On the Infeasibility of Entanglement Generation in Gaussian Quantum Systems via Classical Control

Abstract

This technical note uses a system theoretic approach to show that classical linear time invariant controllers cannot generate steady state entanglement in a bipartite Gaussian quantum system which is initialized in a Gaussian state. The technical note also shows that the use of classical linear controllers cannot generate entanglement in a finite time from a bipartite system initialized in a separable Gaussian state. The approach reveals connections between system theoretic concepts and the well known physical principle that local operations and classical communications cannot generate entangled states starting from separable states.

Full text

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On the infeasibility of entanglement generation in
Gaussian quantum systems via classical control
Hendra I. Nurdin, Ian R. Petersen, and Matthew R. James
Abstract—This paper uses a system theoretic approach to show that
classical linear time invariant controllers cannot generate steady state
entanglement in a bipartite Gaussian quantum system which is initialized
in a Gaussian state. The paper also shows that the use of classical
linear controllers cannot generate entanglement in a finite time from a
bipartite system initialized in a separable Gaussian state. The approach
reveals connections between system theoretic concepts and the well known
physical principle that local operations and classical communications
cannot generate entangled states starting from separable states.
I. INTRODUCTION
Entanglement is a unique feature of quantum mechanical systems
not found in classical systems and is responsible for some of their
predicted counterintuitive behavior, as exemplified by the famous
Einstein-Podolsky-Rosen paradox [1]. Entanglement gives rise to
experimentally verifiable non-classical correlations among measure-
ment statistics [2] that cannot be explained by the usual classical
probability models. One well known application of entanglement
is quantum teleportation, the process of transferring the unknown
state of one quantum system to another whilst destroying the state
of the former, without the two quantum systems ever interacting
directly with one another [3]. This process is at the heart of quantum
communication schemes.
A bipartite quantum system is the composite of two quantum
systems. The state of such a system will be referred to as a bipartite
state and is represented by a density operator ρ1. Suppose that the
system is composed of a quantum system A with underlying Hilbert
space HA and a quantum system B with underlying Hilbert space
HB. A state ρ is said to be separable if it can be decomposed as
ρ =?
tensor product of operators. If a bipartite state is not separable, then
it is said to be entangled. For a pure state density operator ρ (i.e.,
tr(ρ2) = 1), it can be easily determined if it is separable; e.g., see
[4]. However, determining the separability of a mixed state bipartite
density operator ρ (i.e., tr(ρ2) < 1), is far from straightforward
and a complete characterization is only known for certain types
of bipartite systems, such as for bipartite systems on the finite
kpkρA
k⊗ ρB
k, with ρA
k and ρB
k being density operators on
HA and HB, respectively, for k = 1,2,.... Here ⊗ denotes the
Supported by the Australian Research Council and Air Force Office of
Scientific Research (AFOSR). This material is based on research sponsored
by the Air Force Research Laboratory, under agreement number FA2386-
09-1-4089. The U.S. Government is authorized to reproduce and distribute
reprints for Governmental purposes notwithstanding any copyright notation
thereon. The views and conclusions contained herein are those of the authors
and should not be interpreted as necessarily representing the official policies
or endorsements, either expressed or implied, of the Air Force Research
Laboratory or the U.S. Government.
H. I. Nurdin is with the Research School of Engineering, The Aus-
tralian National University, Canberra, ACT 0200, Australia. Email: Hen-
dra.Nurdin@anu.edu.au. Research supported by the Australian Research
Council.
I. R. Petersen is with the School of Information Technology and Electrical
Engineering, University of New South Wales at the Australian Defence Force
Academy, Canberra, ACT 2600, Australia. Email: i.r.petersen@gmail.com.
M. R. James is with the ARC Centre for Quantum Computation and
Communication Technology, Research School of Engineering, The Aus-
tralian National University, Canberra, ACT 0200, Australia. Email: Matthew.
James@anu.edu.au. Research supported by the Australian Research Council
and AFOSR Grant FA2386-09-1-4089 AOARD 094089.
1ρ is a self-adjoint, positive semidefinite operator on an underlying Hilbert
space with tr(ρ) = 1; e.g., see [1].
dimensional Hilbert space C2⊗ C2. In fact, the general problem
of determining the separability of a given mixed quantum state is
known to be NP-hard [5]. Another class of bipartite systems for
which a complete characterization of separability is known is the
class of bipartite Gaussian systems [6], [7], [8], [9]. These systems
are commonly encountered in the field of quantum optics. For such
systems, the underlying Hilbert space is the tensor product of two
quantum harmonic oscillator Hilbert spaces; e.g., see [10, Chapter
III]. Also, the separability of a state can be completely determined
from the (symmetrized) covariance matrix of the canonical position
and momentum operators of the system [6]. The class of systems
considered in this paper is the class of bipartite Gaussian systems. In
particular, we analyze dynamical bipartite Gaussian quantum systems
whose covariance matrices evolve in time. Hence, the separability
or entanglement of these systems also evolves in time. In quantum
optics, these dynamical bipartite Gaussian systems correspond to a
class of linear quantum stochastic systems [11], [12], [13] that are
driven by Gaussian bosonic fields and with a density operator initially
in a Gaussian state. The dynamics of such systems can be represented
by linear quantum stochastic differential equations (QSDEs) in the
canonical position and momentum operators and this makes them
suitable for a system-theoretic analysis.
We study the problem of entanglement generation using classical
finite dimensional (linear time-invariant (LTI) and time varying) con-
trollers from a system-theoretic point of view. The main contribution
of the paper is the use of system theoretic arguments and methods
to show that the application of a classical linear dynamic controller
cannot induce entanglement in a dynamical bipartite Gaussian system
which is initially in a separable state. Our result is in agreement
with the fundamental physical principle that Local Operations and
Classical Communication (LOCC) cannot generate entanglement
between initially separable states; e.g., see [4] for a proof of this
result. One motivation for the results of this paper is that they provide
a starting point for investigating connections between systems theory
and quantum physical principles. The no-go results for Gaussian
quantum systems considered here are in a similar spirit to other no
go results that have previously been obtained in [14], showing that
linear modulation of a beam cannot create out-of-loop squeezing, and
[15], showing that neither in-cavity squeezing nor output squeezing
can be created using linear modulation of the cavity field.
II. PRELIMINARIES
A. Notation
We will use the following notation: i =√−1,∗denotes the adjoint
of a linear operator as well as the conjugate of a complex number.
If A = [ajk] then A#= [a∗
matrix transposition. ?{A} = (A + A#)/2 and ?{A} =
A#). We denote the identity matrix by I whenever its size can be
inferred from context and use In to denote an n×n identity matrix.
Similarly, 0 denotes a matrix with zero entries whose dimensions
can be determined from context. diag(M1,M2,...,Mn) denotes a
block diagonal matrix with square matrices M1,M2,...,Mn on its
diagonal, and diagn(M) denotes a block diagonal matrix with the
square matrix M appearing on its diagonal blocks n times. Also, we
will let J =
−1
jk], and A†= (A#)T, whereTdenotes
1
2i(A −
?
01
0
?
.
B. The class of linear quantum stochastic systems
In this paper, we are concerned with a class of quantum stochastic
models of open (i.e., quantum systems that can interact with an
environment) Markov quantum systems that are widely used in the
area of quantum optics. Such models have been used in the physics
arXiv:1107.3174v1 [cs.SY] 15 Jul 2011

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and mathematical physics literature since at least the 1980’s; e.g., see
[16], [17], [18], [19], [20]. We focus on the special sub-class of linear
quantum stochastic models (e.g., see [21, Section 7.2], [20, Section
6.6], [19, Sections 3, 3.4.3, 5.3, Chapters 7 and 10], [22], [23, Section
5], [11], [12], [13], [24]) that describe the Heisenberg evolution of the
(canonical) position and momentum operators of several independent
open quantum harmonic oscillators that are coupled to external
coherent bosonic fields, such as coherent laser beams. These linear
stochastic models describe quantum optical devices such as optical
cavities [25, Section 5.3.6][21, Chapter 7], linear quantum amplifiers
[19, Chapter 7], and finite bandwidth squeezers [19, Chapter 10].
Following [11], [12], [13], we will refer to this class of models as
linear quantum stochastic systems.
Suppose we have n independent quantum harmonic oscillators.
The jth quantum harmonic oscillator has position and momentum
operators qj and pj with underlying Hilbert space L2(R); see, e.g.,
[10, Chapter III]. The position and momentum operators satisfy the
canonical commutation relations [qj,pk] = 2iδjk, [qj,qk] = 0,
and [pj,pk] = 0, where δjk denotes the Kronecker delta and
[·,·] denotes the commutation operator. The quantum harmonic
oscillators are assumed to be coupled to m external independent
quantum fields modelled by bosonic annihilation field operators
A1(t),A2(t),...,Am(t) which are defined on separate Fock spaces
(over L2(R)) for each field operator [16], [18]. For each annihilation
field operator Aj(t), there is a corresponding creation field operator
A∗
adjoint of Aj(t). The field operators are adapted quantum stochastic
processes with forward differentials dAj(t) = Aj(t+dt)−Aj(t) and
dA∗
[18]: dAj(t)dAk(t)∗= δjkdt; dA∗
dA∗
We collect the position and momentum operators in the col-
umn vector x defined by x = (q1,p1,q2,p2,...,qn,pn)T. Note
that we may write the canonical commutation relations as xxT−
(xxT)T
= 2iΘ with Θ = diagn(J). We take the composite
system of n quantum harmonic oscillators to have a quadratic
Hamiltonian H given by H =1
symmetric matrix. The quantum harmonic oscillators are coupled
to the k-th quantum field via the singular interaction Hamiltonian
Hk
= i(Lkη∗
(with Kk ∈ C1×2n) is a linear coupling operator describing the
linear coupling of the quantum harmonic oscillator position and
momentum operators to ηk(t). Here ηk(t) is a quantum white noise
process [17], [19] satisfying the relation Ak(t) =
We now collect the coupling operators L1,L2,...,Lm together in
one linear coupling vector L = (L1,L2,...,Lm)T= Kx, with
K = [ KT
1
KT
2
...KT
together as A(t) = (A1(t),A2(t),...,Am(t))T. Then the joint
evolution of the oscillators and the quantum fields is given by a
unitary adapted process U(t) satisfying the Hudson-Parthasarathy
QSDE [16], [18]:
j(t), which is defined on the same Fock space and is the operator
j(t) = A∗
j(t+dt)−A∗
j(t) that have the quantum Itˆ o products [16],
j(t)dAk(t) = dAj(t)dAk(t) =
j(t)dA∗
k(t) = 0.
2xTRx, where R is a real 2n × 2n
k(t) − L∗
kηk(t)) [17], [19], where Lk
= Kkx
?t
0ηk(s)ds.
m ]T, and collect the field operators
dU(t)=
?
tr?(S − I)TdΛ(t)?+ dA(t)†L − L†SdA(t)
−(iH +1
2L†L)dt
?
U(t),
where S ∈ Cm×mis a complex unitary matrix (i.e., S†S = SS†=
I) called the scattering matrix, and Λ(t) = [Λjk(t)]j,k=1,...,m, with
Λkj(t) = Λjk(t)∗. The processes Λjk(t) for j,k = 1,...,m are
adapted quantum stochastic processes referred to as gauge processes,
and the forward differentials dΛjk(t) = Λjk(t+dt)−Λjk(t) j,k =
1,...,m have the quantum Itˆ o products:
dΛjk(t)dΛj?k?(t)
dΛjkdAl(t)∗
with all
Aj(t),A∗
For any adapted processes X(t) and Y (t) satisfying a quantum
Itˆ o stochastic differential equation, we have the quantum Itˆ o rule
d(X(t)Y (t)) = X(t)dY (t) + (dX(t))Y (t) + dX(t)dY (t); e.g.,
see [16], [18]. Using the quantum Itˆ o rule and the quantum Itˆ o
products given above, as well as exploiting the canonical commu-
tation relations between the operators in x, the Heisenberg evolution
x(t) = U(t)∗xU(t) of the canonical operators in the vector x can be
obtained. Then x(t) satisfies the QSDE (see [22], [23], [11], [13]):
=δkj?dΛjk?(t), dAj(t)dΛkl(t)=δjkdAl(t),
δkldA∗
remaining cross
k(t),Λj?k?(t) being 0.
=
j(t),
other productsbetween
dx(t) = d(U(t)∗xU(t)),
= Aox(t)dt + Bo
?
dA(t)
dA(t)#
?
;x(0) = x,
dY (t) = d(U(t)∗A(t)U(t)),
= Cox(t)dt + DodA(t),
with Ao = 2Θ(R + ?{K†K}), Bo = 2iΘ[ −K†S
Co = K, and Do = S. Here, Y (t) = (Y1(t),...,Ym(t))T=
U(t)∗A(t)U(t) is a vector of output fields that results from the
interaction of the quantum harmonic oscillators and the incoming
quantum fields A(t). Note that the dynamics of x(t) is linear, and
Y (t) depends linearly on x(t). We refer to n as the number of degrees
of freedom of the linear quantum stochastic system.
In this paper it will be convenient to write the dynamics in
quadrature form as in [11]:
(1)
KTS#],
dx(t) = Ax(t)dt + Bdw(t); x(0) = x.
dy(t) = Cx(t)dt + Ddw(t),
(2)
with
w(t)=2(?{A1(t)},?{A1(t)},...,?{Am(t)},?{Am(t)})T;
2(?{Y1(t)},?{Y1(t)},...,?{Ym(t)},?{Ym(t)})T.
Here, the real matrices A,B,C,D are in a one to one correspondence
with the matrices Ao,Bo,Co,Do. Also, the quantity w(t) satisfies
the Itˆ o relationship dw(t)dw(t)T= Fwdt where Fw ≥ 0; see
[11]. Furthermore, we define the matrix Sw =
the differential commutation matrix Tw =
boson fields that we consider here, Tw is necessarily of the form
Tw = idiagm(J). The symmetric matrix Swis then such that Fw ≥ 0
and for the Gaussian boson fields that are of interest here this matrix
reflects the statistics of the field. For instance, Sw = I corresponds
to a vacuum Gaussian boson field.
Fig. 1 shows an example of a two degree of freedom linear
quantum stochastic system connected to a classical controller. The
linear quantum stochastic system consists of two independent optical
cavities [25], [21], [13] denoted by G1 and G2. The two optical
cavities are connected to the classical controller via a homodyne
detector (HD) which measures one of the quadratures of the output
field Y1(t) from G1, and an electro-optic modulator (MOD) which
modulates the quantum field A3(t) with the controller output signal
u(t) and then sends the resulting field A2(t) to G2.
Remark 1: For the remainder of this paper, we will consider the
case where n = 2, corresponding to two degree of freedom linear
quantum stochastic systems, with the quantum harmonic oscillators
y(t)=
1
2
?Fw+ F#
w
?
and
1
2
?Fw− F#
w
?. For the

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G1
G2
Optical cavity
Modulator
A (t)
Y (t)
2
A (t)
A (t)
1
11
Y (t)
11
MOD
HD
Classical
controller
HD
Homodyne detector
MOD
Y (t)
22
Y (t)
1
A (t)
22
2
A (t)
3
m (t)
1
Partially transmitting mirror

Fully refecting mirror
u(t)
Fig. 1.The interconnection of two optical cavities G1and G2via a classical controller.
being initialized in a Gaussian state2. Also, for such a linear quan-
tum stochastic system G, we define a (linear) dynamical bipartite
Gaussian quantum system (corresponding to G) as the open quantum
system obtained from G by tracing out (averaging) the bosonic fields.
III. SIMPLE SYSTEM-THEORETIC PROOF OF THE HEISENBERG
UNCERTAINTY PRINCIPLE
For a system of the form (2), the corresponding symmetric covari-
ance matrix defined by
P(t) =1
2tr(ρ(0)(x(t)x(t)T+ (x(t)x(t)T)T))
varies with time. Here, ρ(0) is the initial density operator of the
overall composite closed system. In this section, we will assume that
the matrix A in (2) is Hurwitz. Then the steady-state symmetrized
covariance matrix P = limt→∞P(t) satisfies the real Lyapunov
equation (see, e.g., [20, p. 327], [12, Section 4]):
AP + PAT+ BSwBT= 0.
(3)
On the other hand, since the commutation relations are preserved, we
also have that [11]:
AΘ + ΘAT− iBTwBT= 0.
(4)
Defining the complex Hermitian matrix˜P = P + iΘ, we see
from combining (3) and (4) that˜P satisfies the complex Lyapunov
equation: A˜P +˜PAT+BFwBT= 0, where Fw = Sw+Tw. Since
Fw ≥ 0 and A is Hurwitz, it follows that˜P ≥ 0; e.g., see [27].
Equivalently, in terms of P and Θ we have that: P + iΘ ≥ 0. This
matrix inequality is a version of the Heisenberg Uncertainty Principle
that must be satisfied by all Gaussian quantum systems; e.g., see [6],
[7] for this alternate form of the Heisenberg Uncertainty Principle.
IV. CLASSICAL LTI CONTROLLERS CANNOT GENERATE
STEADY-STATE BIPARTITE ENTANGLEMENT IN LINEAR GAUSSIAN
QUANTUM SYSTEMS
A. Separability criterion for dynamical bipartite Gaussian systems
It has been shown in [6] that the separability of a bipartite
Gaussian density operator ρ can be completely determined from a
complex linear matrix inequality (LMI) involving the (symmetrized)
covariance matrix P =1
Lemma 2 ([6], [8], [9]): A bipartite Gaussian density operator ρ
is separable if and only if the corresponding covariance matrix P
satisfies the LMI P + idiag(J,−J) ≥ 0.
Note here that without loss of generality, we can assume that x
has zero mean because the mean of x plays no role in determining
the separability of the associated density operator. Now, in the case
2tr(ρ(xxT+ (xxT)T)); see also [8], [9].
2A state with density operator ρ, is said to be Gaussian if tr(ρeiλTx) =
eiλTm−1
matrix satisfying G + iΘ ≥ 0 with Θ as defined previously; see, e.g., [26],
[19], [18].
2λTGλfor all λ ∈ R2n, where m ∈ R2nand G is a real symmetric
of a dynamical bipartite Gaussian quantum system corresponding to
a linear quantum stochastic system, the covariance matrix can vary
with time and is given by
P(t)=
1
2tr(ρo(t)(xxT+ (xxT)T))
1
2tr(ρ(t)(xxT+ (xxT)T))
1
2tr(ρ(0)(x(t)x(t)T+ (x(t)x(t)T)T)),
=
=
(5)
where ρ(t) is the density operator at time t ≥ 0 of the overall
composite closed system, while ρo(t) is the reduced density operator
of the two quantum harmonic oscillators at time t obtained by tracing
out the bosonic fields; e.g., see [19]. Note that the second equality in
(5) follows from the definition of the partial trace (e.g., see [18, p.
102]) since the elements of x are operators on the bipartite quantum
harmonic oscillator Hilbert space. Also, the final equality in (5)
follows by switching from the Schr¨ odinger picture (in which ρ(t)
evolves in time) to the Heisenberg picture (in which x(t) evolves in
time and the overall density operator is fixed as ρ(0)). Thus, to check
whether the system is separable at any time t ≥ 0, it is equivalent to
check if the LMI P(t) + idiag(J,−J) ≥ 0 is satisfied at that time
t.
B. Separability of dynamical bipartite Gaussian systems coupled via
a classical LTI controller
Let G1 = (A1,B1,C1,D1) and G2 = (A2,B2,C2,D2) define
two linear quantum stochastic systems of the form (2). We form a
linear quantum stochastic system G = (A,B,C,D) of the form (2)
from G1 and G2, with A = diag(A1,A2), B = diag(B1,B2),
C = diag(C1,C2), and D = diag(D1,D2). We then obtain a
dynamical bipartite Gaussian quantum system corresponding to G
(see Remark 1). The quantum system G is connected to a finite
dimensional classical controller as shown in Fig. 2 to form a quantum
feedback control system. In this quantum feedback control system,
some of the output fields from G1 and G2 are measured and fed to
the classical controller that processes these measurements linearly to
produce control signals that are fed back into G1 and/or G2. Here,
control actuation can be facilitated in two ways:
1) Modulating the Hamiltonian of Gk by a classical 2 × 1 signal
vector u1,k. If the canonical operators of Gk are represented
by a vector of operators xk = (qk,pk)T, this means that the
quadratic Hamiltonian Hk of Gk is augmented by adding a
linear (time varying) Hamiltonian term Hl,k(t) of the form
Hl,k(t) = u1,k(t)TMkxk, where Mk is a real 2 × 2 matrix.
Thus, the total Hamiltonian for Gk becomes Hk + Hl,k(t).
The signal u1,k(t) is classical and can depend linearly on the
classical controller internal variables (i.e., its state) as well as
the measurement results. This actuation can be implemented in
different ways, for instance, as described in the Appendix of
[28].

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2) Modulating (or displacing) an input field of Gkwith a classical
control signal u2,k(t). This can be implemented by an electro-
optic modulator.
In the quantum feedback control system shown in Fig. 2, the vector
of quantum input fields Ak(t) for the system Gk is partitioned into
two parts: some of which will be the components of Ak1(t) while
the others will be components of Ak2(t). Here Ak1(t) represents the
input fields of Gk that will not be modulated by the controller, while
Ak2(t) represents the input fields that are modulated by the controller.
Part of the output vector of quantum signals, Yk2(t), of Gk(k = 1,2)
is passed through a network of static optical components (as listed
in [13, Section 6.2]) and homodyne detectors (labelled as HDN in
the diagram) that produces the set of classical measurements signals
mk(t) which drive the controller. The controller produces two sets
of classical control signals (k = 1,2): one set, uk1(t), modulates the
linear Hamiltonian term H1,k, and another set, uk2(t), is modulated
by a network of (possibly electro-optic) modulators (denoted in the
diagram by MOD) to produce the quantum signal Ak2(t) as one of
the input fields into Gk. The signals Vjk(t), j,k = 1,2 are any
additional quantum noises required for the operation of HDN and
MOD (they may be suitably absorbed into the definition of w1 or
w2).
Classical
controller
MOD
HDN
HDN
G1
Y12(t)
u11(t)
u22(t)
A12(t)
A11(t)
A22(t)
A21(t)
Y22(t)
m1(t)m2(t)
u21(t)
V22(t)
V12(t)
V11(t)
V21(t)
G2
MOD
u12(t)
Y11(t)
Y21(t)
Fig. 2.Interconnection of G1and G2via a classical controller.
The assumptions that we will use regarding this quantum feedback
control system are:
1) The control u(t) = (u1,1(t),u2,1(t),u1,2(t),u2,2(t))Thas
been generated by a finite dimensional linear (time invariant
or time varying) system.
2) The quantum signals coming into G1 and G2 come from
independent sources. Therefore, Fw = diag(Fw1,Fw2) and
hence, Sw = diag(Sw1,Sw2), Tw = diag(Tw1,Tw2).
Note that the systems G1and G2are not directly connected to one
another. That is, no output field from G1 is passed directly to G2
and vice-versa. They are only indirectly connected via the classical
controller. Note also that the overall closed-loop system is then a
mixed classical-quantum linear stochastic system as described in [11].
Let z(t) denote the controller internal state which is classical
in nature and of arbitrary dimension nc. We could also allow the
classical controller to be driven by an additional classical Wiener
noise source wc(t) that is not derived from the measurement signals.
However, this additional noise may be absorbed into w1 or w2;
see [11] for details. Now, let x(t) = (x1(t),x2(t),z(t))Tand
w(t) = (w1(t),w2(t))Twhere x1(t) represents the vector of system
variables for the quantum system G1 and w1(t) represents the vector
of quantum noise inputs for G1. Also, x2(t) represents the vector of
system variables for the quantum system G2 and w2(t) represents
the vector of quantum noise inputs for G2. Now, since G1 and G2
each only interact with the controller, it follows that the dynamics of
the closed-loop system can be written in the form:
dx(t)=
˜ Ax(t)dt +˜Bdw(t);x(0) = x,
(6)
where the real matrices˜ A and˜B have the special structure:

A31
A32
A33
˜ A=

A11
0
0A13
A23
A22

;
˜B =


B11
0
B31
0
B22
B32

,
with A11 = A1, A22 = A2,B11 = B1; B22 = B2. Our main result
in this section is the following.
Theorem 3: Consider any classical LTI controller that is connected
to the linear quantum system G such that ˜ A is Hurwitz in the
closed loop system (6). Then the resulting closed-loop dynamical
bipartite Gaussian quantum system is separable at steady state. Thus,
a classical LTI controller cannot generate an entangled steady state
from any initial Gaussian state.
Proof: Since the controller state z(t) is classical, the commuta-
tion matrix Θ for x(t) will be degenerate canonical [11, Sections II,
III, and III C] of the form Θ =


Θ1
0
00
0Θ2
0nc×2
0nc×2
0nc×nc

,
where Θ1 = Θ2 = J. Suppose that the controller state z(t)
is of arbitrary dimension nc. We have that the closed-loop mixed
quantum-classical system satisfies the constraint [11, Theorem 3.4]
:˜ AΘ + Θ˜ AT− i˜BTw˜BT= 0, with Tw = diag(Tw1,Tw2). This
equation is equivalent to the following:

A31Θ1− iB31Tw1BT

A11Θ1+ Θ1AT
11− iB11Tw1BT
0
11
11
0
A22Θ2+ Θ2AT
A32Θ2− iB32Tw2BT
Θ1AT
Θ2AT
−iB32Tw2BT
22− iB22Tw2BT
22
22

31− iB11Tw1BT
32− iB22Tw2BT
31
32
32
= 0.
(7)
Multiplying the (2,2), (2,3), (3,2), and (3,3) elements of this
matrix equation by −1, yields

A31Θ1− iB31Tw1BT

A11Θ1+ Θ1AT
11− iB11Tw1BT
0
11
11
0
22+ iB22Tw2BT
−A22Θ2− Θ2AT
−A32Θ2+ iB32Tw2BT
Θ1AT
−Θ2AT
iB32Tw2BT
22
22

31− iB11Tw1BT
32+ iB22Tw2BT
31
32
32
= 0.
(8)
LettingˆΘ = diag(Θ1,−Θ2,0nc×nc) andˆTw = diag(Tw1,−Tw2),
this matrix equality can be written as:˜ AˆΘ+ˆΘ˜ AT−i˜BˆTw˜BT= 0.
We now use the fact that the closed-loop matrix˜ A is Hurwitz. Then,
as discussed in Section III, the symmetrized steady state covariance
matrix P satisfies:˜ AP + P˜ AT+˜BSw˜BT= 0. Defining˜P =
P + iˆΘ and˜Fw = Sw+ˆTw, we have that˜P satisfies the complex
Lyapunov equation:˜ A˜P+˜P˜ AT+˜B˜Fw˜BT= 0. Recalling that Sw =
diag(Sw1,Sw2) and Fw ≥ 0, we note that Fwi= Swi+ Twi≥ 0
for i = 1,2. Then, we note that since˜Fw = diag(Fw1,F#
have that˜Fw ≥ 0; see, e.g., [27]. Therefore, since˜ A is Hurwitz, we
have that˜P ≥ 0. Partitioning P according to the partitioning of x(t)
into its quantum and classical components as P =
w2), we
?
P11
P21
P12
P22
?
,
the property˜P = P +iˆΘ ≥ 0 implies that P11+idiag(Θ1,−Θ2) =
P11+idiag(J,−J) ≥ 0. Therefore it follows from Lemma 2 that the
dynamical bipartite Gaussian quantum system is separable at steady-

Page 5

5
state. Thus, a classical LTI controller cannot generate an entangled
steady state from any initial Gaussian state.
Removing the classical controller by defining all its system ma-
trices to be zero, a special case of Theorem 3 shows that two
independent and unconnected systems G1 and G2, with A1 and A2
Hurwitz, which are initially entangled become separable in the steady
state.
V. CLASSICAL FINITE DIMENSIONAL LINEAR CONTROLLERS
CANNOT GENERATE ANY ENTANGLEMENT IN BIPARTITE
GAUSSIAN QUANTUM SYSTEMS IN FINITE TIME
In the previous section, we have shown that starting from any state,
separable or entangled, a classical LTI controller cannot generate or
maintain entanglement at steady state in a linear dynamical bipartite
Gaussian quantum system. In this section, by a slight modification
of the arguments of the previous section, we will show that classical
finite dimensional linear controllers cannot generate bipartite entan-
glement in a finite time for any initially separable linear dynamical
bipartite Gaussian quantum system. Moreover, for this finite time
analysis, we may drop the requirement that the controller is chosen
so that the closed-loop matrix˜ A is Hurwitz.
We follow the notation and set up of the last section. Instead
of considering the steady state covariance matrix P, we now
consider the symmetrized finite time covariance matrix P(t) =
1
2tr?ρ(0)(x(t)x(t)T+ (x(t)x(t)T)T)?, 0 ≤ t < ∞ satisfying the
˙P(t)=
˜ AP(t) + P(t)˜ AT+˜BSw˜BT,
Lyapunov differential equation:
P(0) = P0.
Theorem 4: Suppose that a linear dynamical bipartite Gaussian
quantum system is initially separable. Then it remains separable for
all t ≥ 0 under the action of any classical LTI controller.
Proof: Since the system is initially separable, P(0)+iˆΘ ≥ 0 by
Lemma 2. Let˜P(t) = P(t) + iˆΘ. Then following the same lines of
argument as in the proof of Theorem 3, and by using standard results
on Lyapunov differential equations, we find that since˜P(0) ≥ 0 and
˜Fw ≥ 0 (hence also˜B˜Fw˜BT≥ 0) that˜P(t) ≥ 0, for all t ≥ 0
regardless of the values of˜ A and˜B. Similarly partitioning P(t)
according to the partitioning of x(t) into its quantum and classical
components as P(t) =
P12(t)
idiag(J,−J) ≥ 0, for all t ≥ 0. This shows that when the bipartite
Gaussian system is initially in a separable state, then under the action
of a classical LTI controller it will remain so for all times.
Remark 5: Note that it is straightforward to extend the proof of
the above theorem to allow for linear time-varying controllers rather
than LTI controllers.
Example 6: Consider the quantum optical system shown in Fig. 1.
Suppose that the two optical cavities are identical with the partially
reflecting mirror on each cavity having coupling coefficient 0.01.
Also, let the position and momentum operators of cavity Gi be
(qi,pi), and let ˜ x = (q2,p2,q1,p1)T. Let wi1(t) = 2?{Ai(t)}
and wi2(t) = 2?{Ai(t)}, i = 1,2,3. Then the dynamics of the
two degree of freedom linear quantum stochastic system (without
the controller, homodyne detector and modulator attached) is given
by:
?
P11(t)P12(t)
P22(t)
?
, it follows that P11(t)+
d˜ x(t) = −0.005˜ x(t)dt − 0.1d(w21(t),w22(t),w11(t),w12(t))T,
dyo(t) = 0.1˜ x(t)dt + d(w21(t),w22(t),w11(t),w12(t))T.
Let yo(t) = (yo
yo
used as the (stochastic) input m1(t) = yo
1(t),yo
2(t),yo
3(t),yo
4(t))T. The amplitude quadrature
3(t) of yo(t) is measured using the homodyne detector and is
3(t) to a first order LTI
controller that produces a two dimensional output signal u(t) =
(u1(t),u2(t))T. The dynamics of the controller is:
dz(t) = Az(t)dt + Bdm1(t), z(0) = 0,
u(t) =?
where z(t) denotes the state of the controller, and A = −1, B =
C1 = C2 = 1. The output signal u(t) is passed through an electro-
optic modulator and sent to the partially reflecting mirror of cavity
G2. Let x(t) = (˜ x(t),z(t))T. We then have that the interconnection
of the controller with the two cavities via the homodyne detector and
electro-optic modulator is a mixed quantum-classical system with
dynamics of the form (6) defined by the matrices


000.1B


00B
CT
1
CT
2
?Tz(t),
˜ A =




−0.005
0
0
0
00
0
0
0
0
−0.1C1
−0.1C2
0
0
A
−0.005
0
0
−0.005
0
−0.005
0






;
˜B =


−0.1
0
0
0
00
0
0
0
0
−0.1
0
0
−0.1
0
−0.1
0


,
and is driven by the noise (w31(t),w32(t),w11(t),w12(t))T. Sup-
pose that the bipartite state of the two cavities is in an initially
entangled bipartite Gaussian state with covariance matrix P11(0)
given below in (9)


Wetake asour measureof
negativity EN
[29], [4], [28]. Partitioning P11(t) into 2 ×
2
blocksas
P11,2(t)T
P11,3(t)
by
EN(P11(t))= max(0,−ln(2ν(t))),
1
√2
det(P11,3(t))−2det(P11,2(t)). Note that the logarithmic negativity
is always nonnegative and has a value of zero if and only if the
state is separable [29], [4], otherwise the state is entangled, with a
higher value of EN indicating a higher degree of entanglement. The
initial value of the logarithmic negativity is EN(P11(0)) = 0.1054.
The solid line in Fig. 3 shows that under the action of this classical
controller, the logarithmic negativity steadily decreases and finally
goes to zero in a finite time. At this point, the state becomes separable
and remains so for all future times. If we instead start at an initially
separable state with covariance matrix P11(0) as given in (10)


then the oscillators’ joint state remains separable, as shown in the
dashed line in Fig. 3.
P11(0) =

0.5028
0
−0.0528
0
0
−0.0528
0
0.5028
0
0
0.5028
0
0.0528
0.0528
0
0.5028

.

(9)
entanglement
?
the logarithmic
?
P11,1(t)P11,2(t)
,
EN(P11(t))
where
is given
ν(t)=
?
˜∆(t) −
?
˜∆(t)2− 4det(P11(t)) and˜∆(t) = det(P11,1(t))+
P11(0) =

0.5704
0
0.0034
0.0562
00.0034
0
0.6203
0.0499
0.0562
0.0528
0.0499
0.6203
0.5704
0
0.0528

,

(10)
VI. CONCLUSIONS
By employing system-theoretic arguments and methods, we were
able to give a systems theory proof of the fact that classical LTI con-
trollers cannot generate steady state entanglement in linear dynamical
bipartite Gaussian quantum systems. Furthermore, we also give a
systems theory proof of the fact that classical linear controllers cannot
generate entanglement in a dynamical bipartite Gaussian system