Journal of Quantum Informatio n Science, 2011, 1, 116-120
doi:10.4236/jqis.2011.13016 Published Online December 2011 (http://www.SciRP.org/journal/jqis)
Practical Stabilization of Counterfactual
Quantum Cryptography
Musheng Jiang, Shihai Sun, Linmei Liang*
Departme nt of Physics , National University of Defense Technology, Changsha, China
Received August 14, 2011; revised Se p tember 23, 2011; acc e pt ed September 30, 2011
Abstract
A novel counterfactual quantum key distribution scheme was proposed by T.-G. Noh and a strict security
analysis has been given by Z.-Q. Yin, in which two legitimate geographical separated couples may share se-
cret keys even when the key carriers are not traveled in the quantum channel. However, there are still plenty
of practical details in this protocol that haven’t been discussed yet, which are of significant importance in
physical implementation. In this paper, we will give a practical analysis on such kind of counterfactual
quantum cryptography in the aspects of quantum bit error rate (QBER) and stabilization. Furthermore, modi-
fied schemes are proposed, which can obtain lower QBER and will be much more robust on stabilization in
physical implementation.
Keywords: Counterfactual Quantum Cryptography, Quantum Bit Error Rate, Practical Stabilization
1. Introduction
Combining with one time pad, quantum key distribution
(QKD) [1,2] based on the fundamental principles of
quantum mechanics can in principle offer the uncondi-
tionally secure private communications between two
users, Alice and Bob. Many successful QKD experi-
ments [3-9] have been realized during the past decade,
which must transmit key carriers in a public quantum
channel. Besides, an entirely different scheme based on
the quantum counterfactual effect was proposed recently,
named as counterfactual quantum cryptography [10].
Since this counterfactual quantum cryptography is based
on polarization-multiplexing, we call it polarization-mul-
tiplexed counterfactual quantum cryptography (PCQC) in
this paper. In this scheme, the task of a secret key distri-
bution can be accomplished without transmitting any
particle carrying secret key information in the quantum
channel. A photon that carries secret key information has
been confined from its birth to death within Alice’s se-
cure zone, and Eve can never access the photon, but Bob
still can extract a secret key from the nondetectable
events, which is a surprising counterintuitive fact. Fur-
thermore, the PCQC protocol provides clear security
advantages when taking photon number splitting attack
(PNS) [11] into account. More recently, a strict security
analysis of the PCQC protocol has also been given in an
ideal situation [12]. However, some practical factors are
unconsidered. In this pap er, we will give a practical ana-
lysis on the PCQC protocol in the aspects of quantum bit
error rate (QBER) and stabilization. Furthermore, some
modified schemes are proposed, which can obtain lower
QBER and will be much more robust on stabilization in
physical implementation.
2. Stabilization Analysis of Counterfactual
Quantum Cryptography
To begin with, let us review the PCQC protocol briefly.
As shown in Figure 1 (cited from [10]), Alice randomly
encodes a single photon in one of the two orthogonal
Figure 1. (Color online) Schematic diagram of the original
polarization-multiplexed counterfactual quantum crypto-
graphy system (cited from Ref. [10]).

117 M. S. JIANG ET AL.
polarization states to represent his bit value. The encoded
single-photon pulse entering a Michelson-interferometer
is split into two pulses by a beam splitter (BS) and trav-
els through two modes a and b. In mode a, the pulse is
reflected back by a Faraday mirror (FM), and always
confined within Alice’s secure zone. In mode b, the pulse
travels from Alice’s site to Bob’s site. Bob also ran-
domly chooses one of the two orthogonal polarizations
representing his bit value by blocking the corresponding
polarization state, named as polarization-selection. If an
optical pulse incident to Bob’s site is horizontally polar-
ized, it passes thro ugh the polarizin g beam splitter (PBS)
and goes directly to the high-speed optical switch (SW).
However, if the pulse is vertically polarized, it is first
reflected by the PBS, passes through the optical loop
(OL), and then goes to the SW. Therefore, through accu-
rate control of the switch timing, Bob can effectively
switch the selected polarization state to the detector D3,
while the other was reflected back to Alice’s BS. Thus, if
the bit values chosen by Alice and Bob are different, the
split pulse going through mode b is not blocked by Bob,
and the two split pulses are recombined in the BS, and
the single photon is detected at detector D2 certainly as a
result of quantum interference. On the other hand, if the
two bit values are equal, the split pulse going through
mode b is blocked by detector D3. Then, the photon can
be detected at detector D1 with a finite probability,
which is caused by the breakage of the interference. In
this case, the photon has been completely confined
within Alice’s secure zone, and Eve can never access the
photon, as it has only traveled through mode a. Alice and
Bob can then establish a sifted key by selecting only the
events for which D1 clicks alone. In summary, the proc-
ess can be described as follows: 1) when the bit values of
Alice and Bob are different, D2 clicks with probability
1/2; 2) when the bit values of Alice and Bob are equal,
D1 clicks with probability RT2, D2 clicks with prob-
ability 2
R2, and D3 clicks with probability T2. The
events for which D1 clicks alone are used to extract a
secret key, and the other events are used to detect the
latent eavesdropper (Eve). Here and T1 are
the reflectivity and transmissivity of the BS, respectively.
RR
Now, we analyze the PCQC protocol in the aspects of
QBER and stabilization. It is mentioned in [10] that it
may be hard to stabilize a long-armed interferometer,
which is related to QBER and stabilization. For a long-
armed interferometer, the symmetry of the interferometer
relies sensitively on th e environmental disturbances such
as temperature fluctuations. The breakage of this sym-
metry will cause a variation of the interference, for ex-
ample, phase drift may even completely destroy the in-
terference. Ideally, it is easily seen that th e single photon
is detected at detector D2 with certainty when the bit
values of Alice and Bob are different, as a result of
quantum interference. But in fact, we can never keep the
interference perfect for a long-armed interferometer un-
der environmental disturbances. The extreme result is
that the optical path difference of the interferometer is
larger than the coherence length of the light source be-
cause of fiber length drift. Consequently, the interference
is completely destroyed, that is, mode a and mode b of a
single photon are not coherent any more. In this case, a
single photon can be detected at detector D1 with prob-
ability 1/2, which is an error ev ent and add s an add ition al
QBER in the raw keys. Generally speaking, the inter-
ferometer can be stabilized using feedback control. Here
we assume that, once the bit va lues of Alice and Bob are
different, mode a and mode b of a single photon are al-
ways coherent under feedback control. However, error
events may still happen with some probability as a resu lt
of phase drift. Here we note this event as phase-crosstalk,
and the corresponding probability as
p
hase . Note that in
PCQC protocol, the events for extracting a secret key
have a probability of RT/2, and the probability of error
events caused by phase-crosstalk is
C
phase
C2. Thus, the
additional QBER caused by phase-crosstalk can be writ-
ten as
phase
phase phase
C2
QBER C2RT 2 (1)
Furthermore, since Bob must perform polarization-
selection through the PBS to represent his bit value, in-
stability due to the polarization mode dispersion effects
in long-distance single-mode fiber should also be con-
sidered. In fact, a long-distance single-mode fiber should
be considered as a birefringent medium as a result of its
intrinsic imperfection and environmental disturbances.
When a single photon passes through such a birefringent
medium, the polarization mode dispersion effect is visi-
ble, which will result in the instability of polarization.
Therefore, well performed polarization compensation is
needed to compensate for the instability of polarization;
otherwise the polarization-selection will not perform
well, resulting in another additional QBER in the raw
keys. But in fact, the polarization of a single photon will
of course drift away from its original state more or less
after traveling from Alice’s site to Bob’s site, regardless
of how well the polarization compensation is performed.
In the process of polarization-selection, if a single photon
enters Bob’s PBS with horizontal polarization, it passes
through the PBS and goes directly to the SW; and if the
single photon is vertically polarized, it is firstly reflected
by the PBS, passes through the OL, and then goes to the
SW. However, polarization dr ift may sometimes bring in
unexpected result. For example, when the bit values of

M. S. JIANG ET AL.
118
Alice and Bob are different, the split pulse going through
mode b should not be blocked by Bob but was reflected
back to Alice’s BS, and the two split pulses interfere in
the BS; unfortunately, the polarization of a single photon
may drift away from its original state when the photon
enters Bob’s PBS and then lose its way. Therefore, the
split pulse going through mode b may be blocked by Bob
and D1 clicks with some probability, which is an error
event too. Similarly, we note this event as polarization-
crosstalk, and the corresponding probability as
p
olarization . Then the additional QBER caused by polari-
zation-crosstalk can be expressed as
C
polarization
polarization polarization
RTC 2
QBER RTC2RT 2  (2)
Finally, the total QBER of the system can be deduced
when we consider all the potential factors: phase-
crosstalk, polarization-crosstalk and the dark-counts of
detector D1 D1

.


polarizationD1 phase
total
phase polarization
RT C2C2
QBER CRTC RT2
 

(3)
Besides, phase-crosstalk and polarization-crosstalk
also bring in other problems. On the one hand, to detect
Eve’s intervention in PCQC protocol, Alice and Bob
monitor the operation of the interferometer. They tell
each other whether or not each of the detectors clicked
for a photon, and obtain the probability for each event
over a period of time. If D2 or D3 clicks, they also an-
nounce both the detected polarization state and the initial
polarization states that were chosen. However, the opera-
tion of the interferometer may also be affected by phase-
crosstalk and polarization-crosstalk: 1) the probability
for each event may be changed by both the phase-
crosstalk and the polarization-crosstalk; 2) when D3
clicks, the detected polarization state and the initial po-
larization states may be different as the result of polari-
zation-crosstalk. Therefore, Alice and Bob may be hard
to detect Eve’s intervention in a way. And the following
process of quantum crypto graphy such as privacy ampli-
fication should be based on the worst condition to guar-
antee the security. In other words, phase-crosstalk and
polarization-crosstalk impair the security of the protocol.
On the other hand, the control of stabilization is crucial
to compensate for both the phase-crosstalk and the po-
larization-crosstalk. However, since it is a multivariable
feedback control problem, it is hard to operate continu-
ously alongside key distribution, which will result in key
rate reduction. And in practical implementation, the com-
plexity may be terrible while the precision of stabiliza-
tion control may be limited, compared to single-variable
feedback control.
3. Improvement on Stabilization
If we can avoid the phase-crosstalk or polarization-
crosstalk, the system can obtain lower QBER and will be
much more robust on stabilization in physical imple-
mentation, and the advantage of counterfactual quantum
cryptography will attract more attention in physical im-
plementation. Here we present a modified scheme of
PCQC protocol, in which we have eliminated the polari-
zation-crosstalk simply by moving the PBS from Bob’s
site to Alice’s site. Figure 2 shows the schematic of our
PCQC system. In fact, our modified scheme of PCQC
protocol is similar to the original one and we focus on
the differences here. Within Alice’s secure zone, the po-
larization-encoded single-photon pulse will choose a
path according to its polarization state before entering
the Michelson-interferometer. If an optical pulse is en-
coded horizontally polarized, it passes through the PBS
and goes directly into the interferometer. However, if the
pulse is encoded vertically polarized, it is first reflected
by the PBS, passes through the OL, and then enters the
interferometer. Therefore, the encoded single-photon
pulses with different polarization enter the interferometer
with different time nodes, and arrive at Bob’s SW with
different time nodes in mode b. Thus, Bob can directly
switch the selected polarization state to the detector D3
while the other was reflected back to Alice’s BS.
It is worthwhile to point out here that this modified
PCQC protocol provides equivalent function as the origi-
nal one in an ideal situation. Evidently, all the processes
of PCQC can be implemented by the modified scheme.
And since Eve can only access one subsystem (path b)
while she can never access the other subsystem (path a),
the security of the modified scheme is also guaranteed by
no-cloning principle for orthogonal states: if reduced
density matrices of an available subsystem are nonor-
thogonal and the other subsystem is not allowed access,
it is impossible to distinguish two orthogonal quantum
states without disturbing them [10]. Moreover, in physi-
cal implementation, the modified PCQC not only inherits
all the advantages of the original one, but also has a sig-
Figure 2. (Color online) Schematic diagram of the modified
polarization-multiplexed counterfactual quantum cryptog-
raphy system.

119 M. S. JIANG ET AL.
nificant improvement on QBER and stabilization. Since
the encoded single-photon pulses with different polariza-
tion are separated just after their emission, the polariza-
tion-crosstalk due to polarization mode dispersion effects
in long-distance single-mode fiber is eliminated. Then
the polarization-selection can be performed without error
through accurate control of the switch timing in Bob’s
site. That is to say, all the error events caused by the po-
larization-crosstalk do not exist in our modified PCQC.
Therefore, the additional QBER due to polarization-
crosstalk is discarded. And the total QBER of the system
can be expressed as

D1 phase
total
phase
RTC 2
QBER CRT
 
2
(4)
which is smaller than that in Equation (3). Moreover, the
security and the stabilization are also improved because
of the elimination of the polarization-crosstalk. And sin-
gle-variable feedback control can meet the requirement
of stabilization control, which is much simpler and more
precise.
We also present another modified counterfactual quan-
tum cryptography based on wavelength-multiplexing,
named as wavelength-multiplexed counterfactual quan-
tum cryptography (WCQC). Figure 3 shows the sche-
matic of our WCQC system. Alice randomly send s out a
single photon of wavelength 1

or 2

to represent his
bit value. The photons of different wavelengths are cou-
pled to the Michelson-interferometer via wavelength-
division multiplexin g WDM. In mode b, the pulses travel
from Alice’s site to Bob’s site and will be separated in
time by group velocity dispersion, then transmitted to
SW with different time nodes. For example, the group
velocity dispersion of single-mode fiber is 17D

ps/nm·km, supposed 12 20

  L nm, and the
fiber length between Alice and Bob is km, then
the group delay in optical transmission is 30
10.2DL

  ns. Therefore, Bob can also ran-
domly choose one of the two wavelengths representing
his bit value by accurate control of the switching time,
blocking the single photon with corresponding wave-
Figure 3. (Color online) Schematic diagram of the Wave-
length-multiplexed counterfactual quantum cryptography
system.
length. Similar to PCQC protocol, all the processes of
counterfactual quantum cryptography can be imple-
mented effectively.
In fact, Alice can even randomly encode the single
photon pulses in different time nodes to represent his bit
value, by random control of SW1 shown in Figure 4.
Similarly in mode b, single-photon pulses encoded in
different time nodes will be transmitted to SW2 in Bob’s
site with different time nodes. And counterfactual quan-
tum cryptography can be carried out similarly. Since time-
multiplexing is used, this modified scheme is named as
time-multiplexed counterfactual quantum cryptography
(TCQC).
It is easily seen that both the TCQC protocol and the
WCQC protocol are not involved in the polarization of
the single-photons, so does the polarization-crosstalk.
The total QBER of these systems can also be expressed
by Equation (4). Therefore, they are much more robust
on QBER, security and stabilization in physical imple-
mentation.
4. Discussion and Conclusions
We have given a practical analysis on counterfactual
quantum cryptography, and have proposed some kinds of
modified schemes, in which the single-photons are en-
coded in orthogonal states of different degrees of free-
dom. Here we can characterize the counterfactual quan-
tum cryptography as follows: key carriers can be en-
coded into either orthogonal states or nonorthogonal
states in any degrees of freedom; Alice and Bob extract
secret keys in the encoded degrees of freedom (such as
polarization, time and wavelength), while another de-
grees of freedom (usually the phase) is used to detect the
disturbance of Eve; Alice and Bob can extract secret
keys without transmitting any particle carrying secret key
information in the quantum channel. The key point is
that these two kinds of freedom are encoded in one
physical carrier. These characteristics make the counter-
factual quantum cryptography completely different from
the previous protocols of QKD, BB84 protocol for ex-
ample, in which nonorthogonal states must be used with
Figure 4. (Color online) Schematic diagram of the time-
multiplexed counterfactual quantum cry ptography system.

M. S. JIANG ET AL.
120
basis reconciliation, signal particle transmission is needed,
and both the extract of secret keys and the detection of
the latent Eve are operated in the same degree of free-
dom. Furthermore, these characteristics have provided
security advantages, especially when considering the
PNS attack.
In summary, we have analyzed the phase-crosstalk and
the polarization-crosstalk of PCQC protocol in the as-
pects of QBER and stabilization. We find that the phase-
crosstalk and the polarization-crosstalk not only bring in
QBER but also impair the security and stabilization of
the protocol. And a modified scheme of PCQC protocol
without polarization-crosstalk has been proposed in this
paper. Moreover, another two protocols of counterfactual
quantum cryptography, TCQC and WCQC, were pro-
posed. Time-multiplexed and Wavelength-multiplexed
are used in these protocols, which are independent of the
polarization of the single-photons. Since we have avoided
the polarization-crosstalk in all of these modified schemes,
they have a significant improvement on QBER, security
and stabilization. However, the phase-crosstalk is a re-
mained problem and single-variable feedback control is
still needed to guarantee the stabilization of these modi-
fied protocols of counterfactual quantum cryptography.
Further study on counterfactual quantum cryptography
may find out another practical choice of QKD.
5. Acknowledgements
This work is supported by National Natural Science
Foundation of China Grants No.61072071. Shi-Hai Sun
is supported by H unan Provincial Innovation Fo undation
for Postgraduate No.CX2010B007, and Fund of Innova-
tion, Graduate School of NUDT No.B100203.
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