Since the focus of this thesis is on Quantum Correlations, I specify this section to describe the well-known forms of quantum correlations and the way to quantify them especially in Gaussian states.
2.11.1 Entanglement and non-locality
When two physical systems have an interaction with each other, some form of cor- relation with a quantum nature is created between them, which remains even when the two systems get specially separated. This suggests that, if one performs a mea- surement on a local observable on the first system, the state of the second system, no matter where it is, is modified instantaneously. This phenomenon, named by Ein- stein, Podolsky and Rosen as "spooky action at a distance" [3], is calledentanglement which is the non-classical and non-local quantum correlation. However, non-locality and entanglement are a bit different. It can be understood from the general frame- work of no-signalling theories which demonstrate more non-local features than the quantum mechanics [33, 36]. According to Bell [33, 37], non-locality is a channel in nature which allows one to distribute correlations between distant parties, in a way that the correlations are not pre-determined at the source, and the correlated random variables can be generated when distant parties perform local measurements on their subsystems. Quantum mechanics describes this channel as an entangled pair [33]. As I will describe in the following chapters this characteristic of nature is harnessed in secure quantum communications.
Considering the importance of quantum entanglement, it is desirable to have an op- erational criterion to examine if a given state is entangled or not. As I will show shortly, forpure statesof the composite quantum system, it is relatively easy to quan- tify entanglement. However, the situation is more complicated withmixed states, as a mixture can be in many different ways where no one can extract all the information it contains [33].
2.11.2 Entanglement Criteria for Pure Bipartite States
A pure quantum state |ψi ∈ H = H1⊗H2 is entangled if it cannot be written as a
§2.11 Quantum Correlations 27
|ψi= |φi1⊗ |χi2 ≡ |φ,χi. (2.83) where|φi1 ∈H1and|χi2∈ H2.
In order to quantify entanglement one can write a pure quantum state in its unique Schmidt decomposition [41] as follows [33] :
|ψi= d
∑
k=1 λk|uk,vki, (2.84) where d=min{d1,d2} , (2.85) λk ≥0, d∑
k=1 λ2k =1. (2.86)The local bases {|uki} ∈ H1 and {|vki} ∈ H2 are the Schmidt bases, the positive
numbers {λk}are the Schmidt coefficients and the number d of the non-zero terms
in the Schmidt number. It can be seen that the product states |ψi = |φ,χi can be automatically written in the Schmidt form whend= 1. In other words if a state can be written as Schmidt decomposition with only one coefficient, then it is necessarily a product state. Hence, a pure state|ψiof a bipartite system is entangled if and only ifd>1 [33].
2.11.3 Entanglement Criteria for Mixed States
A mixed state can be written as a convex combination of pure states : ρ=
∑
k
pk|ψkihψk| (2.87)
Now the problem is that this decomposition is not unique, unless ρ is already a pure state. This means that the mixed states can be prepared in many different ways which makes the entanglement’s quantification very difficult. Considering the ambiguity on the state preparation, no one knowsa prioriif the correlations between the subsystems arose from a quantum interaction or were induced by means of LOCC (Local Operations and Classical Communications) which causes classical correlations [33].
A mixed quantum state of a bipartite system, described on the Hilbert space H=H1⊗H2, is separable if and only if it can be written as follows :
ρ=
∑
k
pk(σk⊗τk). (2.88)
tangled . However, this is a very impractical way of checking if a state is entangled or not. Since deciding entanglement or separability according to this definition would require one checks all the infinitely many decomposition of a state ρand look for at least one of them to be in the form of 2.88. Hence, several operationalcriteria have been developed to check entanglement in mixed quantum states [33]. In addition, developing a criterion for checking the inseparability will be dramatically simplified if we restrict ourselves to the certain class of quantum states. Considering that the focus of this thesis is on the continuous-variables and particularly on Gaussian states, here I only present two entanglement criteria applicable to the two-mode Gaussian states. The first one is theDuan inseparability criterionintroduced by Duanet al. [47], which provides a necessary and sufficient condition for the inseparability of two- mode Gaussian states. The second one is the EPR-paradox criterion introduced by Reid [46] which quantifies the degree of EPR paradox of a state.
2.11.4 Duan Inseparability Criterion
The Duan inseparability criterion quantifies the strength of entanglement of a quan- tum state. For the case of two-mode quadrature entangled state, It is defined as [48]: I = q ∆2xˆ a±b∆2pˆa±b (2.89) where∆2Oˆ
a±b=minh(δOˆa±δOˆb)2i/2 . IfI <1, the state is inseparable. I =0 cor-
responds to the best possible entanglement obtained from combining two perfectly squeezed beams [48].
2.11.5 EPR Paradox Criterion
EPR criterion measures the degree of the EPR paradox, introduced by Einstein, Podolsky and Rosen in 1935 [3]. It is based on the ability of a state to produce an apparent violation of the Heisenberg uncertainty relation between two conjugate variables. A quantifying measure of an EPR violation for the continuous variables was introduced by Reid in 1988 [45, 46]. This criterion which is more restrictive than the Duan inseparability criterion, is defined as the product of the conditional variances of the phase and amplitude quadratures as follows:
eab =∆qˆa|b∆pˆa|b <1 (2.90)
eba =∆qˆb|a∆pˆb|a <1 (2.91) The conditional variances are defined as :
∆qˆa|b =∆qˆa−
|σabqq|
∆qˆb