PHYSICAL REVIEW A 83, 012313 (2011)
Optimal unambiguous comparison of two unknown squeezed vacua
Stefano Olivares,
1,2,*
Michal Sedl
´
ak,
3,4,†
Peter Rap
ˇ
can,
3
Matteo G. A. Paris,
1,2
and Vladim
´
ır Bu
ˇ
zek
3,5
1
CNISM UdR Milano Universit
`
a, I20133 Milano, Italy
2
Dipartimento di Fisica, Universit
`
a degli Studi di Milano, I20133 Milano, Italy
3
Research Centre for Quantum Information, Institute of Physics, Slovak Academy of Sciences, D
´
ubravsk
´
a cesta 9,
SK845 11 Bratislava, Slovak Republic
4
QUIT group, Dipartimento di Fisica “A. Volta”, via Bassi 6, I27100 Pavia, Italy
5
Faculty of Informatics, Masaryk University, Botanick
´
a 68a, CZ602 00 Brno, Czech Republic
(Received 4 October 2010; published 24 January 2011)
We propose a scheme for the unambiguous state comparison (USC) of two unknown squeezed vacuum states
of the electromagnetic ﬁeld. Our setup is based on linear optical elements and photonnumber detectors, and it
achieves optimal USC in an ideal case of unit quantum efﬁciency. In realistic conditions, i.e., for nonunit quantum
efﬁciency of photodetectors, we evaluate the probability of getting an ambiguous result as well as the reliability
of the scheme, thus showing its robustness in comparison to previous proposals.
DOI: 10.1103/PhysRevA.83.012313 PACS number(s): 03.67.−a, 42.50.Ex
I. INTRODUCTION
The possibility of creating physical systems with identical
properties is crucial to any physical theory that is veriﬁable
by experiments. Comparison of preparators—a procedure of
determining whether they prepare the same objects or not—is
one of the basic experiments we would like to do when testing
a theory, because it allows us to operationally deﬁne the equiv
alence of such devices for their further use. In the framework of
classical physics, we can in principle measure and determine
the state of the system perfectly without disturbing it. Thus, to
compare the states of two systems it sufﬁces to measure each
system separately. However, in quantum theory, due to its
statistical nature, we cannot make deterministic conclusions
or predictions even for the simplest experimental situations.
Therefore, the comparison of quantum states is different from
the classical situation.
Imagine we are given two independently prepared quantum
systems of the same physical nature (e.g., two photons or
two electrons). We would like to determine unambiguously
whether the (internal) states of these two systems are the same
or not. If we have just a single copy of each of the states and
we possess no further information about the preparation, then
a measurement performed on each system separately cannot
determine the states precisely enough to allow an errorfree
comparison. In this case, all other strategies would also fail,
because our knowledge about the states is insufﬁcient [1],
e.g., if each of the systems can be in an arbitrary mixed
state, then it is impossible to unambiguously test whether the
states are equal or not. However, there are often situations
in which we have some additional a priori information on
the states we want to compare. For example, we might know
that each system has been prepared in a pure state. This kind
of scenario has been considered in Ref. [2] for two qudits
and in Ref. [3] for the comparison of a larger number of
systems. Thereafter, the comparison of coherent states and
its application to quantum cryptography has been addressed in
*
stefano.olivares@mi.infn.it
†
fyzimsed@savba.sk;on leave from Bratislava.
Ref. [4]. Sedl
´
ak et al. [5] analyzed the comparison with more
copies of the two systems and proposed an optimal comparator
for coherent states, which, on this subset, outperforms the
optimal universal comparator [2] working for all pure states.
In the present paper we analyze the unambiguous quantum
state comparison (USC) of two unknown squeezed vacuum
states, that is, we would like to unambiguously determine
whether two unknown squeezedvacuum states are the same
or not. The conclusion has to be drawn from a procedure
using only a single copy of the states. At the end of
the procedure, using only the outcome of the measurement,
we have to decide whether the two states given to us have
been the same or different or that we don’t know which
of the former conclusions is true. We strive to ﬁnd an
optimal procedure, i.e., one maximizing the probability of
correctly judging the equivalence of the compared squeezed
states.
Our proposal relies on the interference of two squeezed
states at a beam splitter and on the subsequent measurement
of the difference between the number of detected photons at
the two output ports. In Ref. [4], the unambiguous comparison
of coherent states has been considered in detail and a short
remark is devoted to the comparison of squeezed vacua. In
the setup of Ref. [4], after interference at a beam splitter,
one needs to measure the parity of the detected number of
photons: a detection of an odd number of photons indicates the
difference between the inputs. As a consequence, the quantum
efﬁciency of the detectors is a critical parameter and plays
a crucial role in the robustness of t he scheme. As we will
show, this problem is less relevant in our case, since our setup
requires the measurement of the difference of the detected
number of photons. Our conﬁguration also allows us to prove
the optimality of our setup.
The plan of the paper is as follows. In Sec. II we
introduce our scheme to compare two squeezed vacuum states,
whereas the proof of the optimality of the setup is given
in Sec. III. The performance of our scheme, also in the
presence of imperfections at the detection stage, is investigated
in Sec. IV, together with its reliability in the presence of
noise. Section V close the paper with some concluding
remarks.
0123131
10502947/2011/83(1)/012313(7) ©2011 American Physical Society
STEFANO OLIVARES et al. PHYSICAL REVIEW A 83, 012313 (2011)
II. COMPARISON OF SQUEEZED VACUUM STATES
Our goal is the comparison of two squeezed vac
uum states ξ " ≡ S(ξ)0" and ζ " ≡ S(ζ )0", where S(γ ) =
exp[
1
2
γ (a
†
)
2
−
1
2
γ
∗
a
2
] is the singlemode squeezing operator,
ξ,ζ,γ ∈ C [6]. We let ξ = re
iψ
and ζ = se
iϕ
, where r = ξ ,
ψ = arg(ξ), s = ζ , ϕ = arg(ζ ). We recall that a comparator
is a measuring device with two systems at the input and two
or more possible outcomes, aimed at determining whether
the two systems have been prepared in the same state. The
setup we propose for the comparison of the two squeezed
vacuum states is composed of a phase shifter, beam splitter,
and photoncounting detectors and can be implemented with a
current technology. The basic idea is sketched in Fig. 1(a): we
start from the two squeezed vacuum states we wish to compare,
S(ξ )0"and S(ζ )0". At the ﬁrst stage of our protocol, one of the
two states, say S(ξ )0", undergoes a phase shift U (π/2), i.e.,
U(π/2)S(ξ ) 0" = S(−ξ )0"; then we mix the states, having
now orthogonal phases, at a balanced beam splitter (BS). If
ξ = ζ , i.e., the input states are equal, then the output state is
the twomode squeezed vacuum state of radiation (twinbeam
state, TWB) [7], namely,
'
out
(ξ,ξ )"" = U
BS
S(ξ ) ⊗ S(−ξ)0" ≡ S
2
(ξ)0" (1)
=
!
1 − λ(ξ )
2
∞
"
n=0
λ(ξ )
n
n"n", (2)
where n"n" ≡ n" ⊗ n", U
BS
is the unitary operator describ
ing the action of the BS, S
2
(ξ) = exp(ξa
†
b
†
− ξ
∗
ab) is the
twomode squeezing operator acting on the two modes a and
b, respectively, and λ(γ ) = e
i arg(γ )
tanh γ . One ﬁnds perfect
correlations in the photon number of the two beams, which
can be detected, e.g., by measuring the difference between
the number of photons at the outputs (see Fig. 1), which, in
this case, is always equal to zero. On the contrary, if ξ (= ζ , a
different number of photons can be detected in the two beams,
as we are going to show in the following.
Though the setup in Fig. 1(a) works for generic ξ and ζ ,
in the following we address the scenario in which the two
squeezing phases are unknown but equal, i.e., arg(ξ) = arg(ζ )
BS
(b)
(a)
BS
S(r
+
)0!
U(π/2)
S(r
−
)
S(r
−
)
Ψ
out
(ξ, ζ)!!
S
2
(r
+
)0!
S(r
+
)0!
Ψ
out
(ξ, ζ)!!
S(ζ)0!
S(ξ)0!
U(π/2)
FIG. 1. (a) Schematic diagram of a setup for USC of the squeezed
vacuum states S(ξ )0" and S(ζ )0". (b) Scheme leading to the same
output state '
out
(ξ,ζ )"" as in (a) when the two squeezing phases are
unknown but equal, i.e., arg(ξ) = arg(ζ ). We deﬁned r
±
= (ξ ± ζ )/2.
See the text for comments and details.
[note that in Eq. (2) a twomode squeezed vacuum state results
if and only if the two input squeezed states are the same, no
matter the value of their phase]. This allows us to write the
output state '
out
(ξ,ζ )"" in a simple form that will turn out to
be useful for characterizing our setup. Now, the same result of
the evolution as in Fig. 1(a) can be obtained considering the
scheme displayed in Fig. 1(b) (see Appendix A). Here the two
input states with squeezing parameters ξ and ζ are substituted
with two squeezed vacuum states having the same squeezing
parameter amplitude r
+
= (ξ + ζ )/2. It is also worth noting
that the scheme in Fig. 1(b) cannot be directly used for the
comparison, since it requires a prior knowledge about the
squeezing parameters to assess r
±
. Now, after the mixing
at the BS, the outgoing modes undergo two local squeezing
operations with amplitude r
−
= (ξ − ζ )/2. In the formula, one
has the (formal) equivalence:
U
BS
S(ξ ) ⊗ S(−ζ )0" = S(r
−
) ⊗ S(r
−
) S
2
(r
+
)0". (3)
Since S
2
(r
+
)0" =
!
1 − λ(r
+
)
2
#
n
λ(r
+
)
n
n"n", we obtain
'
out
(ξ,ζ )"" =
!
1 − λ(r
+
)
2
∞
"
n=0
λ(r
+
)
n
ψ
n
"ψ
n
", (4)
where we deﬁned the new basis ψ
n
" = S(r
−
)n". Finally, the
probability of measuring h and k photons in the two beams,
respectively, is given by
p(h,k) = )h)k'
out
(ξ,ζ )""
2
, (5)
with
)h)k'
out
(ξ,ζ )""
=
!
1 − λ(r
+
)
2
"
n
λ(r
+
)
n
[S(r
−
)]
hn
[S(r
−
)]
kn
, (6)
where [S(r
−
)]
lm
= )lS(r
−
)m" are the matrix elements of the
squeezing operator, whose analytical expressions are given,
e.g., in Ref. [8]. If ξ = ζ and h (= k, then )h)k'
out
(r,r)"" = 0
and p(h,k) = 0, as one can see from Eq. (2). Thus, the
probability p(h,k) for h (= k can be nonzero only if ξ (= ζ ,
that is, only if the input states are different.
In the ideal case (unit quantum efﬁciency of the detectors)
the measurement apparatus we want to use gives two possible
outcomes: zero or nonzero photoncounting difference. Thus,
the positive operatorvalued measure (POVM) describing
the measurement is deﬁned by the effects E
0
and E
D
,
corresponding to the “zero” and “nonzero” photoncounting
events, respectively, given by
E
0
=
∞
"
n=0
n")n ⊗ n")n, E
D
= I − E
0
. (7)
The occurrence of the “D” event implies that the incident
squeezedvacuum states could not have been identical [see
Eqs. (4) and (6)]. The occurrence of the “0” event, on
the other hand, implies nothing, as each possible pair of
squeezedvacuum states leads to a nonzero overlap with any
of the states n"n". Thus, event “D” unambiguously indicates
the difference of the compared squeezed states, whereas “0”
is an inconclusive outcome.
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OPTIMAL UNAMBIGUOUS COMPARISON OF TWO UNKNOWN . . . PHYSICAL REVIEW A 83, 012313 (2011)
III. PROOF OF THE OPTIMALITY OF THE SETUP
In this section we prove the optimality of the proposed
setup for two situations: (i) the restricted scenario in which
the squeezing phases of the compared states are unknown but
equal for both states and (ii) for the general situation, when no
assumption on the squeezing phases is taken. We ﬁrst tackle
the former scenario, which is considered in most of the paper,
and at the end of the proof we comment on the differences in
proving scenario (ii).
Let us denote by S
ϕ
≡ {S(re
iϕ
)0"; r ∈ R} the set of
squeezed states from which we randomly chose the states
to be compared. We also deﬁne the sets S
ϕ
S
≡ {S(re
iϕ
)0" ⊗
S(re
iϕ
)0"; r ∈ R}and S
ϕ
D
≡ S
ϕ
⊗ S
ϕ
\S
ϕ
S
, composed by pairs
of identical and different squeezed vacuum states, respectively.
We assume a generic measurement with three outcomes
(“same”, “different”, and “don’t know”) described by the
POVM )
S
+ )
D
+ )
0
= I, and we optimize the overall
probability
P = z
S
$
S
ϕ
S
d* p
S
(*))*)
S
*"
+z
D
$
S
ϕ
D
d* p
D
(*))*)
D
*", (8)
where z
D
and z
S
= 1 − z
D
are the a priori probability of being
different or the same, and p
S
(*), p
D
(*) are the probability
densities of choosing *" from S
ϕ
S
, S
ϕ
D
, respectively. We also
impose the noerror constraints
Tr()
S
*")*) = 0, ∀*" ∈ S
ϕ
D
, (9a)
Tr()
D
*")*) = 0, ∀*" ∈ S
ϕ
S
, (9b)
which guarantee the unambiguity of the results. From the
mathematical point of view, the constraints (III) restrict
the support of the operators )
S
and )
D
. The fact that
the possible states in S
ϕ
form a continuous subset of
pure states is responsible for the impossibility to
unambiguously conﬁrm that the compared states are
identical. The proof of this statement can be found in
Appendix B and essentially states that due to the noerror
conditions (III), we must have )
S
= 0. Thus, the measurement
actually has only two outcomes, the effective POVM is given
by )
D
,)
0
= I − )
D
, and it is clear that increasing the
eigenvalues of )
D
without changing its support increases the
ﬁgure of merit and leaves the noerror conditions satisﬁed.
This is true independently of the distribution p
D
, and thus the
optimal measurement is formed by )
D
being a projector onto
the biggest support allowed by the noerror condition (III) and
)
0
being a projector onto the orthocomplement. Moreover,
the quantity that completely characterizes the behavior of
the squeezedstates comparator is p(Dr,s) = )*)
D
*",
i.e., the conditional probability of obtaining the outcome )
D
if different squeezed states *" = S( re
iϕ
)0" ⊗ S(se
iϕ
)0"
(r (= s) are sent to the comparator. It is worth noting that
in what follows one does not need to know the actual value
of ϕ. Summarizing, in order to ﬁnd an optimal comparator
of squeezed states from S
ϕ
we need to reﬁne the deﬁnition
of the largest allowed support of )
D
hidden in the noerror
condition (9b). To do this we equivalently rewrite Eq. (9b) as
Tr(W )
D
W
†
W *")*W
†
) = 0 ∀*" ∈ S
ϕ
S
, (10)
which, by denoting E
D
≡ W )
D
W
†
and choosing W to be
the unitary transformation performed by the proposed setup
from Fig. 1(a), becomes
Tr[E
D
'
out
(r,r)""))'
out
(r,r)] = 0 ∀r ∈ R. (11)
The optimality of the proposed setup is proved by showing
that the biggest support allowed by the previous condition
coincides with the support of the projective measurement E
D
we use, see Eq. ( 7).
From the expression of '
out
(r,r)"", Eq. (2), it is clear
that for any operator E
D
with the s upport orthogonal to
the span of n"n", with n ∈ N, the unambiguous noerror
condition (11) holds. Hence, if any such operator E
D
is a
part of a POVM, then the emergence of the outcome related
to it unambiguously indicates the difference of the squeezing
parameters. We now proceed to show that the support of such
E
D
cannot be further enlarged. Now let us assume that a vector
v"" =
#
∞
h,k=0
d
hk
h"k" with at least one nonzero coefﬁcient
d
ii
is in the support of E
D
. As a consequence of the required
noerror condition (11), the overlap
))v'
out
(r,r)"" =
!
1 − λ(r)
2
∞
"
n=0
d
∗
nn
λ(r)
n
(12)
has to be vanishing for all values of r. Equation (12) is
vanishing if and only if
))v'
out
(r,r)""
!
1 − λ(r)
2
=
∞
"
n=0
d
∗
nn
λ(r)
n
(13)
vanishes for all r. The sum on the righthand side of Eq. (13)
can be seen as a polynomial in λ(r) and should vanish for all
possible values of λ(r), i.e., for all λ(r) < 1. Polynomials of
this type on a ﬁnite interval form a vector space with linearly
independent basis vectors λ(r)
k
, with k ∈ N. Thus the sum in
Eq. (13) vanishes ∀r ∈ )0,∞) only if d
nn
= 0, ∀n ∈ N. This
contradicts our assumption about the vector v", and therefore
the largest support an operator E
D
, unambiguously indicating
the difference of the squeezing parameters, can have is the
orthocomplement of the span of vectors n"n", with n ∈ N .
This concludes the proof.
In the case (ii) of compared states with completely arbitrary
phases of the complex squeezing parameters, the proof can be
done in the same way as before, up to deﬁning accordingly the
set of pairs of the same states.
IV. PERFORMANCE OF THE SETUP
In this section we give a thorough analysis of the statistics of
our setup also in the presence of nonunit quantum efﬁciency at
the detection stage in order to assess its reliability in Sec. IV C.
A. Probability of revealing the difference
The conditional probability of revealing the difference of
compared states with ξ (= ζ [but arg(ξ ) = arg(ζ ) = ϕ, though
unknown], which is the probability to obtain an E
D
outcome,
reads
p(Dξ,ζ ) = 1 − p(0ξ,ζ ), (14)
0123133
STEFANO OLIVARES et al. PHYSICAL REVIEW A 83, 012313 (2011)
with
p(0ξ,ζ ) = ))'
out
(ξ,ζ )E
0
'
out
(ξ,ζ )""
= [1 − λ(r
+
)
2
]
∞
"
n,m=0
[λ(r
+
)]
n
[λ
∗
(r
+
)]
m
×
∞
"
k=0
{[S(r
−
)]
kn
}
2
{[S
†
(r
−
)]
mk
}
2
, (15)
where '
out
(ξ,ζ )"" is given in Eq. (4). For ξ → ζ we correctly
obtain p(0ξ,ξ) = 1. By noting that [8]
[S(γ )]
hk
∝
%
exp
&
i
'
h−k
2
(
θ
)
for h,k odd or even,
0 otherwise,
(16)
where γ = γ e
iθ
, it is straightforward to see that Eq. (15) does
not depend on the (equal) phase ϕ of ξ and ζ . Thus, in order
to investigate the performances of the optimal squeezedstates
comparator, we may set ϕ = 0 and let ξ = r and ζ = s, with
r,s ∈ R, without loss of generality. Furthermore, it is possible
to show by numerical means that the probability p(Dr,s) does
not depend on the sum of the squeezing parameter δ
+
= r + s,
but only on the difference δ
−
= r − s. In Fig. 2 we plot
the probability p(Dr,s) given in Eq. (15) as a function of
δ
−
= r − s, and we compare it with the possible use of the
universal comparator [2], which works unambiguously for all
pure states leading to
p
UC
(Dω) =
1
2
(1 − ω
2
), (17)
where ω = )ψ
1
ψ
2
" = (cosh δ
−
)
−1/2
is the overlap between
the two squeezed vacuum states.
B. Inﬂuence of nonideal detectors
In a realistic scenario, in which the photonnumber resolv
ing detectors have nonunit quantum efﬁciency η, we should
modify the POVM by replacing the projectors n")n in Eq. (7)
by the f ollowing operators [9,10]:
)
n
(η) = η
n
∞
"
k=n
(1 − η)
k−n
*
k
n
+
k"
)
k

, (18)
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.05
0.10
0.15
0.20
p
D r, s
δ
FIG. 2. (Color online) Conditional probability of revealing the
difference of two squeezed vacuum states ψ
1
" = S(r)0" and ψ
2
" =
S(s)0" in the ideal case (η = 1) as a function of δ
−
= r − s. Solid
lines, from top to bottom, correspond to the optimal squeezedstates
comparator (blue) and the universal comparator (red line). The dashed
line is the upper bound on the probability in the case of only two
possible squeezings. See Sec. IV C for details.
namely (we assume that the two detectors have the same
quantum efﬁciency),
E
0
(η) =
∞
"
n=0
)
n
(η) ⊗ )
n
(η), (19a)
E
D
(η) = I − E
0
(η). (19b)
The performance of this kind of detector and its reliability to
resolve up to tens of photons have been recently investigated
in Ref. [11]. Equation (18) shows that the single projector
is turned into a (inﬁnite) sum of projectors. This could be
a relevant issue for protocols that rely on the discrimination
between even and odd number of photons [4], as we mentioned
in Sec. I, since it becomes challenging to detect the actual parity
of the number of incoming photons, )
n
(η) being a sum over
both even and odd number of photons. For what concerns our
setup, as one may expect, the presence of nonunit quantum
efﬁciency no longer guarantees the unambiguous operation.
In principle, the effect of an imperfect detection could be
taken into account while designing the comparison procedure.
However, this would be mathematically challenging and
most probably would not provide an unambiguous procedure
anyway because of the form of the noise in realistic detectors.
Alternatively, one could try to maximize the reliability (con
ﬁdence) of the outcomes (for maximum conﬁdence in state
discrimination see [12]), nevertheless this would require one
to make some particular choice of the prior probabilities z
S
and z
D
and of the probability distributions p
S
(*) and p
D
(*)
(see Sec. III). By following this type of approach, an optimal
comparison of coherent states in realistic conditions can be
improved by employing a linear ampliﬁer [13]. On the other
hand, as we are going to show in the next section, the reliability
of the difference detection of our proposal is quite close to
unambiguity if the detector efﬁciency is high enough.
The conditional probability p
η
(Dξ,ζ ) for the detectors
with nonunit quantum efﬁciency η reads
p
η
(Dξ,ζ ) = 1 − p
η
(0ξ,ζ ), (20)
with
p
η
(0ξ,ζ ) = ))'
out
(ξ,ζ )E
0
(η)'
out
(ξ,ζ )""
= [1 − λ(r
+
)
2
]
∞
"
n,l,m=0
η
2n
[λ(r
+
)]
l
[λ
∗
(r
+
)]
m
×
∞
"
h,k=n
(1 − η)
h+k−2n
*
h
n
+*
k
n
+
×[S(r
−
)]
kl
[S(r
−
)]
hl
[S
†
(r
−
)]
mk
[S
†
(r
−
)]
mh
,
(21)
which, in the case of ξ = ζ , reduces to
p
η
(0ξ,ξ ) = ))'
out
(ξ,ξ )E
0
(η)'
out
(ξ,ξ )""
= [1 − λ(ξ )
2
]
∞
"
n=0
η
2n
λ(ξ )
2n
×
2
F
1
[1 + n,1 + n,1,(1 − η)
2
λ(ξ )
2
], (22)
where
2
F
1
are hypergeometric functions and [S(r
−
)]
lm
are
the matrix elements of the squeezing operator as in Eq. (6).
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OPTIMAL UNAMBIGUOUS COMPARISON OF TWO UNKNOWN . . . PHYSICAL REVIEW A 83, 012313 (2011)
0.2 0.4 0.6 0.8 1.0
r
0.2
0.4
0.6
0.8
1.0
p
η
( r‚ r )
FIG. 3. (Color online) p
η
(0r,r) (solid lines) and p
η
(Dr,r) (dot
dashed lines) as functions of r for different values of the efﬁciency
η; from top to bottom (solid) and from bottom to top (dotdashed):
η = 0.999 (red), 0.99 (green), 0.90 (blue), 0.50 (magenta).
Because of Eq. (16), the probabilities (20) and (21) are still
independent of the unknown value of ϕ, thus, from now on,
we set ϕ = 0 and put ξ = r and ζ = s, with r,s ∈ R, without
loss of generality. In Fig. 3 we plot p
η
(0r,r) and p
η
(Dr,r) for
different values of η. If r . 1, then Eq. (22) can be expanded
up to the second order in r, obtaining
p
η
(0ξ,ξ ) / 1 − 2η(1 − η)r
2
. (23)
C. Reliability of the setup
To assess the reliability of our setup, we address the
scenario in which only two squeezing parameters for each
of the squeezed vacua are possible. In such case one knows
that the two squeezing parameters are either {(r,r),(s,s)} or
{(r,s),(s,r)} with the same prior probability. Our squeezed
states comparator may not be optimal in this case. However,
as one can see in Fig. 2, the performance of our setup is
nearly as good as if it were optimized also for this restricted
scenario. In particular, the dashed line in Fig. 2 refers to the
optimal measurement, unambiguously detecting the difference
in the case of only two possible squeezing parameters, in the
formula [14]
p
max
(Dω) =
1 − ω
2
1 + ω
2
. (24)
We deﬁne the reliability R
D
of the scheme in revealing
the difference of the squeezing parameters r and s as the
conditional probability of the two squeezed vacuum states
being different if the outcome E
D
is found, i.e. (we assume
equal prior probabilities),
R
D
(η; r,s) =
p
η
(Dr,s) + p
η
(Ds,r)
#
u,v=r,s
p
η
(Du,v)
. (25)
In the ideal case, i.e., η = 1, we have p
η
(Dr,r) = 0 and, thus,
R
D
(η; r,s) = 1, which is guaranteed by the construction of
the setup. On the other hand, if η < 1, then p
η
(Dr,r) (= 0
and, consequently, the conclusion based on the outcome D
is not unambiguous anymore. The actual value of R
D
can be
numerically calculated starting from Eqs. (20) and (21). The
reliability R
D
(η; r,s) is plotted in the upper panel of Fig. 4 as
a function of δ
−
= r − s. Note that differently from the case
η = 1, for η < 1 the probability p
η
(Dr,s) depends not only on
the difference δ
−
= r − s but also on the sum δ
+
= r + s.
The dependence on δ
+
is shown in the the lower panel of
0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.5
0.6
0.7
0.8
0.9
1.0
R
D
0.2 0.4 0.6 0.8 1.0
0.5
0.6
0.7
0.8
0.9
1.0
R
D
δ
δ
FIG. 4. (Color online) Top: Reliability R
D
(η; r,s) as a function of
δ
−
for ﬁxed δ
+
= 1.0 and different values of the efﬁciency. Bottom:
Reliability R
D
(η; r,s) as a function of δ
+
for difference δ
−
= 0.2 and
different values of the efﬁciency. In both plots, from top to bottom:
η = 0.999 (red), 0.99 (green), 0.90 (blue), 0.50 (magenta).
Fig. 4, where we plot R
D
(η; r,s) as a function of δ
+
for ﬁxed
difference δ
−
= 0.2.
V. CONCLUDING REMARKS
In this paper we have addressed the comparison of two
squeezed vacuum states of which we have a single copy
available. We have suggested an optical setup based on a
beam splitter, a phase shifter, and two photodetectors which is
feasible with the current technology. Even though we analyzed
the scenario with an equal, though unknown, phase of the
compared states, our setup is able to operate unambiguously
with ideal detectors irrespective of the squeezing phases, and
without the knowledge of the relative phases of the squeezed
states. We have proved the optimality of our scheme for
arbitrary phases and ideal detectors, and we analyzed its
performance and reliability also in the presence of nonunit
quantum efﬁciency at the detection stage in the case of equal
phases. As one may expect, the detection efﬁciency strongly
affects the reliability; nevertheless we have shown that for
small energies and not too low quantum efﬁciency, the setup
is still robust.
Our scheme may be employed not only for the comparison
of two squeezed vacua, but also for a more general scenario in
which the input states ξ " and ζ "are known to be transformed
by two ﬁxed known local unitaries U and V , respectively
(namely, U ξ " ⊗ V ζ ") or by any ﬁxed known global unitary
transformation W (namely, W ξ " ⊗ ζ "): now it is enough to
apply the inverse of the transformation before processing the
state with our setup.
ACKNOWLEDGMENTS
Fruitful discussions with M. Ziman are acknowledged. This
work has been supported by the project INQUEST APVV
SKIT000708 within the “Executive programme of scientiﬁc
and technological cooperation between Italy and Slovakia,”
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