Entanglement condition via su(2) and su(1,1) algebra using
  Schr{\"o}dinger-Robertson uncertainty relation
Entanglement condition via su(2) and su(1,1) algebra using Schrödinger-Robertson
uncertainty relation
Hyunchul Nha∗
ARC Center of Excellence for Quantum Computer Technology,
University of Queensland, Australia
School of Computational Sciences,
Korea Institute for Advanced Study, Korea
(Dated: August 25, 2021)
The Schrödinger-Robertson inequality generally provides a stronger bound on the product of
uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as
such, it can yield a stricter separability condition in conjunction with partial transposition. In this
paper, using the Schrödinger-Robertson uncertainty relation, the separability condition previously
derived from the su(2) and the su(1,1) algebra is made stricter and refined to a form invariant with
respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant
separability condition.
PACS numbers: 03.67.Mn, 03.65.Ud, 42.50.Dv
I. INTRODUCTION
When a quantum system is subject to measurements
corresponding to two noncommuting observables {A, B},
the product of uncertainties in measurement outcomes,
〈(∆A)2〉〈(∆B)2〉, has a certain lower bound. The Heisen-
berg uncertainty relation (HUR)[1], which is most widely
used, provides the bound as
〈(∆A)2〉〈(∆B)2〉 ≥
|〈[A,B]〉|2. (1)
On the other hand, the Schrödinger-Robertson rela-
tion(SRR) [2, 3] in general provides a stronger bound
〈(∆A)2〉〈(∆B)2〉 ≥
|〈[A,B]〉|2 + 〈∆A∆B〉2S , (2)
where the cross correlation 〈∆A∆B〉S is defined in a sym-
metric form as
〈∆A∆B〉S ≡
〈∆A∆B +∆B∆A〉. (3)
The SRR can be derived from the Cauchy-Schwartz in-
equality, 〈f |f〉〈g|g〉 ≥ |〈f |g〉|2, where |f〉 = ∆A|Ψ〉 and
|g〉 = ∆B|Ψ〉 for a generic quantum state |Ψ〉[4]. The
HUR describes a special case of the SRR under the con-
dition 〈∆A∆B〉S = 0, which is of course not always met.
Recently, one of the important issues in quantum in-
formatics has been to obtain conditions by which one
can distinguish entangled states from separable ones.
Some of such entanglement criteria derived so far have
relied on the bounds set by various forms of uncertainty
relations[5, 6, 7, 8], and remarkably for certain cases,
in explicit conjunction with partial transposition(PT)[9,
10, 11]. More precisely, separable states can represent a
certain physical state even under PT[12] and all uncer-
tainty relations must therefore be satisfied by separable
states under PT. The uncertainty relations in combina-
tion with PT can thereby provide necessary conditions
for separability.
For continuous variables (CVs), earlier works were
focused on Gaussian entangled states[13, 14, 15], but
considerable attention has also been directed to non-
Gaussian entangled states[16]. Most of all, the separa-
bility conditions applicable to non-Gaussian entangled
states have recently emerged[8, 9, 10, 11], and in partic-
ular, Refs. [8, 10, 11] employed the su(2) and the su(1,1)
algebra to derive such entanglement criteria. Using the
HUR along with those two algebras, Nha and Kim have
particularly derived the optimal separability condition
among a certain class of inequalities[11]. This condi-
tion has also been proposed to detect multipartite en-
tanglement of photonic W states and shown to be robust
against the detector inefficiency[17].
In this paper, it is our aim to refine the separability
condition in Refs. [10, 11] by employing the SRR instead
of the HUR. By doing this, we obtain a stricter separa-
bility condition given by a form invariant with respect to
local phase shifts. This invariance is a very adequate
attribute as entanglement condition, for entanglement
property must be invariant under any local unitary oper-
ations. Furthermore, we propose how to experimentally
test this invariant condition using linear optics and also
discuss the practical connection of the previous condition
in [10, 11] to the present one.
II. SEPARABILITY CONDITION
First, we briefly introduce how to derive the separabil-
ity condition via the uncertainty relations in the su(2)
and the su(1,1) algebra[11]. The su(2) algebra deals
with the angular momentum operators Jx, Jy and Jz,
which obey the commutation relations [Ji, Jj ] = iǫijkJk
(i, j, k = x, y, z). This algebra can be represented by two
http://arxiv.org/abs/0704.1939v2
bosonic operators a and b, as
a†b+ ab†
a†b− ab†
a†a− b†b
. (4)
On the other hand, the operators Kx,Ky and Kz in the
su(1,1) algebra can be represented by
a†b† + ab
a†b† − ab
a†a+ b†b+ 1
, (5)
which results in the commutation relations, [Kx,Ky] =
−iKz, [Ky,Kz] = iKx, and [Kz,Kx] = iKy, different in
sign from those of the su(2) algebra.
Specifically, the commutator [Kx,Ky] = −iKz in the
su(1,1) algebra gives the uncertainty relation via the
HUR as
〈(∆Kx)
2〉〈(∆Ky)
|〈Kz〉|
2, (6)
which must be satisfied by any quantum states. Most
importantly, the inequality (6) must be satisfied under
PT by every separable state, since it can still describe
a certain physical state[12]. That is, one obtains the
separability condition as
〈(∆Kx)
2〉PT〈(∆Ky)
2〉PT ≥
|〈Kz〉|
PT, (7)
where the subscript PT means that the quantum average
is calculated after taking partial transposition. Using a
general relation
〈a†manb†pbq〉ρPT = 〈a
†manb†qbp〉ρ (8)
between the quantum average for the partially trans-
posed density operator ρPT and that for the original den-
sity operator ρ[11], the inequality (7) can be recast to give
the separability condition expressed as
+ 〈(∆Jx)
+ 〈(∆Jy)
[1 + 〈N+〉]
, (9)
where N+ = a
†a + b†b is the total excitation number.
Note that the inequality (9) is the optimal condition de-
rived in [11], where the HUR was employed in a sum form
to obtain a class of separability conditions[18].
III. STRICTER SEPARABILITY CONDITION
In this section, let us now start from the SRR for the
commutator [Kx,Ky] = −iKz, i.e.,
〈(∆Kx)
2〉〈(∆Ky)
|〈Kz〉|
2 + 〈∆Kx∆Ky〉
S , (10)
instead of the HUR, then follow the same steps as below
Eq. (6). Using the relation
〈∆Kx∆Ky〉S,PT = 〈∆Jx∆Jy〉S (11)
via Eq. (8), we obtain a separability condition stricter
than the one in (9) as
+ 〈(∆Jx)
+ 〈(∆Jy)
[1 + 〈N+〉]
+ 〈∆Jx∆Jy〉
S .(12)
Compared with the inequality (9), the new inequal-
ity (12) prodvides a stronger condition for separabil-
ity as long as the off-diagonal covariance 〈∆Jx∆Jy〉S is
nonzero. As an example, consider the two-photon en-
tangled states of the type |Ψ〉 = cos θ|2, 0〉+ i sin θ|0, 2〉.
All these states satisfy the inequality (9), but violate the
stricter one in (12), regardless of the parameter θ. There-
fore, only the inequality (12) can detect entanglement for
those two-photon states.
We next show that the inequality (12) is invariant with
respect to local phase shifts. Let us consider a 2 × 2
covariance matrix C of which elements are defined as
Cij ≡
〈∆Ji∆Jj +∆Jj∆Ji〉, (13)
where {i, j} = {x, y}. The inequality (12) is then ex-
pressed as
Det{C}+
Tr{C} ≥
2 + 2〈N+〉
, (14)
where Det{} and Tr{} denote the determinant and the
trace of a matrix. If one takes a local phase shift for
mode b as b′ = be−iφ, the su(2) operators Jx and Jy are
transformed into
cosφ sinφ
− sinφ cosφ
. (15)
The determinant and the trace of a matrix are unchanged
under rotation, and the total photon number 〈N+〉 is
also preserved through passive optical elements. The in-
equality (14) is therefore invariant with respect to local
phase shifts. This is an attribute very adequate as entan-
glement condition, for entanglement should be invariant
under local unitary operations. Note that a phase shift is
the only local unitary operation that preserves the total
photon number.
IV. MEASUREMENT SCHEME
We now discuss how the separability condition (12)
can be tested in experiment. In Ref.[11], a linear optical
scheme was proposed to measure the observables Jx, Jy
and 〈N+〉 for the inequality (9), as depicted in Fig. 1.
The mode b first undergoes a phase shift by φ and the
FIG. 1: Experimental scheme for measuring the quantities
necessary to test the inequality (12). All the quantum av-
erages in (12) can be measured by detecting the photon
number difference at the output, N{−,φ} ≡ c
c − d
−iφ + ab†eiφ, with four different phase shifts, φ = 0, π
and −π
. (See the main text.) BS: 50:50 beam-splitter, PS:
phase-shifter, and PD: photo detector.
two modes a and b are then injected to a 50:50 beam
splitter. The modes c and d at the output are given by
c = 1√
(a+ be−iφ) and d = 1√
(−a+ be−iφ), respectively.
One needs to measure the photon number difference at
the output, i.e.,
N{−,φ} ≡ c
†c− d†d = a†be−iφ + ab†eiφ, (16)
which becomes 2Jx (2Jy) for φ = 0 (φ =
). (See
Eq. (4).) The total photon number 〈N+〉 is simply given
by the sum, c†c+ d†d, at the output.
In the present inequality (12), in addition to Jx, Jy and
〈N+〉, one also needs to measure the off-diagonal covari-
ance 〈∆Jx∆Jy〉S . Note that 〈∆Jx∆Jy〉S =
〈JxJy +
JyJx〉 − 〈Jx〉〈Jy〉, where
JxJy + JyJx =
a†2b2 − a2b†2
{−,φ=π
{−,φ=−π
. (17)
Thus, by choosing two different phase shifts φ = π
φ = −π
in Fig. 1, the quantum average 〈JxJy + JyJx〉
can be measured in two pieces as shown in Eq. (17). In
summary, the single experimental setup in Fig. 1 can be
used to measure all the quantities necessary to test the
inequality (12).
Finally, we discuss how the inequality (9) can be re-
garded as ”equivalent” to the stricter inequality (12). Us-
ing the relation in Eq. (15) implemented by a local phase
shift, one has the covariance in the rotated frame as
〈∆J ′x∆J
y〉S =
sin 2φ
〈(∆Jy)
2〉 − 〈(∆Jx)
+cos 2φ〈∆Jx∆Jy〉S . (18)
Thus, by choosing the phase shift as
tan 2φ =
2〈∆Jx∆Jy〉S
〈(∆Jx)2〉 − 〈(∆Jy)2〉
, (19)
the covariance in the rotated frame can be made vanish.
In this situation, the inequality (12) is reduced to the
inequality (9). In other words, as long as one is allowed
to perform a local phase shift, which does not alter the
entanglement property at all, the two inequalities can be
interpreted as equivalently useful. However, this relies
on the capability of measuring all the covariances and
of performing a phase shift very accurately required by
Eq. (19). It is then of no practical advantage to adhere
to the inequality (9): One can simply test the inequal-
ity (12) if one is able to measure the off-diagonal covari-
ance 〈∆Jx∆Jy〉S in addition.
V. SUMMARY
In this paper, we have derived a stricter separability
condition via the su(2) and the su(1,1) algebra using the
Schrödinger-Robertson inequality instead of the Heisen-
berg uncertainty relation. It has been shown that this
refined condition is expressed in a form invariant with
respect to local phase shifts. A linear optical setup has
been proposed to test the invariant separability condition
and the practical connection of the previously obtained
condition to the present one was also discussed.
Note added in proof. Recently, the author has learned
that a similar linear optical method was proposed to mea-
sure the same quantities as the ones in this paper, but in
a different context[19].
VI. ACKNOWLEDGMENT
This work was supported by the University of Queens-
land.
*email:phylove00@gmail.com
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http://arxiv.org/abs/quant-ph/9711028
