CUADRO 3.15 LA DESIGUALDAD EN EL CONO SUR DE SUDAMÉRICA, 1870 Y

whereσ_abqq is the covariance and∆qˆa and∆qˆbare the variances (please see 2.44). Sim-

ilar relation applies for other quadrature ˆp or the other direction of the conditional variances. The conditional variance quantifies the reduced uncertainty on subsystem afollowing a measurement on subsystemb. This gives a measure where the subsys- temsa andbare not perfectly correlated, but by performing a measurement on one subsystem some information can be gained about the other subsystem.

A state demonstrates an EPR paradox if the product of the conditional variances of the two orthogonal quadratures is below one. Satisfying this criterion is a sufficient but not the necessary condition for a state to be entangled.

2.11.6 Quantum Discord

While it was thought previously that the absence of entanglement implies the classi- cality of the states, recent research has shown that some form of quantum correlations can exist even in the separable states. This notion of quantumness which was first discovered by Ollivier and Zurek [30] is calledquantum discord. This is an important form of quantum correlations that attract lots of attentions during recent years. This concept will be elaborated in the next chapter.

2.12

Summary

In this chapter I reviewed the key concepts of quantum optics and information the- ory which will be used though out this thesis. This includes introducing the optical field quadratures, different quantum states of light, Wigner function, Gaussian states and symplectic transformations, quantum measurement and the detection schemes, phase and amplitude modulations which are used in the continuous-variable quan- tum optics experiments. In the information theory part, I covered the concepts of Shannon entropy, relative entropy, conditional entropy, mutual information, von Neumann entropy, quantum mutual and conditional entropy and Holevo bound. In addition, in the section of Quantum Correlations, I discussed Entanglement and non-locality, the criterion of entanglement in pure states and two important entan- glement criteria in the two-mode Gaussian states. I mentioned "Discord" here as a form of quantum correlations that can exists even in the separable states. However, this concept will be discussed thoroughly in the next chapter.

Chapter3

Experimental Verification of

Quantum Discord in

Continuous-Variable States

3.1

Introduction

Quantum systems can be correlated in a superior way than the classical systems. These correlations are particularly important as the main resource for quantum infor- mation processing and quantum communication. Due to the importance of quantum correlations, quantifying the classical and quantum part of the correlations has been the subject of many studies during the last decade. As mentioned in the previous chapter, entanglement was considered before as the only form of non-classical corre- lation and the main resource that causes quantum computation and communication outperform the classical counterparts. Any correlation in the absence of entangle- ment was thought to be purely classical. This idea was also supported by the funda- mental Bell’s inequality [5], which cannot be violated by any classical or quantum superposition and requires genuine entanglement to be violated. However, nowa- days the accepted definition of a classical bit, is in one of two fully-distinguishable states. According to this definition, general quantum states form a subset of the sep- arable states, meaning that some separable states are quantum correlated [43]. This suggests that entanglement no longer is the only source of quantumness. In addi- tion, quantum computation models with no or very little entanglement were sug- gested which can achieve much higher efficiency than classical computers [49, 50]. These models and other studies during the last decade have convinced physicists that entanglement and classical correlation do not exhaust all the possible correla- tions. Ollivier and Zurek were the first to characterize this extra quantum correlation beyond entanglement and named it "Discord" [30]. Quantum discord has attracted so much attention during the last decade and was suggested as a figure of merit for characterizing the quantum resources in a computational model [51]; it was also introduced as a resource for quantum state merging [52, 53], and for encoding in- formation onto a quantum state [54]. Besides it has been shown as the necessary resource for quantum remote state preparation, outperforming entangled states [55]

This measure of nonclassical correlation has been extended to continuous-variable systems to study quantum correlations in Gaussian states [35, 56] and certain non- Gaussian states [57]. In addition, resilience of quantum illumination, a paradoxical technique that employs entanglement to detect reflecting objects in noisy environ- ments, has also been demonstrated to be due to quantum discord [58]. It also de- termines the interferometric power of quantum states used in metrology [59]. Since quantum discord is more robust than entanglement, it has been shown that it can serve well in Device-Dependent quantum key distribution, where the trusted noise is so high that prevents the distribution and distillation of entanglement [60]. The relation between quantum coherence (superposition) and quantum discord [61], and incoherent operation and discord-type quantum correlation [62] has also been stud- ied recently.

Considering the importance of quantum discord, of particular interest is to experi- mentally verify discord for an unknown quantum system. Methods have been pro- posed to test for nonvanishing quantum discord of bipartite discrete-variable quan- tum states [63, 64, 65, 66, 67, 68, 69], some of which have been experimentally imple- mented in nuclear-magnetic-resonance systems [70, 71] and in an optical system [72]. A more general measurement-based method for verifying quantum discord was in- troduced in ref [73], which can be applied to both discrete- and continuous-variable systems. We have introduced and demonstrated experimentally a simple and effi- cient technique to verify quantum discord in unknown Gaussian sates and a certain class of non-Gaussian states, which I will elaborate it here.

The structure of this chapter is as follows. At first, I will review the definition of quantum discord as the measure of quantum correlation beyond entanglement in section 3.2. Then I mention the Gaussian quantum discord and present the general form of it in section 3.3. After that, in section 3.4, I will then discuss the general method for verification of quantum discord proposed in ref [73]. The main part of this chapter which describes our simple experimental technique to verify quantum discord in an unknown Gaussian states and certain class of non-Gaussian states, will appear in section 3.5. It itself consists of two other sections, where in the section 3.6, I will explain the theoretical development of our technique, and in the section 3.7, I will detail our experimental implementation and results. This survey has been published in the " Journal of Physics B " with the title and author list as follows:

"S. Hosseini, S. Rahimi-Keshari, J. Y. Haw, S. M. Assad, H. Chrzanowski, J. Janousek, T. Symul, T. C. Ralph and P. K. Lam, "Experimental Verification of Quan- tum Discord in Continuous-Variable States", J. Phys. B: At. Mol. Opt. Phys. 47, 025503 (2014)."

This research has been carried out in the "CQC2T Center of Excellence". The the- oretical development has been done by Saleh Rahimi-Keshari and Timothy C Ralph at the University of Queensland. The experiment which consisted of four parts; one Gaussian state and three non-Gaussian states, was implemented in our group at the Australian National University. Here I will present the theoretical development in order to make the experimental implementation and results understandable.