Optical homodyne detection in view of joint probability distribution
Toru Kawakubo and Katsuji Yamamoto
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan
(Dated: July 12, 2010)
Optical homodyne detection is examined in view of joint probability distribution. It is usually
discussed that the relative phase between independent laser elds are localized by photon-number
measurements in interference experiments such as homodyne detection. This provides reasoning to
use operationally coherent states for laser elds in the description of homodyne detection and optical
quantum-state tomography. Here, we elucidate these situations by considering the joint probability
distribution and the invariance of homodyne detection under the phase transformation of optical
elds.
PACS numbers: 03.65.Wj, 42.55.Ah, 03.65.Ta
I. INTRODUCTION
Laser technology plays essential roles in a wide variety
of elds in physics. Since the rst observation of inter-
ference fringes between two independent laser elds [1],
it has been common to employ a coherent state [2]
ji = ejj
2=2
1X
n=0
n
n!
jni (1)
to describe the quantum state of a laser eld, which in-
volves a coherent superposition of photon number states
jni with a denite phase in the complex amplitude .
This quantum coherence provides indispensable resources
for quantum information and communication. On the
other hand, it is widely accepted [3, 4] that by consider-
ing the driving mechanism the steady state of eld inside
a laser cavity should be a mixed state as
^jj = e
jj2
1X
n=0
jj2n
n!
jnihnj =
Z
d'
2
jei'ihei'j; (2)
which has no coherence, lacking a denite phase. It is,
however, shown by a numerical simulation [5] that two
cavity elds without denite phases even exhibit interfer-
ence when continuously monitored by photon detectors.
This provides a typical example for the apparent rele-
vance of the coherent state as the laser eld. Then, there
have been a lot of debates concerning the quantum state
of laser and optical coherence (see [5{13], and references
therein). The essential problem is whether the use of the
coherent state of Eq. (1) in various applications is valid
or not instead of the mixed state of Eq. (2) inside the
laser cavity.
In quantum optics with the rotating wave approxima-
tion, which is usually employed when describing matter-
eld interaction, one can implement only the photon-
number measurement, without observing the absolute
phases of elds. Then, the U(1) invariance appears nat-
urally in quantum optics [8], namely the photon number
 kawakubo@nucleng.kyoto-u.ac.jp
operator n^ = a^ya^ of each optical mode is invariant under
the phase transformation, a^y ! a^yei' and a^ ! a^ei'
for the creation and annihilation operators a^y and a^, re-
spectively, inducing ji ! jei'i for the coherent state.
This phase transformation has intimate relation to the
fact that the phase of a single mode solely has no phys-
ical relevance, i.e., there is no absolute reference frame
for the optical phases [13]. In this sense, it is trivial that
there appears no signicant dierence between the co-
herent state of Eq. (1) and the mixed state of Eq. (2) as
long as U(1)-invariant operations and measurements are
performed starting only with a single-mode optical eld.
The real issue to be claried is rather the interference
between two independent optical elds, which are mu-
tually incoherent without denite phases as seen in Eq.
(2). It has been discussed that photon-number measure-
ments induce localization of the relative phase when two
mutually incoherent elds interfere [8, 11, 12]. Actually,
after many photons are detected in an interference exper-
iment, the relative phase is eventually localized around
a certain value, and the remaining state gets projected
to have a some denite relative phase. Thus, the later
measurement outcomes exhibit an interference pattern.
The aim of this paper is to elucidate these situations
in optical interference experiments, where laser elds are
treated operationally as coherent states. Specically, we
consider homodyne detection and quantum-state tomog-
raphy. In order to illustrate the apparent relevance for
the use of coherent states, we consider the joint proba-
bility distribution of the measurement outcomes and the
resultant empirical measure determining the quadrature
distribution. In this examination we adopt the proper
description of the output eld of laser [7]. We also note
that the homodyne detection or generally photon-number
detections are invariant under the rotation of the phase
frame over optical elds.
This paper is organized as follows. In Sec. II, we con-
sider repeated homodyne detections for independent sig-
nal and local oscillator elds to see the localization of
the relative phase. In Sec. III, we introduce the joint
probability distribution of the outcomes of the homo-
dyne detections, and discuss the apparent relevance to
use the coherent state as the laser eld in the description
ar
X
iv
:1
00
7.
15
96
v1
[
qu
an
t-p
h]
9
Ju
l 2
01
0

2FIG. 1. Schematic diagram of homodyne detection.
of homodyne detection. In Sec. IV, we consider the opti-
cal quantum-state tomography based on these arguments
on the homodyne detection, and examine the quantum
states of laser elds. Sec. V is devoted to summary.
II. PHASE LOCALIZATION BY HOMODYNE
DETECTIONS
It has been argued [8, 11, 12] that the relative phase
gets localized by measurements when two mutually inco-
herent elds interfere. Here, we consider the homodyne
detection as a typical example of interference experiment
to discuss the phase localization, which provides a reason
why a coherent state may be adopted operationally for
the local oscillator (LO) eld as the reference.
The homodyne detection is a scheme to measure the
eld quadratures and their probability distribution (see
Fig. 1). A signal eld and a local oscillator are injected
together into a 50:50 beam splitter, and then the dier-
ence of the photon counts in the output modes is mea-
sured. The quadrature of the signal eld is given with a
coherent state LO ji (  eijj) by
x^ = (a^e
i + a^yei)=
p
2; (3)
which is approximately proportional to the photon-
number dierence
n^ '
p
2jjx^ for jj  1 [18]. This
quantity is sensitive to the relative phase between the
signal and LO.
In usual homodyne experiments, the signal and LO
are derived from a common laser eld to compensate the
phase
uctuation. In such a case, the laser eld can be re-
garded as a coherent state since its absolute phase, which
is inherited equally by the signal and LO, is irrelevant (or
unobservable) in the homodyne detection sensitive to the
relative phase. Instead, we here consider the case where
the signal and LO come from independent sources, which
are thus mutually incoherent.
We adopt an ideal continuous-wave (CW) laser as the
LO. The quantum state of the CW laser eld is described
properly according to a formalism in Ref. [7]. [A pulsed-
wave (PW) laser (not phase-locked one) will also be con-
sidered in Sec. IV to discuss the optical quantum-state
tomography though the phase localization is unavailable
for the PW case.] Specically, the output eld of laser
is separated into a sequence of wave packet modes, each
with the same duration. By assuming the mixed state as
given in Eq. (2) for the eld inside the laser cavity and
the linear coupling between the modes inside and outside
the cavity, the sequence of N output packets is described
as
^CW =
Z
d'
2
P (')

jei'ihei'j


N
; (4)
where  > 0 is determined in terms of the eld intensity
inside the cavity, the cavity leakage rate and the packet
duration. This is a mixture of tensor products of coher-
ent states over their unknown phases. Here, the phase
distribution P (') is introduced generally, which may be
non-uniform, while the Poissonian photon-number dis-
tribution is maintained. As discussed in Ref. [7], this
form for the output state of laser is exchangeable among
the packets to meet the quantum de Finetti theorem
[14, 15]. The phase distribution changes apparently as
P (')! P (' ) under the phase rotation U^CWU
y
 .
We may take a tensor product of N packets as ^
N
for the signal eld. That is, many identical copies of ^
are prepared and measured repeatedly in the homodyne
detection. Then, the initial state of the system is given
by
^
N
^CW =
Z
d'
2
P (')

^
jei'ihei'j


N
: (5)
[Similar argument is also applicable for a phase-mixture
signal, as given in Eq. (4), as long as the relative phase
between the signal and LO is concerned. This case will
be considered explicitly in Sec. IV.] After the rst packet
is measured, the remaining packets are projected to
Z
d'
2
P (')q'(x)

^
jei'ihei'j


(N1)
(6)
up to the normalization. The quadrature value x is de-
termined from the detected photon-number dierence
n
by
x =
n=(
p
2): (7)
The quadrature distribution q'(x) for the signal ^ with
a pure coherent state LO jei'i is given by
q'(x)[^]p
2
=
X
nm=
n
hn;mjB

^
jei'ihei'j

jn;mi;
(8)
where B represents the unitary transformation by the
50:50 beam splitter. Specically, for the strong LO eld
we have approximately [18]
q'(x) = hx^' = xj^jx^' = xi (!1); (9)

3where jx^' = xi is an eigenstate of the quadrature x^' with
an eigenvalue x.
Repetition of M detections with outcomes x1; : : : ; xM
leads the state of the LO eld to be
Z
d'
2
"
P (')
MY
i=1
q'(xi)
#

jei'ihei'j


(NM)
: (10)
Typically, for a coherent state signal ^ = jihj ( > 0)
we have the quadrature distribution (!1)
q'(x) =
1
p

exp
"
22

cos'
x
p
2
2
#
: (11)
Then, the phase distribution of the LO eld after the M
detections is modied from P (') by the factor
MY
i=1
q'(xi) /
(
exp
"
22

cos'
xMp
2
2
#)M
; (12)
where xM is the average over the M outcomes. This
provides a sharp Gaussian distribution around cos' =
xM=(
p
2) with the standard deviation 1=(2
p
M) for
M  1. That is, the phase of the LO eld gets localized
to '0 =  arccos(xM=
p
2) by the repeated homodyne
detections. Generally, as long as q'(x) is non-uniform
with respect to the phase ' of the LO (more precisely
the relative phase between the signal and LO), the phase
localization takes place after a large number of detections
as (up to the normalization)
MY
i=1
q'(xi)  (' '0): (13)
The phase localization is usually discussed in the case
of single measurement (M = 1) with detection of large
numbers of photons under strong sources (;  ! 1)
[8, 11, 12]. Here, we note that the phase localization
takes place by repeated measurements (M  1) even
for the source packets with weak amplitudes (;   1).
This indeed provides a process of aligning the reference
frames between the signal and LO [13] by updating the
relative phase according to the Bayesian rule in Eq. (10)
[7, 12, 16, 17]. The localized phase '0 may apparently
take multiple values, re
ecting a specic symmetry of
the signal state, though they are physically equivalent.
For example, in the case of ^ = jihj we have '0 =
 arccos(xM=
p
2) for q'(x) = q'(x) under the phase
re
ection '! '.
Once the phase of the LO eld is localized to a par-
ticular value '0 in Eq. (13), the state of the LO eld
conditioned on the outcomes x1; : : : ; xM gets projected
as
^(x1;:::;xM )CW 

jei'0ihei'0 j


(NM)
: (14)
Thus, we may conclude that the pure coherent state
jei'0i is provided as the LO for the subsequent detec-
tions in the same way as the usual homodyne detection.
The apparent exception of the phase localization is the
case that the signal state ^ is invariant under the phase
transformation, including the number states and their
mixtures. Nevertheless, the quadrature distributions for
such a state with the pure coherent state LO jei'i in
Eq. (1) and the mixed state LO in Eq. (2) are identical
as q'(x) = q(x) independently of '. Thus, even in this
case without phase localization, the laser eld can be re-
garded as the coherent state. This point will be claried
further in view of the joint probability distribution in the
following sections.
III. JOINT PROBABILITY DISTRIBUTION IN
HOMODYNE DETECTIONS
We have seen that after repeated detections, the state
of the LO eld turns into the product of coherent states
in Eq. (14) due to the phase localization. This provides
reasoning to use the coherent state in the standard de-
scription of homodyne detection. In order to get further
understanding of this point from a viewpoint of proba-
bility theory, we here consider the joint probability dis-
tribution of homodyne detections.
In quantum theory, measurements of a physical quan-
tity yield probabilistic outcomes. Then, from the rela-
tive frequency of outcomes we can infer the probability
distribution for the physical quantity. This argument is
based on the assumption that the outcomes are indepen-
dent and identically distributed (i.i.d.) in repeated mea-
surements for an ensemble of identically prepared quan-
tum states. Specically, in the optical quantum-state
tomography [20] the quadrature distributions are deter-
mined from the outcomes of homodyne detections. In the
standard description, a product of pure coherent states

jei'0ihei'0 j


N
with a common phase '0 is adopted
as the LO packets when the homodyne detections are
performed repeatedly for an ensemble of signal states as
^
N . Then, the joint probability distribution of the out-
comes x1; : : : ; xM is given by
p(x1; : : : ; xM ) =
MY
i=1
q'0(xi); (15)
where q'0(x) is the quadrature distribution of the signal
in Eq. (8). In this case the outcomes are really i.i.d.,
namely they are obtained probabilistically according to
the product of identical quadrature distributions. Thus,
owing to the Glivenko-Cantelli theorem the original dis-
tribution q'0(x) is properly inferred as the relative fre-
quency of outcomes for M !1.
This argument for the standard homodyne detection
with the pure coherent state LO can be extended for
the case of the real output eld of a CW laser whose
quantum state is the mixture as given in Eq. (4). The
(unnormalized) state of the LO eld after M detections is
given in Eq. (10). By tracing out the remaining packets
as Tr

(jei'ihei'j)
(NM)

= 1, the joint probability

4distribution of the outcomes is calculated as
p(x1; : : : ; xM ) =
Z
d'
2
P (')
MY
i=1
q'(xi): (16)
Even in this extended case, where the joint probability
distribution in Eq. (16) appears as a phase-mixture of
the i.i.d. products in Eq. (15), we can infer the origi-
nal quadrature distributions from the measurement out-
comes as described in the following.
Consider a sequence of random real variables (mea-
surement outcomes),
~xM  (x1; x2; : : : ; xM ): (17)
The empirical measure ~xM , or relative frequency of the
M outcomes, is dened as a probability measure on R by
~xM =
1
M
MX
i=1
xi ; (18)
where x denotes the Dirac measure on R:
x(A) =
(
1 if x 2 A  R,
0 otherwise.
(19)
That is, if the number of xi's which have values in A is
k, then ~xM (A) = k=M . In the i.i.d. case of Eq. (15),
the Glivenko-Cantelli theorem ensures that the empiri-
cal measure ~xM converges to the original distribution
q'0(x) for M ! 1. As for the actual homodyne detec-
tion with the LO of a CW laser eld, the joint probabil-
ity distribution in Eq. (16) represents a mixture of the
i.i.d. variables (or i.i.d.'s shortly). Even in this case, by
repeating the detection many times (M  1) the empir-
ical measure provides the quadrature distribution with a
certain random phase '0,
lim
M!1
~xM (x) = q'0(x) (20)
(see Ref. [19] for the mathematical details). This implies
that the outcomes x1; : : : ; xM appear as if they were i.i.d.,
in the same way as the case with the pure coherent state
LO. Therefore, in each sequence of homodyne detections
we may regard the LO of the CW laser eld as a co-
herent state jei'0i while the phase '0 is determined a
posteriori by the localization.
We have made numerical simulations for the joint prob-
ability distributions of homodyne detections, conrming
Eq. (20). A sequence of M outcomes are obtained ac-
cording to Eq. (16) representing the mixture of i.i.d.'s.
Specically, the i-th outcome xi is generated under the
conditional probability distribution,
p(xijx1; : : : ; xi1) =
p(x1; : : : ; xi)
p(x1; : : : ; xi1)
=
Z
d'
2
P(x1;:::;xi1)(')q'(xi) (21)
0
0.02
0.04
0.06
0.08
0.1
-8 -6 -4 -2 0 2 4 6 8
Λ~ x
M

v
er
su
s
q ϕ
0
x
Λ~xMqϕ0
FIG. 2. Typical results of the empirical measure ~xM (x)
(solid lines) are shown for the squeezed state signal ^ =
jr; ihr; j, where M = 10000 outcomes are generated for each
operation of repeated homodyne detections. The parameters
for the LO and signal is taken as  =
p
15, er =
p
3
and r = 1. The resolution of the quadrature is given by

x = 1=(
p
2) = 1=
p
30. The resultant random phases
'0 are estimated from the average xM of the outcomes as
'0 = 3:05 rad, 1:94 rad, 0:43 rad from the left to right. The
quadrature distributions q'0(x) (dashed lines) are also plotted
for these values of '0, showing good agreement with ~xM (x).
with
p(x1; : : : ; xi) =
Z
dxi+1


dxMp(x1; : : : ; xM ): (22)
(A similar analysis is made for the spatial interference
of Bose-Einstein condensates [21].) Here, the phase dis-
tribution is updated by the Bayesian rule [7, 12, 16, 17]
upon the preceding quadrature outcomes x1; : : : ; xi1 as
P(x1;:::;xi1)(') =
i1Y
j=1
q'(xj)
,Z
d'
2
i1Y
j=1
q'(xj) (23)
with the U(1)-invariant initial P (') = 1 for deniteness.
The empirical measure ~xM (x) is then calculated from
the M outcomes with Eq. (18). In this numerical anal-
ysis, the original quadrature distribution q'(x) is calcu-
lated precisely from Eq. (8) without taking the limit of
strong laser intensity. Statistically, a large number of
detections should be made to infer the quadrature dis-
tribution. Thus, we realize in Eq. (23) that in the early
portion of the M detections (M  1) the phase ' is
almost localized to '0 providing Eq. (20).
Fig. 2 shows typical results of the empirical mea-
sure ~xM (x) (solid lines) for the squeezed state signal
^ = jr; ihr; j, where M = 10000 outcomes are gen-
erated for each operation of repeated homodyne detec-
tions. The parameters for the LO and signal is taken as
 =
p
15, er =
p
3 and r = 1. The resolution of
the quadrature x is given by
x = 1=(
p
2) in Eq. (7)
with
n = 1. We note that the expectation value of the

5quadrature is calculated with q'(x) as
p
2er cos' in-
dependently of  > 0 for the coherent state LO. Since
it should agree with the average xM of the outcomes
for M ! 1, the resultant random phases '0 are esti-
mated as '0 = 3:05 rad, 1:94 rad, 0:43 rad from the left
to right in Fig. 2. Then, the quadrature distributions
q'0(x) (dashed lines) are plotted for these values of '0
for comparison. These results really show good agree-
ment of ~xM (x) and q'0(x), as expected in Eq. (20).
IV. OPTICAL QUANTUM-STATE
TOMOGRAPHY
In the view of joint probability distribution for homo-
dyne detection, as described in the previous section, we
now consider the optical quantum-state tomography, and
discuss the quantum state of a laser eld. The signal
state (e.g., Wigner function) is reconstructed from the
quadrature distributions q(x) for various phase shifts ,
which are obtained as the empirical measures from the
outcomes of homodyne detections. The change of  is
realized by applying a phase shifter on the LO.
A. Tomography with a common source oscillator
We rst consider the usual setup for optical tomogra-
phy, where the signal and LO are supplied by splitting
a single oscillator, as done in many actual experiments.
The output state of a CW laser for the original oscillator
(;  > 0) is given as
^0CW =
Z
d'
2
P (')

j(+ )ei'ih(+ )ei'j

N
: (24)
The signal and LO, which share the common random
phase ', are derived from this laser eld as
^SLCW =
Z
d'
2
P (')
h
^(')
jei('+)ihei('+)j
i
N
(25)
with
^(') = Ejei'ihei'jEy = U'^(0)U
y
': (26)
Here, the phase of the LO is shifted by , and the oper-
ation E such as squeezing is applied for the signal, which
commutes with the phase transformation U'. We see
below that this setup reproduces the standard descrip-
tion of homodyne tomography with a pure coherent state
ji as the LO. The joint probability distribution of the
quadrature outcomes (x1; : : : ; xM )  ~xM (M  1) for
M packets is calculated as
p(~xM )[^
SL
CW] =
Z
d'
2
P (')
MY
i=1
q'+(xi)[^(')]
=
MY
i=1
q(xi)[^(0)]: (27)
Here, we have considered the fact that the homodyne de-
tection is an invariant operation under the simultaneous
phase rotation U' for the signal and LO as jei('+)i !
jeii and jei'i ! ji, which implies
q'+(x)[^(')] = q(x)[^(0)]; (28)
i.e., it is sensitive only to the relative phase . This
p(~xM ) turns out to be independent of the phase distribu-
tion P (') for the LO, that is the possible U(1) violation
in the laser eld is not observable in this scheme. The
quadrature outcomes are i.i.d. in Eq. (27), and the LO
appears as if it is the coherent state jeii with ' = 0.
Thus, by repeating independently the sequence of M de-
tections on ^SLCW from the common source with the vary-
ing phase shift  for the LO, the set of quadrature distri-
butions q(x)[^(0)] is obtained to reconstruct the signal
state through tomography as
^rec = ^(0) = E(jihj)E
y; (29)
which is irrespective of the unknown phase '.
Alternatively, we may adopt a PW laser for the orig-
inal oscillator, providing N copies of a phase-mixture of
coherent states,
^0PW =
Z
d'
2
P (')j(+ )ei'ih(+ )ei'j

N
: (30)
The combination of signal and LO is derived as
^SLPW =
Z
d'
2
P (')^(')
jei('+)ihei('+)j

N
:
(31)
Then, the same joint probability distribution is obtained
as Eq. (27) for the CW case,
p(~xM )[^
SL
PW] =
MY
i=1
Z
d'i
2
P ('i)q'i+(xi)[^('i)]
=
MY
i=1
q(xi)[^(0)]; (32)
providing again the reconstruction of the signal state as
^(0) = E(jihj)Ey. Therefore, as long as the common
laser eld is used for the signal and LO, we nd no actual
dierence between the CW and PW cases. In either case,
the use of a pure coherent state as the LO is relevant for
the standard description of the optical quantum-state to-
mography, without need to discuss the phase localization.
The tomography with the common source just character-
izes the process given by the operation E rather than the
signal state [8].
B. Tomography with independent signal and LO
We next consider the case that the signal and LO are
prepared independently, which may be more faithful in

6the sense of tomography to reconstruct an \unknown"
quantum state. The LO is supplied with the output state
^CW of a CW laser as given in Eq. (4). An ensemble of
repeatedly prepared identical states for the signal may
be given generally as
^S =
Z
d'0
2
PS('
0)^('0)
N ; (33)
where
^('0)  U'0 ^(0)U
y
'0 (34)
with certain ^(0). The phase distributions P (') and
PS('0) (with period 2) may not be invariant under the
rotation of phase frame. In the case of a U(1)-invariant
state ^(0) = ^('0) for any '0, namely a mixture of num-
ber states [8], we simply have ^S = ^(0)
N without
PS('0).
The joint probability distribution of homodyne detec-
tions is calculated as
p(~xM )[^S
^CW] =
Z
d'
2
PS(')
MY
i=1
q(0)' (xi) (35)
with a convoluted phase distribution
PS(') =
Z
d'0
2
PS('
0)P ('+ '0): (36)
Here, we have considered the invariance of homodyne
detection under the phase transformation, implying the
relation for the quadrature distributions as
q'(x)[^('
0)]  q('
0)
' (x) = q
(0)
''0(x) (37)
with the periodicity q'+2(x) = q'(x). We nd that this
p(~xM ) represents a mixture of i.i.d.'s, as discussed in Sec.
III. Then, a quadrature distribution q(0)'0 (x) is obtained as
the empirical measure with the probability distribution
PS('0) for the random phase '0. Here, we note that the
U(1)-invariant LO with P (') = 1 provides PS(') = 1, ir-
respective of any PS(') for the original signal. Contrarily,
if any deviation of PS(') from the uniform distribution
is found for the various values of ' = '0 in experiments
(provided '0 is determined in a certain situation, e.g.,
from the average of outcomes for the coherent or squeezed
state signal), that is
PS(') 6= 1! P ('); PS(') 6= 1; (38)
then it might indicate the violation of U(1) symmetry, or
the presence of some implicit phase reference common to
the signal and LO.
The measurement of the M -packet sequence may be
repeated independently. Then, the quadrature distribu-
tions q(0)'0 (x) are obtained with various random phases
'0. It is, however, impossible in general to know the
actual values of '0 without some prior knowledge about
the signal state. These unknown random phases '0 for
q(0)'0 (x) thus can not substitute for the phase shift  of
the LO in tomography. The phase shift  of the LO in
each of the independent M -packet sequences is actually
ineective since it is hidden in the random phase '0.
Instead, in order to realize eectively the phase shift of
the LO, we should extend the single M -packet sequence
to K M -packet sequences in a single operation of to-
mography:
^CW !
Z
d'
2
P (')
KY
k=1

jei('+k)ihei('+k)j

M

;
(39)
where the phase shift k is applied for the LO in each M -
packet sequence. Then, the joint probability distribution
is given as
p(~x(1)M ; : : : ; ~x
(K)
M ) =
Z
d'
2
PS(')
KY
k=1
"
MY
i=1
q(0)'+k(xi)
#
:
(40)
This provides the sequence of empirical measures upon
homodyne detections, determining the quadrature distri-
butions for tomography with varying phases k (M  1):

~x(1)M
= q(0)'0+1(x); : : : ;~x(K)M
= q(0)'0+K (x); (41)
where the original phase of the LO is xed to a certain
value '0 according to the localization.
Provided there is no way to know the value of '0, we
may set '0 = 0 operationally (as a convenient choice of
the phase frame), or consider the relation
q(0)'0+k(x) = q
('0)
k
(x): (42)
Then, the set of quadrature distributions q('0)k (x) for
the pure coherent states LO jeiki with the phase shifts
k ( > 0 and ' = 0) provides the tomographic recon-
struction as
^rec = ^('0): (43)
In each operation of tomography, the reconstructed state
^('0) appears probabilistically as a random rotation
of ^(0). Due to the lack of the absolute phase reference,
however, ^(0) and ^('0) should be regarded equivalent,
and the ensemble of signal states is properly inferred as
^S in Eq. (33) with PS('0). These arguments illustrate
the actual relevance for the use of the pure coherent state
as the LO in the description of optical quantum-state ho-
modyne tomography. The random phases '0 and their
distribution PS('0) may be estimated relatively by com-
paring the rotations for the resultant Wigner functions
obtained from many runs of tomography for the same ^S.
Note, however, that PS('0) = 1 for the U(1)-invariant
LO with P (') = 1, irrespective of the actual PS('0).

7C. CW eld versus PW eld
We also consider the case that the quantum state of
LO is given by a product of mixed states, which may
be prepared with a simple PW laser (not a phase-locked
one):
^PW =
Z
d'
2
P (')jei'ihei'j

N
: (44)
In the PW case as the LO, the joint probability distribu-
tion is calculated for the signal state in Eq. (33) as
p(~xM )[^S
^PW] =
Z
d'0
2
PS('
0)
MY
i=1
q'0(xi); (45)
where
q'0(x) 
Z
d'
2
P (')q('
0'+'0)
'0 (x): (46)
with the relation q('
0)
' (x) = q
('0'+'0)
'0 (x) under the
phase rotation U'+'0 . This smeared quadrature dis-
tribution q'0(x) is reproduced with the LO state jei'
0
i
for the signal of a phase-mixed state
^mix =
Z
d'
2
P (')^('0 '+ '0); (47)
which generally does not coincide with ^(0) or its phase-
rotation ^('0), except for the U(1)-invariant ^(0). Thus,
we nd that the optical quantum-state tomography does
not work rightly by using the PW eld ^PW in Eq. (44)
as the independent LO. The phase-mixture of product
coherent states ^CW in Eq. (4), which is derived from
a CW laser, is required for the successful tomography.
As an interesting case, we may implement the homodyne
tomography for the CW and PW elds with the inde-
pendent CW eld as the LO. Then, we will obtain in the
reconstruction the coherent state in Eq. (1) for the CW
signal, and the mixed state in Eq. (2) for the PW signal,
respectively. In this way, we can distinguish the quantum
states of laser elds.
V. SUMMARY
We have examined the repeated optical homodyne de-
tections and quantum-state tomography in view of the
joint probability distribution of the measurement out-
comes. By adopting the real output state of a CW
laser as the LO, which is independent of the signal eld,
the joint probability distribution represents a mixture of
i.i.d.'s. Then, the original quadrature distribution of the
signal is obtained as the empirical measure, or relative
frequency of the outcomes, with a random phase for the
coherent state LO determined a posteriori by the phase
localization according to the Bayesian rule. This justies
the operational use of the coherent state as the LO in the
standard description of homodyne detection and tomog-
raphy. We have also discussed that the quantum states of
CW and PW lasers are distinguishable by the quantum-
state tomography with the independent CW eld as the
LO. That is, the CW and PW lasers will appear as the
coherent state and the mixed state, respectively. On the
other hand, both of them will be recognized as the co-
herent state indistinguishably if the tomography is imple-
mented with the signal and LO derived from the common
source oscillator, as usually made in optical experiments.
ACKNOWLEDGMENTS
We thank K. Fujii for valuable discussions. T. K. was
supported by the JSPS Grant No. 22.1355.
[1] G. Magyar and L. Mandel, Nature 198, 255 (1963).
[2] R. J. Glauber, Phys. Rev. 131, 2766 (1963).
[3] M. Sargent, III, M. O. Scully, and W. E. Lamb, Jr., Laser
Physics (Addison-Wesley, Reading, 1974).
[4] D. F. Walls and G. J. Milburn, Quantum Optics
(Springer-Verlag, Berlin, 1994).
[5] K. Mlmer, Phys. Rev. A 55, 3195 (1997); J. Mod. Opt.
44, 1937 (1997).
[6] T. Rudolph and B. C. Sanders, Phys. Rev. Lett. 87,
077903 (2001).
[7] S. J. van Enk and C. A. Fuchs, Phys. Rev. Lett. 88,
027902 (2001); Quant. Inf. Comput. 2, 151 (2002),
arXiv:quant-ph/0111157.
[8] B. C. Sanders, S. D. Bartlett, T. Rudolph, and P. L.
Knight, Phys. Rev. A 68, 042329 (2003).
[9] H. M. Wiseman, J. Opt. B 6, S849 (2004).
[10] J. A. Smolin, arXiv:quant-ph/0407009.
[11] H. Cable, P. L. Knight, and T. Rudolph, Phys. Rev. A
71, 042107 (2005).
[12] F. Neri, Phys. Rev. A 72, 062306 (2005).
[13] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev.
Mod. Phys. 79, 555 (2007).
[14] R. L. Hudson and G. R. Moody, Z. Wahrs. 33, 343 (1976).
[15] C. M. Caves, C. A. Fuchs, and R. Schack, J. Math. Phys.
43, 4537 (2002); 49, 019902 (2008).
[16] V. Buzek, R. Derka, G. Adam, and P. L. Knight, Ann.
Phys. (NY) 266, 454 (1998).
[17] R. Schack, T. A. Brun, and C. M. Caves, Phys. Rev. A
64, 014305 (2001).
[18] T. Tyc and B. C. Sanders, J. Phys. A 37, 7341 (2004).
[19] D. J. Aldous, in Ecole d' Ete de Probabilites de Saint-
Flour XIII | 1983, Lecture Notes in Mathematics, Vol.

81117, edited by P. L. Hennequin (Springer-Verlag, Berlin,
1985) pp. 1{198.
[20] A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys. 81,
299 (2009), and references therein.
[21] J. Javanainen and S. M. Yoo, Phys. Rev. Lett. 76, 161
(1996).