Effects of noise in continuous variables quantum communication and measurement

vi CONTENTS
collectively known as decoherence. Though noise and dissipation are unavoidable, many strategies
to suppress their effects have been proposed: for example a quantum counterpart of the classical
protocols of error correction does exist and manages, in principle, to allow Bob for a faithful
recovery of the damaged information. The common feature of such protocols is the redundancy
of the sent information and the identification of decoherence free subspaces. In turn, this is their
weakness, because our ability to produce the high dimensional signals needed is very limited.
In the communication process a key role is played by the protocol chosen to actually send the
information: the two main strategies are direct transmission and quantum teleportation : the main
difference between the two is that the former has a classical counterpart and couples directly the
signal state to the environment, while the latter, which is purely quantum mechanical, makes use
of an intermediate entangled state to accomplish communication. In this work, we are mainly
interested in the effect of noise in the process of quantum teleportation. We start by analyzing
some mathematical techniques useful in quantum optics, when dealing with continuous variables
systems, in Chapter 1. We proceed by studying the evolution of a quantum system coupled to an
environment in Chapter 2, where we present the derivation of the Master equation describing the
approach to equilibrium in presence of the most general Gaussian environment. The derivation is
obtained in two completely different ways the first, more standard, is based on some reasonable
approximations on the properties of the environment; the other makes use of the path integral
tool to write an exact form of the Master equation that, upon imposing the aforementioned ap-
proximations, reduces to the first. In Chapter 3, after introducing the entangled source used in
continuous variable teleportation, we review the quantum teleportation protocol both in the Fock
basis and using the Wigner function formalism. Then, by utilizing the Master equation, we study
the evolution of the twin beam propagating a a squeezed-thermal channel in Chapter 4, where we
find an explicit mathematical expression for the separability time as a function of the environment’s
parameters. In doing this we make use of the positive partial transposition (PPT) condition for
separability applied to the continuous variables Gaussian states, that we review at the beginning
of the Chapter. The results are then used in Chapter 5 to study the performances of quantum tele-
portation when the environment is a squeezed-thermal one. It turns out that there exist a class of
squeezed states, whose exact form depends on the environment, that maximizes the teleportation
fidelity, i.e. improves the performances of the protocol. To see to what extent the improvement
is better than directly transmitting the optimized squeezed state, we compare the fidelity of the
two protocols and find the regimes in which one is better than the other. Finally, in Chapter 6,
we switch to the problem of characterizing the quantum state, at least partially. We analyze a
method for reconstructing the photon statistics by the utilization of on/off photodetection alone,
performed at different values of the detector’s quantum efficiency. At the beginning we allow for
many different values of the quantum efficiency, ranging from zero to nearly unity, to be present,
and find that the photon statistics can be retrieved by making use of the Maximum-likelihood
method, in couple with the Expectation-Maximization algorithm. Being the last one an iterative
procedure, we study in detail, by numerically simulated experiments, its convergence properties
when the experimental parameters are varied and find that it manages to yield sound results,
CONTENTS vii
provided that the experimental data are good enough. Then we proceed allowing the values of
quantum efficiency to be few and low, thus representing a more realistic situation. We find that
reliable photon statistics can be retrieved, provided that we slightly modify the previous method,
and add the Maximum Entropy principle to fill in the blanks of our ignorance.
The results of my research during the PhD studies, have been published in the following articles:
• S. Olivares, M. G. A. Paris and A. R. Rossi, Optimized teleportation in Gaussian noisy
channels, Phys. Lett. A 319, 32 (2003).
• A. R. Rossi, S. Olivares and M. G. A. Paris, Degradation of continuous variable entanglement
in a phase-sensitive environment, J. Mod. Opt. vol. 51, 1057 (2004).
• A. R. Rossi, S. Olivares and M. G. A. Paris, Photon statistics without counting photons,
Phys. Rev. A 70, 055801 (2004).
• A. R. Rossi and M. G. A. Paris, A two-step MaxLik-MaxEnt strategy to infer photon distri-
bution from on/off measurement at low quantum efficiency, Eur. Phys. J. D., in press.
• A. R. Rossi and M. G. A. Paris, About distillability of depolarized states, J. Mod. Opt. vol.
51, 1037 (2004).
Chapter 1
Quantum optics for quantum
information and communication
In quantum information and communication, a set of quantum states ρ
i
is used to encode the
information and then is sent toward the receiver through a quantum noisy channel. In the fol-
lowing we will always assume that the ρ
i
’s are chosen between the states of the electromagnetic
field. There are a number of reasons for this. First of all, photons are easily produced, both in
coherent and incoherent beams, by lasers and thermal sources while, exploiting the properties of
nonlinear crystals, it is possible to achieve the production of entangled photons, thus solving the
problem of having the entangled resource needed to perform the tasks outlined in the introduction.
Second, photons are easily manipulated: beam splitters, high reflectivity mirrors and rotating wave
plates are well within today’s technology reach. The third, and most important property, is that
photons does not interact among themselves, so that we are guaranteed that, if we produce a state
with defined characteristics, and manipulate it in a carefully unitary way, it will retain all of its
properties, as long as it propagates in a not too “aggressive” environment. The last requirement,
moreover, is usually quite well satisfied in the optical region because, in standard conditions, the
average number of thermal photons is very small.
In this Chapter we give a review of the mathematical methods and detection schemes used in
quantum optics. We review the notion of Wigner function, i.e. the map between the operators used
to describe quantum object, and the set of quasi-probability functions on the complex plane. This
map is of fundamental importance to reduce the almost always unmanageable operator equations,
needed to study the evolution of a system coupled to an environment, to more amenable partial
differential equations, that usually take the form of a Fokker-Planck equation.
As stated in the introduction, the possibility for Bob to precisely measure a received quantum
state, is fundamental to establish reliable quantum communication channels. This is due to the
fact that the set of quantum states used by Alice to encode the information to be sent, need
not to be necessarily a set of orthogonal states and, even if they are, the interaction with the
environment, though possibly reduced to a minimum, can destroy their orthogonality. From this
fact arises the need for Bob, to discriminate in a reliable way between the states he gets. A
3
4 Chapter 1. Quantum optics for quantum information and communication
first theoretical step in this direction is the extension of the projection-valued measures (PVMs),
introduced by Von Neumann, to the positive operator-valued measures (POVMs); this is the more
general theoretical tool to identify the received state but, as exemplified in Section 1.2.2, could
not be always conclusive. Of course the theoretical tool must match a practical implementation
of the POVM, so we analyze in detail, in Section 1.3, three different kind measurement, that is
photodetection, homodyne detection and heterodyne detection, suitable to gain different pieces of
information about the quantum system under investigation.
1.1 The Wigner function
In this Section we review some simple formulas that connect the generalized Wigner functions for
s-ordering with the density matrix, and vice-versa. These formulas prove very useful for quan-
tum mechanical applications as, for example, for connecting Master equations with Fokker-Planck
equations, or for evaluating the quantum state from Monte Carlo simulations of Fokker-Planck
equations, and finally for studying positivity of the generalized Wigner functions in the complex
plane.
Since Wigner’s pioneering work [4], generalized phase-space techniques have proved very useful
in various branches of physics [5]. As a method to express the density operator in terms of c-
number functions, the Wigner functions often lead to considerable simplification of the quantum
equations of motion, as for example, for transforming Master equations in operator form into
manageable Fokker-Planck differential equations (see, for example, reference [6]). Using the Wigner
function one can express quantum-mechanical expectation values in form of averages over the
complex plane (the classical phase-space), the Wigner function playing the role of a c-number
quasi-probability distribution, which generally can also have negative values. More precisely, the
original Wigner function allows to easily evaluate expectations of symmetrically ordered products of
the field operators, corresponding to the Weyl’s quantization procedure [7]. However, with a slight
change of the original definition, one defines generalized s-ordered Wigner function W
s
(α, α
∗
), as
follows [8]
W
s
(α, α
∗
)=
∫
C
d
2
λ
pi
2
e
αλ
∗
−α
∗
λ+
1
2
s|λ|
2
Tr{D(λ)ρ} , (1.1)
where α
∗
denotes the complex conjugate of α, the integral is performed on the complex plane with
measure d
2
λ = d
e[λ] dm[λ], ρ being the density operator, and D(α) ≡ exp{αa
†
−α
∗
a} denotes
the displacement operator, where a and a
†
([a, a
†
] = 1) are the annihilation and creation operators
of the field mode of interest, respectively. The Wigner functions in equation (1.1) allow to evaluate
s-ordered expectation values of the field operators through the following relation
Tr{:(a
†
)
n
a
m
:
s
ρ} =
∫
C
d
2
αW
s
(α, α
∗
)α
∗n
α
m
, (1.2)
where s is a real number. The particular cases s = −1, 0, 1 correspond to anti-normal, symmetrical,
and normal ordering, respectively. In these cases the generalized Wigner function W
s
(α, α
∗
) are
1.1. The Wigner function 5
usually denoted by the following symbols and names
1
pi
Q(α, α
∗
) for s = −1(Q-function);
W (α, α
∗
) for s = 0 (usual Wigner function);
P (α, α
∗
) for s =1(P -function).
(1.3)
For the normal (s = 1) and anti-normal (s = −1) orderings, the following simple relations with
the density matrix are well known
Q(α, α
∗
)≡〈α|ρ|α〉 , (1.4a)
ρ =
∫
C
d
2
αP(α, α
∗
) |α〉〈α| , (1.4b)
|α〉 being a coherent state. Among the three particular representations (1.3), the Q-function is
positively definite and infinitely differentiable (it actually represents the probability distribution
for ideal joint measurements of position and momentum of the harmonic oscillator). On the other
hand, the P -function is known to be possibly highly singular and the only pure states for which
it is positive are the coherent states [9]. Finally, the s = 0 Wigner function has the remarkable
property of providing the probability distribution of the quadratures of the field in form of marginal
distribution, namely
∫
∞
−∞
dm[α]W (αe
iϕ
,α
∗
e
−iϕ
)=
ϕ
〈
e[α]| ρ |
e[α]〉
ϕ
, (1.5)
where |x〉
ϕ
denotes the (unnormalizable) eigenstate of the field quadrature
X
ϕ
=
a
†
e
iϕ
+ ae
−iϕ
2
(1.6)
with real eigenvalue x. Notice that any couple of quadratures X
ϕ
, X
ϕ+pi/2
is canonically conjugate,
namely [X
ϕ
,X
ϕ+pi/2
]=i/2, and it is equivalent to position and momentum of a harmonic oscillator.
Usually, negative values of the Wigner function are viewed as a signature of a non-classical state, the
most eloquent example being the Schro¨dinger-cat state [10], whose Wigner function is characterized
by rapid oscillations around the origin of the complex plane. From equation (1.1) one can notice
that all s-ordered Wigner functions are related to each other through Gaussian convolution
W
s
(α, α
∗
)=
∫
C
d
2
βW
s
′
(β,β
∗
)
2
pi(s
′
− s)
exp
{
−
2|α− β|
2
s
′
− s
}
(1.7)
= exp
{
s
′
− s
2
∂
2
∂α∂α
∗
}
W
s
′
(α, α
∗
) , (s
′
>s) . (1.8)
Equation (1.7) shows the positivity of the generalized Wigner function for s≤−1, as a consequence
of the positivity of the Q function. From a qualitative point of view, the maximum value of
s, keeping the generalized Wigner functions as positive, can be considered as a measure of the
nonclassicality of the physical state [11].
An equivalent expression for W
s
(α, α
∗
) can be derived as follows. We rewrite equation (1.1) as
W
s
(α, α
∗
)=Tr{ρD(α)
ˆ
W
s
D
†
(α)} , (1.9)
6 Chapter 1. Quantum optics for quantum information and communication
where
ˆ
W
s
=
∫
C
d
2
λ
pi
2
e
1
2
s|λ|
2
D(λ) . (1.10)
Through the customary Baker-Campbell-Hausdorff (C.2) formula
e
A
e
B
= exp
{
A + B +
1
2
[A, B]
}
, (1.11)
which holds when [A, [A, B]] = [B, [A, B]] = 0, one writes the displacement in normal order, and,
integrating with respect to arg[λ] and |λ|, one obtains
ˆ
W
s
=
2
pi(1− s)
∞
∑
n=0
1
n!
(
2
s− 1
)
n
(a
†
)
n
a
n
=
2
pi(1− s)
(
s +1
s− 1
)
a
†
a
, (1.12)
where we used the normal-ordered forms
:(a
†
a)
n
:=(a
†
)
n
a
n
= a
†
a(a
†
a− 1) ...(a
†
a− n +1), (1.13)
and the identity
:e
−xa
†
a
: =
∞
∑
l=0
(−x)
l
l!
(a
†
)
l
a
l
=(1− x)
a
†
a
. (1.14)
The density matrix can be recovered from the generalized Wigner functions using the following
expression
ρ =
2
1+s
∫
C
d
2
αW
s
(α, α
∗
)e
−
2
1+s
|α|
2
e
2α
1+s
a
†
(
s− 1
s +1
)
a
†
a
e
2α
∗
1+s
a
. (1.15)
For the proof of equation (1.15) the reader is referred to reference [12]. In particular, for s = 0 one
has the inverse formula
ρ =2
∫
C
d
2
αW(α, α
∗
)D(2α)(−)
a
†
a
, (1.16)
whereas for s = 1 one recovers equation (1.4b) that defines the P -function.
Given the Wigner function (1.1) with s = 0, it is now possible to show [6] that, in a Master
equation such as (2.41), the action of mode operators a and a
†
on the density matrix ρ corresponds
to partial derivatives and multiplication by α, α
∗
:
aρ ←→ (α +
1
2
∂
α
∗
)W (α, α
∗
) (1.17a)
a
†
ρ ←→ (α
∗
−
1
2
∂
α
)W (α, α
∗
) (1.17b)
ρa ←→ (α−
1
2
∂
α
∗
)W (α, α
∗
) (1.17c)
ρa
†
←→ (α
∗
+
1
2
∂
α
)W (α, α
∗
) (1.17d)
allowing to reduce the operatorial relations involved in the Master equations to partial differential
equations. This will be exploited in Chapter 2
1.2. Positive operator-valued measure (POVM) 7
1.2 Positive operator-valued measure (POVM)
1.2.1 Formal definition
Let X be a self-adjoint operator with a continuous spectrum x ∈ S ⊂ R, X|x〉 = x|x〉, {|x〉} being
a complete ortho-normal system (|〈x|x
′
〉|
2
= δ
x,x
′
). Using the spectral theorem we can write
X =
∫
S
dx x |x〉〈x| , (1.18)
where |x〉〈x|≡P (x) is the (orthogonal) projector onto the state |x〉.Ifρ is the state of a system
and X is considered as an observable, then the Born’s rule says that the probability density p(x)
of obtaining as outcome x after a measurement of X is given by
p(x) ≡ Tr{ρP(x)} . (1.19)
Moreover, we can define the map
x −→ P (x) (1.20)
which associates the projector P (x) to the eigenvalue x. The map is called projection-valued
measure (PVM).
Let us consider the map
x ∈ S
′
⊂ C −→ Π(x) , (1.21)
and define
p˜(x) ≡ Tr{ρΠ(x)} . (1.22)
The class of operators Π(x) such that p˜(x) is a probability distribution, are characterized by the
requirements
p˜(x) ≥ 0=⇒ Π(x) ≥ 0 , (1.23)
∫
S
′
dx p˜(x)=1=⇒
∫
S
′
dxΠ(x)=I . (1.24)
In this way, from the conditions on p˜(x), we obtain that Π(x) should be semi-definite positive
(equation (1.23)) and a resolution of the identity (equation (1.24)). A map fulfilling the conditions
(1.23) and (1.24) is called positive operator-valued measure (POVM).
POVMs are more general than the PVMs. For example the map α → Π(α) ≡
1
pi
|α〉〈α|, |α〉 being
a coherent state, fulfills (1.23) and (1.24) but Π(α) are not orthogonal projectors. Notice that Π(α)
is the POVM describing heterodyne detection (see Section 1.3) and Tr{ρΠ(α)} =
1
pi
〈α|ρ|α〉≡Q(α),
i.e. the Q-function.
1.2.2 An example: non-orthogonal photon polarizations
Let us consider two non-orthogonal states of photon polarization, namely
|ψ
1
〉 = |H〉 and |ψ
2
〉 =
1
2
(|H〉+ |V 〉) , (1.25)
8 Chapter 1. Quantum optics for quantum information and communication
where |H〉 and |V 〉 stand for horizontal and vertical polarization, respectively. A simple von
Neumann-type projective measurement cannot conclusively distinguish between these two states.
On the other hand, this task can be achieved using the following POVM
Π
1
≡
√
2
1+
√
2
|H〉〈H| , (1.26a)
Π
2
≡
√
2
1+
√
2
I−|H〉〈V |−|V 〉〈H|
2
, (1.26b)
Π
3
≡ I−Π
1
−Π
2
. (1.26c)
In fact, a measurement outcome Π
1
or Π
2
says that the state is |ψ
2
〉 or |ψ
1
〉, respectively; a
measurement outcome Π
3
does not determine the state. An implementation of this POVM was
proposed in reference [13].
This simple example explicitly shows the advantage of a POVM: it may allow the extraction
of more information with respect to the usual von Neumann-type projective measurement.
1.3 Detection of radiation
1.3.1 Photodetection
Light is revealed by exploiting its interaction with atoms/molecules or electrons in a solid, and,
essentially, each photon ionizes a single atom or promotes an electron to a conduction band, and the
resulting charge is then amplified to produce a measurable pulse. In practice, however, available
photodetectors are not ideally counting all photons, and their performances is limited by a non-unit
quantum efficiency ζ, namely only a fraction ζ of the incoming photons lead to an electric signal,
and ultimately to a count: some photons are either reflected from the surface of the detector, or
are absorbed without being transformed into electric pulses.
Let us consider a light beam entering a photodetector of quantum efficiency ζ, i.e. a detector
that transforms just a fraction ζ of the incoming light pulse into electric signal. If the detector is
small with respect to the coherence length of radiation and its window is open for a time interval
T , then the Poissonian process of counting gives a probability p(m;T ) of revealing m photons that
writes [14]
p(m;T)=Tr
[
ρ:
[ζI(T )T ]
m
m!
exp[−ζI(T )T ]:
]
, (1.27)
where ρ is the quantum state of light, ::denotes the normal ordering of field operators, and I(T )
is the beam intensity
I(T)=
2
0
c
T
∫
T
0
E
(−)
(r,t) ·E
(+)
(r,t)dt , (1.28)
given in terms of the positive (negative) frequency part of the electric field operator E
(+)
(r,t)
(E
(−)
(r,t)). The quantity p(t)=ζTr [ρI(T )] equals the probability of a single count during the
time interval (t, t+ dt). Let us now focus our attention to the case of the radiation field excited in
1.3. Detection of radiation 9
a stationary state of a single mode at frequency ω. Eq. (1.27) can be rewritten as
p
η
(m)=Tr
[
ρ :
(ηa
†
a)
m
m!
exp(−ηa
†
a):
]
, (1.29)
where the parameter η = ζc
ω/V denotes the overall quantum efficiency of the photodetector.
Using Eqs. (1.13) and (1.14) one obtains
p
η
(m)=
∞
∑
n=m
ρ
nn
(
n
m
)
η
m
(1− η)
n−m
, (1.30)
where
ρ
nn
≡〈n|ρ|n〉 = p
η=1
(n) . (1.31)
Hence, for unit quantum efficiency a photodetector measures the photon number distribution of
the state, whereas for non unit quantum efficiency the output distribution of counts is given by a
Bernoulli convolution of the ideal distribution.
The effects of non unit quantum efficiency on the statistics of a photodetector, i.e. Eq. (1.30)
for the output distribution, can be also described by means of a simple model in which the realistic
photodetector is replaced with an ideal photodetector preceded by a beam splitter of transmissivity
τ ≡ η. The reflected mode is absorbed, whereas the transmitted mode is photo-detected with unit
quantum efficiency. In order to obtain the probability of measuring m clicks, notice that, apart
from trivial phase changes, a beam splitter of transmissivity τ affects the unitary transformation
of fields
(
c
d
)
≡ U
†
τ
(
a
b
)
U
τ
=
(
√
τ −
√
1− τ
√
1− τ
√
τ
)
(
a
b
)
, (1.32)
where all field modes are considered at the same frequency. Hence, the output mode c hitting the
detector is given by the linear combination
c =
√
τa−
√
1− τb , (1.33)
and the probability of counts reads
p
τ
(m)=Tr
[
U
τ
ρ⊗|0〉〈0|U
†
τ
|m〉〈m|⊗1
]
=
∞
∑
n=m
ρ
nn
(
n
m
)
(1− τ)
n−m
τ
m
. (1.34)
Eq. (1.34) reproduces the probability distribution of Eq. (1.30) with τ = η. We conclude that
a photo-detector of quantum efficiency η is equivalent to a perfect photo-detector preceded by a
beam splitter of transmissivity η which accounts for the overall losses of the detection process.
When our detector is only able to tell us whether some photons are present or not, without
specifying how many photons are present, we call this measuring process on/off photodetection.
It is described by the POVM:
Π
off
(η)=(1− η)
a
†
a
=
∞
∑
n=0
(1− η)
n
|n〉〈n| (1.35a)
Π
on
(η)=I−Π
off
. (1.35b)