Quantum Filtering of Optical Coherent States
Quantum Filtering of Optical Coherent States
C. Wittmann1,∗ D. Elser1, U.L. Andersen1,2, R. Filip1,3, P. Marek4, and G. Leuchs1
Institut für Optik, Information und Photonik, Max-Planck Forschungsgruppe,
Universität Erlangen-Nürnberg, Günther-Scharowsky-Straße 1, 91058, Erlangen, Germany
Department of Physics, The Technical University of Denmark, 2800 Kongens Lyngby, Denmark
Department of Optics, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic
School of Mathematics and Physics, Queens University, Belfast BT7 1NN, United Kingdom
(Dated: August 23, 2021)
We propose and experimentally demonstrate non-destructive and noiseless removal (filtering) of
vacuum states from an arbitrary set of coherent states of continuous variable systems. Errors i.e.
vacuum states in the quantum information are diagnosed through a weak measurement, and on
that basis, probabilistically filtered out. We consider three different filters based on on/off detection
phase stabilized and phase randomized homodyne detection. We find that on/off detection, optimal
in the ideal theoretical setting, is superior to the homodyne strategy in a practical setting.
PACS numbers: 03.67.-a, 03.67.Hk
I. INTRODUCTION
Ultra-low noise quantum channels transmitting dis-
crete or Continuous-Variable (CV) quantum informa-
tion are prerequisite for the successful execution of many
quantum information protocols. For example, the secu-
rity and the secret key rate of quantum key distribution
critically depend on the amount of excess noise added
to the quantum state during transmission [1, 2]. All
realistic quantum channels are afflicted by such noise:
In fiber channels, for example, light scattering by ther-
mal phonons causes Gaussian phase noise. On the other
hand, noise sources important in atmospheric transmis-
sion, such as time jitter and beam pointing noise [3], show
a characteristic non-Gaussian behavior.
In order to retain security, the errors imposed by the
noisy channels must be corrected. Various methods have
been developed to combat noise in CV quantum com-
munication, examples being entanglement distillation [4]
and quantum error correction coding [5], which are rely-
ing on highly non-classical resources and complex pro-
cessing. An alternative is quantum filtering which is
a protocol that probabilistically rejects erroneous quan-
tum states through detection. The simplest approach
is a classical measure-prepare strategy based on optimal
state discrimination using the Neyman-Pearson criterion
[6] followed by state recreation. Helstrom found that
by using a tailored detection process, it is possible to
identify a pure target state in a noisy mixture [7] (see
also [8, 9, 10]). Takeoka et al. generalized this strategy
and named it unambiguous quantum state filtering since
it unambiguously filters out a specific signal from the
noise [11]. However, only a single a priori known state
is resurrected, which is done destructively and therefore
not suitable for quantum communication.
In this paper, we propose and experimentally realize
∗Electronic address: cwittmann@optik.uni-erlangen.de
a quantum state filter protocol specially tailored to non-
Gaussian noise as in atmospheric transmission. The pro-
tocol filters a coherent state alphabet non-destructively
and noiselessly, i.e. the quantum states are not com-
pletely destructed and no excess noise is added by our
filter. Our protocol is based on a weak measurement of
the corrupted signal followed by a post selection of the
remaining part of the signal. We investigate two different
weak measurement strategies, namely homodyne detec-
tion and on/off detection and compare their efficiencies
in filtering out noise. We find that optimum filtering
is obtained by the use of an ideal on/off detector. The
scheme presented in this paper provides the first imple-
mentation of a CV error detection protocol enabled by a
photon counting detector.
An exemplary application of such a filter is shown in
Fig. 1. Suppose a signal is conveyed through two differ-
ent quantum channels each possessing different kinds of
noises (e.g. a free space channel and a fiber channel). If
the first channel is inflicted by the non-Gaussian on/off
noise and the following channel by Gaussian noise, the
on/off noise might be completely masked by the Gaus-
sian noise and cannot easily be filtered out at the receiv-
ing station. In order to circumvent a mixing of the two
noise sources, the filtration station could be placed be-
tween the two channels thus removing the on/off noise
before the signal enters the Gaussian noise channel.
The filtration protocol can be also used to improve the
security of a quantum key distribution scheme based on a
coherent state alphabet and heterodyne detection. This
is proven at the end of the paper.
II. DESCRIPTION OF THE PROTOCOL
Let us consider the protocol in detail. Information is
encoded into quantum states taken from a coherent state
alphabet with a possibly unknown probability distribu-
tion. The quantum state is subsequently sent through
the quantum channel where it is subject to time jitter
http://arxiv.org/abs/0704.1918v3
mailto:cwittmann@optik.uni-erlangen.de
on/off
noise
excess
noise
X X X
P P P
classical channel
freespace
channel
fiber
channel
FIG. 1: Application of the quantum filter device. The
filter F is placed between two quantum channels connect-
ing sender S and receiver R. We assume, that the channels
have non-Gaussian on/off (first part) and Gaussian proper-
ties (last part). The on/off behavior of the first channel
will be masked by excess noise in the second channel (e.g.
|α〉〈α| → |0〉〈0| → ρth). However, a quantum filter in the in-
termediate station can sense the channel break and reject the
noisy state by sending information over a classical channel to
or beam positioning noise. Such non-Gaussian noise oc-
curs when the detection time is longer than the signal
but shorter than the jitter time or when the aperture of
the receiver is larger than the beam but much smaller
than the beam pointing noise. This noise can be approx-
imated by a mixture of the sent coherent state |α〉 and
the vacuum state:
ρ(α) = p|α〉〈α|+ (1 − p)|0〉〈0|, (1)
where p is an unknown probability for perfect transmis-
sion. The task is now to find a protocol that unambigu-
ously filters out the vacuum state, while only attenuating
the coherent state, e.g. |α〉 → |γα〉, γ < 1.
To accomplish a state independent weak measurement
adding no excess noise the signal system must be coupled
unitarily and phase insensitively to a meter system in
which the actual measurement takes place. Due to these
requirements, the coupling can be enabled by a beam
splitter with the meter system being in the vacuum state
before interaction [12]. The signal-meter coupling can
therefore simply be described by the transformation
ρ(α) ⊗ ρ(0) → ρ(
1−Rα)⊗ ρ(
Rα), (2)
where R is the reflectivity of the beam splitter. After
this interaction, the presence or absence of the vacuum
contribution is correlated in the two systems. Thus by
detecting the vacuum state in the meter system, filtering
of the vacuum in the signal system can be performed
by post selecting on the correlated state. The strategy
is illustrated in Fig. 1. The next step is thus to find the
measurement strategy that optimally and unambiguously
detects the vacuum contribution.
Let us assume that we use the measurement operators
Π⊥ and Π to discriminate the vacuum state and the un-
known signal state. We seek a strategy that maximizes
the probability 〈
Rα|Π|
Rα〉 of measuring |
Rα〉 un-
FIG. 2: Schematic illustration of a coherent state quantum
filter for the non-Gaussian channel: (a) Filter device with ver-
ification measurement; (b left) Filter using APD as a detec-
tor, (b right) using homodyne detection with a local oscillator
(LO).
der the condition that the vacuum state is never de-
tected incorrectly, that is E = 〈0|Π|0〉 = 0. Such de-
cision problem was first encountered by Neyman and
Pearson [6] and was further elaborated upon by Hel-
strom [7] and Holevo [13]. They found that the maxi-
mum probability of detecting the signal correctly (also
called the acceptance probability) with no error detec-
tions (E = 0) is given by P (
Rα) = 1 − exp(−R|α|2)
(Note that E = P (0)). We readily find that measurement
operators satisfying these conditions are Π⊥ = |0〉〈0| and
Π = 1 − |0〉〈0| for rejecting and accepting the state, re-
spectively. Therefore, using these measurement opera-
tors the signal states can be unambiguously detected in
the meter system and thus perfectly filtered out in the
signal system. We stress that since this optimized mea-
surement is independent of the signal amplitude and the
reflection coefficient, it is the optimal strategy for every
coherent state.
The physical implementation of these measurement op-
erators is known to be an ideal avalanche photodiode
(APD) operating in the break down voltage mode. Prac-
tical APDs are, however, lossy and possess dark counts
which results in a reduced success probability and gives
rise to errors, that is, E > 0. Therefore, in addition
to the APD, we investigate in the following the filtering
performance using a homodyne detector for the decision
problem; quadrature values larger than a certain a pri-
ori specified threshold value are assumed to stem from
the unknown signal state, smaller values from the vac-
uum state. Note that a similar strategy was proposed
in Ref. [14] and experimentally realized in Ref. [15, 16]
to purify non-classical resources. We also note that the
incorporation of an APD in a CV system has been im-
plemented in previous experiments on state preparation
and estimation [17, 18].
In order to quantify the performance of the filtering
protocol using different detection methods, we introduce
−3 −2 −1 0 1 2 3
X quadrature measurement (shot noise units)
vacuum
noisy signal
filtered signal
APD, η=1
HDS, η=1
FIG. 3: Marginal distribution for the perturbed state (p =
0.02) (circles), the vacuum state (crosses) and the filtered
state using an APD filter (triangles). The solid and the dotted
dashed line correspond to the theoretical performance of a
filter with APD (ηAPD = 1) and with homodyne detector
(ηHDS = 1) respectively. The mean photon number in the
filter is R|α|2 = 1.65 and the error probabilities are identical
EAPD = EHDS = 5.3 · 10
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Mean photon number(R|α|2)
FIG. 4: Acceptance probability for mean photon number
R|α|2 impinging on the filter detector. The triangles, circles
and squares show experimental data for APD and homodyne
detection with and without stabilized LO, respectively. The
solid lines are theoretical predictions for detectors with unit
quantum efficiency. EAPD = EHDS = EHDR = 5.3 · 10
two appropriate functions: the sensitivity S and the gain
G. The sensitivity quantifies the filtering efficiency near
the vacuum state and we define it as
d|α|2P (
Rα)|α=0. (3)
Since the probablity P must be minimal when α = 0, the
sensitivity S is a measure for how quickly the probability
increases around α = 0. For the ideal filter we easily find
S = R, thus we will be using S/R as the figure of merit.
The other parameter that we will use to quantify the per-
formance of the filter is the gain G = p′/p where p′ is the
probability for the coherent state to occur in the mix-
ture after filtering: ρ′ = p′|
Tα| + (1 − p′)|0〉〈0|.
The success probability for positive filter outputs is PS =
Rα)+(1−p)P (0) and the gain can thus be written
1− (1 − p) E
. (4)
Note that the sensitivity S depends solely on the filter
implementation. Thus, it is independent of the chan-
nel. In contrast, the gain is a signal-, channel- and filter-
dependent parameter, and therefore describes the joint
action of channel and filter.
III. EXPERIMENTAL SETUP AND RESULTS
We proceed with the experimental demonstration of
quantum filtering using an APD-based filter. The setup
is shown in Fig. 2(a). The source is a diode laser emitting
light at 810nm, characterized with a coherence time of
1µs and measured to be shot noise limited in the detected
bandwidth. The statistical mixture of the coherent sig-
nal states and vacuum states is prepared in a computer
controlled electro-optical modulator (EOM). The EOM
therefore generates the signal and simulates the noisy
channel. To obtain a small excitation of the coherent
state, the beam is heavily attenuated after modulation.
We set the coherent state probability to p = 2%, thus
the probability for vacuum to occur being 1 − p = 98%.
The signal duration is defined within 800ns time win-
dows, while the rate of the signal preparation is set to
100kHz.
We investigate the performance of the APD-based fil-
ter by characterizing the state with homodyne detection
before and after filtering. First we demonstrate the prin-
ciples of the protocol using the APD as a filter and the
homodyne detector for characterization. The probability
distribution of the mixed input state is shown in Fig. 3
by circles and for comparison the pure vacuum state is
shown by crosses. Next, the mixed state is passing the fil-
ter beam splitter and the homodyne quadrature data are
selected based on the measurement outcomes of the APD
detector. The resulting probability distribution is shown
by the dotted curve and should be compared with the
ideally filtered signal (solid line) and two other curves:
the one expected using homodyne detection with equal
error probability and one using unit quantum efficiency
as filter (dotted dashed line).
In the following, we fully characterize the quantum fil-
ters. The mixed state is split on a 50/50 beam splitter
and subsequently directed to two different detector units:
a fiber coupled APD (Perkin-Elmer SPCM CD3017) and
a homodyne detector. We therefore simultaneously mea-
sure the acceptance probabilities for the two different de-
tection schemes; here each detector represented a filter.
increasing B
increasing R| |α
FIG. 5: In all figures the triangles are experimental data
points for a filter using APD, the circles and squares show
data for filters using homodyne detection with and without
stabilized LO respectively; the solid lines are theoretical pre-
dictions for filters using unit quantum efficiency detectors,
η = 1 (APD: S/R = (1 − pd)
2 for η = 1). (a) Sensitivity as
a function of error probability. The dashed line should guide
the eye to the error rate where the detectors are compared
EAPD = EHDS = EHDR = 5.3 ·10
−3. (b) Gain G as a function
of mean photon number R|α|2 impinging on the filter detec-
tor. Signal probability fixed to p = 2%. (c) Parametric plot of
Gain G and success probability PS for R|α|
2 ∈ [0, 1.65]. The
performance decreases from APD to HDS and HDR. The er-
ror bars show the statistical errors (three-sigma error bars)
and seem to be much smaller than the experimental errors.
Let us first discuss the APD-based filter (see Fig. 2(b
left)). A gate option is used to precisely determine the
detection time. The quantum efficiency is estimated
to be ηAPD = 63 ± 3%, while the dark count rate is
180cts/s. Due to these imperfections, the expected ac-
ceptance probability is
PAPD(β) = 1− (1− pd) exp
−ηAPD(1− pd)|β|2
where β =
Rα and pd is the dark count probability.
The expected error probability is EAPD = PAPD(0) = pd.
We measure the acceptance probability by comparing the
actual decision (based on the filter measurement out-
come) with the a priori known preparation of the state.
The results are presented in Fig. 4 as a function of R|α|2.
Note that R should be tailored to the actual amplitude
α to optimize the performance. The error probability is
found to be E = 5.3 · 10−3, which is limited not by the
dark count probability 1.4 ·10−4 but by the imperfections
in preparing the vacuum state.
Next we discuss the filter based on homodyne detec-
tion (see Fig. 2(b right)). The detector’s bandwidth is
10MHz. Using a local oscillator (LO) power of about
5mW, the shot noise to electronic noise ratio is 18dB.
The detection efficiency, including the mode matching
efficiency and the quantum efficiency of the photodiodes
is ηHD = 84 ± 3%. We investigate two different kinds
of homodyne detectors: one with a phase stabilized lo-
cal oscillator (HDS) and one with a phase randomized
local oscillator (HDR). The latter scheme should be used
when the input alphabet of coherent states is rotationally
symmetric in phase space whereas the former scheme is
superior if e.g. a binary phase encoding is used where
the absolute direction of the displacement is known a
priori. The hypothesis whether a signal or a vacuum
state was measured is based on the absolute value of the
measured quadrature; if it is above a certain threshold
value, denoted B, we estimate the state to be |α〉, if not
|0〉. Knowing that the signal is encoded into |α〉, the ex-
pected acceptance probability for a phase stabilized local
oscillator is
PHDS(β) =
Erfc[
2(B + a)] + Erfc[
2(B − a)]
where a = ηHDβ, while for the phase randomized local
oscillator
PHDR(β) =
2(B − a cos θ)
dθ. (7)
The error is identical for the two approaches and given
by EHDR = EHDS = Erfc[
A classical signal is appended to the pulse trains to
estimate the phase difference of signal and LO at a
given time. Measurement data, acquired with random-
ized phase of the LO, is subsequently directly used to
evaluate the performance of the scheme based on ran-
dom LO. However, due to the appended classical signal
the relative phase is known and we selected the data as-
sociated with a phase difference of zero corresponding to
the phase stabilized case. Using these data and the above
hypothesis, we find the acceptance probability for vari-
ous excitations R|α|2 and various threshold values B. In
Fig. 4 we plot the acceptance probability as a function
of R|α|2 with the threshold value set such that the error
probability matches the one of the APD. It is clear from
the plot that the APD performs better than the homo-
dyne detector despite the much higher quantum efficiency
of the latter one.
The sensitivity of the two filters is obtained by fitting
curves to the measured acceptance probabilities corre-
sponding to various thresholds (for the homodyne case)
and subsequently using equation (3). The results are
plotted in Fig. 5(a), where the triangle represents the
APD-based filter whereas the circles and squares are as-
sociated with the phase randomized and phase locked
LO, respectively. For post selection thresholds B close to
the coherent state variance the sensitivity of homodyne
detectors is maximal. As evident from the plot, for iden-
tical error probability the sensitivity of the APD-based
filter is much larger than that of the homodyne-based
filter.
From the measurements we also calculate the gain
for different mean photon numbers and different success
probabilities as shown in Fig. 5(b) and 5(c), respectively.
The former figure clearly shows the superior performance
of the APD filter compared with the homodyne filters.
From Fig. 5(c) we clearly see that the behavior of the gain
as a function of the success probability follows the same
curve for the three filters if the error rates are tailored to
be the same (by adjusting the error thresholds appropri-
ately). This is also what is expected from Eq. (4).
IV. FILTERING IN A QUANTUM KEY
DISTRIBUTION SCHEME
In the final part, we consider the filtering action in
a CV quantum key distribution scheme using Gaussian
modulated coherent states [1]. To estimate a lower bound
for secure transmission we use the recent work of Garcia-
Patron and Cerf [19] (see also ref [20]), showing that the
lower bound can be directly computed from the covari-
ance matrix of the joint state between the sender of infor-
mation, Alice, and the receiver, Bob. The lower bound
(for reverse reconciliation) is thus given by
Klower = I
ab − χGbE (8)
where χGbE is the maximum information between Bob
and the eavesdropper, Eve, corresponding to the Holevo
bound and IGab is the Gaussian mutual information be-
tween Alice and Bob. These quantities are computed
solely from the covariance matrix of the joint state of
Alice and Bob.
The joint state is found by using the equivalence be-
tween the coherent state scheme and an entanglement-
based protocol. In an ideal entanglement based proto-
col, Alice generates Gaussian two-mode squeezed states
and measures one mode using heterodyne detection. The
other mode is then being prepared in a coherent state
with a Gaussian distributed displacement. A two-mode
squeezing variance of V results in a displacement variance
of σ = (V + 1/V )/2− 1. The prepared coherent state is
sent to Bob through the erasure channel and produces
the following density matrix
ρAB = p (|V 〉〈V |)AB + (1− p)ρA ⊗ (|0〉〈0|)B , (9)
where |V 〉AB represents a two-mode squeezed state and
ρA is a thermal state having a variance (V +1/V )/2. The
lower bound for secure communication is found numeri-
cally by solvingKlower < 0 using the covariance matrix of
the joint state ρAB and Eqn. (8). We find security cannot
be guaranteed if p < 0.87 (corresponding to Klower < 0).
Klower is maximized over the variance V .
Let us now consider the security when the filtering pro-
tocol is implemented. Assuming first the APD to be
ideal, the state after filtering is
ρfilter =
ΠUBSρABTU
Tr(ΠUBSρABTU
where UBS is the unitary beam splitter operation,
ρABT = ρAB ⊗ |0〉〈0| and Π is the measurement oper-
ator of the tap T. The structure of the covariance matrix
for this state is given by CV ′AB =
(CVAB −P 0CV 0AB),
where CVAB is the covariance matrix of the state right
after the beam splitter after tracing out the tap mode,
ρBS = TrT (UBSρABTU
), and CV 0AB is the covari-
ance matrix of the state if the measurement outcome of
the filter measurement is associated with Π⊥ = 1 − Π,
ρ⊥ = Π⊥UBSρABTU
Π⊥/Tr(Π⊥UBSρABTU
). P 0 is
the probability for getting a measurement result associ-
ated with Π⊥ and PS = 1 − P 0 is the filtering success
rate. Using the covariance matrix of the filtered state,
CV ′AB , we again compute the lower bound numerically,
and find that the probability p for which secure commu-
nication can take place is now just required to be larger
than zero. This can be also investigated analytically in
the limit of weak two-mode squeezing (V ≈ 1), corre-
sponding to a small Gaussian alphabet. In this case, the
lower bound on the secure key rate is approximately
Klower ≈ pPS
T (V − 1)2, (11)
which is positive for any T > 0 and V 6= 1. We have
thus shown that the usages of an ideal filter reestablishes
the security of the quantum key distribution system in-
dependent on the amount of vacuum noise.
We now consider the realistic case where the APD is
nonideal. Using an APD with nonzero quantum effi-
ciency is not a hindrance for obtaining secure communi-
cation, but it will result in a lower success rate. A limiting
factor on the security, however, is the presence of dark
counts which limits the minimum noise probability pmin
for which security can be proven. For a given dark count
rate, the transmission T of the filtering beam splitter and
the modulation variance σ can be numerically optimized
to maximize Klower for which security can be guaran-
teed in the protocol with respect to the noisy channel.
Using the experimental parameters of the APD used in
our experiment (η = 0.63, pd = 0.005) we find that secure
communication can be guaranteed if p > pmin = 0.222.
By reducing the dark counts to pd = 5 × 10−4 the mini-
mum probability is reduced to pmin = 0.028 and for even
lower pd = 5× 10−5, the threshold is only pmin = 0.003.
V. CONCLUSION
In summary, we have investigated a filtering protocol
that successfully filters out vacuum states from a set of
coherent states of a continuous variable system. A weak
measurement scheme consisting of a beam splitter and
an optimized measurement was employed to probabilis-
tically filter out the unwanted vacuum states. Different
measurement strategies based on homodyne and on/off
detection were investigated and compared. We therefore
provided the first direct comparison between an APD-
based and a homodyne-based protocol, and found that
the ideal on/off detection is optimal and that the prac-
tical (that is non-ideal) on/off detector is superior to
homodyne detection despite the much higher quantum
efficiency of the latter one. The protocol will be advan-
tageous in continuous variable quantum communication
wherever beam positioning and time jitter noise are the
main obstacles for faithful and secure transmission.
Acknowledgments
This work was supported by the EU project COV-
AQIAL (project no. FP6-511004), the EU-IST network
SECOQC, COMPAS project no. 212008, the project
202/07/J040 of the GACR and project MSM6198959213
of Czech Ministry of Education. R.F. acknowledges sup-
port from the Alexander von Humboldt foundation. P.M.
acknowledges support from the European Social Fund.
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