Modern tools for quantum technology : non-Gaussianity, local quantum estimation theory and their applications to quantum optical systems

Università degli Studi Roma TRE

e

Consorzio Nazionale Interuniversitario per le Scienze Fisiche

della Materia

Dottorato di Ricerca in Scienze Fisiche della Materia

XXIII ciclo

Modern tools for quantum technology: non-Gaussianity, local quantum estimation

theory and their applications to quantum optical systems

Tesi di dottorato del dott. Marco Giovanni Genoni

Relatore

Coordinatore Dottorato

Prof. Matteo Paris

Prof. Settimio Mobilio

Acknowledgments i

Introduction v

1 Preliminary concepts 1

1.1 Elements of quantum mechanics . . . 2

1.1.1 Discrete variable systems – qubits and qudits . . . 3

1.1.2 Density Operator . . . 4

1.1.3 Positive operator valued measure (POVM) . . . 8

1.1.4 Completely positive map . . . 10

1.2 Distances and measures of distinguishability for quantum states . . . . 12

1.2.1 Trace distance . . . 12

1.2.2 Hilbert-Schmidt distance . . . 14

1.2.3 Fidelity and Bures distance . . . 14

1.2.4 Quantum relative entropy . . . 16

1.3 Quantum entanglement in DV systems . . . 18

1.3.1 Quantifying entanglement . . . 19

1.4 Continuous-variable systems . . . 23

1.4.1 Symplectic transformations . . . 25

1.4.2 Two-mode mixing: the beam splitter . . . 27

1.4.3 Squeezing operators . . . 28

1.4.4 Displacement operator and coherent states . . . 34

1.4.5 Characteristic function and Wigner function . . . 37

1.5 CV Gaussian states . . . 41 I

1.5.1 Definition and some properties of Gaussian states . . . 41

1.5.2 Entanglement of Gaussian states . . . 49

1.5.3 Gaussian and non-Gaussian quantum channels . . . 52

1.5.4 Gaussian and non-Gaussian quantum measurements . . . 57

2 Non-Gaussianity in quantum information 63 2.1 Non-Gaussianity of a classical probability distribution . . . 66

2.1.1 Kurtosis . . . 67

2.1.2 Negentropy . . . 67

2.2 Quantum nonG measures: definitions and properties . . . 68

2.2.1 Measuring the non-Gaussianity using Hilbert-Schmidt distance . 69 2.2.2 Measuring the non-Gaussianity using quantum relative entropy 72 2.2.3 A measure of nonG based on the Wehrl entropy . . . 78

2.2.4 Non-Gaussianity of a quantum operation . . . 79

2.3 Non-Gaussianities of specific families of quantum states . . . 80

2.3.1 Fock states and superpositions . . . 80

2.3.2 Mixtures of Fock states . . . 81

2.3.3 Schr¨odinger cat states . . . 83

2.4 Gaussification and de-Gaussification processes . . . 84

2.4.1 Loss mechanism . . . 85

2.4.2 Phase-diffusion evolution . . . 87

2.4.3 Self-Kerr interaction . . . 88

2.5 NonG and distillation of entanglement . . . 90

2.6 NonG and quantum communication . . . 95

2.6.1 The Holevo Bound . . . 95

2.6.2 The quantum mutual information . . . 96

2.6.3 The quantum conditional entropy . . . 96

2.7 NonG and parameter estimation . . . 98

2.8 NonG and the quantum central limit theorem . . . 101

2.9 Experimentally friendly lower bounds to QRE nonG measure . . . 104

2.10 Experimental generation and characterization of non-Gaussian states . 109 2.10.1 Photon subtraction on classically correlated beams . . . 111

2.10.3 Photon addition on coherent states . . . 132

3 Quantum estimation for quantum technology 139 3.1 Local quantum estimation theory . . . 141

3.1.1 Estimability of a parameter . . . 145

3.1.2 Pure state model and unitary families . . . 146

3.1.3 Multiparametric models and reparametrization . . . 147

3.1.4 Geometry of quantum estimation . . . 149

3.2 Entanglement Estimation . . . 150

3.2.1 Two-qubit systems . . . 150

3.2.2 Two-qutrit bound entangled states . . . 155

3.2.3 Gaussian states . . . 157

3.3 Estimation of parameters in quantum optics . . . 163

3.3.1 Estimation of displacement via Gaussian states and Kerr interaction164 3.3.2 Estimation of squeezing via Gaussian states and Kerr interaction 166 3.3.3 Non-Gaussianity as an overall indicator of precision enhancement 168 3.3.4 Estimation of the environment temperature via Gaussian states . 169 3.4 Phase Estimation under phase-diffusion evolution - the qubit case . . . 174

3.4.1 Phase-shift estimation in qubit systems . . . 175

3.4.2 Phase-shift estimation by spin measurement . . . 180

3.4.3 Phase-shift estimation for polarization qubit . . . 183

3.4.4 Monte Carlo simulated experiments . . . 184

3.4.5 Experimental setup and results . . . 185

3.5 Phase Estimation under phase-diffusion evolution - the CV case . . . . 190

Conclusions and outlooks 199

A Uncertainty principle i

B Operator ordering theorems iii

First of all, I would like to thank my supervisor Matteo Paris. I consider myself very fortunate for having the opportunity to work with him: he has been an invaluable teacher and a true friend. I thank him for his suggestions, his encouragements and his trust throughout all these years.

I would like to thank also Stefano Olivares, for his friendship and his helpfulness: his door was always open to discuss both scientific and non-scientific problems.

A big thank goes also to the friends who shared with me the experience of taking a PhD in Milan, in particular Matteo Bina, Carmen Invernizzi and Berihu Teklu with which I had also the opportunity of working together, but also Eugenio Cinquanta, Vera Bernardoni, Davide Sangalli, Davide D’Elia, Renato Coretti, Margherita Ghezzi, Chiara Paganelli, Andrea Smirne and Sheik Sergio M’Baye. Thank you for the cofees and lunches we had together at the Bar di Fisica and at the Union Club during these years.

During the last year I visited for six months the Quantum Information Group at University College London. I’m grateful to Alessio Serafini for his hospitality, his friendship, for sharing with me his ideas and for showing me a different perspective on several problems related to my thesis. I would like to thank all the people of the group at UCL, in particular Guillermo Cordourier Maruri, Hannu Wichterich, Tommaso Tuffarelli, Paolo Barletta, Simone Severini, Hulya Yadsan-Appleby and Sai-Yun Ye. A thank goes also to Gerardo Adesso and Mauro Paternostro for inviting me during this period to visit the University of Nottingham and the Queen’s University Belfast respectively, and for the useful discussions we had in those days.

Beside the ones already mentioned above, I would like to thank all the people with which I had the opportunity to work throughout these years. In no particular order I would like to mention Paolo Giorda, Konrad Banaszek, Federico Casagrande, Alfredo

Lulli, Simone Cialdi, Davide Brivio, Stefano Vezzoli, Alessia Allevi, Federica Beduini, Maria Bondani, Alessandra Andreoni, Nicol`o Spagnolo, Marco Barbieri and all the Quantum Optics group led by Philippe Grangier at Universit`e Paris XI. I also thank my examiner Nicolas Cerf for his comments and suggestions for this final version of the thesis.

I would like to thank my parents and my brother for all their help and their example, it is because of them if I could decide to start this work.

Finally, I want to dedicate this thesis to my future wife Silvia: thank you for your love and support during all these years!

In any protocol aimed at manipulating or transmitting information, symbols are en-coded in states of some physical system. If this system is quantum mechanical rather than classical, as is bound to become common if the miniaturisation of processing units persists at the current rate, the laws of information processing are different than the ones governing ordinary classical devices. Quantum Information (QI) science investi-gates these laws [NC], and explores the novel, potentially revolutionary, possibilities offered by quantum mechanical systems across the whole spectrum of information tech-nologies. In the last two decades, we have witnessed the rise of QI science from the realm of theoretical conjecture to that of actual technological application, with the prominent example of operating quantum cryptographic systems [GRZ02, Pet al.09, GG02]. One of the strengths of QI is the versatility of its theoretical framework, which is applicable to a number of technological substrates. Quantum information has thus been encoded in the degrees of freedom of different physical systems: the first examples have been the polarisation of light and the excitation of a two-level atom, both abstractly described as qubits, that is quantum states living in a two-dimensional Hilbert space. Successively, discrete variable (DV) physical systems with higher dimensions (qudits) have been con-sidered and, more recently, great advances have been achieved for infinite-dimensional systems, the so-called continuous-variable (CV) systems, such as light modes or mo-tional degrees of freedom of trapped particles. The aim of this thesis is to characterize at a quantum level physical systems and operations as a resources for quantum informa-tion processing. In particular, we will deal with two different main topics in quantum information: the analysis of the role of two quantities as non-Gaussianity in CV quan-tum information and the study of different quanquan-tum estimation problems for quanquan-tum technology purposes, with a main attention on quantum optical implementation.

The first part of the thesis is then focused on the concept of non-Gaussianity (nonG). v

In particular we will address the quantification of the nonG of quantum states and operations in the quantum information framework. We will illustrate in details the properties and the relationships of the two measures of non-Gaussianity we recently proposed, based on the Hilbert-Schmidt distance and on the quantum relative en-tropy (QRE) between the state under examination and a reference Gaussian state. We will then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behaviour on most of the examples taken into account. However, we also show that they introduce a different relation of order, i.e. they are not strictly monotone each other. We exploit the non-Gaussianity measures for states in order to introduce a measure of non-Gaussianity for quantum operations, to assess Gaussi-fication and de-GaussiGaussi-fication processes, and to investigate in details the role played by non-Gaussianity in entanglement distillation protocols. Besides, we will exploit the QRE-based non-Gaussianity measure to provide new insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information and the Holevo bound and in the framework of the quantum central limit theorem. We will also deal with parameter estimation and present a theorem connect-ing the QRE nonG to the quantum Fisher information. Finally, we will firstly derive some experimentally friendly lower bounds to the QRE based nonG for some class of states and by considering the possibility to perform on the states only certain efficient or inefficient measurements, and then we will present and characterize some experi-mental protocols that generate non-Gaussian states by means of photon-subtraction and photon-addition operations.

If one wants to completely exploit a resource, one should be able to estimate its value. However, many quantities of interest in physics are not always directly ac-cessible and this is particularly true for quantum mechanical systems where several relevant quantities do not correspond to proper quantum observables. In these situa-tions one should resort to indirect measurements and infer the value of the quantity of interest by inspecting a set of data coming from the measurement of different observ-ables. This is a parameter estimation problem which may be properly addressed in the framework of quantum estimation theory. The goal of an estimation problem is not only retrieve the actual value of the unknown parameter, but obtain this information with the minimum uncertainty. The second part of the thesis is thus dedicated to

this topic, and we will focus in particular on the estimation of resources for quantum technology and of noise parameters affecting the resources themselves, i.e. we will evaluate the corresponding Quantum Fisher Information (QFI) and thus the ultimate precision posed by quantum mechanics in the estimation of these parameters, looking for optimal probes and optimal measurements protocols able to attain these bounds. We will start by considering, both in DV and CV systems, the estimation of the key-ingredient for quantum information, i.e. the entanglement. Then we will move to the CV realm, addressing the estimation of quantities characterizing single-mode Gaussian operations, as the displacement and squeezing parameters. We will also highlight the non-Gaussianity induced by Kerr interaction as a resource for the estimation of these parameters. Finally we will consider the paradigmatic problem in quantum estimation theory: the estimation of the quantum phase. Here, both in the qubit and in the CV case, we will consider quantum states affected by a phase-diffusive noise. We will look for the ultimate bounds on the estimation, finding the optimal probe states and the optimal measurements, and we will also show an experimental verification of these results, demonstrating an optimal estimation protocol for optical qubits.

The thesis is thus organized in three main chapters. In the first chapter we will introduce all the preliminary notions that are necessary to deal with the arguments treated through the thesis. Then, in the following chapters we will dedicate our study to the non-Gaussianity and to quantum estimation problems, respectively as a resource and a tool for quantum technology.

The results presented in this thesis are based on the following publications:

1. M. G. Genoni, P. Giorda and M. G. A. Paris, Optimal estimation of entanglement, Phys. Rev. A 78, 032303 (2008)

2. M. G. Genoni, M. G. A. Paris and K. Banaszek, Quantifying the non-Gaussian character of a quantum state by quantum relative entropy, Phys. Rev. A 78, 060303 (Rapid Communication) (2008)

3. M. G. Genoni and M. G. A. Paris, Non-Gaussianity and purity in finite dimen-sion, Int. Jour. Quant. Inf. 7, 97 (2009).

4. M. G. Genoni, C. Invernizzi and M. G. A. Paris, Enhancement of parameter estimation by Kerr interaction, Phys. Rev. A 80, 033842 (2009)

5. D. Brivio, S. Cialdi, S. Vezzoli, B. Teklu, M. G. Genoni, S. Olivares and M. G. A. Paris, Experimental estimation of one-parameter qubit gates in the presence of phase diffusion, Phys. Rev. A 81, 012305 (2010).

6. A. Allevi, A. Andreoni, M. Bondani, M. G. Genoni and S. Olivares, Reliable source of conditional states from single-mode pulsed thermal fields by multiple-photon subtraction, Phys. Rev. A 82, 013816 (2010).

7. M. G. Genoni, F. A. Beduini, A. Allevi, M. Bondani, S. Olivares and M. G. A. Paris, Non-Gaussian states by conditional measurements, Phys. Scr. T140 014007 (2010).

8. B. Teklu, M. G. Genoni, S. Olivares and M. G. A. Paris, Phase estimation in the presence of phase diffusion: the qubit case, Phys. Scr. T140 014062 (2010). 9. A. Allevi, A. Andreoni, M. Bondani, F. A. Beduini, M. G. Genoni, S. Olivares

and M. G. A. Paris, Conditional measurements on multimode pairwise entangled states from spontaneous parametric downconversion, arXiv:1007.0446v1 [quant-ph] - in press for Eur. Phys. Lett..

10. M. G. Genoni and M. G. A. Paris, Non-Gaussianity in quantum information, arXiv:1008.4243 [quant-ph] - in press for Phys. Rev. A.

11. M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier and R. Tualle-Brouri, Non-Gaussianity of quantum states: an experimental test on single-photon added coherent states - submitted.

Other publications:

1. M. Bina, F. Casagrande, M. G. Genoni, A. Lulli and M. G. A. Paris, Dynam-ical description of state mapping and discontinuous entanglement transfer for tripartite systems, Eur. Phys. Lett. 90, 30010 (2010).

2. M. Bina, F. Casagrande, M. G. Genoni, A. Lulli and M. G. A. Paris, Tripartite quantum state mapping and discontinuous entanglement transfer in cavity QED open system, Phys. Scr. T140 014015 (2010).

Chapter

1

Preliminary concepts

In this chapter we introduce the notions and the tools that will be used throughout the thesis. We start in Section 1.1 reformulating the postulates of quantum mechanics in terms of density operator, positive operator valued measure and completely-positive quantum map. At the same time we introduce the basic notions about qubits and in general of discrete-variable (DV) quantum systems, defining concepts as the von Neu-mann entropy and the Schmidt decomposition. In Section 1.2 we review the principal distances and measures of distinguishability for quantum states useful in quantum in-formation theory. In Section 1.3 we introduce the concept of entanglement along with some entanglement measures for bipartite quantum states. Most part of the thesis deals with continuous-variable (CV) quantum systems, the mathematical formalism needed to deal with them is reported in Section 1.4. We describe in details the single-mode and two-mode symplectic operations (two-mode mixing, single-mode and two-mode squeezing, and displacement operators). We then provide a phase-space representation of CV quantum states by defining their characteristic and Wigner functions along with their principal properties. Finally, in Section 1.5 we give the definition and the most relevant properties of the most important class of CV quantum states: the Gaussian states. In particular we describe the criteria to detect entanglement and the some use-ful entanglement measures. Then we will describe, giving some paradigmatic examples, Gaussian and non-Gaussian quantum channels and quantum measurements.

1.1

Elements of quantum mechanics

Quantum mechanics provides a mathematical description of physical systems at a fun-damental level. To provide such description what we need is to specify states, measure-ments and evolutions. These concepts are given via postulates that are here outlined:

• Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product !φ|ψ". The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems.

• A state is a complete description of a physical systems and corresponds to a normalized vector |ψ" in the Hilbert space associated with the physical system (such that _{!ψ|ψ" = 1)}

• An observable, that is a property of a physical system that in principle can be measured, corresponds to a self-adjoint operator A = !_jaj|aj"!aj|, such that

!aj|ak" = δj,k (with δj,k denoting the Kronecker delta) and where aj ∈ are the

possible values taken by the observable.

Its (orthogonal) eigenstates define a projector-valued measure (PVM). Given a quantum state |ψ", the probability of observing the result aj is

p(aj) = !ψ|Pj|ψ" with Pj =|aj"!aj|

and the state after the measurement becomes

|ψj" =

Pj|ψ"

" p(aj)

• The time evolution of a quantum state is given by a unitary operator Ut acting

on the quantum state |ψ":

|ψ(t)" = Ut|ψ".

The unitary evolution Ut is generated by a self-adjoint operator H, called the

of the state is thus governed by the Schrodinger equation d

dt|ψ(t)" = −iH|ψ(t)".

This postulates provide a complete formulation of the quantum theory. Anyway they are valid until we intend to describe the entire universe, or at least a completely isolated system. If we want to limit our observations to a small part of a much larger quantum system, the postulates must be reformulated. In the following we will present the mathematical objects we have to consider when we deal with open quantum systems, i.e. density operators to describe physical systems, positive operator valued measure (POVM) to describe observables/measurements, and quantum operations (CP-maps) for what concerns evolutions. To help the presentation we will introduce here the concept of discrete-variable quantum system, along with the definition of qubit. Then we will introduce the objects mentioned above by means of a simple example of indirect measurement [NC, Preb].

1.1.1

Discrete variable systems – qubits and qudits

To introduce the mathematical objects which provide the postulates of quantum me-chanics for open quantum systems, we will start by considering discrete-variable (DV) quantum systems, that is systems associated with a finite d-dimension Hilbert space H, spanned by a basis {|i", i = 0, . . . , d − 1}.

The most simple example of DV systems is given by two-dimensional systems, whose state-vectors are called qubit, being the “unit” of quantum information. The qubit is indeed the quantum counterpart of the indivisible unit of classical information, the bit. While the bit can take one of two possible values_{{0, 1}, the most general qubit can be} written as

|ψ" = a|0" + b|1" (1.1) where a and b are complex number satisfying the normalization condition_|a|2_+|b|2 _{= 1.}

So qubits can be represented as quantum superpositions of the states |0" and |1" and this property is at the basis of most of the famous quantum algorithms proposed so far [Sho94, Gro96] that gave birth and fame to quantum information and quantum

computation.

1.1.2

Density Operator

Let us consider a quantum state _|ψ"AB living in a bipartite Hilbert space HAB =

HA⊗HB (where⊗ denotes the tensor product symbol). Now let us suppose to measure

an observable A = !_jaj|aj"!aj| corresponding to the first Hilbert space. On the

composite system, the observable reads A⊗ B where B denotes the identity operator

on the Hilbert space _HB. Because of the postulates given before, the probability to

obtain the result ak from the measurement can be evaluated as

p(ak) = AB!ψ|(|ak"!ak| ⊗ B)|ψ"AB (1.2)

= Tr[$A|ak"!ak|] (1.3)

where

$A = TrB[|ψ"AB AB!ψ|] (1.4)

and with Trj[˙] denoting the partial trace operation on the degrees of freedom

corre-sponding to the Hilbert space Hj.

One can also verify that the expectation value of A reads

!A" =AB!ψ|(A ⊗ B)|ψ"AB (1.5)

= Tr[$AA]. (1.6)

The operator $A defined in Eq. (1.4) is called density operator of the physical system

A, and as we have seen by considering a generic observable living in the same Hilbert space, fully characterizes the quantum system associated with the Hilbert space _HA.

For this reason the density operator can be considered as the more general object describing a quantum system. In general, a density operator $ living in a Hilbert space H has to satisfy the following properties

1. $ is positive, that is_{∀ |ψ" ∈ H, then !ψ|$|ψ" > 0.} 2. Tr[$] = 1 (normalization condition).

3. Tr[$2_{]≤ 1.}

Notice that from the first property follows that $ is also self-adjoint, and thus it can be diagonalized in an orthonormal basis and written as a convex sum of pure states

$ =#

j

pj|j"!j|,

where the condition!_jpj = 1 holds because of the second property. Moreover one has

that Tr[$2_{] = 1 iff $ = |ψ"!ψ|, and in this case is called pure state; if Tr[$}2_{] < 1, $ is}

referred to as a mixed state. The quantity µ = Tr[$2_{] is then called purity of the state.}

One can demonstrate that the partial trace is the only map that transforms a quantum state of a composite system_|ψ"AB into an operator $Aliving in the first subsystem and

fulfilling the properties just oulined. The qubit case, the Bloch sphere

The density operator of a qubit is a self-adjoint 2× 2 matrix. The most general self-adjoint 2_{×2 matrix has four real parameters and can be expanded in the Pauli-matrices} basis{ , σx, σy, σz} as

$ = a0 +

#

j

ajσj. (1.7)

where here denotes the 2× 2 identity matrix and in the basis that diagonalizes σz

we have σx =   0 1 1 0   σy =   0 −i i 0   σz =   1 0 0 −1   (1.8)

Since each σj is traceless, to fulfill the condition Tr[$] = 1 we have to choose a0 = 1/2

and thus we obtain

$ = 1

2(1 + n· σ) (1.9)

where n = (n1, n2, n3)T and σ = (σx, σy, σz)T denotes respectively the vector of the

components and of the Pauli matrices. We can compute the determinant of $ obtaining det[$] = 1₄(1− n · n). Therefore a necessary condition for $ to have non-negative

eigenvalues is det[$] ≥ 0 and thus n · n ≤ 1. This is also a sufficient condition for $; since the condition Tr[$] = 1, it is not possible to have two negative eigenvalues. Thus there is a one-to-one correspondence between the density operator of a qubit and a 3-ball defined by the vector n that satisfies the condition 0 _{≤ |n| ≤ 1. This ball is} usually called the Bloch sphere while the vector n is usually referred to as the Bloch vector.

The density operators that correspond to points on the boundary (|n| = 1) are pure states _{|ψn". A useful parametrization for pure states is the following}

|ψn" = cos

θ

2|0" + e

iφ_sin θ

2|1" (1.10)

where_{{|0", |1"} corresponds to the eigenvectors of σ}z such that σz|0" = |0" and σz|1" =

−|1". In this case the Bloch vector reads n = (sin 2θ cos φ, sin 2θ sin φ, cos 2θ) and the angles θ and φ clearly identify a point on the Bloch sphere as depicted in Fig. 1.1. The

Figure 1.1: Representation of a qubit on the Bloch sphere

interior of the Bloch sphere, the open Bloch ball, represents the mixed states and the most mixed state $ = /2 corresponds to the center of the ball.

Von Neumann Entropy

The von Neumann entropy is the generalization to the quantum realm of the Shannon entropy defined in the framework of classical information theory.

Given a random variable X = {x1, . . . , xn} along with its probability distribution

P (X = xi) = p(xi), the Shannon entropy is defined as

H(X) =₋#

i

p(xi) log2p(xi). (1.11)

In quantum mechanics, given a density operator $ = !_ipi|i"!i|, the von Neumann

entropy is given by the formula1

S_{($) =−Tr[$ log}

2$] (1.12)

=−#

i

pilog2pi

The von Neumann entropy is extremely important in quantum information science. Firstly it replaces the Shannon entropy for classical information as the fundamental unit for the quantification of information in quantum information theory (see [NC] for more details about the quantum versions of the Shannon’s theorems). Moreover, as we will see in the following sections, it is fundamental for the quantification of entanglement and in general of relevant quantities for quantum information (mutual information, non-Gaussianity, etc...).

Pure states are the only states with zero entropy. On the other hand, mixed states have von Neumann entropy higher than zero, and the most mixed state in a d-dimensional Hilbert space, that is $ = /d, has the maximum value of entropy, S($) = log2d.

Bipartite states and Schmidt decomposition

Let us consider now a bipartite physical system corresponding to the composite Hilbert space_HA⊗ HB, with dimensions dim(HA) = dA and dim(HB) = dB.

Given two basis of the two Hilbert spaces, we can write a generic pure state _|ψAB" as

1

When we will deal with continuous-variable systems, that is corresponding to Hilbert space with infinite dimension (see Sec. 1.4) the logarithm of base two log2 in the definition, will be replaced by

follows |ψAB" = d#A−1 i=0 d#B−1 j=0 χi,j|ui"A|vi"B. (1.13)

A statement in linear algebra says that, for every pure state |ψ"AB, there exist bases

{|αi"A} and {|βi"B} such that

|ψ"AB = ˜ d−1 # i=0 ψi|αi"A|βi"B. (1.14)

where ˜d = min(dA, dB) and where ψiare positive and real numbers such that!_iψi2 = 1.

This formula is known as Schmidt decomposition of the pure state |ψ"AB.

The Schmidt coefficients ψi correspond to the square root of the eigenvalues of the two

partial traces $A= TrB[|ψ"AB AB!ψ|] = ˜ d−1 # i=0 ψ_i2_|αi"!αi| (1.15) $B = TrA[|ψ"AB AB!ψ|] = ˜ d−1 # i=0 ψ2 i |βi"!βi| (1.16)

As we will see in the following, the Schmidt decomposition will be useful for the char-acterization of the entanglement of quantum states.

1.1.3

Positive operator valued measure (POVM)

Let us consider two subsystems A and B described by two density operators $Aand σB

belonging to the Hilbert spacesHAandHB respectively. The entire system is described

by the density operator $A⊗ σB belonging to the Hilbert space HA⊗ HB. Suppose

that the two systems are coupled by a unitary operation U and, after the interaction a measure of an observable described by the self-adjoint operator O = !_jOi|i"!i| is

of the operator O). The probability of measuring the eigenvalue Oi is given by

pi = TrAB[U$A⊗ σBU† A⊗ |i"B B!i|]

= TrAB[$A⊗ σBU† A⊗ |i"B B!i|U]

= TrA[$AΠi] (1.17)

where in (1.17) we have introduced the operator Πi

Πi = TrB[ A⊗ σBU† A⊗ |i"B B!i|U] (1.18)

Therefore, neglecting the subsystem B we found a more general operator that permits us to obtain the statistic of the measure without taking into account the entire quantum system.

This kind of operators defines a more general measure than the projector valued measure (PVM), called positive operator valued measure (POVM).

In general we define a POVM as a measure M

M : Oi → Πi (1.19)

that associates the results of the measure (i.e. the eigenvalues Oi) to operators Πi

which satisfy the following properties:

Πi ≥ 0 (1.20)

#

i

Πi = 1. (1.21)

In this sense we say that a set of operators {Πi} defines a POVM.

Notice that the properties (1.20) and (1.21) are the unique necessary properties to assure that the quantities pi = Tr[$Πi] define a probability distribution.

The concept of POVM has been here introduced considering an orthogonal measure in a bigger space than _HA. The following theorem assures that is always possible

considering the inverse process.

{Πi} in a Hilbert space HA, then

• ∃ an Hilbert space HB.

• ∃ a self-adjoint operator O on the Hilbert space Ha⊗ HB ( with real eigenvalues

Oi and with corresponding projectors POi).

• ∃ a density operator σ belonging to HB

such that, for any density operator $ belonging to HA we have

pi = TrA[$Πi] = TrAB[$⊗ σ POi] (1.22)

or, equivalently Πi = TrB[ A⊗ σ POi]. That is, for every POVM we can find a PVM,

i.e. a self-adjoint observable, in a larger Hilbert space.

In particular the theorem assures that infinite extensions (Naimark extensions) of this kind exist. For this reason we can affirm that POVM are, physically, more important objects than PVM since they represent via the Naimark theorem the whole class of observables of a quantum system.

1.1.4

Completely positive map

Let us consider a system described by a factorized state $⊗ |µ"!µ| belonging to the Hilbert space_HA⊗HB. This state evolves by means of a unitary evolution U. If we are

interested only in the evolution of the subsystem corresponding to the Hilbert space HA we perform a partial trace on the other subsystem, obtaining

$" = TrB[U$⊗ |µ"!µ|U†] = # s !s|U|µ"$!µ|U†_|s" = # s Ms$Ms† = E($) (1.23)

where Ms =!s|U|µ" and where {|s"} is a basis of the Hilbert space HB. It’s easy to

verify that the operators Ms satisfy the following property:

#

s

MsMs† = A. (1.24)

The map E which gives the evolution of $ is not a unitary operator, and belongs to a more general class of transformations called quantum operations, quantum channels or completely positive maps (CP maps).

In general we define _{E : $ → E($) a quantum operation if} • E is linear

• E is trace-preserving: Tr[E($)] = Tr[!sMs$Ms†] = Tr[$

!

sMs†Ms] = Tr[$].

• If A is a self-adjoint operator, then E(A) is self-adjoint.

• E is completely positive, i.e. considering every possible extension of the Hilbert spaceHA in a Hilbert space HA⊗ HB, the map E ⊗ B is a positive map for all

HB.

As we pointed out before, quantum operations are not necessarily unitary operations, i.e., even in this case, ignoring the subsystem B we have found a more general math-ematical object that describes the evolution of quantum state. Again, as for POVM, infinite possible unitary extensions correspond to a quantum operation _{E. Moreover} quantum operations are in general not invertible operators.

Eq. (1.23) is known as Kraus-Sudarshan decomposition [Kra] of a quantum operation. A unitary transformation is obtained if the decomposition is made by a single operator, and, in this case, we have also that the operator is invertible.

1.2

Distances and measures of distinguishability for

quantum states

The distance _D[$1, $2] between two arbitrary elements $1 and $2 belonging to a Hilbert

space has to satisfy the following axioms: (i) positive semi-definiteness

D[$1, $2] ≥ 0 ∀ $1, $2 (1.25)

D[$1, $2] = 0 ⇔ $1 = $2 (1.26)

(ii) symmetry

D[$1, $2] =D[$2, $1] (1.27)

(iii) triangualr inequality

D[$1, $2]≤ D[$1, $] +D[$, $2] (1.28)

All these axioms are automatically satisfied if one employs an arbitrary norm _{-A- and} defines the distance as

D[$1, $2] = -K($1− $2)

-where K is a multiplying costant.

Different distances and in general measures of distinguishability between quantum states have been used in quantum information for different purposes. Here we will show the definitions and the properties of some of these distances: trace distance based on the trace norm, the Hilbert Schmidt distance, based on the Hilbert-Schmidt norm, the Bures distance based on the fidelity between two quantum states and finally the quantum relative entropy that though not being a proper distance has been widely used in quantum information for its other relevant properties.

1.2.1

Trace distance

The trace norm is defined by

-A-1 ≡ Tr[

√

where the positive square root of A†_{A is considered.}

The trace distance is defined by

DT r[$1, $2] ≡ ( ( ( ( 1 √ 2($1 − $2) ( ( ( ( 1 (1.30) = 1 2Tr|$1− $2|

The trace distance, besides satisfying the three properties assured by the norm defini-tion, owns the following additional properties:

- 0≤ DT r[$1, $2]≤ 1.

- invariance under unitary transformations, i.e.:

DT r[U$1U†, U$2U†] =DT r[$1, $2]

- contractivity under generic quantum maps, i.e.

DT r[E($1),E($2)]≤ DT r[$1, $2]

- subadditivity under tensor product, i.e.

DT r[$1 ⊗ σ1, $2⊗ σ2]≤ DT r[$1, $2] +DT r[σ1, σ2]

Moreover the trace distance has an important physical operational meaning, as a mea-sure of state distinguishability. Suppose Alice prepares a quantum system in the state $ with probability 1/2 and in the state σ with probability 1/2. Bob receives the system and can perform a POVM measure to distinguish the two states. It can be shown that Bob’s probability of correctly identifying which state Alice prepared is

pc =

1

2(1 +DT r[$, σ]). This physical interpretation follows from the relation

DT r[$, σ] = max

where the maximum is taken over all positive operators P satisfying P − ≤ 0. The trace distance has been used to evaluate non-classicality of states in quantum optics [Hil87], however its evaluation is often challenging, in particular for infinite-dimensional quantum states, since it requires the diagonalization of the operator $1− $2.

1.2.2

Hilbert-Schmidt distance

The most simple and natural norm in quantum theory is the Hilbert-Schmidt norm

-A-2 ≡

"

Tr[A†_A].

The related Hilbert-Schmidt (HS) distance between two quantum states is given by:

DHS[$1, $2] ≡ -1 √ 2($1− $2)-2 = ) 1 2Tr[($1− $2) 2_] = ) Tr[$2 1] + Tr[$22]− 2Tr[$1$2] 2 = ) µ[$1] + µ[$2]− 2κ[$1, $2] 2 (1.32)

where µ[$] = Tr[$2_{] denotes the purity of the quantum state and κ[$}

1, $2] = Tr[$1$2]

the overlap betweeen $1 and $2.

It’s easy to verify that the Hilbert-Schmidt distance, besides the properties guaran-teed by the Hilbert-Schmidt norm, is bounded to [0, 1] and is preserved under unitary transformations.

In addition, the Hilbert-Schmidt distance is suitable for explicit calculations, since it involves only the evaluations of overlaps between quantum states, whereas diagonal-ization of these states is not required and for these reasons has been widely used, e.g. to define measures of non-classicality of quantum states [DMMW00].

1.2.3

Fidelity and Bures distance

Let us define the fidelity between two quantum states $1 and $2 as [NC]

F[$1, $2] ≡ * Tr +,_√ $1$2√$1 -.2 (1.33)

that for pure states $1 =|ψ1"!ψ1| and $2 =|ψ2"!ψ2| reduces to

F[|ψ1", |ψ2"] = |!ψ2|ψ1"|2, (1.34)

i.e. to the probability of transition from a quantum state_|ψ1" to |ψ2".

We can define a distance, the Bures distance, as

DB[$1, $2] ≡

,

2(1₋"_F[$1, $2]) (1.35)

It’s possible to see that (1.35) reduces to the Hilbert-Schmidt distance when pure states $1 =|ψ1"!ψ1| and $2 =|ψ2"!ψ2| are considered.

Theorem 2 (Ulhman) [NC]: Given $1 and $2 belonging to a Hilbert spaceHA , then

F[$1, $2] = max

|χ$$,|φ$$|!!φ|χ""|

2 _(1.36)

where _{|χ"" and |φ"" are respectively purifications of the states $}1 and $2, i.e. states

belonging to a Hilbert space_HA⊗ HB such that

$1 = TrB[|χ""!!χ|]

$2 = TrB[|φ""!!φ|]

By means of this theorem the three properties required for a distance between quan-tum states can be demonstrated. Moreover, because of the properties of the fidelity F[$1, $2], also the Bures distance is bounded to [0, 2], invariant under unitary

opera-tors and contractive under generic quantum maps. In Sec. 3.1.4 we will show that the Bures distance owns other properties related to the geometry in the Hilbert space and to the estimation properties of quantum parameters.

1.2.4

Quantum relative entropy

Given two quantum states $1 and $2, the quantum relative entropy (QRE) is defined

as2

S_{($1-$2) = Tr[$1(log}

2$1− log2$2)] (1.37)

As for its classical counterpart, the Kullback-Leiber divergence, it can be demonstrated that 0≤ S($1-$2) <∞ when it is definite, i.e. when the support of the fist state in the

Hilbert space supp $1 ⊆ supp $2, is contained in that of the second one. In particular

S_{($1-$2) = 0 iff $1 ≡ $2. This quantity, though not defining a proper metric in the} Hilbert space (it is not symmetric in its arguments, and the triangle inequality does not hold), has been widely used in different fields of quantum information as a measure of statistical distinguishability for quantum states [Ved02, SW02] because of its nice properties and its operational meaning.

Let us consider two quantum states $1 and $2and suppose to perform N measurements

on $1. The probability of confusing $1 with $2 after the measurements is (for large N)

PN($→ τ) ∼ exp{−NS($-τ)}.

Besides this operational interpretation, the QRE owns the following properties - invariance under unitary transformations, i.e.:

S_(U$1U†_-U$2U†_{) = S($1-$2)} - contractivity under generic quantum maps, i.e.

S_{(E($1)-E($2))≤ S($1-$2)} - additivity under tensor product, i.e.

S_{($1⊗ σ1-$2⊗ σ2) = S($1-$2) + S(σ1-σ2)}

2

As stated at the definition of the von Neumann entropy, when we will deal with infinite dimen-sional Hilbert spaces, the log2 will be replaced by the natural logarithm written simply as log.

- convexity in both arguments: S₍# i pi$i -# i piσi)≤ # i piS($i-σi) where!_ipi = 1.

1.3

Quantum entanglement in DV systems

Quantum entanglement is one of the most odd features of quantum mechanics, de-scribed for the first time in 1935 by Einstein, Podolski and Rosen [EPR35] and by Schr¨odinger [Sch35], and nowadays is considered the key resource for QI processing [HHHH09]. It occurs in composite systems as a consequence of the superposition prin-ciple and of the fact that the Hilbert space that describes a composite quantum system is the tensor product of the Hilbert spaces associated to each subsystems. If the en-tangled subsystems are spatially separated nonlocality properties may arise, showing its incompatibility with a classical world-view.

In the following we will give its definition, we will introduce a criterion for the entan-glement of a generic bipartite state and we will briefly discuss how entanentan-glement may be quantified introducing some entanglement measures that will be useful in the rest of the thesis. We will focus only on bi-partite entanglement (that is between pairs of subsystems), and for the sake of simplicity we will give examples mainly referring to qubit systems.

A pure states on the bipartite Hilbert space H1⊗ H2 is entangled when it cannot

be written as a product state, i.e. as _{|ψ" = |0"}1 ⊗ |0"23. In terms of the Schmidt

decomposition introduced in Eq. (1.14) pure states are entangled iff the number of non-zero Schmidt coefficients is greater than one.

A mixed state is entangled whether it cannot be written as a separable state

$sep = # i pi|ψi"!ψi| ⊗ |φi"!φi| # i pi = 1, (1.38)

that is as a convex sum of product states.

This definition for mixed state is motivated by an operational definition of entanglement given in terms of Local Operations and Classical Communication (LOCC) [Wer89]. Let us assume that the two subsystems are spatially separated and that it is possible to perform local operations, including measurement, on each subsystem and that they are also connected to each other via a “classical” communication channel, such as a telephone connection. The separable states are those which, starting from subsystems in a product state, can be generated by LOCC, while the entangled states are those that

3

Where there will be no problems of misunderstanding we will omit the suffixes and the tensor product symbol_⊗

cannot. In this sense entanglement can be seen as a kind of non-classical correlation between the subsystems.

A natural question that may arise is the following: given a quantum state, how can I check if it is entangled or not? Fortunately a simple test, which is always sufficient but not necessary for all the physical systems, exists. This is known as the Positivity of Partial Transpose (PPT) criterion, introduced by Peres [Per96] and by Horodecki [HHH96] and is based on the fact that the partial transposition operation is not a completely positive operation. Let us consider the separable state in Eq. (1.38) and perform a transposition only on the second subsystem obtaining

$PT

sep =

#

i

pi|ψi"!ψi| ⊗ (|φi"!φi|)T (1.39)

Since transposition is positive, (|φi"!φi|)T is still a positive operator, thus corresponds

to a physical state and the global state $PT

sepis positive as well. The same argument does

not hold when the state is entangled and thus entangled states may have a negative partial transpose. For this reason a negative partial transpose (NPT) is a sufficient condition for entanglement whereas it is not a general necessary condition. However for certain system, as pairs of qubits , systems of one qubit and a qutrit (three-dimensional system), as well as bipartite continuous-variable Gaussian states (see Sec. 1.5.2 for de-tails), the NPT is both a sufficient and necessary condition for entanglement. Quantum states that do not belong to these classes and that are entangled though having a nega-tive partial transpose are called bound entangled states and has been fully characterized in Ref. [Hor97].

1.3.1

Quantifying entanglement

Since as mentioned before, entanglement is the key resource for several protocols in QI processing, a big effort has been done to quantify it in a proper way.

Quantifying bipartite entanglement in pure states is rather simple. Indeed if a pure bipartite state is not entangled, both its partial traces are pure states as well, while whether the bipartite state is entangled, its partial traces will be necessarily mixed. Then it is natural to associate the degree of entanglement with the degree of mixedness of the partial trace. The measure of mixedness chosen is the von Neumann entropy of

the reduced state:

EvN = S($A) =−Tr[$Alog₂$A] (1.40)

where $A= TrB[|ψ"AB AB!ψ|] is the reduced state of one subsystem 4.

The pure state entanglement measure EvN, that is often called entropy of entanglement,

satisfies the following important properties 1. It is zero for separable (product) states.

2. It is invariant under local unitary transformation.

3. It is not increased on average under LOCC on the two subsystems.

A quantity that satisfies these three properties is called entanglement monotone [PV07] and this notion will be crucial in the following to define properly entanglement measures for mixed states.

According to this measure we can identify also the maximally entangled states. For qubit systems, the maximally entangled states are given by the so-called Bell states

|φ±" = |0"|0" ± |1"|1"√ 2

(1.41) |ψ±_{" =} |0"|1" ± |1"|0"_√

2

which in fact have as partial trace the most mixed state $ = 2/2. In general, a

bipartite quantum state describing two d-dimensional systems is maximally entangled iff it is unitarily equivalent to the state

|φ+ d" =

|0"|0" + |1"|1" + · · · + |d − 1"|d − 1" √

d (1.42)

being its partial trace $d = d/d and its degree of entanglement EvN = log₂d (in the

following we will denote the corresponding density operator as Φ(d) =|φ+

d"!φ+d|). It is

also worth to notice that the state_|φ+_{d" is totally equivalent, via a change of the basis, to}

4

Notice that because of Eqs. (1.16) and (1.15), the von Neumann entropies of the two reduced states of a bipartite pure states are always equal and thus one can also choose the von Neumann entropy of !B = TrA[|ψ"AB AB!ψ|]

the factorized state of log2d copies of the Bell state|φ+"⊗ log2d. Since entanglement can

not increase under LOCC, maximally entangled states are the only states from which any other state can be obtained (asymptotically) by LOCC alone [Nie99]. Moreover the maximally entangled states of two qubit play a central role in quantum information being the golden standard for several QI processing, and thus the target state of every entanglement distillation protocols. As regards pure states one can demonstrate that every other entanglement measure, additive, continuous and entanglement monotone is necessarily a monotonic function of EvN. One of these measures that is often used,

being usually easy to be evaluated, is the linearized entropy of entanglement defined as

EL(|ψ"AB AB!ψ|] = d(1 − Tr[$2A]) (1.43)

For mixed states the picture becomes a little bit more complicated and different en-tanglement monotones have been proposed giving a different ordering of states [PV07]. Two of the main important entanglement measures, for their operational meaning, are the entanglement cost and the distillable entanglement. In their both definitions the role of two-qubit maximally entangled states5 _{is crucial.}

The entanglement cost [HHT01] of a given state $ quantifies the maximal possible rate r at which one can convert blocks of 2-qubit maximally entangled states into output states that approximate many copies of $, such that the approximations become van-ishingly small in the limit of large block size. In formula, if we denote the general trace preserving LOCC operations as _{L, we have}

EC($) = inf / r : lim n→∞ 0 inf L DT r[$ ⊗n_,L(Φ(2rn_))]1 _{= 0}2_{. (1.44)}

One can consider the reverse problem, that is evaluate at what rate we can obtain maximally entangled states from an input made of many copies of a given state $. This process is known in literature as entanglement distillation. The corresponding entanglement measure is called distillable entanglement and as for the entanglement cost, we can express it in the following formula:

ED($) = sup / r : lim n→∞ 0 inf L DT r[L($ ⊗n_{), Φ(2}rn_)]1 _{= 0}2_{. (1.45)} 5

Common synonyms for two-qubit maximally entangled states include singlet states, Bell pairs . Even though these terms strictly mean different things, we will follow this abuse of terminology.

ED($) is an important measure because in two-party quantum information protocols

usually entanglement is required in the form of maximally entangled states, and thus ED($) tells us the rate at which noisy mixed entangled states can be converted back

into singlet states by LOCC.

One would now ask whether the two measures EC($) and ED($) coincides, i.e. if

entanglement transformations are reversible. It is possible to show that in general this is not the case, but when we consider bipartite pure states we have the important result

EC(|ψ"!ψ|) = ED(|ψ"!ψ|) = EvN(|ψ"!ψ|), (1.46)

that is the two measures coincides and are identical to the entropy of entanglement defined above.

Unfortunately, given an arbitrary state, particularly of high dimension, these quantities can be very difficult to calculate. For this reason two other measures inspired by the PPT criterion described above have been introduced, namely the negativity EN and the

logarithmic negativity ELN [ZHSL98, EP99, VW02]. They provide a way of quantifying

how much this criterion is violated, in formula

EN($) = -$ ΓA

-1− 1

2 (1.47)

ELN($) = log2-$ΓA-1 (1.48)

where $ΓA denotes the partial transposition of $ respect to the subsystem A. The trace

norm -A-1 introduced in Eq. (1.29) is equal to the sum of the absolute value of the

eigenvalues, and thus also singular values of A. Singular value decompositions of large matrices may be accomplished, and thus EN and ELN can be calculated even for high

dimensional systems. Moreover they both fulfil the three criteria for an entanglement monotone, and in particular the logarithmic negativity is also additive and an upper bound to the distillable entanglement. As we will see in Sec. 1.5.2, they can be easily evaluated also for bipartite Gaussian states in infinite dimensional systems. For all these reasons EN and ELN are so widely used in quantum information and as we will

1.4

Continuous-variable systems

In this chapter we briefly introduce some basic concepts and notations we need through our study to deal with continuous-variable (CV) systems. We analyze Cartesian de-composition of mode operators in the phase space. The covariance matrix and the sympletic transformation are defined as well as the unitary evolutions corresponding to complex rotations and to displacement and squeezing operators [FOP]. Moreover we introduce the characteristic function and the Wigner function of an operator in a infinite dimensional Hilbert space, along with their basic properties [CG69b, CG69a].

Let us consider a system of n bosons described by the mode operators ak, k = 1 . . . n,

satisfying the commutation relations [ak, a†j] = δkj , where δij is the Kronecker delta,

and is the identiy operator in the Hilbert space_{H = ⊗}n

k=1Hk. The free Hamiltonian of

the system is H =!n_k=1(a†_kak+1₂ ). The canonical position and momentum operators

are given by:

qk = 1 √ 2(ak+ a † k), pk= 1 i√2(ak− a † k)

and the corresponding commutation relations are [qj, pk] = iδjk . Introducing the

vector R = (q1, p1, . . . , qn, pn)T, (. . . )T noting the transposition operation, the

commu-tation relations become

[Rk, Rj] = iΩkj (1.49)

where Ωkj are the elemets of the sympletic matrix

Ω = n 3 k=1 ω, ω=   0 1 −1 0   . (1.50)

We can now define for a quantum state of n bosons, described by the density matrix $, the covariance matrix σ = σ[$], of elements σkj as follows

σkj =

1

2!{Rk, Rj}" − !Rj"!Rk", (1.51) where _{{A, B} = AB + BA denotes the anticommutator, and !O" = Tr[$O] is the} expectation value of the operator O.

Moreover if $ is a density matrix, and thus defined as follows $ =# i pi|i"!i|, # i pi = 1, pi ≥ 0 ∀i, (1.52)

we can express the positivity of $ and rewrite the Heisenberg-Robertson uncertainty relations (see Appendix A) in the following compact form

σ+ i

2Ω≥ 0 . (1.53)

The vacuum state of n bosons is a pure and separable state |0" = 4nk=1|0"k

charac-terized by the covariance matrix σ0 = 1₂ 2n, 2n being the 2n× 2n identity matrix.

A system at thermal equilibrium is described by the density operator ν =4n_k=1ν(Nk)

where ν(Nk) = e−βa†ka Tr[e−βa†ka] = 1 1 + Nk ∞ # m=0 * Nk Nk+ 1 .m |m"kk!m| . (1.54)

Nk = (eβ − 1)−1 is the average number of thermal quanta at equilibrium in the k−th

mode and {|m"k}m∈ are the eigenstates of the number operator a†kak which form a

basis, namely the number (or Fock) basis, of each Hilbert space _Hk.

The covariance matrix of a thermal state ν is given by

σ[ν] = Diag * N1+ 1 2, . . . , Nn+ 1 2 . (1.55)

where Diag(s1, s2, . . . , sn) denotes the diagonal matrix with elements sk, k = 1, . . . , n.

Consider a linear transfomation applied on the system, represented by a n_{× n} matrix F such that

R! _{= F R. (1.56)} Then, since R_k"R"_l=# ps FkpRpFlsRs= # ps FkpRpRs 5 FT6_sl, (1.57) and according to definition (1.51), the covariance matrix evolves as follows

1.4.1

Symplectic transformations

Let us consider a classical system described by the canonical coordinates (q1, . . . , qn)

and conjugated momenta (p1, . . . , pn). Denoting by ˙x the time derivative, the classical

equations of motions are:

˙qk = ∂H ∂pk , ˙pk=− ∂H ∂qk , (1.59)

where H is the Hamiltonian of the system.

Equations (1.59) for a system of n particles can be rewritten as ˙

Rk = Ωks

∂H ∂Rs

(1.60)

where Ωks are the elements of the symplectic matrix (1.50). Let us consider the linear

transformation F R! _{= F R, F} ks = ∂R" k ∂Rs , (1.61)

we obtain that the new equation of motions are given by ∂R" k ∂t = ∂R" k ∂Rs ∂Rs ∂t = FksΩsp ∂H ∂Rp = FksΩsp ∂R" l ∂Rp ∂H ∂R" l = FksΩspFlp ∂H ∂R" l . (1.62)

Therefore the Hamilton equations are left unchanged if and only if F satisfies

FksΩspFlp = Ωkl or F ΩFT = Ω (1.63)

which characterize symplectic classical transformations in the phase space.

Turning back to a quantum state of n bosons: a given linear mode transformation R! _{= F R, leaves the kinematics invariant, if it preserves the canonical communication}

(1.63).

Since ΩT = Ω−1 = _{−Ω, from (1.63) one obtains that Det[F ]}2 _{= 1, and thus F}−1

exist. Moreover we can directly verify that if F , F1 and F2 are symplectic, then also

F−1, FT and F1F2 are symplectic. Thus these matrices form a group, the symplectic

group Sp(2n, ) with dimension n(2n + 1). If we consider even the translations in the phase spaces, i.e. R! _{= R + Λ where Λ is a real vector, we have the affine}

(inhomogeneous) symplectic group ISP(2n, ). In our study we deal also with two-mode bipartite systems, therefore we are interested in the Sp(4, ) group. The matrix S _{∈ SP(4, ) is a 4×4 matrix with only ten free parameters due to the constraint (1.63).} A relevant subgroup of Sp(4, ) is the six parameter subgroup Sp(2, )_{⊗Sp(2, ) ⊂} Sp(4, ) corresponding to local linear canonical transformations on the two subsystems. The symplectic matrix Slocal belonging to this group is a direct sum block matrix:

Slocal =   S1 0 0 S2   , (1.64)

where S1 and S2 belong to the group Sp(2, ).

We may ask which physical transformations correspond to symplectic transforma-tions or, in other words, which interaction Hamiltonians give rise to the whole group of symplectic transformations. It can be demonstrated that the most general Hamiltonian of this type is at most bilinear in the field modes:

Hbil = n # k=1 g(1)_k a†_k+ n # k>l=1 g_kl(2)a†_kal+ n # k,l=1 g_kl(3)a†_ka†_l + h.c. , (1.65) where h.c. means hermitian conjugate. Moreover any Hamiltonian generated by sym-plectic transformations is described by (1.65). Transformations induced by Hamiltonian (1.65) U(S) = exp{−iHbil} belong to the unitary representation of the affine

symplec-tic group ISP(2n, ), which is called the metaplecsymplec-tic representation. In parsymplec-ticular U(S) acts on a density operator $ as

$" = U(S)$U†(S). (1.66) Through our study we use different unitary operators which correspond to different parts of the Hamiltonian (1.65).

In the special case of bipartite systems H = H1⊗ H2, the unitary operator U(Slocal)

can be written in the factorized form

U(Slocal) = U(S1)⊗ U(S2) (1.67)

where U(S1) and U(S2) act respectively to the Hilbert spaces H1 and H2.

1.4.2

Two-mode mixing: the beam splitter

The simplest example of two-mode interaction is the linear mixing described by the part of Hamiltonian (1.65) of the form H ∝ a†_{b + b}†_{a, where the operators a and b}

denote repsectively the first and the second mode operators. Since the total number of quanta is conserved, this interaction is called passive, and the corresponding device, the beam splitter (BS), is said to be a passive device. The evolution operator can be written as

U(ζ) = exp_{ζa†b_{− ζ}∗ab†_} ζ _∈ (1.68) with ζ = φeiθ_{. We can use the Schwinger two mode boson representation of SU(2)}

algebra to make the indentifications: J+= a†b, J− = ab†, J3 = 1₂[J+, J−] = 12(a†a−b†b).

Then, using the operator ordering theorem (B.3) (see appendix B) we can write

U(ζ) = exp{ζa†_{b− ζ}∗_ab†_}

= exp7eiθtan φ a†b8 5cos2φ6b†b−a†aexp7−e−iθ_{tan φ ab}†8

= exp7−e−iθ_{tan φ ab}†8 5_cos2_φ6a†a−b†b_exp7_eiθ_{tan φ a}†_b8 _(1.69)

The Casimir of this algebra is N = a†_a+b†_{b, that is the number of photons is conserved}

or, in other words, the energy is a constant of motion.

We define the transmissivity of the beam splitter as T = cos2_{φ. We refer to as a}

balanced beam splitter if φ = π/4 and hence T = 1/2. Using (1.69), the following Baker-Hausdorf formula

eλABe−λA = B + λ[A, B] + (λ)

2

2! [A, [A, B]] + +_{· · · +} (λ)

n

and the commutation relations of SU(2) algebra, we obtain the Heisenberg evolutions of the modes a and b

  a" b"   = U†_(ζ)   a b   U(ζ) = Sζ   a b   (1.71) with Sζ = 

 cos φ eiθsin φ −e−iθ_{sin φ cos φ}



 . (1.72)

that is a (two-mode) complex rotation of the field modes. From (1.71) we can obtain the two-mode mixing symplectic matrix FU (ζ), namely

FU (ζ) =   Re[Sζ] −Re[Sζ] Im[Sζ] Im[Sζ]   (1.73)

where Re[λ] and Im[λ] denote respectively the real part and the imaginary part of a complex number λ.

In the case of a real rotations U(φ) = R(φ) = exp{φ(a†_b−ab†_{)}, the symplectic matrix,}

by using the transmissivity parameter T = cos2_{φ, reduces to}

FR(T )=         √ T 0 √1− T 0 0 √T 0 √1_{− T} −√1− T 0 √T 0 0 ₋√1_{− T} 0 √T         . (1.74)

Notice that, since the number of quanta is a constant of motion, the vacuum state |0" = |0" ⊗ |0" remains invariant under the action of U(ζ), i.e. U(ζ)|0" = |0".

1.4.3

Squeezing operators

In this section we focus our attention on two unitary operators belonging to the quadratic subclass of the Hamiltonians (1.65): single-mode and two-mode squeezing.

Single-mode squeezing and the squeezed vacuum state

Consider the Hamiltonian H _{∝ (a}†₎2_+a2_{, the corresponding unitary evolution operator}

is the single mode squeezing operator

S(ξ) = exp ; 1 2[ξ(a †₎2 − ξ∗a2] < , ξ _∈ (1.75)

where ξ = reiψ_{. As in the previous case, we can disentangle the operator S(ξ) using}

(B.6) and making the identification with the two-boson representation of the SU(1, 1) algebra K+ = 1₂(a†)2, K− = (K+)† and K3 =−1₂[K+, K−]. We set µ = cosh r (µ∈ )

and ν = eiψ_{sinh r (ν ∈ ), obtaining}

S(ξ) = exp ; 1 2[ξ(a †₎2_{− ξ}∗_a2_] < = exp ; ν∗ 2µa †2 < µ−(a†a+12) exp ; − ν 2µa 2 < = exp ; − ν 2µa 2 < µ(a†a+12) exp ; ν∗ 2µa †2 < . (1.76)

By means of (1.76) and (1.70), we obtain the Heinsenberg evolutions of the field mode operators

S†(ξ)aS(ξ) = µa + νa†, S†(ξ)a†S(ξ) = µa†+ ν∗a (1.77) that in matrix form read

  a" a"†   = S†_(ξ)   a a†   S(ξ) = S   a a†   (1.78) with S =   µ ν ν∗ _µ   . (1.79)

The symplectic matrix FS(ξ) obtained from (1.77) is

FS(ξ) =   µ + Re[ν] Im[ν] Im[ν] µ_{− Re[ν]}   (1.80)

which, in the case of real squeezing, i.e. for ψ = 0 and hence ν ∈ , reduces to FS(ξ) = Diag(er, e−r).

The single mode squeezer does not conserve the energy and, in contrast to the single-mode mixing, is said to be an active device.

If we apply the squeezing operator S(ξ) to the vacuum state|0", we obtain the following state |ξ" = S(ξ)|0" = √1_µ ∞ # k=0 * ν 2µ .k√ 2k! k! |2k" (1.81) called squeezed vacuum state. Its average photon number is _!ξ|a†_{a|ξ" = |ν|}2 _which

represents the squeezing energy. In general, if $" _{= S(ξ)$S}†_{(ξ) is the state after the}

squeezer, the total number of photon is given by

!a†a_")! = sinh2r + (2 sinh2r + 1)!a†a"₎+ sinh(2r)!a2eıψ+ a†2eiψ"₎. (1.82)

We introduce the generic quadrature operator

xφ =

1 2(ae

−iφ_{+ a}†_eiφ_{). (1.83)}

By means of (1.77) we can evaluate the expectation value of xφ and its variance,

obtaining

!xφ" = 0

!(∆xφ)2" =

1

4[1 + 2 sinh

2_{r + 2 sinh r cosh r cos(θ− 2φ)]. (1.84)}

If we consider φ = 0 and φ = π/2 we have

x1 = xφ=0 = 1 2(a + a †_{), x} 2 = xφ=π/2= 1 2i(a− a †_{) (1.85)}

whose variances, for θ = 0 are

!(∆x1)2" = 1 4e 2r _(1.86) !(∆x2)2" = 1 4e −2r_{. (1.87)}

state for the pair of observable x1 and x2. The term squeezed in fact refers to the

stretching of quadratures’ variances.

Two-mode squeezing and the twin-beam state

Two mode squeezing unitary transformations, corresponding to Hamiltonians H ∝ a†_b†_{+ ab, are realized by the following operator}

S2(ξ) = exp{ξa†b†− ξ∗ab} (1.88)

with ξ = reiψ_{. As in the single-mode squeezing case, we can disentangle the operator}

considering a different two bosons realization of the algebra SU(1, 1): K+ = a†b†,

K₋= (K+)† and K3 =−1₂[K+, K−]. Using (B.6) we obtain

S2(ξ) = exp{ξa†b†− ξ∗ab} = exp ; ν∗ µa †_b† <

µ−(a†a+b†b)exp ; −ν µab < = exp ; −ν µab < µ(a†a+b†b+1)exp ; ν∗ µa †_b† <

where we set µ = cosh r and ν = eiψ_{sinh r. The Heinsenberg evolution of the field}

modes is given by   a" b"†   = S† 2(ξ)   a b†   S2(ξ) = S   a b†   (1.89) with S =   µ ν ν∗ _µ   . (1.90)

As for single mode squeezing, the number of quanta is not conserved. The Casimir of SU(1, 1) algebra in terms of field operators is given by a†_{a − b}†_{b, thus only the}

difference in mean photon number is conserved by S2(ξ). For this reason also the

two-mode squeezer is an active device.

The two-mode squeezing operator symplectic matrix is

FS2(ξ)=   µ 2 T2 T2 µ 2   , (1.91)

with T2 =   Re[ν] Im[ν] Im[ν] Re[ν]   . (1.92)

In the special case ξ = r ∈ , ψ = 0, we obtain

FS2(ξ)=   cosh r 2 sinh rσ3 sinh rσ3 cosh r 2   , σ3 =   1 0 0 ₋₁   . (1.93)

Using (1.88) we can evaluate the action of the two mode squeezing operator to the vacuum state _{|0" = |0" ⊗ |0", namely}

S2(ξ)|0" = |ξ"" = 1 µ ∞ # k=0 * ν µ .k |k" ⊗ |k", (1.94)

with ==₌ν_µ==_{= ≤ 1, or, equivalently}

|ξ"" ="1_{− |λ|}2 ∞

#

k=0

λk_{|k" ⊗ |k",} (1.95) being λ = ν_µ. The state _{|ξ"" will be also denoted as |Λ"" where Λ is the operator} Λ ="1− |λ|2_λa†_a

.

Let us introduce the superposition quadrature operators

z1 = 1 232 (a + a†+ b + b†), z2 = 1 i232 (a− a†_{+ b− b}†_{), (1.96)}

we obtain mean values on the state (1.94) _!zi" = 0, and variances

!(∆z1)2" = 1 4e 2r , !(∆z2)2" = 1 4e −2r _(1.97)

in analogy with what we have found for the single-mode squeezed vacuum state. The two mode squeezed vacuum state (1.94) is known as twin beam state of radiation (TWB), since it is an eigenstate of the photon number difference operator na − nb,

where na = a†a and nb = b†b, with eigenvalue zero:

Since na− nb is a constant of motion, TWB shows perfect correlation in the photon

number. The mean photon number of each mode is given by

!na" = !nb" = |ν|2 = sinh2r, (1.99)

while the variances are

!(∆na)2" = !(∆nb)2" =

1 4sinh

2_{2r. (1.100)}

Since!(∆ni)2" > !ni" both modes are described by super-Poissonian photon statistics,

i.e. broader than the Poissonian coherent state distribution.

If we neglect one mode of the twin beam state, operation realized throught a partial trace, we obtain the two reduced density operators $a and $b:

$i = ∞ # n=0 |ν|2n µ2n+2|n"ii!n|, i = a, b (1.101)

so that the probability Pi(n) of finding n photons in the mode i (i = a, b), using (1.99),

results

Pi(n) = !ni" n

(1 +_!ni")n+1

(1.102) which is the thermal distribution with average photon number !ni". That is, the

reduced state $i obtained measuring only one of the two mode of the TWB is exactly

the thermal state in Eq. (1.54)

It can be demonstrated that, considering the evolution of a balanced beam splitter U(π 4e iθ_{) (1.69),} U†(π 4e iθ_)S 2(ξ)U( π 4e

iθ_{) = S(ξe}iθ_{)⊗ S(−ξe}−iθ_{). (1.103)}

Hence from Eq. (1.103) it’s easy to verify that the TWB can be generated mixing two squeezed states at a balanced beam splitter, i.e.,

U(π 4e

iθ_)|ξeiθ_{" ⊗ | − ξe}−iθ_{" = U(}π

4e

iθ_{) S(ξe}ıθ_{)⊗ S(−ξe}−iθ_)|0"

= U(π 4e

iθ_{) S(ξe}ıθ_{)⊗ S(−ξe}−iθ_{) U}†₍π

4e

iθ_)|0"

= S2(ξ)|0"