(1) Denote the polarization basis for the kn -mode (n = 1, 2) by {eh (kn ) , ev (kn )}.

For a rotation of each basis around kn by the angle θ, i.e.

eh (kn ) = cos θ eh (kn ) + sin θ ev (kn ) ,

ev (kn ) = ’ sin θ eh (kn ) + cos θ ev (kn ) ,

derive the corresponding transformation of the creation operators a† n h , a† n v and

k k

show that the Hamiltonian in eqn (20.33) has the same form in the new basis.

(2) The (2, 0)-events in which two photons are present in the k1 v-mode are counted

by letting the output fall on a beam splitter with detectors at each output port.

A coincidence count shows that two photons were present. For an ideal balanced

beam splitter and 100% detectors, show that the probability of a coincidence count

is 1/2. Use this to explain the discrepancy between eqn (20.42) and the baseline

data in Fig. 20.3.

20.8 Wave plates

A polarization-dependent retarder plate (wave plate) is made from an anisotropic

crystal, with indices of refraction nF and nS for light polarized along the fast axis eF

and the slow axis eS respectively (Saleh and Teich, 1991, Sec. 6.1-B).

Consider a classical ¬eld with amplitude E = Eh eh + Ev ev propagating in the

z-direction, that falls on a retarder plate of thickness ∆z lying in the (x, y)-plane.

½

Exercises

(1) By discarding an overall phase factor show that the output ¬eld E = Eh eh + Ev ev

is related to the input ¬eld by col (Eh ,Ev ) = Tξ (‘) col (Eh ,Ev ), where the Jones

matrix Tξ (‘) is given by

’ sin ‘ cos ‘ 1 ’ eiξ

cos2 ‘ + sin2 ‘eiξ

Tξ (‘) = ,

’ sin ‘ cos ‘ 1 ’ eiξ sin2 ‘ + cos2 ‘eiξ

‘ is the angle between eh and eF , and ξ = (nS ’ nF ) ω∆z/c.

(2) Evaluate the Jones matrix for ξ = π/2 (the quarter-wave plate) and ξ = π (the

half-wave plate).

(3) For ‘ = 0 and a 45—¦ -polarized input, i.e. Eh = Ev , what is the output polarization

state? Answer the same question if ‘ = π/4 and the input ¬eld is h-polarized.

20.9 Quantum dense coding

The unitary operators used by Bob for quantum dense coding are de¬ned by

U ‘»/4 , ‘»/2 = Tπ/2 ‘»/4 Tπ ‘»/2 , where Tξ (‘) is given by the result of the previ-

ous exercise. As explained in the text, this operator only acts on the second argument

of |sA , sB .

(1) For the general state

|˜ = chh |hA , hB + chv |hA , vB + cvh |vA , hB + cvv |vA , vB

determine the expansion coe¬cients for which U (0, 0) |˜ = |Ψ+ .

(2) Find three other sets of values ‘»/4 , ‘»/2 such that U ‘»/4 , ‘»/2 |˜ is equal

(up to a phase factor) to the remaining Bell states.

20.10 Bell states incident on a balanced beam splitter

For the Bell states in eqns (20.52) and (20.53) use the method described in Section

8.4.1 to show that the scattered states produced by a balanced beam splitter are

Ψ’ = Ψ’ ,

1

= √ |hA , vA + (A ” B) ,

Ψ+

2

i

¦± = {|hA , hA ± |vA , vA } + (A ” B) .

2

20.11 Rotated polarization basis

Consider the 45—¦ -rotated polarization basis de¬ned by eqn (20.48).

(1) Derive

√ √

a† = a † ’ a † / 2 , a † = a † + a † / 2 ,

γv

γv γv

γh γh γh

where γ ∈ {A, B}.

¾ Quantum information

(2) Show that

1 1

’ √ hA , v A ,

|hA , hA = hA , hA + |v A , v A

2 2

1 1

+ √ hA , v A .

|vA , vA hA , hA + |v A , v A

=

2 2

(3) Starting with eqn (20.56), derive eqns (20.57) and (20.58).

20.12 Insu¬cient information

Consider Alice™s attempt to give Bob instructions for making an approximate copy of

her unknown qubit |γ .

(1) Given the unit vector n and the eigenvalue of n · σ, explain why Bob™s best

estimate for the unknown state |γ is given by eqn (20.62).

(2) Why cannot Alice get more information for Bob by making further measurements?

(3) Suppose that the sender of Alice™s qubit, who does know the state |γ , is willing

to send her an endless stream of qubits, all prepared in the same state. Alice™s

research budget, however, limits her to a ¬nite number of measurements. Can

Alice supply Bob with enough information to permit an exact reproduction (up

to an overall phase) of |γ ?

20.13 Teleportation of qubits

(1) Express the basis states |u, v AT (u, v = 0, 1) as linear combinations of the Bell

states, and then derive eqn (20.66).

(2) Show that the Pauli matrices are unitary as well as hermitian, and use this fact

to construct unitary operators for the phase-¬‚ip and the bit-¬‚ip.

(3) Suppose that Alice does her Bell state measurement, but that Eve intercepts the

message to Bob. Calculate the reduced density operator ρB that Bob must use in

this circumstance, and comment on the result.

(4) Now suppose that Alice misunderstands the theory, and thinks that she should

make a measurement that projects onto the basis vectors |u, v AT . After Alice

tells Bob which of the four possibilities occurred, what information does Bob have

about his qubit?

20.14 Teleportation of photons

Consider the application of the teleportation protocol to photons.

(1) Write out the explicit expressions for the Bell states in the A“T subsystem.

(2) Derive the photonic version of eqn (20.66).

(3) Give explicit forms for the action of the unitary transformations Upf (phase-¬‚ip)

and Ubf (bit-¬‚ip) on the creation operators.

¿

Exercises

Quantum logic gates—

20.15

(1) Show that the X, Z, and Hadamard gates are unitary operators.

(2) Use the representation |γ = γ0 |0 + γ1 |1 of a general qubit to express all three

gates as 2 — 2 matrices. Explain the names for the X and Z gates by relating them

to Pauli matrices.

(3) For a spin-1/2 particle, the operator for a rotation through the angle ± around the

axis directed along the unit vector u is (Bransden and Joachain, 1989, Sec. 6.9)

±

±

’ i sin u·σ.

Ru (±) = cos

2 2

Combine this with the Poincar´-sphere representation

e

θ θ

|γ = cos |0 + eiφ sin |1

2 2

for qubits to show that the X, Z, and Hadamard gates are respectively given by

iRux (π), iRuz (π), and iRh (π), where ux , uy , uz are the coordinate unit vectors

√ √

and h = ux / 2 + uz / 2.

(4) Show that the control-NOT operator CNOT , de¬ned by eqn (20.72), is unitary.

Use the basis {|0, 0 , |0, 1 , |1, 0 , |1, 1 } to express CNOT as a 4 — 4 matrix.

Single-photon gates—

20.16

Identify the polarization states of a single photon with the logical states by |h ” |0

and |v ” |1 . Use the results of Exercise 20.8 to show that the Z and Hadamard gates

can be realized by means of half-wave plates.

Quantum circuits—

20.17

Work out the gates required for the outcomes |¦’ and |Ψ’ in the computation

discussed in Section 20.5.1-A and draw the corresponding quantum circuit diagrams.

Controlled-NOT gate—

20.18

For the nondeterministic control-NOT gate sketched in Section 20.5.3, use the notation

aCh , aCv , aT h , aT v for the control and target modes and b1h , b2h for h-polarized

vacuum ¬‚uctuations in the Vac-1 and Vac-2 channels. Devise a suitable notation for

the operators associated with the internal lines in Fig. 20.9, and carry out the following

steps.

(1) Write out the scattering relations for each of the optical elements in the gate. For

this purpose it is useful to impose a consistent convention for assigning the ± s

√

to the asymmetric beam splitters, e.g. assign ’ R for re¬‚ection from the lower

surface of a beam splitter.

(2) Explain why the vacuum v-polarizations b1v , b2v can be omitted.

(3) Use the scattering relations to eliminate the internal variables and thus ¬nd the

overall scattering relations (aCh , aCv , . . .) ’ (aCh , aCv , . . .) which de¬ne the ele-

ments of the scattering matrix for the gate.

Quantum information

(4) Employ the general result (8.40) to determine the action of the gate on each input

state in the coincidence basis, and thus show that

1 1

|hC , hT ’ ’ 3 ) R |hC , hT ’ 3 ) R |hC , vT + ··· ,

( ( +

1 2 1 2

2 2

1 1

|hC , vT ’ ’ 3 ) R |hC , hT

1 ( 2 ’ 3 ) R |hC , vT + · · · ,

1( 2 + +

2 2

1 1

|vC , hT ’ [(2 ’ 2 3 ) R ’ 1] |vC , hT + [1 ’ (2 + 2 3 ) R] |hC , vT + · · · ,

2 2

1 1

|vC , vT ’ [1 ’ (2 + 2 3 ) R] |vC , hT + [(2 ’ 2 3 ) R ’ 1] |hC , vT + · · · .

2 2

Determine the value of R and the assignment of the s needed to de¬ne a control-

NOT gate.

Appendix A

Mathematics

A.1 Vector analysis

Our conventions for elementary vector analysis are as follows. The unit vectors cor-

responding to the Cartesian coordinates x, y, z are ux , uy , uz . For a general vector

v, we denote the unit vector in the direction of v by v = v/ |v|.

The scalar product of two vectors is a · b = ax bx + ay by + az bz , or

3

a·b= ai b i , (A.1)

i=1

where (a1 , a2 , a3 ) = (ax , ay , az ), etc. Since expressions like this occur frequently, we will

use the Einstein summation convention: repeated vector indices are to be summed

over; that is, the expression ai bi is understood to imply the sum in eqn (A.1). The

summation convention will only be employed for three-dimensional vector indices. The

cross product is

(a — b)i = ijk aj bk , (A.2)

where the alternating tensor is de¬ned by

ijk

§

⎪1 ijk is an even permutation of 123 ,

⎨

= ’1 ijk is an odd permutation of 123 , (A.3)

ijk

⎪

©

0 otherwise .

A.2 General vector spaces

A complex vector space is a set H on which the following two operations are de¬ned.

(1) Multiplication by scalars. For every pair (±, ψ), where ± is a scalar, i.e. a complex

number, and ψ ∈ H, there is a unique element of H that is denoted by ±ψ.

(2) Vector addition. For every pair ψ, φ of vectors in H there is a unique element of H

denoted by ψ + φ.

The two operations satisfy (a) ±(βψ) = (±β) ψ, and (b) ± (ψ + φ) = ±ψ + ±φ. It

is assumed that there is a special null vector, usually denoted by 0, such that ±0 = 0

and ψ + 0 = ψ. If the scalars are restricted to real numbers these conditions de¬ne a

real vector space.

Mathematics

Ordinary displacement vectors, r, belong to a real vector space denoted by R3 . The

set Cn of n-tuplets ψ = (ψ1 , . . . , ψn ), where each component ψi is a complex number,

de¬nes a complex vector space with component-wise operations:

±ψ = (±ψ1 , . . . , ±ψn ) ,

(A.4)

ψ + φ = (ψ1 + φ1 , . . . , ψn + φn ) .

Each vector in R3 or Cn is speci¬ed by a ¬nite number of components, so these spaces

are said to be ¬nite dimensional.

The set of complex functions, C (R), of a single real variable de¬nes a vector space

with point-wise operations:

(±ψ) (x) = ±ψ (x) , (A.5)

(ψ + φ) (x) = ψ (x) + φ (x) , (A.6)

where ± is a scalar, and ψ (x) and φ (x) are members of C (R). This space is said to

be in¬nite dimensional, since a general function is not determined by any ¬nite set

of values.

For any subset U ‚ H, the set of all linear combinations of vectors in U is called

the span of U, written as span (U). A family B ‚ H is a basis for H if H = span (B),

i.e. every vector in H can be expressed as a linear combination of vectors in B. In this

situation H is said to be spanned by B.

A linear operator is a rule that assigns a new vector M ψ to each vector ψ ∈ H,

such that

M (±ψ + βφ) = ±M ψ + βM φ (A.7)

for any pair of vectors ψ and φ, and any scalars ± and β. The action of a linear operator

M on H is completely determined by its action on the vectors of a basis B.

A.3 Hilbert spaces

A.3.1 De¬nition

An inner product on a vector space H is a rule that assigns a complex num-

ber, denoted by (φ, ψ), to every pair of elements φ and ψ ∈ H, with the following

properties:

(φ, ±ψ + βχ) = ± (φ, ψ) + β (φ, χ) , (A.8a)

—

(φ, ψ) = (ψ, φ) , (A.8b)

(φ, φ) < ∞ ,

0 (A.8c)

(φ, φ) = 0 if and only if φ = 0 . (A.8d)

An inner product space is a vector space equipped with an inner product. The

inner product satis¬es the Cauchy“Schwarz inequality:

2

|(φ, ψ)| (φ, φ) (ψ, ψ) . (A.9)

Two vectors are orthogonal if (φ, ψ) = 0. If F is a subspace of H, then the orthogonal

complement of F is the subspace F⊥ of vectors orthogonal to every vector in F.

Hilbert spaces

The norm ψ of ψ is de¬ned as ψ = (ψ, ψ), so that ψ = 0 implies ψ = 0.

Vectors with ψ = 1 are said to be normalized. A set of vectors is complete if

the only vector orthogonal to every vector in the set is the null vector. Each complete

set contains a basis for the space. A vector space with a countable basis set, B =

φ(1) , φ(2) , . . . , is said to be separable. The vector spaces relevant to quantum theory

are all separable. A basis for which φ(n) , φ(m) = δnm holds is called orthonormal.

Every vector in H can be uniquely expanded in an orthonormal basis, e.g.

∞

ψn φ(n) ,

ψ= (A.10)

n=1

where the expansion coe¬cients are ψn = φ(n) , ψ .

A sequence ψ 1 , ψ 2 , . . . , ψ k , . . . of vectors in H is convergent if

ψ k ’ ψ j ’ 0 as k, j ’ ∞ . (A.11)

A vector ψ is a limit of the sequence if

ψ k ’ ψ ’ 0 as k ’ ∞ . (A.12)

A Hilbert space is an inner product space that contains the limits of all convergent

sequences.

A.3.2 Examples

The ¬nite-dimensional spaces R3 and CN are both Hilbert spaces. The inner product

for R3 is the familiar dot product, and for CN it is

N

—

(ψ, φ) = ψn φn . (A.13)

n=1

If we constrain the complex functions ψ (x) by the normalizability condition

∞

dx |ψ (x)|2 < ∞ , (A.14)

’∞

then the Cauchy“Schwarz inequality for integrals,

∞ ∞ ∞

2

— 2 2

dx |ψ (x)| dx |φ (x)| ,

dxψ (x) φ (x) (A.15)

’∞ ’∞ ’∞

is su¬cient to guarantee that the inner product de¬ned by

∞

dxψ — (x) φ (x)

(ψ, φ) = (A.16)

’∞

makes the vector space of complex functions into a Hilbert space, which is called

L2 (R).

Mathematics

A.3.3 Linear operators

Let A be a linear operator acting on H; then the domain of A, called D (A), is the

subspace of vectors ψ ∈ H such that Aψ < ∞. An operator A is positive de¬nite

0 for all ψ ∈ D (A), and it is bounded if Aψ < b ψ , where b is a

if (ψ, Aψ)

constant independent of ψ. The norm of an operator is de¬ned by

Aψ

A = max for ψ = 0 , (A.17)

ψ

so a bounded operator is one with ¬nite norm.

If Aψ = »ψ, where » is a complex number and ψ is a vector in the Hilbert space,

then » is an eigenvalue and ψ is an eigenvector of A. In this case » is said to belong

to the point spectrum of A. The eigenvalue » is nondegenerate if the eigenvector

ψ is unique (up to a multiplicative factor). If ψ is not unique, then » is degenerate.

The linearly-independent solutions of Aψ = »ψ form a subspace called the eigenspace

for », and the dimension of the eigenspace is the degree of degeneracy for ». The

continuous spectrum of A is the set of complex numbers » such that: (1) » is not

an eigenvalue, and (2) the operator » ’ A does not have an inverse.

The adjoint (hermitian conjugate) A† of A is de¬ned by

—

ψ, A† φ = (φ, Aψ) , (A.18)

and A is self-adjoint (hermitian) if D A† = D (A) and (φ, Aψ) = (Aφ, ψ). Bounded

self-adjoint operators have real eigenvalues and a complete orthonormal set of eigen-

vectors. For unbounded self-adjoint operators, the point and continuous spectra are

subsets of the real numbers. Note that ψ, A† Aψ = (φ, φ), where φ = Aψ, so that

ψ, A† Aψ 0, (A.19)

i.e. A† A is positive de¬nite.

A self-adjoint operator, P , satisfying

P2 = P (A.20)

is called a projection operator; it has only a point spectrum consisting of {0, 1}.

Consider the set of vectors P H, consisting of all vectors of the form P ψ as ψ ranges

over H. This is a subspace of H, since

±P φ + βP χ = P (±φ + βχ) (A.21)

shows that every linear combination of vectors in P H is also in P H. Conversely, let S

be a subspace of H and φ(n) an orthonormal basis for S. The operator P , de¬ned

by

φ(n) , ψ φ(n) ,

Pψ = (A.22)

n

is a projection operator, since

P 2ψ = φ(n) , ψ P φ(n) = φ(n) , ψ φ(n) = P ψ . (A.23)

n n

Thus there is a one-to-one correspondence between projection operators and subspaces

of H. Let P and Q be projection operators and suppose that the vectors in P H are

Hilbert spaces

orthogonal to the vectors in QH; then P Q = QP = 0 and P and Q are said to be

orthogonal projections. In the extreme case that S = H, the expansion (A.10)

shows that P is the identity operator, P ψ = ψ.

A self-adjoint operator with pure point spectrum {»1 , »2 , . . .} has the spectral

resolution

A= »n Pn , (A.24)

n

where Pn is the projection operator onto the subspace of eigenvectors with eigenvalue

»n . The spectral resolution for a self-adjoint operator A with a continuous spectrum

is

A = » dµ (») , (A.25)