measurement of the observableXon subsystemAgiven the knowledge of subsystem B(see sections 2.10.7 and 2.10.8 for the description on von Neumann and conditional entropy).
Since a negative conditional entropy is a sign of entanglement [114], relation (5.3) shows the effect of entanglement in reducing the uncertainty.
One can also consider that the state ρAB could have experienced some decoher-
ence which is purified by an environment, or eavesdropper, in a way that ρAB =
trE(|ABEihABE|). Considering the purity of the overall state i.e. S(AB) = S(E)one
can modify the relation (5.3) to find [20] :
S(XA|E) +S(PA|B)≥log2
1
c (5.4)
Although relation (5.4) is an elegant mechanism to bound the key that Alice and Bob can extract as shown in ref [20], it is only valid for measurement and states in finite-dimensional Hilbert space. For CV-QKD we need an uncertainty relation valid for infinite-dimensional Hilbert spaces and continuous-valued measurements. Particularly, we are interested in homodyne measurements (see section 2.8.1) of the canonically conjugate quadratures (see section 2.3 ).
Fortunately such a relation has been lately derived, based on the previous findings for discrete and finite measurements on infinite dimensional Hilbert spaces [115]. At first it was extended to countably infinite measurements with the possible application for a discretised version of a homodyne detection [26]. By considering the infinite precision limits of these coarse-grained POVM’s, results for the continuous spectra, had previously been extensively investigated for the Shannon entropies, with an analogous procedure for the quantum conditional von Neumann entropy employing by Ferenczi [28] and Furrer et al [26]. An alternative derivation was also presented by Frank and Lieb [27]. The ultimate result is the following relation for the homodyne detection performed on the infinite dimensional Hilbert spaces [26, 27, 28],
S(X|E) +S(P|B)≥log 2π¯h (5.5)
Utilizing relation (5.5) which bounds Eve’s information, we developed a key rate in infinite dimensional Hilbert space. I will describe it in detail in the following sections.
5.3
Quantum Cryptography in Continuous-Variable Regime
Since the focus of this thesis is on continuous-variable QKD, before showing how we use relation (5.5) for QKD purposes, I will briefly describe the most important fami- lies of continuous-variable QKD protocols using Gaussian states and measurements. A generic QKD protocol was previously described in section 4.2.
The most common CV-QKD protocols are the Gaussian protocols in which the infor- mation is encoded in the field quadratures (see section 2.3). In Gaussian protocols one can use either an entangled source (EB) or the equivalent picture; prepare and
measure (P&M) scheme, where Alice by using a random number generator, prepares an ensemble of signal states. In fact, one of the most significant results in CV-QKD was the discovery that the secret key can be generated by using coherent states [117]. It is easier to generate coherent states in the laboratory which opens the door for the real-life implementation of CV-QKD protocols. In this scenario Alice encodes two real variables aq and ap, onto a coherent state. She draws these variables from
a Gaussian distribution of variance Va and zero mean. By considering Va = V−1,
Alice obtains a thermal state of varianceV as an output (see section 3.7.1 and 3.7.4 for more details on the preparation of the coherent states). Alice sends the thermal state to Bob, where for each incoming state he measures either ˆqor ˆpquadrature by performing a homodyne detection (see section 2.8.1) . At the end, Alice has a long string of encoded data with the values (aq,ap)which are correlated with Bob’s ho-
modyne outcomes. After sifting, Alice keeps only the string of data compatible with Bob’s quadrature measurements [83].
The previous protocol can be modified by changing the homodyne detection to the heterodyne detection (see section 2.8.2) where both quadratures are observed simul- taneously. This protocol is famous as "no-switching protocol". The advantage of this protocol is that Alice can keep both real random variables, hence producing higher secret-key rates [83, 118].
Although coherent states are better candidates for CV-QKD protocols, squeezed states are also utilized especially in the early QKD protocols [83, 87, 116]. In this protocol Alice randomly chooses to squeeze and displace either ˆq or ˆp quadrature. When the state received by Bob, he randomly decides to perform a homodyne mea- surement on one quadrature. After sifting, Alice and Bob keep only the data which correspond to the same quadratures. Squeezed-state protocol can also be conducted where Bob performs heterodyne measurement on his received state [83].
In entanglement-based representation, a bipartite entangled state is distributed be- tween Alice and Bob. Here Alice’s preparation is realized by performing a suitable measurement on the entangled source. For (P&M) scheme, either squeezed [87, 116] or coherent [117] states, are respectively equivalent of performing the homodyne or heterodyne measurement on one part of the entangled state.
The communicating parties Alice and Bob, can implement either a direct reconcilia- tion (DR) or the reverse reconciliation (RR) which make a total of 16 possible Gaus- sian protocols.
5.3.1 Virtual Entanglement
In (EB) scheme [46], an entangled state is shared between two communicating parties Alice and Bob, while in (P&M) approach Alice prepares a coherent (squeezed) state and forwards it to Bob. The interchangeability between these two approaches has been demonstrated in a device dependent scenario [84]. This one-to-one analogy between (P&M) and (EB) schemes is famous as "virtual entanglement". This effect is captured in figure 5.1 for heterodyne measurement and coherent states. It can be understood as if the EPR source and measuring apparatus of Alice shown in