GRÁFICO 4.2 INDICADORES DE CRISIS DE BALANZA DE PAGOS, 1950-

figure 5.1 (a) are hidden in a black box. The only things emerging from this box are the measurement values QA and PA, and Q and P which are sent to Bob. The

outputs of this black box are equivalent to the outputs of the other black box depicted in figure 5.1 (b), where QA and PA are chosen by the adequate random number

generator. The physical implementation of (P&M) scheme is easier and cheaper, while the equivalent (EB) scheme is better to use for the mathematical calculations. We will use the equivalence between these two schemes later to prove the key rate for the (P&M) scheme.

Het

EPR Source

Alice

towards Bob Source Modulators towards Bob

Alice

RNG (a) (b) Hom Hom

Figure 5.1: (a) showing an EB scheme where Alice measures both quadratures of her beam by performing a heterodyne (dual homodyne) measurement on her EPR arm. (b) showing the equivalent black box to (a), where a coherent state source generates the beam, which is then displaced in phase space using a modulator. The

two numbers are produced by a random number generator (RNG)

In the next section, I will show how we employed the uncertainty relation (5.5) to lower bound the secret key rate, in the asymptotic regime and taking into account only the Gaussian collective attacks. Asymptotic regime is when two communicating parties exchange infinite number of data to establish a key. Besides the collective at- tacks are shown to be the optimal attack using de Finetti theorem adapted to infinite dimension [119].

5.4

CV-QKD using entropic uncertainty relations

I will describe here the derivation of the key rate for RR protocol. Finding the secret key for DR protocol is straightforward. As mentioned earlier, in CV-QKD the secret key can be extracted from Alice and Bob’s quadrature measurements, symbolized by the random variables X_A(B) with outcomes xA(B) which follow probability distri-

butions p(x_A(B)). Here the detector and reconciliation efficiencies are neglected for

simplicity. These effects will be included later. The asymptotic RR secret key rate is lower bounded by [120, 121] :

where the left white triangle denotes the information flow during reconciliation from Bob to Alice. HereI(XB :XA)denotes the classical mutual information between Alice

and Bob (see section 2.10.6), withH(X)being the continuous Shannon entropy of the measurement strings defined as H(X) = −R

dx pX(x)logpX(x)(see section 2.10.3),

and χ(XB : E) = S(E)−R dxBp(xB)S(E|xB) denotes the continuous Holevo bound

(see section 2.10.9) with S(X) being the von Neumann entropy (see section 2.10.7) andS(A|B)the conditional von Neumann entropy of AgivenB(see section 2.10.8). In the case that systems are classical, i.e. B = XB, the von Neumann entropies may

be replaced by the Shannon entropy.

By substituting the classical mutual information and the Holevo bound in the relation (5.6) we have:

K/ ≥ H(XB)−H(XB|XA)−S(E) +

Z

dxBp(xB)S(E|xB) (5.7)

Using the following definition:

S(XB|E) = H(XB) +

Z

dxBp(xB)S(ρx_EB)−S(E) (5.8) whereρx_EB is the conditional state of E given measurement outcome xB, and combin-

ing the relations (5.7) and (5.8) we have :

K/ ≥S(XB|E)−H(XB|XA) (5.9)

Now using the entropic uncertainty relation (5.5) and changing the places of A and B, one can bound the eavesdropper’s information as follows:

S(XB|E)≥log 2π¯h−S(PB|A) (5.10)

ConsideringS(PB|A)≤ S(PB|PA) = H(PB|PA), we can have:

S(XB|E)≥log 2πh¯ −H(PB|PA) (5.11)

By substituting relation (5.11) in (5.9) and assuming ¯h=2 we have:

K/ ≥log 4π−H(PB|PA)−H(XB|XA) (5.12)

Thus by employing an expression that relies only upon the conditional Shannon entropies, we have bounded the secret key. These conditional Shannon entropies are directly available for Alice and Bob. Besides for any probability distribution p(x), it can be demonstrated via a variational calculation that the analogous Shannon entropy is maximized for a Gaussian distribution of the same variance. In other words, by measuring Bob’s conditional variances, Alice and Bob can bound their secret key rate for this protocol . Substituting the Shannon entropy for a Gaussian distribution HG(XB|XA) = logp2πeV_XB|XA, where VXB|XA = VXB −

hXAXBi2

V_XA is Bob’s

§5.4 CV-QKD using entropic uncertainty relations 67

RR key rate as:

K/≥log( 2 ep

V_XB|XAVPB|PA

) (5.13)

The DR expression is obtained by simply permuting the labels of Alice and Bob. The extension of equation (5.13) to the other Gaussian protocols is straightforward and is given in ref [113]. Due to the importance of the protocol with coherent states and homodyne measurement in this thesis, I will present its key rate calculation here. As mentioned in section 5.3.1, a DR coherent state in EB picture involves Alice mak- ing a heterodyne detection upon her arm of an EPR pair, where she mixes her mode with the vacuum. The resulting modes are A1 and A2 upon which she measures

ˆ

x and ˆp respectively. Bob makes a homodyne detection by switching between the quadratures. The DR key rate is then bounded by:

K. ≥S(XA1|E)−H(XA1|XB) (5.14)

where the right white triangle denotes here the information flow during reconcilia- tion from Alice to Bob. After Alice’s projective measurement uponA2the stateρA1BE

is pure and we can again apply the entropic uncertainty relation :

K. ≥log 4π−S(PA1|B)−H(XA1|XB) (5.15)

≥log 4π−H(PA1|PB)−H(XA1|XB) (5.16)

Although we do not measure ˆpupon modeA1, we trust the device in Alice’s station.

Therefore we can assume H(PA1|PB) = H(PA2|PB) which is measured. Hence we

have: K. ≥log 4π−log q 2πeV_PA2|PB−log q 2πeV_XA1|XB (5.17) =log( 2 ep V_XA1|XBVPA2|PB ) (5.18)

In order to compare with the previous protocol (EB scheme and homodyne detection) we use the fact that mode A is mixed with the vacuum as shown in figure 5.1. Hence we have: XA1 = 1 √ 2(XA+Xv) (5.19) VXA1 = 1 2(VXA +1) (5.20)

whereXv is the vacuum fluctuation, and its variance isVXv =1.

Alice’s variance on Bob’s measurement is as follows: V_XA1|XB =VXA1 −

hXA1XBi 2

VXB

(5.21) Using the expressions (5.21) and (5.20) we have :

V_XA1|XB = (VXA +1) 2 − h_√1 2(XA+Xv)XBi 2 VXB (5.22) = 1 2VXA − 1 2 hXAXBi2 VXB (5.23) 2V_XA 1|XB−1=VXA|XB (5.24)

A similar expression can be written for VPA₂. Hence considering the equation (5.18)

the key rate will be given as follows:

K. ≥log( 4

eq(V_PA|PB+1)(VXA|XB +1)

) (5.25)

In document DESARROLLO, VAIVENES Y DESIGUALDAD (página 177-180)