µ|φ of A, where λ, µ are two complex numbers:

U (|χ⊗|s) =|χ⊗|χ≡|χ,χ. (16.61)

If we develop the left-hand side of Eq. (16.61) we obtain:

U (|χ⊗|s) = U (λ|ψ+µ|φ) ⊗|x

= λ|ψ⊗|ψ+µ|φ⊗|φ

≡ λ|ψ, ψ+µ|φ,φ,

(16.62)

while the right-hand side in Eq. (16.61) yields:

|χ⊗|χ=(λ|ψ+µ|φ) ⊗ (λ|ψ+µ|φ)

= λ

|ψ⊗|ψ+µλ(|ψ⊗|φ+|φ⊗|ψ) + µ

|φ⊗|φ

≡ λ

|ψ, ψ+µλ|ψ, φ+µλ|φ, ψ+µ

|φ,φ.

(16.63)

Equating Eqs. (16.62) and (16.63) yields

λ(λ − 1)|ψ, ψ+µλ|ψ, φ+µλ|φ,ψ+µ(µ − 1)|φ,φ≡0. (16.64)

Assuming that |ψ, |φ are pure states, the above equation implies that µλ = 0 and,

thus, |χ =|ψ or |χ=|φ. This result means that if there exists an operator U that

can clone two pure states |ψ, |φ, this operator cannot clone any of their mixtures

|χ=λ|ψ+µ|φ, which is a quite restrictive conclusion.

We are then left with the open question: does any cloning operator U exist in the

ﬁrst place? The answer is straightforward, but it requires one to use the inner product

of states, which is introduced in Chapter 17. Sufﬁce it here to provide the result: there

always exist a unitary operator U capable of cloning a pure state |ψ, or any pair of

pure states |ψand |

ψ(see Chapter 17 for proof). As we have previously seen, however,

such an operator cannot clone the mixture |χ =λ|ψ+µ|

ψ. The key conclusion is

that, except for the limiting case of pure-state pairs, it is not possible to clone quantum

states in the general case. This fundamental result is known as the noncloning theorem.

In the speciﬁc case of qubits, the two possible bases of pure states are {|s

, |s

} ≡

{|0, |1}, {|+, |−}. With our knowledge of quantum gates, it is trivial that we can ﬁnd

operators capable of “cloning” pure states into each other. Thus the exception about

16.7 Exercises 331

pure states does not weaken in any sense the generality of the noncloning theorem.In

particular, there is no quantum gate capable of executing the equivalent of the classical

FANOUT gate (Chapter 15), for any state other than a pure state.

16.7 Exercises

16.1 (B): Show by two different methods that the Hadamard gate H corresponds to

a unitary transformation.

16.2 (T): Prove the property according to which the three operators R

(γ )(k =

x, y, z) rotate any qubit |q on the Bloch sphere by an angle γ about the axis k,

in the counterclockwise direction.

16.3 (B): Prove the following properties of the rotation operators:

(2θ)R

(−2θ) = I

(2θ), R

(2θ



)] =−2iε

ijk

sin θ sin θ



+ sin(θ −θ



)

(2θ),σ

] = 2ε

ijk

sin θ.

16.4 (B): Determine the parameters n,θ associated with any unitary transformation

U according to the deﬁnition:

U = exp[i( n · σ )θ ].

Clue: use the generic deﬁnition of unitary matrices for U :

U = e

iδ



−



16.5 (M): Prove that any 2 × 2 unitary matrix U can be expressed from the two

rotation operators R

, R

according to the product:

U = e

iδ

(α)R

(β)R

(γ ),

where α, β, γ , δ are real numbers (Euler’s theorem).

16.6 (M): Considering the quantum-gate circuit

CNOT

show ﬁrst that the matrix of the intermediate B

CNOT

gate takes the form

CNOT







1000

0001

0010

0100







332 Quantum bits and quantum gates

assuming the computational base {|0|0, |0|1, |1|0, |1|1}. Then prove by

matrix multiplication that the circuit is equivalent to a CROSSOVER or SWAP

gate.

16.7 (M): Analyze the action of a CNOT gate with control qubit |ain a superposition

of states (assume |a=γ |0+δ|1 and |x=α|0+β|1). Consider then the

two cases γ = δ = 1/

√

2 and γ =−δ = 1/

√

2 for the control qubit. What are

the possible output states? Show that the same result is obtained with all inputs

in pure states and with a Hadamard gate placed on the control input path.

16.8 (T): Prove that the quantum circuit

which includes Hadamard gates (H ), ±π/4-phase gates (T, T

), and a single

π/2-phase gate (S), is an equivalent realization of a CCNOT or Toffoli gate.

16.9 (B): Given the Pauli matrices Y, Z , calculate the tensor product Y ⊗ Z.Given

two qubits |a, |b and their tensor product |a⊗|b, show that

Y ⊗ Z (|a⊗|b) = Y |a⊗Z |b.

16.10 (T): Given the computational base V ={|0, |1} and the Hadamard operator

H, show that the two-fold tensor product H

⊗2

= H ⊗ H satisﬁes

⊗2

|a=



(−1)

a∗b

|b,

where |a=|v

⊗|v

 and |b=|w

⊗|w

 are any elements of V

(with

= 0, 1) and

∗

b = v

+ v

Then prove by induction that, in the general case,

⊗n

|a=

√



(−1)

∗

|b,

with

∗

b = v

+ v

+···+v



i=1

17 Quantum measurements

This chapter is concerned with the measure of quantum states. This requires one to

introduce the subtle notion of quantum measurement, an operation that has no counterpart

in the classical domain. To this effect, we ﬁrst need to develop some new tools, starting

with Dirac notation, a formalism that is not only very elegant but is relatively simple

to handle. The introduction of Dirac notation makes it possible to become familiar with

the inner product for quantum states, as well as different properties for operators and

states concerning projection, change of basis, unitary transformations, matrix elements,

similarity transformations, eigenvalues and eigenstates, spectral decomposition and

diagonal representation, matrix trace and density operator or matrix. The concept of

density matrix makes it possible to provide a very ﬁrst and brief hint of the analog of

Shannon’s entropy in the quantum world, referred to as von Neumann’s entropy,tobe

further developed in Chapter 21. Once we have all the required tools, we can focus

on quantum measurement and analyze three different types referred to as basis-state

measurements, projection or von Neumann measurements, and POVM measurements.

In particular, POVM measurements are shown to possess a remarkable property of

unambiguous quantum state discrimination (UQSD), after which it is possible to derive

“absolutely certain” information from unknown system states. The more complex case

of quantum measurements in composite systems described by joint or tensor states is

then considered. Although of a more abstract and formal character than any previous

ones, this chapter is crucial to the understanding of the rest of these chapters, which are

concerned with the manipulation of qubits. This is not only because of the importance of

being comfortable with the Dirac formalism, but also of the need to have conceptually

assimilated the basics of quantum measurement.

17.1 Dirac notation

In this section, I shall introduce so-called Dirac notation, which are used in quantum

mechanics. When compared with the basic math formalism used in engineering, Dirac

notation looks quite esoteric, if not highly involved and complex. But as we shall see, such

notation is straightforward to assimilate, and quite easy to handle after familiarization.

Dirac notation is used to recapitulate the basic properties of linear operators and their

action on quantum states (here, qubits), which leads to a simple formalization of the

concept of quantum measurement.

334 Quantum measurements

In Chapter 16, we deﬁned the qubit as a 2D vector noted |q. In the orthonormal

basis of pure states V ={|0, |1}, the qubit has complex coordinates α, β, so that

|q=α|0+β|1. As we have seen, higher-dimension qubits can be deﬁned from the

extended basis V

={|x}, where |xis the nth tensor product of the pure state |δ

=|0

or |1, namely:

|x=|δ

⊗|δ

⊗···⊗|δ



≡|δ

,δ

,...,δ

≡|δ

···δ

,

(17.1)

where the two equivalent notations in the right-hand side can be used for lightening

purposes. In this extended space, the qubit |q can, thus, be expanded into

|q=



x∈V

x|x, (17.2)

where x represents the complex coordinate of |q with respect to the n-qubit basis

element |x.

Regardless of any basis dimension n, the qubit or vector |q is also referred to as a

“ket.” Actually, the ket represents the Dirac notation for a quantum state in the space

deﬁned by V ={|0, |1}, which is referred to as Hilbert space. Here, we do not need

to elaborate further on the notions of quantum state (as related to the physical world)

and of Hilbert space (as deﬁning the continuum of such quantum states). As it turns out,

the concept of qubit as a vector, rather than as a quantum state, even as representing an

oversimpliﬁcation, is accurate and wholly sufﬁcient to grasp Dirac formalism.

Inner product

In a 2D vector space, the scalar product of two vectors x = (x

, x

) and y = (y

, y

)is

deﬁned as the real number x · y = x

+ x

. In particular, the self-product x · x =

+ x

≡|x|

corresponds to the vector’s length or modulus. In the qubit space V

(meaning “deﬁned by any orthogonal basis such as V ”), we can introduce the concept of

the inner product of two qubits, |q=α|0+β|1 and |q



=α



|0+β



|1 according

to:

|q·|q



=¯αα



ββ



, (17.3)

where ¯α,

β are the complex conjugates of the coordinates α, β, respectively. In particular,

we have |q·|q=|α|

+|β|

, which as a real number, represents the modulus of the

qubit |q. In vector notation, Eq. (17.3) can be written as a line-vector–column-vector

product:

|q·|q



=|q



=(¯α,

β)





= ¯αα



ββ



, (17.4)

where

|q

is the conjugate-transposed of |q. Here is a ﬁrst opportunity to show

how Dirac notation comes in handy. Indeed, deﬁne the “bra” q| as representing the

17.1 Dirac notation 335

conjugate-transposed vector |q

. With this notation, Eq. (17.4) becomes the “bra-ket:”

q|q



=¯αα



ββ



(17.5)

And, hence, for the qubit modulus, q|q=|α|

+|β|

. As another property, we note

from Eq. (17.5) that q|q



=q



|q, namely that the inner product of qubit states is

noncommutative, unlike the scalar product of 2D vectors with real coordinates.

As we shall see next, there is more in the “bra” notation than just a convenient way to

express the inner product between two qubit states. Indeed, consider n-qubits |q, |q



in

the space V

, with, for |q



, the expansion in Eq. (17.2) and, for the bra, q| (consistently

with deﬁnition):

q|=



x∈V

xx|. (17.6)

We can now develop the inner product q|q



 according to

q|q



=



x∈V

xx|





∈V







x∈V





∈V



x|x



. (17.7)

Since |x, |x



 are pure states from the orthonormal basis V

,wehavex|x



=δ



where δ



is the Kronecker symbol (δ



= 1forx = x



and δ



= 0 otherwise). Hence

the explicit deﬁnition of inner product when the two qubits are expressed in the same

basis V

q|q



=



x∈V





∈V



. (17.7)

Which, for |q



=|q, reduces to the modulus:

q|q=



x∈V

|x|

. (17.8)

In particular, effecting a left product by x| to Eq. (17.2) we have

x|q=





∈V

xx|x



=





∈V

xδ



= x, (17.9)

which shows that, expectedly, x|q=x is the projection of |q over the pure state |x.

Substituting this result into the deﬁnition in Eq. (17.2), we obtain

|q=



x∈V

x|x=



x∈V

x|q|x≡



x∈V

|xx|q, (17.10)

which represents another way of expanding |q over the basis V

In Chapter 16, we have shown that it is generally not possible to “clone” quantum

states, a property referred to as the noncloning theorem. As was mentioned without

336 Quantum measurements

proof, the only exceptions to this theorem concern any pure state, or any pair of pure

states. With the inner product introduced above, it is quite easy to show this.

Projection operators

In view of the expansion of |q in Eq. (17.10), we can introduce U

=|xx| as the

operator projecting any qubit onto the state |x. To show this, we ﬁrst observe that

|x=(|xx|)|x=|xx|x=|xδ

≡|x (17.11)

and

|q=(|xx|)





∈V









∈V



|xx|x



=|x





∈V



≡ x|x,

(17.12)

which shows that |x is invariant by U

and also that |q is projected on its basis

component x|x. With such a deﬁnition of projector operator, we also have from

Eq. (17.10):

|q=



x∈V

|xx|

|q, (17.13)

from which we obtain



x∈V

|xx|=I. (17.14)

The above property, which is referred to as the completeness (or closure) relation,

expresses the fact that the complete sum of all projection operators over the pure states

of V

is the identity operator. More generally, a unitary operator U is a projector if it

satisﬁes the property U

= U (projecting twice is the same as projecting once). In the

case of U

=|xx|, we have, indeed,

= U

= (|xx|)(|xx|)

=|xx|xx|=|xδ

x|=|xx|≡U

(17.15)

Referring to Section 16.6, given three pure states |s, |ψ, |φ, we assume a unitary operator U capable to

achieve the following cloning transformations:

U(|ψ⊗|x) =|ψ⊗|ψ,

U(|φ⊗|s) =|φ⊗|φ.

Taking the inner product of the two right-hand sides and using the properties U

U = I and s|s=1,

yields:

(φ|⊗s|)U

U(|ψ⊗|x) = (φ|⊗φ|)(|ψ⊗|ψ) ↔φ|ψs|s

=φ|ψφ|ψ↔φ|ψ=φ|ψ

which shows that either φ|ψ=1orφ|ψ=0, corresponding to |φ=|ψ  or |φ, |ψ being orthogonal

states.