Quantum entanglement of decohered two-mode squeezed states in absorbing
  and amplifying environment
Quantum entanglement of decohered two-mode squeezed states in absorbing and
amplifying environment
Phoenix S. Y. Poon and C. K. Law
Department of Physics and Institute of Theoretical Physics,
The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China
(Dated: November 5, 2018)
We investigate the properties of quantum entanglement of two-mode squeezed states interacting
with linear baths with general gain and loss parameters. By explicitly solving for ρ from the master
equation, we determine analytical expressions of eigenvalues and eigenvectors of ρTA (the partial
transposition of density matrix ρ). In Fock space, ρTA is shown to maintain a block diagonal
structure as the system evolves. In addition, we discover that the decoherence induced by the baths
would break the degeneracy of ρTA , and leads to a novel set of eigenvectors for the construction of
entanglement witness operators. Such eigenvectors are shown to be time-independent, which is a
signature of robust entanglement of two-mode squeezed states in the presence of noise.
PACS numbers: 03.67.Mn, 03.65.Yz, 42.50.Dv, 42.50.Lc
I. INTRODUCTION
Optical two-mode squeezed vacuum (TMSV) has been
a major source of continuous-variable entanglement for
quantum communication [1]. In recent years, intriguing
applications such as quantum teleportation [2, 3, 4, 5, 6]
and quantum dense coding [7, 8] have been demon-
strated experimentally with TMSV. Theoretically, it is
also known that TMSV maximizes the EPR correlation
when a fixed amount of entanglement is given [9]. In
order to exploit fully the non-classical properties of such
entangled light fields, it is important to understand deco-
herence effects as they propagate through noisy environ-
ments [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. This belongs
to a more subtle topic involving the characterization and
quantification of mixed state entanglement in general.
For bipartite systems, Peres and Horodecki have de-
veloped a powerful criterion of entanglement, which is
known as the PPT (positive partial transposition) crite-
rion [20, 21, 22]. If the partial transposition of a den-
sity matrix (denoted by ρTA) has one or more negative
eigenvalues, then the state is an entangled state. Phys-
ically, the partial transposition for separable states can
be considered as a time-reversal operation, and one can
construct a variety of uncertainty relations serving as in-
dicators of entanglement [23, 24]. For two-mode Gaus-
sian states such as TMSV, PPT provides a necessary and
sufficient condition of separability [25, 26].
The dynamics of disentanglement of TMSV in various
noisy situations has been addressed by several authors
recently [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. The
fact that an amplitude damped TMSV remains Gaus-
sian enables an elegant description of entanglement based
on the properties of covariance matrix associated with
the density operators [27]. In particular, from the time-
dependent solution of Wigner function [15, 16, 17] or
the corresponding characteristic function [19], one can
quantify the degradation of entanglement by calculating
the negativity [28] and relative entropy [29]. It is now
known that for an initial TMSV at a non-zero thermal
bath, quantum entanglement vanishes completely in a fi-
nite time [27].
However, we notice that there are much less investiga-
tions addressing the structure of ρTA directly, and yet ρTA
is what the PPT criterion originally based upon. Since
ρTA could manifest differently in various basis, the study
of ρTA in Fock space, for example, could reveal entangle-
ment properties not easily found by the Wigner function
method [30]. An example we notice is the construction of
entanglement witness operators via the projectors formed
by the eigenvectors of ρTA with negative eigenvalues [31].
Such entanglement witness operators, which correspond
a variety of observables for the detection of entanglement,
are determined by ρTA .
The main purpose of this paper to indicate some key
features of decohered entanglement as revealed by eigen-
values and eigenvectors of ρTA . Our analysis will con-
centrate on the structures of eigenvectors in Fock space,
which is also where interesting non-local correlations of
continuous-variable systems can be observed [32]. For
a TMSV under the influence of amplitude damping (or
gaining in an amplifier), we solve for the time evolution of
ρ and determine the exact eigenvectors and eigenvalues
of ρTA analytically. These eigenvectors are shown to have
a strong correlation in photon numbers, and hence ρTA is
a block diagonal matrix in Fock space. Therefore witness
operators associated with each block involve only a finite
number of Fock vectors, which implies that the detection
of entanglement can only require a small portion of the
Hilbert space. This is in contrast to entanglement detec-
tion based on uncertainty relations in which the entire
Hilbert space is usually involved [23, 24, 25, 26]. In this
sense the eigenvectors of ρTA access the entanglement sig-
natures ‘locally’, which is a complement to ‘global’ char-
acterization (of Gaussian states) using covariance matri-
ces. As we shall see below, as long as the initial state is
a TMSV, the corresponding eigenvectors do not change
with time, indicating that the entanglement carried by
TMSV is robust against amplitude damping.
http://arxiv.org/abs/0704.1965v1
FIG. 1: The sub-matrix structure of ρTA , with initial TMSV,
in the Fock basis.
II. MASTER EQUATION AND SOLUTION
To begin with, we consider the time evolution of
an initial TMSV, each coupled with a separate phase-
insensitive linear bath. In terms of the annihilation op-
erators a and b of the two modes, the master equation
governing the dynamical process is [33]:
ρ̇ = G(2a†ρa− aa†ρ− ρaa† + 2b†ρb− bb†ρ− ρbb†)
+ L(2aρa† − a†aρ− ρa†a+ 2bρb† − b†bρ− ρb†b),
where G and L are the gain and loss parameters respec-
tively, both having a dimension of time−1. Depending
on the values of G and L, the master equation describes
amplifying or damping effects due to the coupling with
the baths. For dissipation in thermal baths, each of
temperature T , we have the parameters G = γ
nth and
L = γ
(nth +1), where nth =
exp(hω/kT )−1 is the average
number of photons in each of the modes (with frequency
ω) at thermal equilibrium, and γ/2 is the decay rate of
the mode amplitudes. In this paper we focus on the ini-
tial TMSV with the squeezing parameter r > 0:
|ψ(0)〉 = exp[r(a†b† − ab)] |00〉 =
1− λ2
λn |nn〉 ,(2)
where λ ≡ tanh r and |00〉 is the two-mode vacuum state.
A. Block structures of ρ
TA in Fock space
To investigate the entanglement properties of the den-
sity matrix ρ, we study its partial transposition ρTA . The
ρTA is of infinite dimension, however, by examining the
master equation in the Fock basis, block structures of
ρTA can be identified. Let us denote the matrix elements
of ρTA by
ρTAn,m,p,q =
nm|ρTA |pq
= 〈pm|ρ|nq〉 , (3)
which is governed by the following differential equation:
ρ̇TAn,m,p,q = G[2
npρTAn−1,m,p−1,q
mqρTAn,m−1,p,q−1
−(n+m+ p+ q + 4)ρTAn,m,p,q]
(n+ 1)(p+ 1)ρTAn+1,m,p+1,q
(m+ 1)(q + 1)ρTAn,m+1,p,q+1
−(n+m+ p+ q)ρTAn,m,p,q]. (4)
ρTAn,m,p,q(t = 0) = δpmδnq(1− λ2)λm+n. (5)
corresponds to the initial condition (2).
It can be seen from Eq. (4) that each el-
ement ρTAn,m,p,q(t) is coupled with elements
ρTAn+l,m+k,p+l,q+k(0) only, for integers l and k.
Therefore ρTAn+l,m+k,m+l,n+k (t) are the only non-zero
elements at any time t > 0 because of the initial con-
dition. By noting that the sum of the first two indices
equal to that of the last two, we can group all non-zero
elements ρTAn+l,m+k,m+l,n+k (t) into sub-matrices MS
according to the sum index S = n + l + m + k, i.e.,
ρTAn+l,m+k,m+l,n+k (t) is contained in Mn+l+m+k. We
can therefore express ρTA in a direct sum of MS as
follows:
ρTA(t) =
MS(t), (6)
where the sub-matrix MS has a dimension of S+1, since
elements inMS have its first two indices as {0, S}, {1, S−
1}, ... , {S, 0}. Fig. 1 shows the sub-matrix structure of
ρTA . Note that characteristic sum S is equal to the total
number of photons that the two modes contain. From
Eq. (4) we observe that probabilistic flow occurs between
elements in neighboring sub-matrices, with emission or
absorption of one photon in one of the modes at one
time.
The time evolution of a typical sub-matrix of ρTA is
illustrated schematically in Fig. 2, which will be dis-
cussed in detail in the later part of the paper. Ini-
tially, only opposite-diagonal elements are present, hav-
ing the magnitude as λS . As time increases, element
flows from neighboring sub-matrices, and disentangle-
ment of the sub-matrices occurs at a critical time t =
tc (Section IIIA). In the case of thermal bath, ρ
evolves into a diagonal form in the long time limit,
settling as the thermal equilibrium state ρTAn,m,p,q =
δnpδmq
(nth+1)2
( nth
nth+1
)n+m.
B. Analytic solution of ρ in position space
To analyze the properties of ρTA , it is more conve-
nient to first determine ρ in position space and then
FIG. 2: (Color online) Schematic diagram showing the evo-
lution of the distribution of elements in a sub-matrix of ρTA ,
assuming the baths are thermal baths. The tc is the critical
time for disentanglement.
make the transformation to Fock space. The position
space method of finding ρ was previously employed in
Ref. [11, 34] in studying entanglement in various oscil-
lator systems. In this subsection, we present an explicit
solution of master equation (1) with an initial TMSV.
We remark that our method is different from that given
in [11], as the latter involves a Fourier transform of the
density matrix, i.e., the momentum space. Here we solve
the density matrix entirely in position space (Appendix
A). This turns out to be more convenient for the real
symmetric Gaussian states considered here, since fewer
differential equations are involved. In addition, the re-
sultant solution is more transparent for further analysis
of eigenvectors in the next section.
Let us denote the ‘position’ operators as x = 1√
(a+a†)
and y = 1√
(b + b†), and define
ρ(x1, y1;x2, y2; t) ≡ 〈x1, y1| ρ(t) |x2, y2〉 , (7)
then the master equation (1) becomes,
ρ̇ = −1
[L(x21 + x
2 − 2x1x2 + y21 + y22 − 2y1y2 − 4
−∂2x1 − ∂
− ∂2y1 − ∂
− 2∂x1∂x2 − 2∂y1∂y2
−2x1∂x2 − 2x2∂x1 − 2y1∂y2 − 2y2∂y1)
+G(x21 + x
2 − 2x1x2 + y21 + y22 − 2y1y2 + 4
−∂2x1 − ∂
− ∂2y1 − ∂
− 2∂x1∂x2 − 2∂y1∂y2
+2x1∂x2 + 2x2∂x1 + 2y1∂y2 + 2y2∂y1)]ρ. (8)
For an initial state (2), ρ(x1, y1;x2, y2; t) takes a Gaussian
form at any time t,
ρ(x1, y1;x2, y2; t) = Ξ(t) exp[−A(t)(x21 + x22 + y21 + y22)
+B(t)(x1y1 + x2y2)
+C(t)(x1x2 + y1y2)
+D(t)(x1y2 + x2y1)], (9)
where A(t), B(t), C(t) and D(t) are real time-dependent
coefficients, and the normalization factor is:
Ξ(t) =
[2A(t)− C(t)]2 − [B(t) +D(t)]2. (10)
By substituting Eq. (9) into the master equation, the
coefficients are found to obey a set of coupled equations
that can be solved analytically (Appendix A). For the
TMSV considered here, we have,
A(t) =
[2A0η +
G− L(η − 1) +
2(〈x2〉2t − 〈xy〉
B(t) =
[B0η +
〈xy〉t
2(〈x2〉2t − 〈xy〉
C(t) =
[2A0η +
G− L(η − 1)−
2(〈x2〉2t − 〈xy〉
D(t) =
[−B0η +
〈xy〉t
2(〈x2〉2t − 〈xy〉
]. (11)
Here A0 =
cosh 2r, B0 = sinh 2r and η(t) = exp[2(G−
L)t] are defined, and the expectation values are given by,
= A0η +
2(G− L) (η − 1),
〈xy〉t =
η. (12)
III. PROPERTIES OF ρ
According to PPT criterion, the appearance of nega-
tive eigenvalues of ρTA is a signature of entanglement. In
this section, we solve the eigenvectors and eigenvalues of
ρTA as the system evolves. Then we discuss how deco-
herence affects the entanglement properties of ρTA . The
eigenvalues and eigenvectors of ρTA are defined by:
ρTA(x1, y1;x2, y2; t)ϕn,m(x2, y2; t)dx2dy2
= ξn,m(t)ϕn,m(x1, y1; t). (13)
Our main technique of solving the eigen-problem is the
use of Mehler formula which expands a double Gaussian
function into a series of orthogonal functions. After some
calculations (see Appendix B), we obtain the expression
of eigenvalues,
ξn,m(t) = Ξ(t)π
β1)n+1
β2)m+1
and the corresponding eigenvectors,
ϕn,m(x1, y1) =
2n+mn!m!π
x1 − y1√
x1 + y1√
× exp[−1
(x21 + y
1)] (15)
where Hn are the Hermite polynomials. In writing
Eq. (14), we have defined
α1(t) =
[2A(t)−B(t) + C(t) +D(t)],
β1(t) =
[2A(t) +B(t)− C(t) +D(t)],
α2(t) =
[2A(t) +B(t) + C(t)−D(t)],
β2(t) =
[2A(t)−B(t)− C(t)−D(t)] (16)
and the solution of A, B, C and D are given by Eq. (11).
Note that in writing Eq. (15) from (B8), we have used
the fact that for TMSV, α1β1 = α2β2 =
for all time
t ≥ 0.
We now transform the eigenvectors from the position
space to the Fock space. Note that ρTA is a basis de-
pendent operation. The eigenvectors of ρTA defined in
two different basis sets do not transform directly. An
exception is the case when the two sets of basis vectors
transform by a real unitary matrix [35], which is the case
here. This allows us to write down the eigenvector |ϕn,m〉
in Fock space from Eq. (15):
|ϕn,m〉 = [
a† − b†√
a† + b†√
)m] |00〉 , (17)
which can be connected to the sub-matrices of ρTA (Fig.
1) via the photon number sum S ≡ m + n to label the
eigenket, so that
|ϕn,S−n〉 =
n!(S − n)!
ΓS,n,j
j!(S − j)! |j, S − j〉 . (18)
Here we have used the abbreviation
ΓS,n,j ≡
min(j,n)
(−1)n−kCS−nj−k C
k , (19)
with CSr ≡ S!r!(S−r)! . In this way |ϕn,S−n〉 and ξn,S−n
(n = 0, 1, ..., S) are the nth eigenvector and eigenvalue of
the block with characteristic sum S.
A. Evolution of negative eigenvalues
By inspecting Eq. (14), we find that all the negative
eigenvalues in each block share the same value at t = 0
(Fig. 3). The same is true also for positive eigenval-
ues. However, such a strong degeneracy is broken by
coupling with the baths. This is illustrated in Fig. 3
where the time-dependence of individual eigenvalues in
various block indices S is shown. Except at t = 0 and at
the critical time tc, we see that the negative eigenvalues
possessing different values.
It is important to observe that the eigenvalues are neg-
ative for odd n, when
β1, or in other words,
B(t) > C(t). All eigenvalues turn zero at the same time,
when we have B(t) = C(t), except for the only eigenvalue
with n = 0 in each block. Such a critical time tc is given
2 (L−G) log
G+ Lλ
G (1 + λ)
which is always positive finite as long as G 6= 0. For
the case G → 0, we have tc → ∞. We remark that the
FIG. 3: (Color online) The eigenvalues of sub-matrices of ρTA
of an amplifier system with G = 1.5γ, L = 0.5γ and initial
squeezing factor r = tanh−1 0.2, with characteristic sum (a)
S = 1, (b) S = 4, (c) S = 7, and (d) S = 10. Red line
indicates the n = 0 eigenvalue which does not turn zero at
tc, while blue line indicates the most negative eigenvalue with
n = 1.
disentanglement time tc was previous obtained in Ref.
[26] for thermal baths, here we obtained a general ex-
pression (20) that applies to linear amplifiers as well. In
particular, in the case when gain and loss parameters are
equal, i.e., G = L, the critical time can be reduced to
We point out that at the time of disentanglement t =
tc, there is only one non-zero eigenvalue (with the index
n = 0) in each sub-matrix (Fig. 3). Therefore ρTA at the
critical time is highly degenerate, and the corresponding
symmetry property of ρTA(t = tc) is indicated in the
relation:
ρTAj,S−j,j,S−j = ρ
j,S−j,S−j,j
= ρTAS−j,j,j,S−j = ρ
S−j,j,S−j,j. (21)
This results in the symmetric distribution of elements as
shown in Fig. 2 schematically.
As a further remark, it is interesting that negative
eigenvalues may not necessarily be monotones over time.
This can be seen by differentiating Eq. (14) and looking
at the initial rate:
ξ̇n,S−n
= (−1)n tanhS r csch rsech3r
×{(G− L)(S − 2n)
+(G+ L)(S − 2n) cosh2r
−[LS +G(2 + S)] sinh 2r}, (22)
which can result in a negative value for certain param-
eters, i.e., some eigenvalues of odd n can become more
negative over time (Fig. 3d). An exceptional case is when
G = 0, where we find that for odd n, the derivative at
t = 0 must be positive by inspecting Eq. (22).
FIG. 4: (Color online) The negativity N of ρTA with different
G, fixing the parameters L = γ and initial squeezing factor
r = tanh−1 0.2.
B. Negativity and sub-negativity
Negativity N serves as a computable measure of en-
tanglement defined by the trace norm of ρTA minus 1
divided by 2 [28]. For separable states, ρTA is still a den-
sity matrix with trace 1 and hence N = 0. However, for
non-separable states with negative ρTA , we have N > 0.
Specifically, N equals the sum of the absolute value of
negative eigenvalues of ρTA [28, 36]. For the system con-
sidered in this paper, N reads,
N = πΞ(t)
). (23)
Alternatively,N can be derived from the symplectic spec-
trum of the covariance matrix associated with the den-
sity operator [27]. In Fig. 4 we show the time evolu-
tion of negativity N for initial TMSV with different G
parameters. An example for the G = 0 case is the zero-
temperature bath dissipation scenario. Fixing L, we ob-
serve that a larger gain G leads to a shorter tc.
We can also calculate the negativity in a sub-matrix
MS , which measures the contribution of entanglement
from the corresponding block that builds up ρTA . Specif-
ically, the sub-matrix negativity NS is defined the same
way as negativity N but restricted to the sub-matrix of
ρTA with the characteristic sum S. To our knowledge,
such a sub-matrix negativity, which requires the calcu-
lations of individual eigenvalues, has not been discussed
before. Explicitly, NS takes the form:
NS = −πΞ(t)
β1)2n+2
S−2n−1
β2)S−2n
where P is defined as the integral part of S−1
. In Fig.
5 the behavior of NS of some blocks is illustrated. It
is surprising that that for higher sub-matrices the corre-
sponding negativity NS may increase over time. Physi-
cally, the increase of NS is due to the probability flow in
FIG. 5: The negativity of sub-matrices NS of ρ
TA , for an am-
plifier system with G = 1.5γ, L = 0.5γ and initial squeezing
factor r = tanh−1 0.2, with characteristic sum (a) S = 1, (b)
S = 4, (c) S = 7, and (d) S = 10.
Fock space arising from damping or amplifying mecha-
nisms. Since each sub-matrix of ρTA does not necessarily
conserve probability (i.e., the trace of MS is not a con-
stant), it is possible that some blocks could have their
negativity increasing with time.
However, the increase of NS does not violate the fact
that the overall negativity N of ρTA is an entanglement
monotone that does not increase under LOCC (the mas-
ter equation corresponds to local operations). Eq. (14)
reveals that eigenvalues of higher blocks are of smaller
order of magnitude. As we see in Fig. 5, although nega-
tivity of individual higher sub-matrices may increase over
time, their contribution for negativity is smaller by sev-
eral orders than the sub-matrices with lower S, and there-
fore the overall negativity is still monotonic decreasing.
C. Robust structure of entanglement witness
An entanglement witness operator W is designed for
the detection of entanglement such that Tr(ρW) < 0
for some non-separable states ρ, but Tr(ρsepW) ≥ 0
for all separable states ρsep [21]. For each eigenvec-
tor |φ〉 of ρTA with a negative eigenvalue, one can con-
struct an W by W = |φ〉〈φ|TA [31] meeting the criteria
above. In our system, we can construct a family of W
from eigenvectors |ϕn,S−n〉 with odd n accordingly, i.e.,
WS,n = |ϕn,S−n〉 〈ϕn,S−n|TA . From Eq. (18), the explicit
form of WS,n reads,
WS,n =
2Sn!(S − n)!ΓS,n,jΓS,n,l
j!(S − j)!l!(S − l)!} |l, S − j〉 〈j, S − l| ,
FIG. 6: Example of entanglement witness W2,1.
which shows that WS,n operates in only a finite dimen-
sion in the Fock space. Note that Tr(ρWS,n) = ξn,S−n,
and ξ1,S−1 is the most negative eigenvalue in each sub-
matrix, therefore WS,1 provides the most significant en-
tanglement detection among all witnesses constructed
from vectors lying within the sub-matrix MS . Some ex-
amples for entanglement witnesses with n = 1 are shown
in Fig. 6 and Fig. 7.
We observe that W also has a block diagonal structure
in which non-zero elements are: 〈j, S − l|WS,n |l, S − j〉.
This allows us to divide W into sub-blocks ΛK , with each
sub-block characterized by a difference K ∈ [−S, S]:
WS,n =
ΛK , (26)
where the element 〈j, S − l|WS,n |l, S − j〉 lies in the sub-
block with K = S − j − l. The explicit form of ΛK is
min(S,S−K)
l=max(0,−K)
2Sn!(S − n)!ΓS,n,S−K−lΓS,n,l
(S −K − l)!(K + l)!l!(S − l)!}
|l,K + l〉 〈S −K − l, S − l| , (27)
having a dimension of S − |K|+ 1. We remark that the
simplest 2×2 entanglement witnessW1,1 was constructed
in [37] using a different approach. Here our general WS,n
applies to all S and n.
Finally, let us emphasize the robustness feature of
TMSV against decoherence. We have seen from Eq. (18)
explicitly that the eigenvectors |ϕn,S−n〉 remain un-
changed with time. The time-independent |ϕn,S−n〉 sug-
gests that TMSV is robust against noise, in the sense that
structure of entanglement witnessWS,n is preserved. The
degradation of entanglement would only affect the eigen-
values. We stress that such a time-independent property
of eigenvectors is specific to initial TMSV, and does not
hold for arbitrary initial states in general. In Appendix
B, we derive the eigenvectors of ρTA evolving from an ini-
tial two-mode symmetric Gaussian states with arbitrary
FIG. 7: Part of the entanglement witness W3,1, showing the
blocks with characteristic difference from -3 to 0. W3,1 is
symmetric about the top-right to bottom-left diagonal.
real coefficients A0, B0, C0 and D0. We find that the
time dependence of the eigenvectors arises solely from
the evolution of the factors α1β1 and α2β2. Because of
the system-bath interactions, these factors are generally
dependent on time. However, the state evolving from
TMSV is an exception in which one can show that the
corresponding α1β1 and α2β2 equal the constant 1/16 at
all times.
IV. CONCLUSION
To summarize, by deriving an exact analytic solution
for ρTA and examining its eigenvectors, we discover sev-
eral important features about the loss of entanglement
of a TMSV suffered from decoherence. Both amplitude
damping and amplification effects have been included in
our analysis. Throughout the decoherence process, the
block diagonal structure of ρTA is shown to be main-
tained in Fock basis. As each block spans only a finite
portion of the Fock space, the existence of negative eigen-
values in a block implies that entanglement information
can ‘survive’ in the corresponding photon-number sub-
space. If quantum entanglement of the system is to be
destroyed completely, then all the blocks have to be made
positive. In other words, by simply mixing the system
with another state involving finite Fock space would not
destroy the entanglement. For the decoherence process
considered in this paper, all the blocks turn positive at
a critical time tc, which agree with the previous analy-
sis based on the covariance matrix. At t < tc, we de-
rived an explicit expression of the negativity as well as
the negativity of sub-matrices in order to characterize
the time-dependence of entanglement. Interestingly, the
negativity of some sub-matrices could increase with time
when G is non-zero, although the effect is weak accord-
ing to our calculations. The signature of entanglement in
photon number subspace can be detected by the witness
operatorsWS,n, and we have constructed WS,n explicitly
in this paper. These witness operators are also block di-
agonal, and most remarkably, they are time-independent
even though the system is under the influence of noise.
We interpret such a property as a kind of robust entan-
glement structure inherited in TMSV.
Acknowledgments
This work is supported by the Research Grants Council
of the Hong Kong SAR, China (Project No. 401305).
APPENDIX A: TIME EVOLUTION OF
GENERAL REAL SYMMETRIC TWO-MODE
GAUSSIAN DENSITY OPERATOR
In this Appendix, we consider the time evolution of
real symmetric two-mode Gaussian states, each coupling
linearly with a separate bath with general gain and loss
parameters. The density matrix is represented in position
space, obeying the master equation as in Eq. (8). For the
solution of the two-mode real symmetric Gaussian state
presented in Eq. (9), a direct substitution leads to the
following coupled differential equations:
Ȧ = (G− L)C + 1
(G+ L)[1− (2A− C)2 − (B +D)2],
Ḃ = −2(G− L)D − 2(G+ L)(2A− C)(B +D),
Ċ = 4(G− L)A+ (G+ L)[1 + (2A− C)2 + (B +D)2],
Ḋ = −2(G− L)B − 2(G+ L)(2A− C)(B +D). (A1)
Without loss of generality, we consider the G 6= L case.
From Eqs. (A1), B − D and 2A + C have the simple
solution,
B(t)−D(t) = (B0 −D0)η(t),
2A(t) + C(t) = (2A0 + C0)η(t) +
G− L [η(t)− 1],
where η(t) ≡ exp[2(G− L)t] and the zero subscripts de-
note the values at t = 0. The other two combinations,
B+D and 2A−C, can be found by noting that they are
related to the second moments 〈x2〉 and 〈xy〉 by:
2A(t)− C(t) =
2[〈x2〉2t − 〈xy〉
B(t) +D(t) =
〈xy〉t
2[〈x2〉2t − 〈xy〉
. (A3)
with the subscript t denoting the value at time t. Under
the condition:
= 0 (A4)
which applies to TMSV, and from the Heisenberg equa-
tions of motions of a and b, the time-dependence of the
second moments are given by:
η(t) +
2(G− L) [η(t)− 1],
〈xy〉t = 〈xy〉0 η(t). (A5)
Here the initial second moments are given by:
2A0 − C0
2[(2A0 − C0)2 − (B0 +D0)2]
〈xy〉0 =
B0 +D0
2[(2A0 − C0)2 − (B0 +D0)2]
. (A6)
Thus the time evolution of the coefficients of the two-
mode Gaussian state solution are as follows:
A(t) =
[(2A0 + C0)η +
G− L (η − 1)
2(〈x2〉2t − 〈xy〉
B(t) =
[(B0 −D0)η +
〈xy〉t
2(〈x2〉2t − 〈xy〉
C(t) =
[(2A0 + C0)η +
G− L (η − 1)
2(〈x2〉2t − 〈xy〉
D(t) =
[−(B0 −D0)η +
〈xy〉t
2(〈x2〉2t − 〈xy〉
], (A7)
where the expectation values are given in Eq. (A5).
In the case of TMSV, the solution is reduced to Eq.
(12) as the initial coefficients satisfy:
C0 = D0 = 0 (A8)
4A20 −B20 = 1. (A9)
In particular, by noting that A0 =
1−λ2 , we have A0 =
cosh 2r.
APPENDIX B: DERIVATION OF
EIGENVECTORS AND EIGENVALUES OF ρ
In position space, ρTA(x1, y1;x2, y2; t) =
ρ(x2, y1;x1, y2; t). The eigenvectors ϕnm and eigenvalues
ξnm of ρ
TA are defined by Eq. (13). Applying the trans-
formation xj = (uj + vj)/
2 and yj = (−uj + vj)/
(j = 1, 2), ρTA becomes a neat product of two double
Gaussians as follows:
ρTA(u1, v1;u2, v2; t)
= Ξ(t) exp[−α1(t)(u1 − u2)2 − β1(t)(u1 + u2)2]
× exp[−α2(t)(v1 − v2)2 − β2(t)(v1 + v2)2], (B1)
where α1 =
(2A−B+C+D), β1 = 14 (2A+B−C+D),
(2A+B +C −D) and β2 = 14 (2A−B −C −D).
We apply Mehler’s Formula twice, one for the u1, u2
double Gaussian, and one for the v1, v2 double Gaussian
function. This would lead to the Schmidt decomposition
on ρTA :
ρTA(u1, v1;u2, v2; t) = Ξ(t)
α1β1α2β2
λnfn(u1)fn(u2)
λ̃mf̃m(v1)f̃m(v2), (B2)
where the Schmidt modes fn(u) and f̃m(u) are:
fn(u) =
2n−1n!
4Hn[2(α1β1)
exp(−2
α1β1u
f̃m(u) =
2m−1m!
4Hm[2(α2β2)
exp(−2
α2β2u
2), (B3)
and the coefficients λn(t) and λ̃m(t)
λn(t) = 2(α1β1)
β1)n+1
λ̃m(t) = 2(α2β2)
β2)m+1
, (B4)
where Hn(u) are the Hermite polynomials. Rearranging
terms, Eq. (B2) gives:
ρTA(x1, y1;x2, y2; t) ≡
ξn,m(t)ϕn,m(x1, y1; t)
×ϕn,m(x2, y2; t), (B5)
where the eigenvectors ϕn,m are
ϕn,m(x1, y1; t)
2n−12m−1n!m!
α1β1α2β2
×Hn[2(α1β1)
x1 − y1√
)]Hm[2(α2β2)
x1 + y1√
× exp[−
α1β1(x1 − y1)2] exp[−
α2β2(x1 + y1)
and the eigenvalues ξn,m of ρ
TA are,
ξn,m(t) = Ξ(t)π
β1)n+1
β2)m+1
These expressions of eigenvectors and eigenvalues are
for any time-dependent coefficients A(t), B(t), C(t) and
D(t), i.e., applicable to states evolving from arbitrary
initial values A0, B0, C0 and D0. The special case with
initial TMSV is given in Eq. (14) and Eq. (18).
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