Y. Guo et al. / Physics Letters A 382 (2018) 372–381 373
the imperfection of the detector at Bob’s side with only slight cost
of transmission distance.
The
performance of proposed scheme in asymptotic limit and
finite-size regime is investigated. In asymptotic limit, we mainly
focus on improvement of secret key rate and the trend of trans-
mission
distance reduction. In detail, we quantify and analyze the
improvement of secret key rate by defining a novel parameter, im-
provement
ratio, and thereby find that the OA has totally different
effects on secret key rate under different transmission distance. As
a result, a parameter called critical transmission distance is used
to define the maximum transmission distance where OA exerts
positive influence upon the performance of eight-state protocol. In
finite-size regime, we obtain more practical result of the proposed
scheme.
This
paper is organized as follows: in Sec. 2 we give the de-
scription
of eight-state protocol and detail the proposed eight-state
scheme with an optic amplifier. Numerical simulation and discus-
sion
are shown in Sec. 3. Conclusion is drawn in Sec. 4.Detailed
derivation of equations is included in the Appendix.
2. Eight-state CVQKD with an optical amplifier
In this section, we mainly elaborate discretely-modulated eight-
state
CVQKD protocol with an optical amplifier. To make the
derivation self-contained, we briefly introduce the original eight-
state
CVQKD protocol first.
2.1. Eight-state CVQKD protocol
Alice first prepares eight coherent displaced states
|β
k
=|αe
kπ
4
, k ∈ Z, (1)
where α is chosen to be a positive real number and Z =
{
0, 1, 2, ..., 7}. Alice then randomly sends one of these eight co-
herent
states to Bob with equal probability through an insecure
quantum
channel controlled by an eavesdropper called Eve. The
quantum channel is characterized by the excess noise and the
transmission efficiency T . The total noise added to Bob’s input by
effects of quantum channel can be expressed by χ
line
=1/T + −1.
After
receiving the state sent by Alice, Bob subsequently per-
forms
homodyne detection or heterodyne detection. Since Bob’s
apparatus is imperfect, detection efficiency η can hardly achieve
1 which denotes the perfect detection. Moreover, Bob’s detector
also introduces some thermal noise υ
el
when Bob measures the
received state. Taking effect of detection efficiency η and thermal
noise υ
el
into account, one can derive a conclusive quantity χ
h
,
where χ
hom
=[(1 − η) + υ
el
]/η and χ
het
=[1 + (1 − η) + 2υ
el
]/η
are the case of homodyne detection and heterodyne detection, re-
spectively.
Consequently, the total quantity of noise χ
tot
can be
described as χ
tot
=χ
line
+χ
h
/T .
Finally,
Alice and Bob share a string of the secret key by using
error correction and privacy amplification.
Since
the PM version is equal to the entanglement-based (EB)
version, which is more convenient for security analysis. In EB ver-
sion,
Alice prepares a pure two-mode state,
|
8
=
1
4
7
k=0
|ψ
k
|β
k
, (2)
where the states
|ψ
k
=
1
2
7
m=0
e
i(4k+1)mπ
4
|φ
k
, k ∈ Z, (3)
are orthogonal non-Gaussian states.
The
state |φ
k
could be described as follows,
|φ
k
=
e
−
α
2
2
√
λ
k
∞
n=0
e
α
(8n+k)
√
(8n+k)!
|8n +k, k ∈ Z (4)
where
λ
0(4)
=
1
4
e
−α
2
[cosh(α
2
) +cos(α
2
) ±2cos(
α
2
√
2
)cosh(
α
2
√
2
)],
λ
1(5)
=
1
4
e
−α
2
[sinh(α
2
) +sin(α
2
) ±2cos(
α
2
√
2
)sinh(
α
2
√
2
)
±
√
2sin(
α
2
√
2
)cosh(
α
2
√
2
)],
λ
2(6)
=
1
4
e
−α
2
[cosh(α
2
) −cos(α
2
) ±2sin(
α
2
√
2
)sinh(
α
2
√
2
)],
λ
3(7)
=
1
4
e
−α
2
[sinh(α
2
) −sin(α
2
) ∓2cos(
α
2
√
2
)sinh(
α
2
√
2
)
±
√
2sin(
α
2
√
2
)cosh(
α
2
√
2
)].
(5)
Alice prepares the entangled state |
8
with variance V =
V
a
+ 1 and V
a
= 2α
2
, where she implements projective measure-
ments
on one of the set |ψ
k
ψ
k
| for k ∈Z to the first half of |
8
and projects the second half of set |ψ
k
ψ
k
| on one of the eight co-
herent
states |β
k
. After modulation, the modulated state is sent to
Bob through an untrusted quantum channel. The covariance matrix
of modulated state can be expressed by
γ
8
=
VI
2
Z
8
σ
Z
Z
8
σ
Z
VI
2
,
(6)
where V = V
a
+ 1, I
2
= diag[1, 1], σ
Z
= diag[1, −1], and the co-
variance
Z
8
= V
a
7
k=0
λ
k−1
3
2
λ
k
−
1
2
.
Bob’s detection efficiency is modeled by a beam splitter with
transmission efficiency η. An EPR state with variance N
d
is used to
model the thermal noise υ
el
that is introduced by the process of
Bob’s detector, where N
d
= ηχ
hom
/(1 −η) is for homodyne detec-
tion
and N
d
=(ηχ
het
−1)/(1 − η) is for heterodyne detection.
After
that, Bob implements the reverse reconciliation and pri-
vacy
amplification to generate the final bit string of secret key
shared with Alice.
2.2. The improved eight-state scheme with an optical amplifier
Due to the imperfection in Bob’s apparatus, the detection pro-
cess
cannot be ideal, thus final secret key rate is lower than ex-
pectation.
Fortunately, the impact of imperfect apparatus can be
reduced by applying an optical amplifier. In what follows, we elab-
orate
the proposed eight-state CVQKD with an optical amplifier
placed at Bob’s side. Fig. 1 shows the Entanglement-based (EB) ver-
sion
of the proposed scheme against Eve’s collective attacks.
Firstly,
Alice prepares an entangled state |
8
. After modula-
tion,
one mode of modulated state with variance V = V
a
+ 1is
sent to Bob through an insecure quantum channel.
After
the quantum channel, the mode go through an optical
amplifier placed in input of Bob’s apparatus and is detected by
Bob’s apparatus. In the viewpoint of calculating secret key rate,
this can be deemed trusted detection noise [28]. Here we mainly
take two types of OA: phase-sensitive amplifier (PSA) and phase-
insensitive
amplifier (PIA) into consideration, as shown in the light
green box of Fig. 1, these two types of optical amplifiers are de-
scribed
as follows.