Minimal energy cost of entanglement extraction
Lucas Hackl
1,2
and Robert H. Jonsson
3
1
Max Planck Institute of Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany
2
Munich Center for Quantum Science and Technology, Schellingstraße 4, D-80799 München, Germany
3
QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
July 15, 2019
We compute the minimal energy cost for
extracting entanglement from the ground
state of a bosonic or fermionic quadratic
system. Specifically, we find the mini-
mal energy increase in the system resulting
from replacing an entangled pair of modes,
sharing entanglement entropy ∆S, by a
product state, and we show how to con-
struct modes achieving this minimal en-
ergy cost. Thus, we obtain a protocol in-
dependent lower bound on the extraction
of pure state entanglement from quadratic
systems. Due to their generality, our re-
sults apply to a large range of physical sys-
tems, as we discuss with examples.
1 Introduction
Entanglement is a hallmark of quantum theory:
On a fundamental level, its very existence has
deep implications for our understanding of na-
ture, e.g., as pertaining to its realism and localism
[1–6]. From an applied point of view, entangle-
ment is a key resource in quantum technologies
and quantum information processing [7–10]. And
moreover, studying entanglement structures pro-
vided deep insights in various fields ranging from
quantum many body systems, over quantum field
theory to quantum gravity [11–17].
The fact that the ground states of local Hamil-
tonians are generally highly entangled gives rise
to the idea that such systems could serve as a
source of entanglement, e.g., for applications in
quantum information. In particular, the idea of
extracting entanglement from the ground state
of a system has been explored in the context
of the vacuum state of a quantum field. Start-
ing with the observation that the entanglement
Lucas Hackl: lucas.hackl@mpq.mpg.de
Robert H. Jonsson: robert.jonsson@math.ku.dk
present in the vacuum fluctuations can violate
Bell’s inequalities [18–22], various works studied
the ability of the quantum field vacuum to entan-
gle systems which couple locally to the quantum
field. In particular, the Unruh-DeWitt particle
detector model has been employed to investigate
aspects ranging from the impact of relativistic
motion [23], spacetime curvature and topology
[24–26], the nature of field states and interactions
[27–32], and potential implementations [33–36].
Previous works on the relation between entan-
glement and energy content of a system focused
mainly on the energy cost of creating entangle-
ment and correlations, or looked at the conversion
of one into the other, e.g., in the context of quan-
tum thermodynamics [37–43]. Also it was shown
that it is possible to use the entanglement present
in the vacuum of a quantum field to teleport en-
ergy from one spacetime region to another. In
quantum energy teleportation one observer per-
forms a local measurement of the field, classically
communicates the result to another observer who
then through local unitary interactions is able to
extract energy from the field [44–47].
However, the systematic study of the energy
cost of entanglement extraction from complex
quantum systems was only initiated recently by
Beny et al. [48]. When entanglement is extracted
from a system, its state is changed and thus its
energy content may change as well. In particular,
the extraction of entanglement from a system’s
unique ground state is inevitably associated with
an energy increase injected into the system when
changing its state.
This interplay of entanglement extraction and
energy cost is not only relevant with respect to
potential implementations and quantum thermo-
dynamics but, in fact, also connects to quantum
field theory on curved spacetimes. For exam-
ple, Jacobson [49] recently showed that the semi-
Accepted in Quantum 2019-07-10, click title to verify 1
arXiv:1904.06246v3 [quant-ph] 12 Jul 2019
classical Einstein equations are equivalent to the
property of the vacuum of a field to locally max-
imize the entanglement entropy. Attempting to
increase a sphere’s entanglement by varying the
field state induces an increase in energy density
which curves spacetime and shrinks the sphere’s
surface, thus reducing the entanglement contribu-
tions proportional to the surface area, and exactly
cancelling out the attempted increase of entangle-
ment.
In this article, we present a first result in the
direction initiated by [48] which applies to large
classes of systems of physical interest: We study
the extraction of entanglement from the ground
state of bosonic and fermionic modes governed
by a quadratic Hamiltonian. This also includes
spin systems that can be mapped to quadratic
fermionic Hamiltonians via the Jordan-Wigner
transform. In particular, we find the minimal en-
ergy cost for the extraction of mode pairs in a
pure entangled state.
An important tool to our analysis is the partner
mode formula, which was developed a in series
of recent papers [50? , 51], mostly in the con-
text of bosonic quantum field theory. We gener-
alize this construction, such that it applies to any
bosonic or fermionic pure Gaussian state. This is
achieved by phrasing the construction in terms
of the complex structure associated with a pure
Gaussian state [52, 53], thus illuminating the ge-
ometric origin of the partner mode construction.
The partner mode construction is based on the
well-known property of pure Gaussian states to
factorize into pairs of entangled mode pairs [54]
(see also [55]). If a system of many modes is
in an overall pure Gaussian state, then for every
mode in that system, which is not in a pure state
on its own, there exists a unique partner mode,
with which it shares all its entanglement. Thus,
the mode and its partner are in a pure entangled
state, that is in a product state with the rest of
the system.
This property is the motivation for extracting
entanglement by unitarily swapping the entan-
gled state of such partner modes with an unen-
tangled state of target modes in some laboratory
system. Thus, the system acts as a source of an
entangled state that is traded with an unentan-
gled state from the target system. This frame-
work was introduced in [? ] where the relation-
ship between energy cost and entanglement ex-
traction was discussed for a harmonic chain.
In this article, we generalize this framework to
arbitrary systems of bosonic or fermionic modes
with quadratic Hamiltonians. We find the min-
imal energy cost for the extraction of partner
mode pairs from their ground state as analytical
functions of the Hamiltonian’s spectrum. Fur-
thermore, we show how to explicitly construct
partner modes that achieve this minimal energy
cost.
The structure of this article is as follows: In
Section 2, we review bosonic Gaussian states
and quadratic Hamiltonians. In Section 3, we
present our approach of extracting entanglement
from partner modes and derive an analytical for-
mula for the minimal energy cost. The following
two Sections 4 and 5 mirror the previous two for
fermionic systems, i.e., we first review fermionic
Gaussian states and quadratic Hamiltonians in
Section 4 and then derive the respective minimal
energy cost for fermions in Section 5. Finally,
Section 6 discusses applications of our results for
both bosonic and fermionic models, before we
conclude with a discussion in Section 7. Ap-
pendix A rederives the pairwise correlation struc-
ture of bosonic and fermionic Gaussian states in
the newly developed language of linear complex
structures. Appendix B reviews relevant results
in linear algebra on how the spectrum of a linear
map changes under restrictions.
The article is structured to be largely self-
contained. To the experienced reader Sections 2
and 4 will mostly serve to establish notation for
bosons and fermions respectively, and to present
a covariant way of computing partner modes.
2 Quadratic bosonic systems and their
partner mode construction
The partner mode construction is a central tool
in our derivation of the minimal energy of en-
tanglement extraction. In this section, we in-
troduce the partner mode formula for arbitrary
pure Gaussian states of bosonic modes (specifi-
cally in Section 2.5). To be largely self-contained
and to establish notation, we begin with a re-
view of quadratic bosonic systems. We follow the
conventions of [56–58], while other comprehensive
reviews include [14, 59–62].
Accepted in Quantum 2019-07-10, click title to verify 2
2.1 Phase space and observables
For N bosonic degrees of freedom, we consider
a classical phase space V ' R
2N
and its dual
V
∗
' R
2N
. We denote vectors v
a
∈ V with an
upper index and dual vectors w
a
∈ V
∗
with a
lower index. As phase space, both V and V
∗
are equipped with symplectic forms, namely Ω
ab
:
V
∗
×V
∗
→ R on V
∗
and its inverse Ω
−1
ab
: V ×V →
R satisfying Ω
ac
Ω
−1
cb
= δ
a
b
.
We can characterize a vector v
a
∈ V by its
coordinate values with respect to a basis as
v
a
≡ (q
1
(v), p
1
(v), ··· , q
N
(v), p
N
(v))
|
∈ R
2N
,
(1)
where “≡” indicates that v
a
is represented by the
column vector with respect to the specified basis.
Typically, we choose coordinate functions
q
i
, p
i
∈ C
∞
(V ) which are linear, i.e., q
i
, p
i
∈ V
∗
,
and which consist of conjugate variables, i.e., sat-
isfying Ω(q
i
, p
j
) = δ
ij
.
With respect to such a basis, the symplectic
form Ω
ab
takes the standard Darboux form, i.e.,
is represented by the block diagonal matrix
Ω
ab
≡ Ω =
Ω
2
0
.
.
.
0 Ω
2
, (2)
where we use boldface symbols Ω for the matrix
representation with respect to a specific basis,
and define the 2 × 2-matrix
Ω
2
=
0 1
−1 0
!
. (3)
In the classical theory, observables correspond
to smooth functions on the phase space V , i.e.,
functions in C
∞
(V ). In particular, a covector
w ∈ V
∗
defines a linear observable through
O
w
: V → R : v
a
7→ w
a
v
a
. (4)
Similarly, a bilinear form h
ab
on V describes
uniquely a quadratic observable
O
h
: V → R : v
a
7→
1
2
h
ab
v
a
v
b
. (5)
In quantum theory, observables correspond to
linear operators on a Hilbert space H. To de-
fine linear and multi-linear observables it is con-
venient to introduce the operator-valued vector
ˆ
ξ
a
referred to as quantization map. It is subject
to the constraint
[
ˆ
ξ
a
,
ˆ
ξ
b
] =
ˆ
ξ
a
ˆ
ξ
b
−
ˆ
ξ
b
ˆ
ξ
a
= {ξ
a
, ξ
b
} = iΩ
ab
1 (6)
where 1 represents the identity operator on H.
With respect to our initial basis where Ω takes
the form of (2),
ˆ
ξ
a
is represented as
ˆ
ξ
a
≡ (ˆq
1
, ··· , ˆq
N
, ˆp
1
, ··· , ˆp
N
)
|
(7)
with the familiar position and momentum op-
erators. From here, we can construct the well-
known creation and annihilation operators ˆa
i
=
1
√
2
(ˆq
i
− iˆp
i
) and ˆa
†
i
=
1
√
2
(ˆq
i
+ iˆp
i
).
The quantum observables associated to the lin-
ear form w
a
and quadratic form h
ab
are then
ˆ
O
w
= w
a
ˆ
ξ
a
and
ˆ
O
h
=
1
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
. (8)
In contrast to classical observables, a quantum
observable constructed from a general tensor
t
a
1
···a
n
depends on the tensor’s non-symmetric
part since the operators ˆq
i
and ˆp
i
do not com-
mute.
Note that in the classical theory, the quantiza-
tion map reduces to the vector-valued function
ξ
a
: V → V : v
a
7→ ξ
a
(v) = v
a
, (9)
which is, in fact, just the identity map on V . It
can be used to represent linear and multi-linear
observables, e.g., through
O
t
: v 7→
1
n!
t
a
1
···a
n
ξ
a
1
(v) ···ξ
a
n
(v) (10)
for a general tensor t
a
1
···a
n
. In terms of a linear
basis of V it is represented as the tupel of func-
tions
ξ
a
≡ (q
1
, p
1
, ··· , q
N
, p
N
)
|
. (11)
Because each component of ξ
a
, i.e., q
i
, p
i
: V →
R, is a function on phase space, the vector val-
ued function encodes the Poisson brackets of the
classical theory through
{ξ
a
, ξ
b
} = Ω
ab
. (12)
This is particularly convenient and allows for a
clean notation when expressing Poisson brackets
of observables abstractly. For instance, for two
linear observables O
w
= w
a
ξ
a
and O
u
= u
a
ξ
a
,
we find immediately
{O
w
, O
u
} = w
a
u
b
{ξ
a
, ξ
b
} = w
a
u
b
Ω
ab
. (13)
Accepted in Quantum 2019-07-10, click title to verify 3
2.2 Bosonic Gaussian states
Bosonic Gaussian states are an important class
of quantum states because they can be described
through powerful analytic methods [59]. They
appear as ground states of quadratic Hamilto-
nians and their wave function representation is
given by multi-dimensional complex Gaussian
distributions. When representing them as quasi-
probability distributions on classical phase space,
they are the only states with positive Wigner
functions [63, 64]. In particular, their Wigner
functions are multi-dimensional real Gaussians
again. In the context of Bogoliubov transfor-
mations, Gaussian states are often identified as
the vacuum associated to a set of annihilation
operators satisfying bosonic commutation rela-
tions. Finally, their n-point correlation func-
tions are completely determined from their 1-
point and 2-point correlation function via Wick’s
theorem [65].
We label a Gaussian state |G, zi by a displace-
ment vector z
a
and a covariance matrix G
ab
z
a
= hG, z|
ˆ
ξ
a
|G, zi , (14)
G
ab
= hG, z|
ˆ
ξ
a
ˆ
ξ
b
+
ˆ
ξ
b
ˆ
ξ
a
|G, zi − 2z
a
z
b
. (15)
The covariance matrix G
ab
is a positive definite
symmetric bilinear form on the dual phase space
V
∗
, i.e., G equips V
∗
with an inner product. In
particular, for each covariance matrix there exists
a basis of the phase space V with respect to which
the covariance matrix is represented by the iden-
tity matrix G
ab
≡ 1, while Ω takes the standard
form (2).
Furthermore, the covariance matrix of a pure
Gaussian state and the inverse symplectic form
satisfy the condition
G
ab
Ω
−1
bc
G
cd
Ω
−1
de
= −δ
a
e
. (16)
In fact, we can use this property to define a linear
map on V
J
a
b
= −G
ac
Ω
−1
cb
. (17)
This map is referred to as linear complex struc-
ture on V because it satisfies
J
2
= (GΩ
−1
)
2
= −1 , (18)
i.e., it acts like the multiplication with the com-
plex number i on the real vector space. For a
given covariance matrix and displacement vector,
the Gaussian state |G, zi ∈ H is then fully speci-
fied by the requirement that
1
2
(δ
a
b
+ iJ
a
b
)(
ˆ
ξ
b
− z
b
) |G, zi = 0 . (19)
Here, the first term corresponds to a projector
onto the space spanned by those annihilation op-
erators that annihilate |G, zi. Describing Gaus-
sian states in terms of linear complex structures
was first introduced to describe unitarily inequiv-
alent Fock space representations [52, 53, 62], but
since then has been used to study entanglement
production [56, 57], typical entanglement of en-
ergy eigenstates [66–68] and to explore circuit
complexity in free field theories [67, 69].
Wick’s theorem provides an efficient way to
compute arbitrary n-point functions
C
a
1
···a
n
n
= hG, z|(
ˆ
ξ
a
1
− z
a
1
) ···(
ˆ
ξ
a
n
− z
a
n
) |G, zi.
(20)
where the subtraction of z
a
simplifies later ex-
pressions. Clearly, knowing C
a
1
···a
n
n
enables us
to compute arbitrary correlations even without
z
a
. With the definitions of G
ab
and z
a
, and the
commutation relations [
ˆ
ξ
a
,
ˆ
ξ
b
] = iΩ
ab
, we find the
2-point function to be
C
ab
2
=
1
2
(G
ab
+ iΩ
ab
) . (21)
With this expression, we can state Wick’s theo-
rem compactly as
C
a
1
···a
n
n
=
X
(contractions of C
ab
2
) (22)
= C
a
1
a
2
2
···C
a
n−1
a
n
2
+ ··· , (23)
where the sum over all contraction refers to writ-
ing C
a
1
···a
n
n
as sum of products of 2-point func-
tions C
ab
2
with all possible inequivalent assign-
ments of ordered indices a
i
. This implies that all
odd n-point functions vanish, because contrac-
tions require an even number of indices. There-
fore, after the 2-point function, the next non-
trivial correlation function is the four-point func-
tion given by
C
abcd
4
= C
ab
2
C
cd
2
+ C
ac
2
C
bd
2
+ C
ad
2
C
bc
2
. (24)
Note the importance of ordering the indices on
each 2-point function C
a
i
a
j
2
in the same way as
on C
a
1
···a
n
n
, i.e., we require i < j.
Accepted in Quantum 2019-07-10, click title to verify 4
Example. In order to illustrate our methods
and to fix conventions, let us look at the ground
state |G, zi of the harmonic oscillator with Hamil-
tonian
ˆ
H =
1
2
(ˆp
2
+ ω
2
ˆq
2
). Clearly, we have
z
a
= 0, because the ground state has zero ex-
pectation value of position ˆq and momentum ˆp.
We represent everything in the basis
ˆ
ξ
a
≡ (ˆq, ˆp)
|
.
Here, we have
Ω
ab
≡
0 1
−1 0
!
and G
ab
≡
1/ω 0
0 ω
!
. (25)
We can use Ω
ab
and G
ab
to compute the matrix
representation of the linear complex structure
J
a
b
= −G
ac
Ω
−1
cb
≡
0 1/ω
−ω 0
!
. (26)
With this, the projection of equation (19) reads
1
2
(δ
a
b
+ iJ
a
b
)
ˆ
ξ
b
≡
1
2
ˆq +
i
ω
ˆp
ˆp − iω ˆq
!
=
q
1
2ω
ˆa
−i
q
ω
2
ˆa
,
(27)
i.e., we project onto the subspace spanned by the
annihilation operator ˆa =
q
ω
2
(ˆq +
i
ω
ˆp).
2.3 Quadratic bosonic Hamiltonians
The most general quadratic Hamiltonian
ˆ
H =
1
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
+ f
a
ˆ
ξ
a
, (28)
contains a linear and a quadratic term, where we
require h
ab
to be positive definite and symmet-
ric. This ensures that
ˆ
H is bounded from below
and its ground state is a Gaussian state |G, zi.
The covariance matrix G
ab
and the linear dis-
placement z
a
can be computed from h
ab
and f
a
:
• Computation of z
a
The ground state minimizes the energy ex-
pectation value E = hG, z|
ˆ
H|G, zi given by
E =
1
4
h
ab
G
ab
+
1
2
h
ab
z
a
z
b
+ f
a
z
a
. (29)
This implies z
a
h
ab
+ f
b
= 0 giving
z
a
= −(h
−1
)
ab
f
b
, (30)
with the inverse form (h
−1
)
ac
h
cb
= δ
a
b
,
which exists since h
ab
is positive definite,
thus non-degenerate.
• Computation of G
ab
We can compute G
ab
directly from h
ab
via
G
ab
= −K
a
c
|K
−1
|
c
d
Ω
db
, (31)
with K
a
b
= Ω
ac
h
cb
. |K
−1
| refers to the lin-
ear map constructed by replacing the eigen-
values of K
−1
with their absolute values.
We can think of these formulas as a projection
from the space of quadratic Hamiltonians onto
the space of Gaussian states. Every Hamiltonian
gives rise to a unique ground state, but for each
state there are plenty of Hamiltonians that share
the same ground state. This redundancy is visible
in (31) where rescaling h
ab
with an overall factor
(or individually in each frequency sector) leaves
G
ab
unchanged.
The energy expectation value E and its vari-
ance Σ
E
for Gaussian states of the form |G
0
, zi,
i.e., with the above z
a
but with arbitrary G
0
, are
readily calculated as
E = hG
0
, z|
ˆ
H |G
0
, zi = Tr(hG
0
)/4 + E
z
, (32)
Σ
E
=
q
hG
0
, z|
ˆ
H
2
|G
0
, zi − E
2
(33)
=
q
Tr(hG
0
hG
0
+ hΩhΩ)/4 , (34)
where E
z
= −
1
2
h
ab
z
a
z
b
= −
1
2
(h
−1
)
ab
f
a
f
b
. This
follows from (30) to rewrite
ˆ
H =
1
2
h
ab
(
ˆ
ξ − z)
a
(
ˆ
ξ − z)
b
+ E
z
. (35)
Example. Let us consider the simple harmonic
oscillator
ˆ
H =
1
2
(ˆp
2
+ ω
2
ˆq
2
) from before. We find
f
a
= 0 and with respect to
ˆ
ξ
a
≡ (ˆq, ˆp)
h
ab
≡
ω
2
0
0 1
!
. (36)
Applying formula (31) yields
K
a
b
≡
0 1
−ω
2
0
!
⇒ G
ab
≡
1/ω 0
0 ω
!
, (37)
which agrees with the result from (25). Here
|K
−1
| ≡
1
ω
0
0
1
ω
!
, (38)
because K
−1
has eigenvalues ±
i
ω
.
Accepted in Quantum 2019-07-10, click title to verify 5
2.4 Entanglement of bosonic Gaussian states
In terms of a quantum system’s Hilbert space H,
a bi-partition of the system corresponds to a ten-
sor product H = H
A
⊗ H
B
. A bi-partition of a
system of N bosonic modes, into subsystems of
N
A
and N
B
modes respectively, induces a decom-
position of the phase space V = A ⊕ B into two
symplectic complements A and B.
Generally a Gaussian state on the full sys-
tem |G, zi contains correlations, i.e., entangle-
ment between A and B. However, there al-
ways exist bases (q
A
1
, p
A
1
, ··· , q
A
N
A
, p
A
N
A
) of A and
(q
B
1
, p
B
1
, ··· , q
B
N
B
, p
B
N
B
) of B, such that the covari-
ance matrix G
ab
takes the standard form [54] re-
derived in appendix A.1
G ≡
ch
1
··· 0 sh
1
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· ch
N
A
0 ··· sh
N
A
0 ··· 0
sh
1
··· 0 ch
1
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· sh
N
A
0 ··· ch
N
A
0 ··· 0
0 ··· 0 0 ··· 0 1
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· 0 0 ··· 0 0 ··· 1
2
, (39)
built from the 2 × 2-matrices, with r
i
≥ 0,
ch
i
=
ch 2r
i
0
0 ch 2r
i
!
, (40)
sh
i
=
sh 2r
i
0
0 −sh 2r
i
!
. (41)
The standard form highlights two important
features of the state’s entanglement structure.
Firstly, the basis modes bring the diagonal blocks,
corresponding to A and B into diagonal form, i.e.,
they provide a normal mode decomposition of the
subsystems’ partial states: The partial states of
A, and B respectively, factorize into a product
state over the modes in each subsystem basis.
Secondly, all entanglement between A and B is
contained in entangled mode pairs: If r
i
> 0, for
a mode (q
A
i
, p
A
i
) in A, then it is entangled with
the mode (q
B
i
, p
B
i
) with which it is in a two-mode
squeezed state.
The total von Neumann entropy S
A
(|G, zi) be-
tween A and B is given by the sum of the mode
pairs’ individual entropies [70, 71]
S
A
(|G, zi) =
N
A
X
i=1
s
b
(ch 2r
i
) (42)
with s
b
(x) =
x+1
2
log
x+1
2
−
x−1
2
log
x−1
2
.
Here, we chose the convention to compute all log-
arithms with respect to base 2.
Note, that the total entropy can be expressed
in a basis invariant way as [56, 58]
S
A
(|G, zi) = Tr
1
A
+ i[J]
A
2
log
1
A
+ i[J]
A
2
,
(43)
where [J]
A
is the restriction of the complex struc-
ture J
a
b
= −G
ac
Ω
−1
cb
to the symplectic subspace
A, i.e., we restrict it to a 2N
A
-by-2N
A
matrix.
The absolute value should be understood in terms
of eigenvalues of [J]
A
which come in conjugate
pairs ±i ch 2r
i
.
2.5 Bosonic partner mode construction
If we pick a single mode from a larger system in a
Gaussian state |G, zi, the single mode is in gen-
eral entangled with the rest of the modes. How-
ever, as implied by the standard form (39), there
exists a second mode, the so called partner mode,
which shares all of the first mode’s entanglement,
i.e., these two modes are in a product state with
the rest of the system.
Accepted in Quantum 2019-07-10, click title to verify 6
Recent works have developed formulae for the
partner mode [50? , 51]. We here recast them
in terms of the complex structure J of the state,
thus illuminating their structure, and generaliz-
ing them to arbitrary Gaussian states. In Ap-
pendix A.1 we derive the partner mode construc-
tion in terms of the complex structure.
Let us assume that the two observables
ˆ
Q
A
=
x
a
ˆ
ξ
a
and
ˆ
P
A
= k
a
ˆ
ξ
a
define a mode, i.e.,
Ω
ab
x
a
k
b
= 1 , (44)
and that they yield the standard form of the mode
according to (39), i.e., for some r > 0
G
ab
x
a
x
b
= G
ab
k
a
k
b
= ch 2r , (45)
G
ab
x
a
k
b
= 0 . (46)
Then the partner mode of this mode is given by
ˆ
Q
¯
A
= ¯x
a
ˆ
ξ
a
and
ˆ
P
¯
A
=
¯
k
a
ˆ
ξ
a
with
¯x
a
= coth(2r)x
a
+
1
sh 2r
(J
|
)
a
c
k
c
, (47)
¯
k
a
= −coth(2r)k
a
+
1
sh 2r
(J
|
)
a
c
x
c
. (48)
2.5.1 Partner modes yield standard form for G
This definition yields a normalized mode
Ω
ab
¯x
a
¯
k
b
= 1, which commutes with the origi-
nal mode and is correlated exactly as predicted
by (39), i.e., as is straightforward to check using
the identities (J
|
)
2
= −1 and GΩ
−1
G = −Ω, we
have
G
ab
¯x
a
¯x
b
= G
ab
¯
k
a
¯
k
b
= ch 2r , (49)
G
ab
x
a
¯x
b
= −G
ab
k
a
¯
k
b
= sh 2r , (50)
G
ab
k
a
¯x
b
= G
ab
x
a
¯
k
b
= G
ab
¯x
a
¯
k
b
= 0 . (51)
Note that repeating the partner mode construc-
tion gives back the original mode again, i.e., the
partner mode’s partner is the original mode.
There is a close relationship between the cor-
relation of other modes with the original mode,
and the commutator of those other modes with
the partner mode: Let x
0
a
and k
0
a
define another
mode in the system, i.e., Ω
ab
x
0
a
k
0
b
= 1. Then
Ω
ab
x
0
a
¯x
b
= coth(2r)Ω
ab
x
0
a
x
b
+
G
ad
x
0
a
k
d
sh 2r
, (52)
Ω
ab
k
0
a
¯x
b
= coth(2r)Ω
ab
k
0
a
x
b
+
G
ad
k
0
a
k
d
sh 2r
, (53)
Ω
ab
x
0
a
¯
k
b
= −coth(2r)Ω
ab
x
0
a
k
b
+
G
ad
x
0
a
x
d
sh 2r
,
(54)
Ω
ab
x
0
a
¯x
b
= −coth(2r)Ω
ab
k
0
a
k
b
+
G
ad
k
0
a
x
d
sh 2r
. (55)
This means that if the mode commutes with the
original mode Ω(x, x
0
) = Ω(x, k
0
) = Ω(k, x
0
) =
Ω(k, k
0
) = 0, then the commutator with the part-
ner mode is proportional to the correlation with
the original mode. In particular, it follows that
with respect to a basis of modes which contains
the original mode and the partner mode as its
first two modes, the covariance matrix G
ab
takes
precisely the standard form.
G ≡
ch 2r 0 sh 2r 0 0
0 ch 2r 0 −sh 2r 0
sh 2r 0 ch 2r 0 0
0 −sh 2r 0 ch 2r 0
0 0 0 0 1
N−2
(56)
This shows that |G, zi is in a product state
consisting of an entangled two-mode state (be-
tween original modes and its partner) and an-
other Gaussian state describing the rest of the
system. Note that the displacement vector z
a
does not affect the entanglement structure of the
state.
2.5.2 Unsqueezing the partner modes
From the matrix representation of G above we
see that partner modes are in a pure two-mode
squeezed state. Hence, they can be viewed as aris-
ing from squeezing two modes which are in a pure
product state with each other. We obtain these
two modes by acting with the inverse squeez-
ing transformation on the two partner modes
(x
a
ˆ
ξ
a
, k
a
ˆ
ξ
a
) and (¯x
a
ˆ
ξ
a
,
¯
k
a
ˆ
ξ
a
). This yields the
modes (y
a
ˆ
ξ
a
, l
a
ˆ
ξ
a
) and (z
a
ˆ
ξ
a
, m
a
ˆ
ξ
a
), given by
y
a
= ch(r)x
a
− sh(r)¯x
a
=
x
a
− (J
|
)
a
c
k
c
2 ch r
,
(57)
l
a
= ch(r)k
a
+ sh(r)
¯
k
a
=
k
a
+ (J
|
)
a
c
x
c
2 ch r
, (58)
z
a
= −sh(r)x
a
+ ch(r)¯x
a
=
x
a
+ (J
|
)
a
c
k
c
2 sh r
,
(59)
m
a
= sh(r)k
a
+ ch(r)
¯
k
a
=
−k
a
+ (J
|
)
a
c
x
c
2 sh r
.
(60)
Since the inverse squeezing is a symplectic trans-
formation these unsqueezed modes form a nor-
malized and commuting basis of the subsystem
Accepted in Quantum 2019-07-10, click title to verify 7
spanned by the two partner modes. In particu-
lar, with respect to them the covariance matrix
is represented by the identity matrix G
ab
≡ 1.
3 Entanglement extraction from
quadratic bosonic systems
In this section, we present the central result of
this paper for bosonic systems. We begin by
defining the entanglement extraction procedure
in Section 3.1. Our proof then follows three steps:
1. Splitting the problem into two parts
In section 3.2, we show that the problem of
finding the minimal energy cost can be bro-
ken up into first solving the problem for a
system of two modes, and then optimizing
how to embed these two modes used for en-
tanglement extraction into the larger system.
2. Solving the two-mode problem
In Section 3.3, we solve the two mode prob-
lem, i.e., we answer the question: What is
the minimal energy cost ∆E
min
of extract-
ing two partner modes with entanglement
entropy ∆S from a system consisting of two
modes with a quadratic Hamiltonian? For
this, we first look at the case of a degenerate
two-mode Hamiltonian in Section 3.3.1, be-
fore tackling the general case in Section 3.3.2.
The result is a general formula for the mini-
mal energy cost as a function of the Hamil-
tonian’s excitation energies
1
and
2
.
3. Solving the embedding problem
In Section 3.4, we derive how to select modes
from a large system that optimize the energy
cost. Not surprisingly, the lowest energy cost
is achieved only if the two modes are chosen
from the subsystem spanned by the two low-
est energy eigenmodes of the Hamiltonian.
Based on this insight, we construct the op-
timal partner modes and the associated ex-
citation energies in the subsystem explicitly.
Furthermore, we generalize the expressions
and constructions to scenarios where the two
partner modes are restricted to lie in a dif-
ferent two-mode subsystem.
(
ˆ
Q
1
,
ˆ
P
1
)
(
ˆ
Q
2
,
ˆ
P
2
)
(
ˆ
Q
B
,
ˆ
P
B
)(
ˆ
Q
A
,
ˆ
P
A
)
S
W
A
P
S
W
A
P
source system
target modes
Figure 1: General framework of entanglement extraction
from a system of bosonic modes. Two target modes
get entangled by swapping their states with a pair of
entangled modes in the source system.
3.1 Pure state entanglement extraction from
bosonic modes
The general framework for entanglement extrac-
tion consists of two target systems and the source
system S. Initially, the total state is assumed to
be a product state
ρ
i
= σ
1
⊗ τ
2
⊗ |ΨihΨ|
S
, (61)
where |Ψi
S
is the initial state of the source sys-
tem, and σ
1
and τ
2
are the initial states of the first
and second target system, respectively. With-
out loss of generality, the interaction between the
source and target systems can be modeled by two
unitary operations U
1
and U
2
, which each couple
one of the targets to the source. The final state
of the two target systems is then
ρ
12
= Tr
S
U
1
U
2
ρ
i
U
†
2
U
†
1
. (62)
In general, the entanglement content of this state
depends on the type of source and target system
and their realizable couplings. In particular, it
also depends on the resources such as the energy
which is available to implement the extraction.
The concrete type of source system we study
in this section, is a system composed of bosonic
modes with a quadratic Hamiltonian. This sys-
tem is supposed to be in its ground state initially,
i.e., the state |Φi
S
is Gaussian. Consequently,
its entanglement structure can be analyzed using
the formalism of partner modes introduced be-
fore. The two target modes are assumed to be
single bosonic modes.
Accepted in Quantum 2019-07-10, click title to verify 8
The model of interaction we use is that the
target modes, with quadratures (
ˆ
Q
1
,
ˆ
P
1
) and
(
ˆ
Q
2
,
ˆ
P
2
), each couple to one mode inside the
source system, with quadratures (
ˆ
Q
A
,
ˆ
P
A
) and
(
ˆ
Q
B
,
ˆ
P
B
). The unitary couplings U
1
and U
2
are
assumed to swap the states of the target modes
and the source modes, i.e.,
U
†
1
ˆ
Q
1
U
1
=
ˆ
Q
A
, U
†
1
ˆ
Q
A
U
1
=
ˆ
Q
1
,
U
†
1
ˆ
P
1
U
1
=
ˆ
P
A
, U
†
1
ˆ
P
A
U
1
=
ˆ
P
1
,
(63)
and U
2
swaps (
ˆ
Q
2
,
ˆ
P
2
) and (
ˆ
Q
B
,
ˆ
P
B
), accordingly.
The only restriction imposed on the source
modes is that they commute according to
h
ˆ
Q
A
,
ˆ
Q
B
i
=
h
ˆ
Q
A
,
ˆ
P
B
i
=
h
ˆ
P
A
,
ˆ
Q
B
i
=
h
ˆ
P
A
,
ˆ
P
B
i
=0.
(64)
This condition ensures that the entanglement of
the two target modes was preexisting in the sys-
tem, but not created by the couplings between
target modes and system [31]. Other than that,
the modes can be chosen freely inside the source
systems.
In particular, the source modes can be chosen
to be partner modes, i.e.,
ˆ
Q
B
=
ˆ
Q
¯
A
and
ˆ
P
B
=
ˆ
P
¯
A
in the notation of Section 2.5. As discussed, this
implies that the ground state of the source system
S is a product state between the two modes and
the rest of the source system
|Φi
S
= |ψi
A
¯
A
⊗ |φi
R
, (65)
where |ψi
A
¯
A
is a two-mode squeezed state, and
|φi
R
is the ground state of the rest of the system.
When the target modes and the source modes are
swapped this puts the total system into the state
ρ
f
= |ψihψ|
12
⊗ σ
A
⊗ τ
¯
A
⊗ |φihφ|
R
, (66)
where now the target modes are in the entangled
state |ψi, whereas the two source modes are in
the initial states of the target modes. The rest of
the source system is unaffected by the extraction
and continues to be in its ground state |φ
R
i.
Choosing partner modes in the source system
is advantageous for two reasons: Firstly, it is the
only way for the target modes to end up in a pure
state. If the system modes are not partners then
after the swap the two target modes are corre-
lated with the source system, i.e., after tracing
out the source system the partial state of target
modes is mixed. Secondly, choosing the modes as
partners maximizes the extracted entanglement.
If the second party chooses a different mode, then
the (mixed state) entanglement between the two
modes is never larger than between the first mode
and its partner. This is because the mixed state
of the two modes is obtained by tracing out part
of a larger state which included the partner mode.
Motivated by this framework and these consid-
erations we study the following concise question:
If in the ground state of a quadratic Hamiltonian
the state of two partner modes sharing entangle-
ment entropy ∆S is replaced by a product state,
by at least what amount ∆E does this increase
the energy expectation value of the system.
While in the following we refer to ∆E as the
energy cost of entanglement extraction, in a re-
alistic implementation of an extraction protocol
additional energy costs are expected: For one, we
do not take into account the energy content of the
target modes outside of the source system. Fur-
thermore, the implementation of the unitary in-
teraction between targets and source may require
energy as well. However, our reason to focus on
the energy ∆E injected into the source system
only, is that the other additional energy costs de-
pend on the details and the design of the specific
extraction protocol implemented.
Another interesting aspect is the dynamics of
the extraction process: In general, the source
modes are not eigenmodes of the Hamiltonian,
i.e., they have a non-trivial time evolution and
their physical localization in the source system
may change over time. This dynamic needs to
be taken into account in the design of the inter-
action of source and target modes. In the scope
of this work, we assume that the swap of target
and source modes happens within a time much
shorter than the time scale of the time evolution
of the target modes. Hence we neglect the dy-
namics and consider the interaction of target a
The vale modes to take part instantaneously. id-
ity of this ind sourcdealization has to be reviewed
with respect to any given implementation of an
extraction protocol. We return to a discussion of
this point at the end of the article.
3.2 Energy cost of entanglement extraction
The energy cost of entanglement extraction, as
introduced above, is given by the difference in
expectation value of the source system’s Hamil-
Accepted in Quantum 2019-07-10, click title to verify 9
tonian before and after the extraction, i.e.,
∆E = Tr
σ
A
⊗ τ
¯
A
⊗ |φihφ|
R
ˆ
H − |ΨihΨ|
S
ˆ
H
.
(67)
The Hamiltonian
ˆ
H =
1
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
+ f
a
ˆ
ξ
a
of the
source system may be coupling all different modes
of the system. Nevertheless, for the calculation
of ∆E, only the part acting on the two selected
partner modes A and B =
¯
A matter.
To see this, we split up the Hamiltonian into a
sum of three terms
ˆ
H =
ˆ
H
A
¯
A
+
ˆ
H
R
+
ˆ
H
A
¯
A,R
(68)
where
ˆ
H
A
¯
A
acts only on the two partner modes,
ˆ
H
R
acts only on the rest of the source system’s
modes, and
ˆ
H
A
¯
A,R
is the part which couples the
partner modes and the rest of the system.
1
The summand
ˆ
H
R
does not contribute to the
energy cost ∆E, because the rest of the system is
in the same state before and after the extraction.
Also the part
ˆ
H
A
¯
A,R
does not contribute to ∆E,
because both the initial state (65) of the system
and its final state (66) are product states with
respect to the partition A
¯
A ⊕ V , hence
Tr
σ
A
⊗ τ
¯
A
⊗ |φihφ|
R
ˆ
H
A
¯
A,R
= 0 . (69)
Therefore, the energy cost of entanglement ex-
traction is determined by
ˆ
H
A
¯
A
acting on the part-
ner modes, only:
∆E = Tr
σ
A
⊗ τ
¯
A
ˆ
H
A
¯
A
− Tr
|ψihψ|
A
¯
A
ˆ
H
A
¯
A
.
(70)
From this follows which initial states of the target
modes minimize the energy cost: Because
Tr
σ
A
⊗ τ
¯
A
ˆ
H
A
¯
A
= Tr
σ
A
ˆ
H
A
+ Tr
τ
¯
A
ˆ
H
¯
A
,
(71)
the initial states σ and τ of the target modes need
to be the ground states of the restrictions
ˆ
H
A
and
ˆ
H
¯
A
of the Hamiltonian
ˆ
H onto the partner
modes. These states are pure Gaussian states.
Equation (70) effectively simplifies finding the
minimal energy cost to a two-mode problem: If
1
Put differently, the operators
ˆ
H
A
¯
A
and
ˆ
H
R
are the
restrictions of
ˆ
H onto the subspace spanned by the partner
modes and the rest of the system, respectively.
ˆ
H
A
¯
A,R
:=
ˆ
H −
ˆ
H
A
¯
A
−
ˆ
H
R
then contains all terms in
ˆ
H that act on
both A
¯
A and R simultaneously.
we know the minimal energy cost for the extrac-
tion of an amount ∆S of entanglement from a
source system consisting of two modes, then ap-
plying this result to
ˆ
H
A
¯
A
and optimizing over dif-
ferent choices of A yields the minimal cost for
extracting ∆S from a larger system of N modes.
3.3 Energy cost for two bosonic modes
In this section, we analyze entanglement extrac-
tion from a source system that consists of only
two bosonic modes and has a quadratic Hamil-
tonian. We show that the minimal energy cost
for the extraction from a pair of partner modes
with a fixed amount of entanglement is a func-
tion only of the Hamiltonian’s excitation energies.
Our derivation is constructive in the sense that we
derive the symplectic transformation which maps
the eigenmodes of the Hamiltonian to a pair of
partner modes which minimize the energy cost.
Since the Hamiltonian is quadratic, we can
choose a basis
ˆ
ξ
a
≡ (ˆq
1
, ˆp
1
, ˆq
2
, ˆp
2
) such that the
quadratic part of the Hamiltonian h
ab
and the co-
variance matrix G
ab
of its ground state are rep-
resented by diagonal matrices h and G given by
h
ab
1,2
≡ h =
1
0 0 0
0
1
0 0
0 0
2
0
0 0 0
2
, (72)
G
ab
1,2
≡ G =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
. (73)
These modes 1 and 2 are not entangled and,
therefore, cannot be used as modes A and
¯
A for
entanglement extraction. Instead, we need to find
a pair of partner modes (
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
) with
respect to which G
ab
is represented in standard
form (56) given by
G
ab
A,
¯
A
≡
˜
G =
ch 2r 0 sh 2r 0
0 ch 2r 0 −sh 2r
sh 2r 0 ch 2r 0
0 −sh 2r 0 ch 2r
(74)
for a fixed r > 0, which we choose based on the
amount of entanglement that we want to extract.
There is an infinite number of choices for such A
Accepted in Quantum 2019-07-10, click title to verify 10
and
¯
A, but we will identify those that minimize
the energy cost. This is, we find the symplec-
tic transformations which map the energy eigen-
modes 1 and 2 to those partner modes A and
¯
A
which are optimal for entanglement extraction.
The two-mode squeezing transformation
M
r
=
ch r 0 sh r 0
0 ch r 0 −sh r
sh r 0 ch r 0
0 −sh r 0 ch r
(75)
brings G into the form from (74) via
˜
G = M
r
GM
|
r
, (76)
i.e., it maps the eigenmodes to a pair of partner
modes with squeezing parameter r. Under the
symplectic transformation M
r
, the Hamiltonian
matrix h transforms under the action of the in-
verse transform M
−1
r
= M
−r
, i.e.,
˜
h = M
|
−r
hM
−r
. (77)
However, M
r
is not the only transformation do-
ing this: We can compose the squeezing M
r
with any symplectic transformation N which
leaves the covariance matrix invariant, i.e., G =
NGN
|
. This yields the most general form
of symplectic transformation T
r
= M
r
N that
transforms the modes 1 and 2 into A and
¯
A
for fixed r. All of these choices share the same
amount of entanglement across A and
¯
A be-
cause the covariance matrix takes the same form
˜
G = T
r
GT
|
r
. However, the Hamiltonian ma-
trix h is in general not left invariant by N, i.e.,
h 6= (N
−1
)
|
hN
−1
and thus the energy cost will
be different.
3.3.1 Degenerate two-mode Hamiltonian (
1
=
2
)
We first consider the case where :=
1
=
2
.
This case is the simplest to solve because the ma-
trix representations h and G are proportional to
each other in the eigenmode basis and, thus, in
all bases. Indeed, any symplectic transformation
N that leaves G invariant, i.e., NGM
|
= G,
also leaves h invariant, i.e., h = (N
−1
)
|
hN
−1
.
Therefore, for any pair of partner modes A and
¯
A where the covariance matrix takes the standard
form
˜
G for fixed r, we find
˜
h =
ch 2r 0 − sh 2r 0
0 ch 2r 0 sh 2r
− sh 2r 0 ch 2r 0
0 sh 2r 0 ch 2r
.
(78)
As discussed following equation (71), the en-
tanglement extraction swaps the state of the two
modes against the product state σ
A
⊗ τ
¯
A
of two
Gaussian states σ and τ. Hence, denoting σ
A
=
|G
0
A
ihG
0
A
| and τ
¯
A
= |G
0
¯
A
ihG
0
¯
A
|, with respect to
partner mode basis, the covariance matrix after
the extraction takes the block diagonal form
˜
G
0
=
˜
G
0
A
0
0
˜
G
0
¯
A
. (79)
To minimize the energy cost, we choose the initial
states of the target modes to be the ground states
of the restricted single-mode Hamiltonians, i.e.,
˜
G
0
A
=
˜
G
0
¯
A
= 1
2
. In terms of this new covariance
matrix, the energy cost (70) of the entanglement
extraction is
∆E =
1
4
Tr
˜
h
|
˜
G
0
−
1
4
Tr (h
|
G) (80)
=
1
4
Tr
˜
h
A
˜
G
A
+
˜
h
¯
A
˜
G
¯
A
− , (81)
where
˜
h
A
=
˜
h
¯
A
= ch 2r1
2
are the diagonal
blocks of
˜
h above.
With these initial states, the minimal energy
cost for a pure two-mode squeezed state with en-
tanglement entropy ∆S = s
b
(ch 2r) (see (42)) is
∆E
min
= 2 sh
2
r . (82)
The final state of the source system after the swap
is not an energy eigenstate anymore. Its energy
variance is
Σ
E
=
r
Tr
˜
h
˜
G
0
˜
h
˜
G
0
+
˜
hΩ
˜
hΩ
4
=
sh 2r
√
2
.
(83)
We notice that for the degenerate Hamiltonian
with
1
=
2
, the energy cost and energy variance
is actually the same for all partner modes with
entanglement entropy ∆S. In the next section,
we will see that this is different for the case
1
<
2
.
Accepted in Quantum 2019-07-10, click title to verify 11
3.3.2 General two-mode Hamiltonian (
1
<
2
)
The case of a general Hamiltonian, with
1
≤
2
,
is more involved to analyze because there ex-
ist symplectic transformations which leave the
covariance matrix of the state invariant but do
change the Hamiltonian matrix and vice versa.
This means that now, before we apply the
squeezing transformation M
r
, we can apply an-
other symplectic transformation N which leaves
G invariant but possibly changes h. All possible
forms of N can be written as a matrix exponen-
tial N = exp (θK) of the generator K subject to
the constraints KG + GK
|
= 0 (leaving G in-
variant) and KΩ + ΩK
|
= 0 (being symplectic).
From this, we find the most general form
K =
0 0 cos φ −sin φ
0 0 sin φ cos φ
−cos φ −sin φ 0 0
sin φ −cos φ 0 0
.
(84)
While N = exp (θK) leaves G invariant, it
changes h. Consequently, T
r
= M
r
N provides
the most general transformation that turns the
covariance matrix from (73) into
˜
G = T
r
GT
|
r
in
the form of (74). Under this transformation, the
Hamiltonian matrix transforms to
˜
h = (T
−1
r
)
|
hT
−1
r
=
h
A
h
X
h
|
X
h
¯
A
, (85)
where the individual blocks are given by
h
A
=
+
ch 2r −
−
(cos 2θ + sin 2θ cos φ sh 2r) −
−
sin 2θ sin φ sh 2r
−
−
sin 2θ sin φ sh 2r
+
ch 2r −
−
(cos 2θ − sin 2θ cos φ sh 2r)
!
, (86)
h
¯
A
=
+
ch 2r +
−
(cos 2θ − sin 2θ cos φ sh 2r)
−
sin 2θ sin φ sh 2r
−
sin 2θ sin φ sh 2r
+
ch 2r +
−
(cos 2θ + sin 2θ cos φ sh 2r)
!
, (87)
h
X
=
−
+
sh 2r +
−
sin 2θ cos φ ch 2r −
−
sin 2θ sin φ ch 2r
−
sin 2θ sin φ ch 2r
+
sh 2r +
−
sin 2θ cos φ ch 2r
!
(88)
with
±
=
1
2
(
2
±
1
). The entanglement extrac-
tion swaps the state of the two modes against the
initial state of the target modes. We denote its
covariance matrix with respect A and
¯
A by
˜
G
0
as
in (79). The energy cost is then given by
∆E =
1
4
Tr
˜
h
A
˜
G
0
A
+
˜
h
¯
A
˜
G
0
¯
A
−
1
+
2
2
. (89)
To minimize the energy cost we choose the initial
states to be the ground states of the single-mode
Hamiltonians of h
A
and h
¯
A
. This yields
∆E =
1
2
(
A
+
¯
A
−
1
−
2
) , (90)
where the excitation energies of the single-mode
Hamiltonians are given by the symplectic eigen-
values of h
A
and h
¯
A
, i.e.,
A
=
1
2
q
4
2
+
ch
2
2r +
2
−
(3 + cos 4θ − (1 − cos 4θ) ch 4r) −8
+
−
cos 2θ ch 2r , (91)
¯
A
=
1
2
q
4
2
+
ch
2
2r +
2
−
(3 + cos 4θ − (1 − cos 4θ) ch 4r) + 8
+
−
cos 2θ ch 2r . (92)
Since the energy cost is independent of the pa- rameter φ, we only need to compute the mini-
Accepted in Quantum 2019-07-10, click title to verify 12
0 1 2 3 4
0.01
0.10
1
10
100
Figure 2: Minimal energy cost ∆E
min
as a function of
the extracted entanglement entropy ∆S from a bosonic,
quadratic two-mode Hamiltonian with excitation ener-
gies
1
and
2
.
mum with respect to θ, in order to find partner
modes which yield the minimal energy cost of en-
tanglement extraction. ∆E only depends on θ
through
A
+
¯
A
and we find that the minimum is
attained for θ =
π
4
+
nπ
2
with n ∈ Z. This value
gives
A
=
¯
A
=
1
2
q
2
1
+
2
2
+ 2
1
2
ch 4r. Hence
the minimal energy cost for the extraction of a
pure squeezed state with entanglement entropy
∆S = S(ch 2r) is given by
∆E
min
=
1
2
q
2
1
+
2
2
+ 2
1
2
ch 4r −
1
−
2
.
(93)
The energy variance of the final system state is
Σ
E
=
s
2
1
2
2
(1 + ch 4r)
2
1
+
2
2
+ 2
1
2
ch 4r
sh 2r . (94)
The minimal energy cost has a couple of in-
teresting properties, which are seen in Figure 2.
First, we note that the expression above correctly
reduces to the degenerate case for
2
=
1
. For
large amounts of extracted entanglement, the en-
ergy cost grows exponentially as
∆E
min
∼ e
2r
∼ e
∆S
as r → ∞, (95)
where we used the asymptotics ∆S ∼ 2r as r →
∞. For small r, we find
∆E
min
∼
4
1
2
1
+
2
r
2
+ O(r
4
) as r → 0 , (96)
which should be seen in the context of the asymp-
totics ∆S ∼ r
2
1 − log
r
2
as r → 0.
Furthermore, the energy cost is a
monotonously increasing function in both
0 1 2 3 4
0.01
0.10
1
10
100
Figure 3: We show the energy variance Σ
E
as a func-
tion of the extracted entanglement entropy ∆S from a
bosonic, quadratic two-mode Hamiltonian with excita-
tion energies
1
and
2
excitation energies. This means, for example
that ∆E
min
increases when
2
is increased and
1
is kept fixed. However, there exists an upper
bound on the energy cost in this case, which is
determined by the lower excitation energy
lim
2
→∞
∆E
min
=
1
(sh 2r)
2
. (97)
Comparing the minimal energy cost for a non-
degenerate Hamiltonian ∆E
min,
2
>
1
with the
cost for a degenerate Hamiltonian ∆E
2
=
1
, we
find that the ratio between them asymptotes to
lim
r→∞
∆E
min,
2
>
1
∆E
2
=
1
=
r
2
1
. (98)
Another interesting figure of merit is to look at
the ratio between a given amount of extracted
entanglement and the minimal energy cost neces-
sary for its extraction ∆S/∆E
min
. Since ∆S =
S(ch 2r) ∼ (1 −2 log(r))r
2
as r → 0, the ratio di-
verges in the limit of small amounts of extracted
entanglement, i.e., we have
lim
r→0
∆S
∆E
min
= ∞. (99)
As the extracted entanglement increases the ratio
is strictly decreasing, and in the limit of large en-
tanglement r → ∞ it decays exponentially. This
means that it requires more energy to extract an
amount of entanglement ∆S from a system once,
than twice time the energy required to extract
half the amount ∆S/2.
3.4 Optimal modes in large source systems
Having understood how to minimize the energy
cost of entanglement extraction from a two-mode
Accepted in Quantum 2019-07-10, click title to verify 13
source system, allows us to also understand how
to optimally extract entanglement from large
source systems.
The partner modes from which entanglement
can be extracted for the least possible energy
cost are linear combinations of the two lowest en-
ergy eigenmodes of the system only. No other
choice of partner modes can have a lower energy
cost, because the excitation energies
1
and
2
of the restricted two-mode Hamiltonian
ˆ
H
A
¯
A
are
restricted by the excitation energies of
ˆ
H by
ω
1
≤
1
≤ ω
N−1
, ω
2
≤
2
≤ ω
N
(100)
as discussed in Appendix B.
These modes can be constructed according to
above derivation. Take the two lowest energy
eigenmodes (ˆq
1
, ˆp
1
, ˆq
2
, ˆp
2
) of the system and mix
them by acting on them with the transformation
exp
π
4
K
=
1
√
2
0
cos φ
√
2
−
sin φ
√
2
0
1
√
2
sin φ
√
2
cos φ
√
2
−
cos φ
√
2
−
sin φ
√
2
1
√
2
0
sin φ
√
2
−
cos φ
√
2
0
1
√
2
(101)
where φ can be chosen freely. The two obtained
modes need to by squeezed by acting with M
r
on them, in order to obtain the globally optimal
partner modes of the source system.
In many scenarios the globally optimal modes
may not be accessible to the experimenter, but
they may be restricted to use a particular pair of
partner modes (
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
) as source modes.
In this case, applying results of the previous sec-
tion to the two-mode restriction
ˆ
H
A
¯
A
yields a
lower bound on the energy cost of entanglement
extraction, as discussed in Section 3.2.
However, the modes (
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
) can only
achieve the minimal energy cost ∆E
min
which the
spectrum of
ˆ
H
A
¯
A
allows for according to (93),
if they exactly correspond to the optimal modes
with respect to
ˆ
H
A
¯
A
as we constructed them in
the derivation of the previous section.
To check if this is the case it is sufficient
to check if
ˆ
H
A
¯
A
acts symmetrically on the two
modes A and
¯
A because this is exactly what hap-
pens for the Hamiltonian matrix in (85) when
θ =
π
4
+
nπ
2
is chosen optimally.
The Hamiltonian matrix of
ˆ
H
A
¯
A
with respect
to the basis (
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
) is conveniently cal-
culated using an inner product on V
∗
which is
induced by h
ab
. To do this, we denote
ˆ
Q
A
= x
a
ˆ
ξ
a
,
ˆ
P
A
= k
a
ˆ
ξ
a
, (102)
ˆ
Q
¯
A
= ¯x
a
ˆ
ξ
a
,
ˆ
P
¯
A
=
¯
k
a
ˆ
ξ
a
. (103)
and assume, without loss of generality, that the
modes are in standard form as in Section 2.5.
With respect to this basis, the quadratic part of
ˆ
H
A
¯
A
is represented by
ˇ
h(k, k) −
ˇ
h(k, x)
ˇ
h(k,
¯
k) −
ˇ
h(k, ¯x)
−
ˇ
h(x, k)
ˇ
h(x, x) −
ˇ
h(x,
¯
k)
ˇ
h(x, ¯x)
ˇ
h(
¯
k, k) −
ˇ
h(
¯
k, x)
ˇ
h(
¯
k,
¯
k) −
ˇ
h(
¯
k, ¯x)
−
ˇ
h(¯x, k)
ˇ
h(¯x, x)
−
ˇ
h(¯x,
¯
k)
ˇ
h(¯x, ¯x)
,
(104)
with the inner product
ˇ
h : V
∗
× V
∗
→ R with
ˇ
h(w, u) = h
ab
Ω
ac
Ω
bd
w
c
u
d
= h
ab
G
ac
G
bd
w
c
u
d
,
(105)
induced by h
ab
of the full Hamiltonian. Denoting
(J
|
w)
a
= (J
|
)
a
b
r
b
, the inner product satisfies
ˇ
h(J
|
w, J
|
u) =
ˇ
h(w, u) , (106)
which just implies that J
|
is orthogonal with re-
spect to
ˇ
h in the same way as J is orthogonal
with respect to h. In particular, this implies
ˇ
h(w, J
|
w) = 0, such that the matrix expressions
simplify
ˇ
h(¯x, ¯x) =
ˇ
h(x, x) + ∆ , (107)
ˇ
h(
¯
k,
¯
k) =
ˇ
h(k, k) + ∆ , (108)
ˇ
h(¯x,
¯
k) = −
ˇ
h(x, k) , (109)
ˇ
h(x, ¯x) =
ch(2r)
ˇ
h(x, x) +
ˇ
h(x, J
|
k)
sh(2r)
, (110)
ˇ
h(k,
¯
k) = −
ch(2r)
ˇ
h(k, k)+
ˇ
h(x, J
|
k)
sh(2r)
, (111)
ˇ
h(x,
¯
k) = −
ch(2r)
sh(2r)
ˇ
h(x, k) = −
ˇ
h(k, ¯x) , (112)
where
∆ =
ˇ
h(x, x) +
ˇ
h(k, k) + 2 ch(2r)
ˇ
h(x, J
|
k)
sh
2
(2r)
.
(113)
For ∆ = 0, the Hamiltonian matrix is symmetric
with respect to the two partner modes. This indi-
cates that the partner modes are in fact optimal
modes inside the two-mode subspace on which
Accepted in Quantum 2019-07-10, click title to verify 14
ˆ
H
A
¯
A
acts and they achieve the minimal energy
cost. Why this is the case is further illuminated
by the following analysis of the case where ∆ does
not vanish
To analyze the general case of ∆ 6= 0 it is use-
ful to reverse-engineer the derivation of the pre-
vious section. There, we showed that all partner
modes in the subspace of
ˆ
H
A
¯
A
are connected to
the eigenmodes of
ˆ
H
A
¯
A
through a transformation
T
r
= M
r
N
θ
which is the product of an orthogo-
nal transformation N
θ
= exp (θK) and a squeez-
ing operation M
r
.
If we apply the first orthogonal transformation
N
θ
to the energy eigenmodes we obtain a pair of
modes which still diagonalize the covariance ma-
trix G. This pair of modes we also obtain by
acting with the inverse squeezing transformation
M
−r
on the partner modes, i.e., they are the un-
squeezed modes (y, l, z, m) as defined in (57).
Now we are able to express the matrix rep-
resenting
ˆ
H
A
¯
A
in two different ways. On the
one hand, we obtain it starting from its diago-
nal form h in (72) for the energy eigenmodes,
and on the other hand we can express its entries
using the form
ˇ
h and the expressions (57) for the
unsqueezed modes.
N
−1
θ
|
hN
−1
θ
=
ˇ
h(l, l) −
ˇ
h(l, y)
ˇ
h(l, m) −
ˇ
h(l, z)
−
ˇ
h(y, l)
ˇ
h(y, y) −
ˇ
h(y, m)
ˇ
h(y, z)
ˇ
h(m, l) −
ˇ
h(m, y)
ˇ
h(m, m) −
ˇ
h(m, z)
−
ˇ
h(z, l)
ˇ
h(z, y) −
ˇ
h(z, m)
ˇ
h(z, z)
.
(114)
From this equation the values of the parameters
φ and θ which yield the transformation N from
the eigenmodes of
ˆ
H
A
¯
A
to the unsqueezed modes
of A and
¯
A as well as the spectrum of
ˆ
H
A
¯
A
can
be determined. The latter is given by
+
=
ˇ
h(x, x) +
ˇ
h(k, k) + ∆
2 ch 2r
, (115)
−
=
v
u
u
u
t
4
ˇ
h(x, k)
2
+
ˇ
h(x, x) −
ˇ
h(k, k)
2
4 sh
2
(2r)
+
∆
2
4
,
(116)
with
±
=
1
2
(
2
±
1
) as before. The parameter
values are determined by
tan φ =
2
ˇ
h(x, k)
ˇ
h(x, x) −
ˇ
h(k, k)
, (117)
tan 2θ =
q
4
2
−
− ∆
2
∆
, (118)
or, alternatively, ∆ = 2
−
cos 2θ.
With these parameters, we can calculate the
minimal energy cost ∆E of entanglement extrac-
tion from a pair of given partner modes (x, k¯,x,
¯
k)
using (90). The energy cost lies in the interval
∆E
θ=π/4
≤ ∆E ≤ ∆E
θ=0
, (119)
which is bounded by the minimal energy cost
∆E
min
, achieved by θ =
π
4
+
nπ
2
,
∆E
θ=π/4
= ∆E
min
=
q
2
+
ch
2
2r −
2
−
sh
2
2r
(120)
and the energy cost for a pair of partner modes
obtained with the least favorable choice for the
parameter θ = 0 +
nπ
2
, which is
∆E
θ=0
= 2
+
sh
2
r . (121)
In fact, the last equality implies an upper
bound on the energy cost of entanglement extrac-
tion with arbitrary partner modes of the system.
Due to (100), we have
∆E
θ=0
≤ 2ω
N
sh
2
r =: ∆E
max
. (122)
This means that for any pair of partner modes
in the system the energy cost ∆E (see (70) and
(71)) of swapping their entangled state against
the product of their one-mode restricted ground
states is upper bounded by the minimal cost for a
degenerate Hamiltonian (see (82)) with only the
system’s largest excitation energy ω
N
in its spec-
trum.
Finally, we note that the analysis above allows
us to construct from a given pair of partner modes
(
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
) the pair of modes which are op-
timal for extracting entanglement from the two-
mode subspace of
ˆ
H
A
¯
A
. For this we only need to
correct for the mismatch between N
θ
correspond-
ing to the given modes, and N
θ=π/4
which yields
the optimal pair. So to obtain the ideal modes
we only need to act with the transformation
M
r
exp
(
π
4
− θ)K
M
−r
(123)
on the modes (
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
).
Accepted in Quantum 2019-07-10, click title to verify 15
4 Quadratic fermionic systems and
their partner mode construction
We review the most important definitions of
quadratic fermionic systems and fermionic Gaus-
sian states. More detailed reviews can be found
in [14, 60, 61], but we follow mostly the con-
ventions introduced in [58, 66]. In particular,
defining fermionic Gaussian states using linear
complex structures emphasizes the similarities to
the bosonic case. The key difference between
bosonic and fermionic Gaussian states is that Ω
and G exchange their roles. From a group theo-
retic perspective, this implies that the symplectic
group Sp(2N, R) preserving Ω (governing com-
mutation relations) is now replaced by the orthog-
onal group O(2N) preserving a positive definite
metric G (governing anti commutation relations).
4.1 Classical and quantum theory
As for bosons, we can formally define a classical
phase space V ' R
2N
and its dual V
∗
' R
2N
for a system with N fermionic degrees of free-
dom. In contrast to bosons, V and V
∗
are now
equipped with a positive-definite bilinear form
G
ab
: V
∗
×V
∗
→ R and its inverse G
−1
ab
: V ×V →
R satisfying G
ac
G
−1
cb
= δ
a
b
.
We can introduce an orthonormal basis of lin-
ear observables q
i
, p
i
∈ V
∗
⊂ C
∞
(V ), such that
G(p
i
, p
j
) = 0 and G(q
i
, q
j
) = G(p
i
, p
j
) = δ
ij
. As
in the bosonic case, q
i
and p
i
yield a component
representation of a vector v
a
∈ V as
v
a
≡ (q
1
(v), p
1
(v), . . . , q
N
(v), p
N
(v)) . (124)
With respect to this basis, the bilinear form G
ab
takes the standard form G
ab
≡ 1. We use ξ
a
:
V → V : v
a
7→ ξ
a
(v) = v
a
in the same way
as for bosons. However, the fermionic classical
observables are not smooth functions on V , but
rather the Grassmann algebra generated by ξ
a
,
i.e., a formal series of anti-symmetrized powers
in ξ
a
.
When we quantized the bosonic system, we
used the classical Poisson brackets encoded in
Ω
ab
. For fermions, we implement the canonical
anti-commutation relations
[
ˆ
ξ
a
,
ˆ
ξ
b
]
+
=
ˆ
ξ
a
ˆ
ξ
b
+
ˆ
ξ
b
ˆ
ξ
a
= {ξ
a
, ξ
b
}
+
= G
ab
,
(125)
where {ξ
a
, ξ
b
}
+
= G
ab
represents the classical
fermionic Poisson brackets.
4.2 Fermionic Gaussian states
Fermionic Gaussian states appear under various
names in the literature, ranging from fermionic
vacua to Slater determinant states. There is no
Gaussian wave function representations of them,
but they share many properties with their bosonic
counterparts. They form the set of ground states
of quadratic Hamiltonians and also—in contrast
to bosonic Gaussian states—their higher excited
eigenstates. In the context of fermionic Bogoli-
ubov transformations, they appear as vacua of
annihilation operators satisfying fermionic anti
commutation relations. Finally, their n-point
correlation functions are completely determined
from their 2-point correlation function.
We label a fermionic Gaussian state |Ωi by its
covariance matrix
Ω
ab
= hΩ|
ˆ
ξ
a
ˆ
ξ
b
−
ˆ
ξ
b
ˆ
ξ
a
|Ωi , (126)
which is antisymmetric, rather than symmetric
as in the bosonic case. Note that there is no lin-
ear displacement for physical fermionic Gaussian
states, i.e., we require hΩ|
ˆ
ξ
a
|Ωi = 0. We can use
Ω to construct a linear complex structure
J
a
b
= Ω
ac
G
−1
cb
= −G
ac
Ω
−1
bc
, (127)
that satisfies J
2
= −1 just as for bosons. More-
over, we have the condition
1
2
(δ
a
b
−iJ
a
b
)
ˆ
ξ
b
|Ωi =
0, analogously to (19).
In contrast to bosons, there is no displacement
vector z
a
for fermions. Hence, fermionic n-point
functions are directly defined as
C
a
1
···a
n
n
= hΩ|
ˆ
ξ
a
1
. . .
ˆ
ξ
a
n
|Ωi . (128)
Apart from this, the definition of the two-point
function is unchanged:
C
ab
2
=
1
2
(G
ab
+ iΩ
ab
) . (129)
Wick’s theorem applies in almost the same way
C
a
1
···a
n
n
=
X
(contractions of C
ab
2
) (130)
= σ
(a
1
,...,a
n
)
C
a
1
a
2
2
···C
a
n−1
a
n
2
+ ··· ,
(131)
however, each contraction enters with a posi-
tive or negative sign, depending on the parity
σ
(a
1
,...,a
n
)
∈ {−1, 1} of its index ordering. This
subtlety needs to be taken into account carefully
when applying Wick’s theorem to fermions.
Accepted in Quantum 2019-07-10, click title to verify 16
Example. Looking at the ground state |Ωi of
the fermionic oscillator
ˆ
H =
i
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
= iˆq ˆp. The
covariance matrix Ω
ab
and the metric G
ab
with
respect to the basis
ˆ
ξ
a
≡ (ˆq, ˆp)
|
are given by
Ω
ab
≡
0 1
−1 0
!
and G
ab
≡
1 0
0 1
!
. (132)
Just as in the bosonic case, we can construct the
linear complex structure
J
a
b
= Ω
ac
G
−1
cb
≡
0 1
−1 0
!
. (133)
With this, the fermionic projector becomes
1
2
(δ
a
b
+ iJ
a
b
)
ˆ
ξ
b
≡
1
2
ˆq + iˆp
ˆp − iˆq
!
=
1
√
2
ˆa
−iˆa
!
,
(134)
i.e., the condition
1
2
(δ
a
b
+ iJ
a
b
) |Ωi = 0 is the
same as for bosons. The projector selects the sub-
space spanned by complex linear combinations of
annihilation operators ˆa =
1
√
2
(ˆq + iˆp).
4.3 Quadratic fermionic Hamiltonians
We are considering the most general quadratic
fermionic Hamiltonian given by
ˆ
H =
i
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
, (135)
where h
ab
is antisymmetric and non-degenerate.
Note that
ˆ
H is always bounded from below.
The ground state of
ˆ
H is given by a Gaussian
state |Ωi. As for bosons, we can compute the
ground state covariance matrix from h
ab
:
• Computation of Ω
ab
Similar to (65), we compute
Ω
ab
= K
a
c
|K
−1
|
c
d
G
db
(136)
with K
a
b
= G
ac
h
cb
.
The energy expectation value E = hΩ
0
|
ˆ
H |Ω
0
i
and its variance Σ
E
for an arbitrary Gaussian
state |Ω
0
i can be efficiently computed as
E = −
1
4
h
ab
Ω
0
ab
= −
Tr(hΩ
0
|
)
4
, (137)
Σ
E
=
p
−Tr(hGhG + hΩ
0
hΩ
0
)
4
. (138)
Again, we note that the symplectic form Ω
ab
and the positive definite metric G
ab
switched
their roles with respect to the bosonic formulae
in Section 2.
Example. We consider the harmonic oscillator
of a single fermionic mode
ˆ
H =
i
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
= iˆq ˆp.
The bilinear form h with respect to
ˆ
ξ
a
≡ (ˆq, ˆp) is
h
ab
≡
0 ω
−ω 0
!
, (139)
which gives via (136) rise to
K
a
b
≡
0 ω
−ω 0
!
⇒ Ω
ab
≡
0 1
−1 0
!
, (140)
where we used |K
−1
| = 1/ω, just like in the
bosonic case. The ground state energy is given
by E = hΩ|
ˆ
H |Ωi = −ω/2 and expressing the
Hamiltonian in creation and annihilation opera-
tors gives
ˆ
H =
ω
2
(ˆa
†
ˆa − 1).
4.4 Entanglement of fermionic Gaussian states
Given a fermionic Gaussian state |Ωi and a
system decomposition into two even dimen-
sional orthogonal complements V = A ⊕
B, we can always choose an orthonormal ba-
sis (q
A
1
, p
A
1
, . . . , p
A
N
A
, q
A
N
A
, q
B
1
, p
B
1
, . . . , p
B
N
B
, q
B
N
B
),
such that the covariance matrix takes the stan-
dard form derived in appendix A.2
Accepted in Quantum 2019-07-10, click title to verify 17
Ω ≡
cos
1
··· 0 sin
1
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· cos
N
A
0 ··· sin
N
A
0 ··· 0
−sin
1
··· 0 cos
1
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· −sin
N
A
0 ··· cos
N
A
0 ··· 0
0 ··· 0 0 ··· 0 A
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· 0 0 ··· 0 0 ··· A
2
, (141)
where A
2
, cos
i
and sin
i
are given by
A
2
=
0 1
−1 0
!
, (142)
cos
i
=
0 cos 2r
i
−cos 2r
i
0
!
, (143)
sin
i
=
0 sin 2r
i
sin 2r
i
0
!
. (144)
This decomposition is analogous to the bosonic
one in (39), where the mode (q
A
i
, p
A
i
) is entangled
with the mode (q
B
i
, p
B
i
). The parameters r
i
can
always be chosen to lie in the interval [0, π/4].
Again, the amount of entanglement is encoded in
the parameter r
i
, but now given by [72]
S
A
(|Ωi) =
N
A
X
i=1
s
f
(cos 2r
i
) (145)
with s
f
(x) = −
1+x
2
log
1+x
2
−
1−x
2
log
1−x
2
.
As for bosons, there exists a compact formula to
express the entanglement entropy in terms of the
linear complex structure J, namely
S
A
(|G, zi) = −Tr
h
1
A
+i[J]
A
2
log
1
A
+i[J]
A
2
i
,
(146)
which closely resembles (43), but where the mod-
ulus is replaced by an overall minus sign.
4.5 Fermionic partner mode construction
The standard form of the fermionic covariance
matrix introduced above implies that also for
a system of fermionic modes in an overall pure
state, each single mode has a unique partner with
which it is entangled.
In fact the partner mode for bosonic modes can
be almost directly carried over to the fermionic
case. Only the hypergeometric functions need to
be replaced by their trigonometric analogues. If
(x
a
, k
a
) define a mode in standard form
G
ab
x
a
x
b
= G
ab
k
a
k
b
= 1 , (147)
G
ab
x
a
k
b
= 0 , (148)
Ω
ab
x
a
k
b
= cos 2r , (149)
then its partner mode is given by
¯x
a
= cot(2r)x
a
+
1
sin 2r
(J
|
)
c
a
k
c
, (150)
¯
k
a
= −cot(2r)k
a
+
1
sin 2r
(J
|
)
c
a
x
c
. (151)
This means that the partner mode is in standard
form itself
G
ab
¯x
a
¯x
b
= G
ab
¯
k
a
¯
k
b
= 1 , (152)
G
ab
¯x
a
¯
k
b
= 0 , (153)
Ω
ab
¯x
a
¯
k
k
= cos 2r , (154)
and the correlations with the original mode are
as required by the standard form
G
ab
x
a
¯x
b
= G
ab
k
a
¯
k
b
= G
ab
x
a
¯
k
b
= 0 (155)
Ω
ab
x
a
¯
k
b
= Ω
ab
k
a
¯x
b
= sin 2r . (156)
Just as in the bosonic case, also for fermionic
modes the partner of the partner mode is the
original mode itself. Also, we observe that any
other mode (x
0
, k
0
) which anti-commutes with
(x, k), has an anti-commutatator with the part-
ner mode proportional to the covariance with the
first mode, i.e.,
G
ab
x
0
a
¯x
b
= cot 2r G
ab
x
0
a
x
b
−
1
sin 2r
Ω
ab
x
0
a
k
b
.
(157)
Accepted in Quantum 2019-07-10, click title to verify 18
This means that any mode which anticommutes
with both partner modes is not correlated with
them. In particular, a basis of modes which con-
tains the two partner modes brings the covariance
matrix of the state into block diagonal form
Ω ≡
˜
Ω 0
0 Ω
R
!
(158)
where
˜
Ω =
0 cos 2r 0 sin 2r
−cos 2r 0 sin 2r
0 −sin 2r 0 cos 2r
−sin 2r 0 −cos 2r
(159)
takes the standard form of (141).
Also fermionic partner modes can be un-
squeezed to find a pair of modes (y, l) and (z, m)
which are in a pure product state and from which
(x, k) and (¯x,
¯
k) are obtained by squeezing. These
modes are given by:
y
a
= cos(r)x
a
− sin(r)¯x
a
=
x
a
− (J
|
)
a
b
k
b
2 cos r
,
(160)
l
a
= cos(r)k
a
+ sin(r)
¯
k
a
=
k
a
+ (J
|
)
a
b
x
b
2 cos r
,
(161)
z
a
= sin(r)x
a
+ cos(r)¯x
a
=
x
a
+ (J
|
)
a
b
k
b
2 sin r
,
(162)
m
a
= −sin(r)k
a
+ cos(r)
¯
k
a
=
−k
a
+ (J
|
)
a
b
x
b
2 sin r
.
(163)
5 Entanglement extraction from
quadratic fermionic systems
In this section, we analyze the minimal energy
cost of pure state entanglement extraction from
the ground state of a system of fermionic modes
with a quadratic Hamiltonian. Our approach is
motivated by the same framework as discussed for
bosonic modes in Section 3. Again, after review-
ing the extraction procedure in section 5.1, our
proof follows exactly the same three steps, split
over the sections 5.2, 5.3 and 5.4.
5.1 Pure state entanglement extraction from
bosonic modes
We assume that the source system consists of
fermionic modes which are in the ground state
|Ψi
S
of an arbitrary quadratic Hamiltonian
ˆ
H,
which is Gaussian. Within the source system,
we select two anti-commuting source modes from
which we want to extract entanglement. For this
we assume that it is possible to swap the state of
the two source modes onto a pair of target modes
outside of the system. We can always use Gaus-
sian evolution, i.e., quadratic coupling terms, to
swap the entangled Gaussian state in the source
system with the unentangled Gaussian state in
the target system. The target modes can be pre-
pared in an arbitrary product state. Hence the
joint initial state of target modes and source sys-
tem is of the form
ρ
i
= σ
1
⊗ τ
2
⊗ |ΨihΨ|
S
. (164)
Just as in the bosonic case, also for fermionic
modes the partner mode structure of entangle-
ment of Gaussian states implies that the only way
to extract a pure entangled state in this way is to
chose the two source modes to be partner modes
which we denote by A and
¯
A. This is also the op-
timal choice in the sense that any entanglement
measure of one mode with with a second non-
partner mode is never larger than of the mode
with its partner.
The ground state of the source system is a
product state |Ψi = |ψi
A
¯
A
⊗ |φi
R
between the
partner modes and the remainder R of the sys-
tem. After the entanglement extraction, i.e., af-
ter the states of the partner modes and the target
modes have been swapped, the source system is
in the state
ρ
0
S
= σ
A
⊗ τ
¯
A
⊗ |φihφ|
R
(165)
where now the partner modes are in the prod-
uct state given by the initial state of the target
modes.
5.2 Energy cost of entanglement extraction
By the minimal energy cost of entanglement ex-
traction we are referring to the minimum value
by which the expectation value of
ˆ
H is increased,
i.e., by how much energy on average is injected
into the system when the modes are swapped.
Accepted in Quantum 2019-07-10, click title to verify 19
Since we have used the same notation, the discus-
sion and formulae from Section 3.2 apply equally
to the fermionic case, as considered here. In par-
ticular, the energy cost for the extraction of a
fermionic pair of partner modes is given by
∆E = Tr
σ
A
⊗ τ
¯
A
ˆ
H
A
¯
A
− |ψihψ|
A
¯
A
ˆ
H
A
¯
A
.
(166)
It results from the restriction
ˆ
H
A
¯
A
of the full
Hamiltonian onto the space spanned by the two
partner modes. Thus, again the problem of find-
ing the minimal energy cost reduces to a two-
mode problem.
Also as before, we find that the final energy
expectation value of the system is
Tr
σ
A
⊗ τ
¯
A
ˆ
H
A
¯
A
= Tr
σ
A
ˆ
H
A
+ Tr
σ
¯
A
ˆ
H
¯
A
(167)
which implies that in order to minimize the en-
ergy cost, the target modes need to be initialized
in the ground states of the single-mode restric-
tions
ˆ
H
A
and
ˆ
H
¯
A
of
ˆ
H onto the individual part-
ner modes. Also for fermionic modes these states
are Gaussian states, as discussed in the previous
section. Thus, we can perform our entire analysis
within the framework of Gaussian states.
5.3 Energy cost for two fermionic modes
This section presents the fermionic analogue to
3.3. We analyze entanglement extraction from a
source system that consists of only two fermionic
modes and has a quadratic Hamiltonian.
Since the Hamiltonian is quadratic, we can
choose a basis
ˆ
ξ
a
≡ (ˆq
1
, ˆp
1
, ˆq
2
, ˆp
2
) of energy eigen-
modes such that the quadratic part of the Hamil-
tonian h
ab
and the covariance matrix Ω
ab
of its
ground state are both represented by block diag-
onal matrices h and Ω given by
h
ab
1,2
≡ h =
0
1
0 0
−
1
0 0 0
0 0 0
2
0 0 −
2
0
, (168)
Ω
ab
1,2
≡ Ω =
0 1 0 0
−1 0 0 0
0 0 0 1
0 0 −1 0
. (169)
The energy eigenmodes are not entangled and,
therefore, cannot be used for entanglement ex-
traction. Instead, we need to find a pair of
partner modes (
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
) with respect to
which Ω
ab
is represented by the matrix corre-
sponding to the standard form (159)
Ω
ab
A,
¯
A
≡
˜
Ω (170)
for some fixed r ∈ [0, π/4], which depends on
the amount of entanglement we want to extract.
Among all choices of partner modes A and
¯
A,
we want to identify those that minimize the en-
ergy cost associated to the entanglement extrac-
tion. Consequently, we will search for the orthog-
onal transformation which maps the energy eigen-
modes 1 and 2 to those optimal partner modes A
and
¯
A for entanglement extraction.
The two-mode squeezing transformation
M
r
=
cos r 0 sin r 0
0 cos r 0 −sin r
−sin r 0 cos r 0
0 sin r 0 cos r
(171)
brings Ω into the form from (159) via
˜
Ω = M
r
ΩM
|
r
, (172)
i.e., it maps the eigenmodes to a pair of partner
modes with squeezing parameter r. Under the
orthogonal transformation M
r
, the Hamiltonian
matrix h transforms under the action of the in-
verse transform M
−1
r
= M
−r
, i.e.,
˜
h = M
|
−r
hM
−r
. (173)
However, M
r
is not the only transformation do-
ing this: We can compose the squeezing M
r
with any orthogonal transformation N which
leaves the covariance matrix invariant, i.e., Ω =
NΩN
|
, to obtain the most general transforma-
tion T
r
= M
r
N that transforms the modes 1 and
2 into A and
¯
A for fixed r. All of these choices
share the same amount of entanglement across A
and
¯
A because the covariance matrix Ω takes the
same form
˜
Ω = T
r
ΩT
|
r
with respect to them.
However, the Hamiltonian matrix h is in general
not left invariant by N, i.e., h 6= (N
−1
)
|
hN
−1
and thus the energy cost can be different.
Accepted in Quantum 2019-07-10, click title to verify 20
5.3.1 Degenerate two-mode Hamiltonian (
1
=
2
)
We first consider the case where :=
1
=
2
.
This case is the simplest to solve because the ma-
trix representations h and Ω are proportional to
each other in the eigenmode basis and, thus, in
all bases. Indeed, any orthogonal transformation
N that leaves Ω invariant, i.e., NΩM
|
= Ω, also
leaves h invariant, i.e., h = (N
−1
)
|
hN
−1
. There-
fore, for any pair of partner modes A and
¯
A where
the covariance matrix takes the standard form
˜
Ω
for fixed r, we find
˜
h =
0 cos 2r 0 − sin 2r
− cos 2r 0 − sin 2r 0
0 sin 2r 0 cos 2r
sin 2r 0 − cos 2r 0
.
(174)
As discussed before, the entanglement extrac-
tion swaps the state of the chosen partner modes
against a product state σ
¯
A
⊗τ
¯
A
. This brings the
covariance matrix into block diagonal form with
respect to the partner modes:
˜
Ω
0
=
˜
Ω
A
0
0
˜
Ω
¯
A
. (175)
In terms of this new covariance matrix, the energy
cost (166) of the entanglement extraction is
∆E = −
1
4
Tr
˜
h
˜
Ω
|
+
1
4
Tr (h Ω
|
) (176)
= −
1
4
Tr
˜
h
A
˜
Ω
A
+
˜
h
¯
A
˜
Ω
¯
A
, (177)
where
˜
h
A
=
˜
h
¯
A
= cos 2r A
2
are the Hamilto-
nian matrices of the restriction of
ˆ
H onto the sin-
gle partner modes respectively, i.e., the diagonal
blocks of
˜
h in (174). To minimize ∆E, the target
modes need to be prepared in the ground state of
˜
h
A
and
˜
h
¯
A
, i.e.,
˜
Ω
0
A
=
˜
Ω
0
¯
A
= A
2
.
With these initial states, the minimal en-
ergy cost for the extraction of a pure two-mode
squeezed state with entanglement entropy ∆S =
S
f
(cos 2r) is
∆E = 2 sin
2
r . (178)
In addition to this elevation of the energy expec-
tation value, the final state of the source system
has an energy variance of
Σ
E
=
r
−Tr
˜
h
˜
G
˜
h
˜
G +
˜
h
˜
Ω
0
˜
h
˜
Ω
0
4
(179)
=
√
2
sin 2r . (180)
5.3.2 General two-mode Hamiltonian (
1
≤
2
)
The case of a general Hamiltonian, with
1
≤
2
,
is slightly more involved than the degenerate case.
Interestingly, there exists a critical value of ex-
tracted entanglement ∆S below which the min-
imal energy cost is independent of the choice of
partner modes. Above the critical amount, it is
optimal to chose partner modes obtained from
squeezing the eigenmodes of
ˆ
H directly.
In the non-degenerate case the orthogonal
transformations which map the energy eigen-
modes to any pair of partner modes is of the gen-
eral form T
r
= M
r
N. This means that before
the squeezing transformation M
r
we can apply a
transformation N which can change h but leaves
Ω invariant. To find all possible forms of N we
write it as a matrix exponential N = exp (θK)
of the generator K subject to the constraints
KG + GK
|
= 0 (leaving G invariant, i.e., be-
ing orthogonal) and KΩ + ΩK
|
= 0 (leaving Ω
invariant, i.e., being symplectic).
2
From this, we
find the most general form
K =
0 0 cos φ −sin φ
0 0 sin φ cos φ
−cos φ −sin φ 0 0
sin φ −cos φ 0 0
,
(181)
giving rise to N = exp(θK).
Consequently, T
r
= M
r
N provides the most
general transformation which brings the covari-
ance matrix
˜
Ω = T
r
ΩT
|
r
in the form of (159).
Such a transformation brings the Hamiltonian
matrix into the general form
˜
h = (T
−1
r
)
|
hT
−1
r
=
h
A
h
X
−h
|
X
h
¯
A
, (182)
with blocks that read, using
±
=
1
2
(
2
±
1
),
2
These are indeed the same conditions as for bosons.
Mathematically speaking, the intersection of the orthogo-
Accepted in Quantum 2019-07-10, click title to verify 21
h
A
=
0
+
cos 2r −
−
cos 2θ
−
+
cos 2r +
−
cos 2θ 0
!
, (183)
h
¯
A
=
0
+
cos 2r +
−
cos 2θ
−
+
cos 2r −
−
cos 2θ 0
!
, (184)
h
X
=
−
sin 2θ sin φ
+
sin 2r +
−
cos φ sin 2θ
+
sin 2r −
−
cos φ sin 2θ
−
sin 2θ sin φ
!
. (185)
The excitation energies of the restricted one-
mode Hamiltonian
ˆ
H
A
and
ˆ
H
¯
A
are thus
A
= |
+
cos 2r −
−
cos 2θ|, (186)
¯
A
= |
+
cos 2r +
−
cos 2θ|. (187)
As discussed above, the entanglement extraction
brings the partner modes A and
¯
A into a pure
Gaussian product state. Since A and
¯
A are
fermionic modes, there are only four different
product states. Their covariance matrices are
A
(k,l)
=
(−1)
k
A
2
0
0 (−1)
l
A
2
, (188)
with k, l ∈ {0, 1}. Which of these states mini-
mizes the energy cost, i.e., is the product of the
ground states of h
A
and h
¯
A
, depends on spectrum
of the Hamiltonian and the squeezing parameter
r. This is because the energy cost arising from
the different A
(k,l)
is
∆E =
+
−
1
4
Tr
˜
h A
(k,l)
=
+
(1 − cos 2r), if (k, l) = (0, 0)
+
−
−
cos 2θ, if (k, l) = (1, 0)
+
+
−
cos 2θ, if (k, l) = (0, 1)
+
(1 + cos 2r), if (k, l) = (1, 1)
. (189)
The fact that for two of the states the energy
cost is independent of θ versus it being indepen-
dent of r for the other two states, has an interest-
ing consequence: It means that for low values of
r, i.e., for low amounts of entanglement, the min-
imal energy cost is independent of θ and given by
∆E
min
= 2
+
sin
2
r which is achieved by the state
A
(0,0)
. However, as soon as cos 2r ≤
−
+
is stisfied,
it is more beneficial to bring the modes into the
nal algebra so(2N ) and the symplectic algebra sp(2N ) is
given by the unitary algebra u(N ).
final state A
(1,0)
which results in an energy cost
∆E =
+
−
−
cos 2θ. In this case, the energy cost
is minimized by the parameter value θ = nπ with
n ∈ Z, i.e., if the partner modes are obtained by
squeezing directly the energy eigenmodes.
In summary, the minimal energy cost for the
extraction of a pair of partner modes sharing
entanglement entropy ∆S = S
f
(cos 2r) from a
fermionic quadratic Hamiltonian with excitation
energies
1
≤
2
, shown in Figure 4, is
∆E
min
=
(
(
1
+
2
) sin
2
r, if cos 2r ≥
2
−
1
2
+
1
1
, if cos 2r ≤
2
−
1
2
+
1
.
(190)
In particular, the minimal energy cost is bounded
from above by the lower excitation energy of the
Hamiltonian
1
. This is because above the criti-
cal squeezing parameter with cos 2r ≤
−
/
+
, the
product state A
(1,0)
in which the partner modes
are left after the extraction is exactly the first
excited state of the Hamiltonian
ˆ
H. In fact, the
covariance matrix A
(0,1)
is left invariant by the
squeezing operation M
r
A
(1,0)
M
|
r
, i.e., the first
excited state of
ˆ
H has the same covariance matrix
with respect to the energy eigenmodes and with
respect to the partner modes A and
¯
A which are
obtained by squeezing the energy eigenmodes.
As a consequence, in the regime cos 2r ≤
−
/
+
, the energy variance Σ
E
of the final sys-
tem state vanishes because the system is in an
energy eigenstate. For lower amounts of entan-
glement, where this is not the case, the variance
of the state which minimizes the energy cost de-
pends on the squeezing parameter. It is plotted
in Figure 5 and reads
Σ
E
=
+
√
2
sin 2r, if cos 2r >
2
−
1
2
+
1
0, if cos 2r ≤
2
−
1
2
+
1
,
(191)
Accepted in Quantum 2019-07-10, click title to verify 22
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 4: Minimal energy cost ∆E
min
of extracting en-
tanglement entropy ∆S from a fermionic, quadratic two-
mode Hamiltonian with excitation energies
1
and
2
.
Note that for cos 2r >
2
−
1
2
+
1
, it is possible to leave
the source system in an energy eigenstate such
that Σ
E
= 0. The energy cost for this is
1
.
Another relevant figure of merit is to look at
the ratio between a given amount of extracted en-
tanglement and its minimal energy cost ∆S/∆E.
Since ∆S = S
f
(cos 2r) ∼ (1−2 log r) r
2
as r → 0,
the ratio diverges as
lim
r→0
∆S
∆E
min
= ∞ (192)
for small amounts of extracted entanglement, just
as we found for bosons in (99). As the extracted
entanglement increases the ratio is strictly de-
creasing and reaches its minimum log(2)/
1
for
r = π/4. Similar to the bosonic case, we thus
find that the price per entanglement bit increases
as we are trying extract a larger total amount.
5.4 Choosing optimal partner modes
The analysis of the previous sections explain how
the increase of energy expectation value of the
source system can be minimized when extracting
a pair of partner modes sharing a given amount of
entanglement: To keep the energy cost as low as
possible the modes need to be constructed from
the two lowest energy eigenmodes of the Hamil-
tonian. Specifically, the partner modes should
be obtained by directly squeezing the two energy
eigenmodes, such that always the lowest possi-
ble energy cost according to (190) is achieved.
Again, as discussed in Appendix B, no other
choice of partner modes yields a lower energy
cost because the energy spectrum of the restricted
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
Figure 5: Energy variance Σ
E
of the state minimizing
the energy cost of extracting partner modes with entan-
glement entropy ∆S from a fermionic, quadratic two-
mode Hamiltonian with excitation energies
1
and
2
.
Hamiltonian is bounded by the spectrum of the
full Hamiltonian.
For a general pair of partner modes A and
¯
A
the minimal energy cost is determined by the
spectrum of the restricted two-mode Hamiltonian
ˆ
H
A
¯
A
, which enters the formula (190). In con-
trast to the fermionic case, an arbitrarily cho-
sen pair of partner modes will in fact achieve
this minimal energy price if the amount of ex-
tracted entanglement is below the critical value,
i.e., if cos 2r ≥
−
+
. If the squeezing is above this
value, according to the analysis of the previous
section only the partner modes obtained by di-
rectly squeezing the energy eigenmodes of
ˆ
H
A
¯
A
do achieve the possible minimal energy cost of
∆E
min
=
1
.
To perform this analysis for a given choice
of partner modes, it is necessary to express the
Hamiltonian matrix of
ˆ
H
A
¯
A
with respect to the
partner mode operators
ˆ
Q
A
,
ˆ
P
A
,
ˆ
Q
¯
A
,
ˆ
P
¯
A
. As
before, we denote these mode operators as
ˆ
Q
A
= x
a
ˆ
ξ
a
,
ˆ
P
A
= k
a
ˆ
ξ
a
, (193)
ˆ
Q
¯
A
= ¯x
a
ˆ
ξ
a
,
ˆ
P
¯
A
=
¯
k
a
ˆ
ξ
a
. (194)
In this basis, the Hamiltonian matrix represent-
ing the restriction
ˆ
H
A
¯
A
of a general quadratic
Hamiltonian
ˆ
H =
i
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
is given by
0
ˇ
h(x, k)
ˇ
h(x, ¯x)
ˇ
h(x,
¯
k)
ˇ
h(k, x) 0
ˇ
h(k, ¯x)
ˇ
h(k,
¯
k)
ˇ
h(¯x, x)
ˇ
h(¯x, k) 0
ˇ
h(¯x,
¯
k)
ˇ
h(
¯
k, x)
ˇ
h(
¯
k, k)
ˇ
h(
¯
k, ¯x) 0
(195)
Accepted in Quantum 2019-07-10, click title to verify 23
where we now define
ˇ
h : V
∗
× V
∗
→ R by
ˇ
h(v, w) = G
ab
G
cd
v
a
h
bc
w
d
, (196)
which for fermionic modes yields a bi-linear anti-
symmetric form:
ˇ
h(v, w) = −
ˇ
h(w, v) . (197)
Using the fact that K as defined in (136) com-
mutes with J, it can be shown that
ˇ
h(J
|
v, J
|
w) =
ˇ
h(v, w). (198)
Which allows to calculate the terms invovling the
partner mode (¯x,
¯
k) as
ˇ
h(x, ¯x) =
ˇ
h(k,
¯
k) =
ˇ
h(x, J
|
k)
sin 2r
, (199)
ˇ
h(x,
¯
k) = −cot 2r
ˇ
h(x, k) +
ˇ
h(x, J
|
x)
sin 2r
, (200)
ˇ
h(k, ¯x) = −cot 2r
ˇ
h(x, k) +
ˇ
h(k, J
|
k)
sin 2r
, (201)
ˇ
h(¯x,
¯
k) =
cot 2r
sin 2r
ˇ
h(x, J
|
x) +
ˇ
h(k, J
|
k)
− (1 + 2 cot
2
(2r))
ˇ
h(x, k) . (202)
Then, by equating the obtained matrix with
(182), we obtain the spectrum of
ˆ
H
A
¯
A
with
±
=
1
2
(
2
±
1
) as
2
+
=
ˇ
h(¯x,
¯
k) +
ˇ
h(x, k)
2
+
ˇ
h(x,
¯
k) +
ˇ
h(k, ¯x)
2
4
,
(203)
2
−
=
ˇ
h(¯x,
¯
k) −
ˇ
h(x, k)
2
+
ˇ
h(x,
¯
k) −
ˇ
h(k, ¯x)
2
4
+
ˇ
h(x, ¯x)
2
4
. (204)
The two parameters φ and θ in the transforma-
tion T
r
(mapping the eigenmodes of
ˆ
H
A
¯
A
to the
partner modes A and
¯
A) are determined by
tan φ =
2
ˇ
h(x, ¯x)
ˇ
h(x,
¯
k) −
ˇ
h(k, ¯x)
, (205)
cos 2θ =
ˇ
h(¯x,
¯
k) −
ˇ
h(x, k)
2
−
. (206)
With this information on the spectrum and
the parameters it is possible to check if a given
pair of partner modes minimizes the energy cost:
If we find that the extracted amount of entan-
glement is low enough such that cos 2r ≥
−
+
then the pair achieves the minimal energy cost of
∆E
min
= 2
+
sin
2
r independent of the value of θ.
However, beyond this regime the minimal energy
cost depends on the value of θ and according to
(189) can be lowered to
∆E
θ
=
+
−
−
|cos 2θ|, (207)
i.e., only if θ ∈ {0, π/2, π, . . . } the minimal en-
ergy cost of ∆E
min
=
+
−
−
=
1
is achieved by
the chosen partner modes A and
¯
A.
Finally, also for fermionic modes we can give
an upper bound on the energy cost of entangle-
ment extraction from arbitrary partner modes of
the system. Combining (190) and (100) (which as
shown in Appendix B applies to fermionic Hamil-
tonians in the same form), we find that the en-
ergy cost to replace the state of an arbitrary pair
of partner modes by the product states of their
one-mode restricted ground states
∆E ≤ 2
+
sin
2
r ≤ 2 ω
N
sin
2
r =: ∆E
max
(208)
is again upper bounded by the energy cost for a
degenerate Hamiltonian with only the system’s
largest excitation energy ω
N
in its spectrum.
6 Applications
In this section, we apply the previously derived
minimal energy cost to concrete physical models.
The main goal is to present a proof of concept
that our general results are easily related to con-
crete scenarios of extracting entanglement from
modes that are accessible in a physical model. In
particular, we present examples for both bosonic
and fermionic systems, the latter being also re-
lated to spin systems via Jordan-Wigner trans-
formation.
6.1 Hamiltonian of dilute Boson gas
We consider the Hamiltonian of the form
ˆ
H =
X
k6=0
ω
k
a
†
k
a
k
+ γ
k
a
†
k
a
†
−k
+ a
k
a
−k
, (209)
with ω
k
= ω
−k
and γ
k
= γ
−k
, which is well known
to describe a weakly interacting dilute Boson gas
(see, e.g., [73]). The sum runs over all modes with
non-zero momentum, which have very low occu-
pation numbers, but it excludes the zero momen-
tum mode, which is macroscopically occupied. In
Accepted in Quantum 2019-07-10, click title to verify 24
this way, the Hamiltonian describes the interac-
tion of excitations in the higher modes with the
condensate of particles in the zero-mode.
Typically, the squeezing term in the Hamil-
tonian is much smaller than the number opera-
tor term, i.e., γ
k
ω
k
. However, as long as
γ
k
< ω
k
/2 the Hamiltonian is bounded from be-
low, as we shall see below.
Inserting a
k
=
1
√
2
(ˆq
k
+ iˆp
k
), we can rewrite
the Hamiltonian as
ˆ
H =
X
k>0
ˆ
H
k
−
X
k6=0
ω
k
2
(210)
with
ˆ
H
k
=
ω
k
2
ˆq
2
k
+ ˆp
2
k
+ ˆq
2
−k
+ ˆp
2
−k
(211)
+ 2γ
k
(ˆq
k
ˆq
−k
− ˆp
k
ˆp
−k
) . (212)
It is evident, that modes with opposite momen-
tum, i.e., pairs of the form (ˆq
k
, ˆp
k
, ˆq
−k
, ˆp
−k
) are
partner modes in the ground state of
ˆ
H because
the blocks
ˆ
H
k
couple them pairwise. Since we as-
sumed ω
k
= ω
−k
each block
ˆ
H
k
is a degenerate
two-mode Hamiltonian. In fact, the Hamiltonian
matrix corresponding to
ˆ
H
k
is
h
k
=
ω
k
0 2γ
k
0
0 ω
k
0 −2γ
k
2γ
k
0 ω
k
0
0 −2γ
k
0 ω
k
, (213)
which matches (78), with the squeezing parame-
ter r and the excitation energy of
ˆ
H
k
being
ch 2r = ω
k
, sh 2r = −2γ
k
,
⇔ =
q
ω
2
k
− 4γ
2
k
, ch 2r =
1
q
1 − 4γ
2
k
/ω
2
k
.
(214)
This means that in order to extract pure state
entanglement from the ground state of the Hamil-
tonian
ˆ
H, one needs to select two modes with
opposite momentum k and −k. Following (42),
these modes share an entanglement entropy of
∆S = s
b
1 − 4γ
2
k
/ω
2
k
−
1
2
. (215)
The minimal energy cost of entanglement extrac-
tion from these modes is, according to (82),
∆E = (ch 2r − 1) = ω
k
1 −
q
1 − 4γ
2
k
/ω
2
k
.
(216)
0.0 0.1 0.2 0.3 0.4 0.5
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Figure 6: Ratio of energy cost to extracted entanglement
from the ground state of a dilute weakly interacting Bo-
son gas (Hamiltonian of (209)) as a function of γ
k
/ω
k
.
It is interesting to see that while the entropy
between the partner modes diverges as γ
k
→
ω
k
/2, the energy cost is upper bounded and ap-
proaches ω
k
. At first this may seem in contradic-
tion to the results of Section 3, but it is due to
the excitation energy of
ˆ
H
k
vanishing, → 0, in
this limit. In particular, this leads to an inter-
esting dependency of the ratio E
min
/∆S between
energy cost and extracted entanglement on the
ratio γ
k
/ω
k
of the Hamiltoian parameters, seen in
Figure 6. The energy that needs to be invested
per bit of extracted entanglement falls off dras-
tically both for very small amounts of entangle-
ment, γ
k
→ 0, and for extremely large amounts
of entanglement as γ
k
→ ω
k
/2. In between the
highest amount of energy per extracted entan-
glement entropy is required at γ
k
/ω
k
≈ 0.389368,
where the entanglement entropy is ∆S ≈ 1.00679
and the energy cost E
min
≈ 0.372648 ω
k
.
An interesting feature of the entanglement
structure of this example system is that it ex-
hibits partner modes which may be realistically
accessible in experiment. In other systems it
seems much more challenging to access both
modes in a pair of partner modes equally: For
example, in the coupled harmonic chain gener-
ally (at least) one of the partner modes are delo-
calized over the complete system. The Hamil-
tonian (209), however, couples and entangles
modes of counter-propagating momentum sym-
metrically such that both partner modes should
be equally accessible in experiment.
Accepted in Quantum 2019-07-10, click title to verify 25
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
Figure 7: Dispersion relation ε
k
of the XY model for
different values of h/J and γ.
6.2 XY spin model
The XY model presents a prime example of a
spin model that can be mapped to a fermionic
quadratic Hamiltonian via the Jordan-Wigner
transformation [74]. We consider the Hamilto-
nian with an external field h given by [75]
ˆ
H
XY
= −
J
2
N
X
j=1
h
(1 + γ)
ˆ
S
X
j
ˆ
S
X
j+1
+(1 − γ)
ˆ
S
Y
j
ˆ
S
Y
j+1
i
− h
ˆ
S
Z
j
,
(217)
where
ˆ
S
X
j
,
ˆ
S
Y
j
and
ˆ
S
Z
j
represent the spin-
1
2
oper-
ators on site j. Ignoring boundary terms that are
negligible in the thermodynamic limit, the XY
model can be mapped to the quadratic fermionic
Hamiltonian given by [76, 77]
ˆ
H
XY
=
X
k∈K
ε
κ
ˆη
†
κ
ˆη
κ
−
1
2
with
ε
κ
=
q
h
2
+ 2hJ cos(κ)+ J
2
+ (γ
2
− 1)J
2
sin(κ)
2
,
(218)
where we defined K = {2πk/N|0 ≤ k < N}.
Figure 7 shows the dispersion relation
κ
as a
function of κ for various parameter choices. The
fermionic annihilation operators ˆη
κ
diagonalize
the Hamiltonian and are related to on-site an-
nihilation operators
ˆ
f
j
=
1
√
N
X
κ∈K
e
−iκ
(u
κ
η
κ
− v
κ
η
†
−κ
) (219)
on site j, where the Bogoliubov coefficients u
κ
and v
κ
are given by
u
κ
=
ε
κ
+ a
κ
p
2ε
κ
(ε
κ
+ a
κ
)
, v
κ
=
ib
κ
p
2ε
κ
(ε
κ
+ a
κ
)
,
a
κ
= −J cos(κ) − h , and b
κ
= γ J sin(κ) .
(220)
Let us emphasize that the Jordan-Wigner trans-
formation relating the spin Hamiltonian (217)
with the fermion Hamiltonian (218) is non-local.
However, it preserve bi-partite entanglement [15]
between localized regions, i.e., the entanglement
entropy associated to a single fermionic site is
equal to the respective entanglement entropy of
the same site in the spin Hamiltonian (217). (The
relationship for general non-local subsystems is
more subtle.) Also, we note that the Hamilto-
nian in (218) is a paradigmatic model in its own
right: Choosing γ = 1, we obtain the transverse
field Ising model [78].
To find the minimal excitation energy, i.e., the
lowest eigenvalue of
ˆ
H
XY
, we need to find the
minimum of the dispersion relation ε
κ
. We find
that the condition cos κ =
h
J(γ
2
−1)
ensures that
ε
2
κ
is minimal, which yields the lowest eigenvalue
min
= min
κ
ε
κ
=
s
γ
2
J
2
+
h
2
γ
2
− 1
. (221)
The model becomes gapless for γ = 0 and for
h = J
p
1 − γ
2
. If the number of sites N is large
enough, there are sufficiently many excitation en-
ergies close to the minimum that we can consider
the spectrum as flat. This means, that by extract-
ing a pair of partner modes from the subspace of
these lowest energy modes, we can achieve the
minimal energy cost
∆E
min
= 2
min
sin
2
r , (222)
with the partner modes’ squeezing parameter r.
The ground state |Ωi of this model is encoded
by the complex structure [68]
Accepted in Quantum 2019-07-10, click title to verify 26
J ≡ J = −
i
N
X
κ∈K
|u
κ
|
2
e
iκ(j−l)
− |v
κ
|
2
e
iκ(l−j)
u
∗
κ
v
∗
κ
e
iκ(l−j)
− e
iκ(j−l)
u
κ
v
κ
e
iκ(l−j)
− e
iκ(j−l)
−|u
κ
|
2
e
iκ(l−j)
+ |v
κ
|
2
e
iκ(j−l)
, (223)
here expressed with respect to the local basis
ˆ
ξ
a
≡ (
ˆ
f
†
1
, . . . ,
ˆ
f
†
N
,
ˆ
f
1
, . . . ,
ˆ
f
N
). We can change to
a hermitian basis defined by ˆq
i
=
1
√
2
(
ˆ
f
†
i
+
ˆ
f
i
) and
ˆp
i
=
i
√
2
(
ˆ
f
†
i
−
ˆ
f
i
), such that G ≡ 1. In this basis,
the only non-vanishing entries of the covariance
matrix are given by
Ω(q
j
, p
l
) =
1
L
X
κ∈K
h
(|v
κ
|
2
− |u
κ
|
2
) cos κ(j − l)
+ 2 Im(u
k
v
k
) sin κ(j − l)
i
.
(224)
Similarly, we find the Hamiltonian matrix to be
ˇ
h(q
j
, p
l
) = hδ
jl
+
J + γ
2
δ
j+1,l
+
1 − γ
2
δ
j−1,l
,
(225)
where we have periodicity N + 1 ≡ 1.
We can now explore how the energy cost of ran-
domly chosen partner modes relate to the min-
imal energy cost ∆E
min
and the upper bound
∆E
max
established above. To this end, we choose
a Haar random mode (x
a
, k
a
) spanning a subsys-
tem A ⊂ V and then apply the following steps:
1. Compute entanglement entropy S
A
(|Ωi).
2. Compute partner mode (¯x,
¯
k).
3. Compute the restriction
ˆ
H
A
¯
A
.
4. Compute the energy cost ∆E for replacing
the partner mode state by the ground state
|Ω
0
A
i ⊗ |Ω
0
¯
A
i of
ˆ
H
A
+
ˆ
H
¯
A
.
Selecting a random mode with respect to the
Haar measure of the orthogonal group can be ac-
complished by generating a random 2N-by-2N
matrix with Gaussian distributed entries and or-
thonormalize its column vectors with respect to
the inner product induced by G
ab
. From here, we
can select the first two column vectors. This is
equivalent to directly generating two vectors and
orthonormalizing them.
As seen in Figure 8, the Haar random sample is
accurately bounded from below by our predicted
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 8: Copmarison of lower and upper bounds for the
energy cost of entanglement extraction with a sample
of 10, 000 Haar randomly generated partner modes. We
consider the XY model with N = 10, and parameter
choices (h = .5, γ = .5) and (h = 0, γ = .8).
minimal energy cost. The upper bound on the
energy cost is due to the fact that the spectrum
of the XY model is also bounded from above by
max
= max
κ
κ
=
p
J
2
+ h
2
+ 2hJ . (226)
We see that our bounds become tighter for nar-
rower excitation spectra. As seen from Figure 7
our spectrum becomes flat for h → 0 and γ → 1.
This is reflected in Figure 8 for (h = 0, γ = .8).
Our analysis of the energy cost for the XY
model did not take into account locality in real
space so far. In particular, the optimal part-
ner modes that accomplish our minimal energy
cost are spanned by the de-localized energy eigen-
modes with lowest energy values. It is there-
fore interesting, to investigate the energy cost in
a more physical scenario where one mode is lo-
calized on a given site, while its partner mode
is localized in the complement of the site. In
this framework the choice of modes is completely
fixed, hence we can scan through the (h, γ) pa-
rameter space of the XY model to compare the
energy cost for this single site extraction to the
Accepted in Quantum 2019-07-10, click title to verify 27
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.90 0.92 0.94 0.96 0.98 1.00
0.4
0.6
0.8
1.0
Figure 9: We consider the XY model with N = 10
and choose the model parameters h/J and γ, such that
the minimal excitation energy
min
is in {.5, .7, .9} as
indicated in the upper panel. For each point in the pa-
rameter space, we then compute the energy cost of en-
tanglement extraction between a single site mode and
its partner. We compare these sample points with our
lower bounds for the given energy.
global lower and upper bounds.
In Figure 9, we show the energy cost of entan-
glement extraction for a single-site mode and its
partner. We scan through the parameter space
along three different paths of points, which are
constrained to yield the same lowest excitation
energy
min
for the Hamiltonian
ˆ
H
XY
, i.e.,
h = ±
1
γ
q
(1 − γ
2
)
γ
2
−
2
min
. (227)
We notice that the entanglement entropy of a sin-
gle site with the rest of the system is close to max-
imal, i.e., ∆S ≈ 1. This is not surprising, as it
is known [66] that every local site becomes max-
imally entanglement in the thermodynamic limit
N → ∞. Furthermore, we see that for
min
= .9,
the actual energy cost from the single site extrac-
tion is comparably close to our lower bound. This
indicates that for certain parameter choices our
minimal energy cost could actually be achieved
in this concrete physical scenario.
7 Discussion and outlook
In this work, we have found the minimal en-
ergy cost of extracting a pair of entangled part-
ner modes from the ground state of a quadratic
Hamiltonian, and we have characterized the
modes which achieve this minimal energy cost.
The minimal energy cost is linear in the spec-
trum of the Hamiltonian restricted to the two-
mode space spanned by the extracted partner
modes. Hence, as one may intuitively expect, the
globally lowest energy cost possible is achieved by
partner modes constructed from the two lowest
energy eigenmodes of the Hamiltonian. In some
applications, access to such less localized energy
eigenmodes may physically not be possible. How-
ever, our analysis of the XY spin model showed
instances of modes that are localized on a single
site but still came relatively close to the global
minimum, with their delocalized partner mode.
Interestingly, the minimal energy cost of en-
tanglement extraction can be used to derive an
upper bound on a Hamiltonian’s spectral gap: If
it is possible to extract a pair of partner modes
sharing entanglement S from the ground state,
while only increasing the system’s energy by an
amount ∆E, then the Hamiltonian’s lowest eigen-
value is at most as high as the value implied by
the minimal energy cost relation derived above.
Apart from the question of the global minimal
energy cost, as dictated by the spectrum of the
full Hamiltonian, it is meaningful to ask whether
a given pair of partner modes minimize the en-
ergy cost locally, i.e., whether they achieve the
minimal energy cost allowed for by the spectrum
of the Hamiltonian restricted to their two-mode
subspace. If not, then there exist symplectic
transformations acting only on the given partner
modes which yield another pair of partner modes
achieving the local minimum.
We showed that for bosonic modes there are
only specific pairs of partner modes which achieve
the local minimal energy cost. These modes are
obtained by squeezing the odd and even superpo-
sitions of the eigenmodes of the restricted Hamil-
tonian. However, for fermionic modes, if the
amount of extracted entanglement lies below a
critical value, all partner modes achieve the lo-
cal minimal energy cost. Only above the critical
value, specific partner modes are required which
are obtained by squeezing the eigenmodes of the
restricted Hamiltonian. The latter is due to the
fact that above the critical value, the lowest ex-
cited energy eigenstate of the two-mode subsys-
tem then coincides with the product of the re-
Accepted in Quantum 2019-07-10, click title to verify 28
stricted ground states of the partner modes.
An intriguing question for future research will
be to analyze the dynamics of the extraction pro-
cess due to the time evolution of the source modes
under the source system’s Hamiltonian. In cer-
tain cases this dynamic may be relatively sim-
ple. For example, the partner modes which min-
imize the energy cost lie within the subspace of
only the two lowest energy eigenmodes of the sys-
tem. Hence their time evolution is restricted to
this subspace. In general, for partner modes con-
structed from a range of energy eigenmodes, a
more complex time evolution is expected. This
would affect how the extraction from such source
modes can be practically implemented, as well as
it affects how the energy injected during the ex-
traction process dissipates inside the source sys-
tem after the extraction process. Indeed, the
prospect that such complex dynamics potentially
could be exploited in the design of entanglement
extraction protocols motivates further study of
the dynamics and time evolution of partner mode
pairs.
7.1 Outlook: Extraction from arbitrary modes
In this work, we assumed that the system modes
which are swapped onto the target modes can be
selected freely from the source system. This led
us to choose a pair of partner modes which is
the only way to obtain a pure final state in the
target modes. Hence, we derived the energy cost
of replacing the state of arbitrary partner modes
in a system’s ground state with the product state
of their restricted ground states. In particular, we
established an upper bound to this cost in terms
of the systems largest excitation energy.
In practical implementations of entanglement
extraction protocols, however, the access to the
source system might be limited. For example,
the modes could be restricted to be localized in a
given subregion of the system, or to lie in the span
of a limited set of modes. In particular, such re-
striction could make it impossible to access pairs
of partner modes. It could also be interesting to
consider randomly chosen modes, see, e.g., [79].
This raises the important question what our re-
sults imply for more general extraction protocols
that are not based on partner modes. We expect
that the minimal energy cost for entanglement
extraction from partner modes, that we derived
here, also constitutes a lower bound for the min-
imal energy cost for swapping pairs of arbitrary
single modes A and B out of the system that are
not necessarily partners.
If the modes A and B are not partner modes
then other entanglement monotones than the en-
tanglement entropy S(ρ
A
) or S(ρ
B
) are necessary
to measure the entanglement between them, such
as logarithmic negativity or entanglement of pu-
rification need to be used instead. This is be-
cause the modes A and B share entanglement
with the remainder of the system, such that their
joint state ρ
AB
, i.e., the state of the subsystem
A ⊕ B, is mixed.
Whichever mixed state entanglement N
AB
measure we employ, we may assume that the
lower entanglement entropy of the two modes
S
min
= min(S(ρ
A
), S(ρ
B
)) implies an upper
bound on N
AB
because the entanglement be-
tween a mode A and its complement is larger
than the entanglement between modes A and B.
In particular, this means that a pair of partner
modes sharing entanglement S
min
is at least as
valuable as the two-mode state ρ
AB
.
However, we expect that the minimal energy
cost for the extraction of a partner mode pair A
and
¯
A with entanglement S
min
is always lower
than the energy cost of extracting modes A and
B. In fact, this can be proven for the case of a
Hamiltonian with degenerate excitation energies
and we have numerical evidence for the general
case [80].
In this way, the derived minimal energy cost
from partner mode extraction actually applies
much more broadly to essentially any entangle-
ment monotone (of Gaussian states) and any two
target modes A and B. Note, however, that this
minimal energy cost will only be saturated if we
are actually using partner modes, which are in
this sense optimal. Where this is prevented by
additional constraints, e.g., locality assumptions,
we are likely to underestimate the actually re-
quired energy cost.
7.2 Outlook: Mixed states
Furthermore, it will be interesting to consider ex-
traction from states other than the ground state
of the source system. Of particular relevance
for practical implementations should be general
mixed states and thermal states. There, we may
encounter situations with vanishing or even neg-
ative energy cost. Similarly, in quantum systems
Accepted in Quantum 2019-07-10, click title to verify 29
with spontaneously broken symmetries, i.e., with
several ground states, one might be able to ex-
tract entanglement by moving from one ground
state to another.
In the context of mixed states, but also in gen-
eral, it should be relevant to consider more gen-
eral methods of entanglement extraction. Instead
of replacing the entangled state of a subsystem
by a product state in one discrete step, one could
consider more general local operations on the sub-
system such as the local operations and classical
communication (LOCC) envisioned in the general
framework of [48]. This may yield an infinitesi-
mal relation between entropy and energy increase
around a given system state.
For these and other future research directions,
the advancement of the partner mode formula
provided in this work should prove instrumen-
tal because it can yield better insight into the
correlation structure of mixed Gaussian states.
In particular, we are interested in extending the
partner mode formula to mixed states. This al-
lows, e.g., for the decomposition of a mixed Gaus-
sian states into chains of k-tuples of modes where
the modes in each k-tuple are not correlated with
each other, and only correlated with modes in
the preceding and subsequent tuple. In fact, this
construction does not only apply to mixed states
but to bi-linear forms in general. Thus, it may be
used to further develop techniques such as [81]
to efficiently describe the interaction of several
quantum systems with a given bath of harmonic
oscillators, with applications ranging from open
system dynamics to fundamental quantum com-
munication between local observers via relativis-
tic fields [82–87].
Acknowledgements
The authors thank Dmytro Bondarenko, Tom-
maso Guaita, Terry Farrelly, and Tommaso
Roscilde for inspiring discussions. LH is sup-
ported by the the Max Planck Harvard Re-
search Center for Quantum Optics and the
Deutsche Forschungsgemeinschaft (DFG, Ger-
man Research Foundation) under Germany’s Ex-
cellence Strategy – EXC-2111 – 39081486. LH
thanks the QMATH center for their hospitality
during several visits related to this project. RHJ
acknowledges support by ERC Advanced grant
321029 and VILLUM FONDEN via the QMATH
center of excellence (grant no.10059).
A Standard forms of Gaussian states
and the complex structure
The linear complex structure associated to
a quantum state is a powerful method to
parametrize Gaussian states. This is because
most properties of the partial, mixed state of sub-
systems, such as the von Neumann entropy and
the full entanglement spectrum, are encoded in
the restriction of the complex structure to the
relevant sub-phase space. Therefore, this ap-
pendix uses the complex structure to rephrase
some known properties of Gaussian states in a
largely basis-independent way. The key proper-
ties of the complex structure are reviewed and de-
rived based on compatibility with the symplectic
form (for bosons) or the positive-definite metric
(for fermions). This appendix is in parts based
on the respective appendix in [58].
Definition 1. Given a finite-dimensional real
vector space V , we refer to a linear map J : V →
V as complex structure if it satisfies J
2
= −1.
The spectrum of a linear complex structure
takes the following simple form.
Proposition 1. A complex structure is diago-
nalizable over the complex numbers with eigen-
values that come in conjugate pairs as ±i with
equal multiplicity. This also implies that V must
be even dimensional.
Proof. The equality J
2
= −1 implies that J is di-
agonalizable and every eigenvalue squares to −1.
Moreover, J is a real map which means that all
non-real eigenvalues come in complex conjugate
pairs of equal multiplicity. Thus, the spectrum
of J is given by ±i with equal multiplicity equal
to half of the dimension of V . The last part im-
plies that such a linear map J can only exist in
an even dimensional real vector space V .
We will be particularly interested in the restric-
tion of linear complex structures.
Proposition 2. Given a complex structure J
over V and a direct sum decomposition of V =
Accepted in Quantum 2019-07-10, click title to verify 30
A ⊕ B, we have the unique decomposition
J =
J
A
J
AB
J
BA
J
B
!
with
J
A
: A → A : a 7→ P
A
(Ja) ,
J
B
: B → B : b 7→ P
B
(Jb) ,
J
AB
: B → A : b 7→ P
A
(Jb) ,
J
BA
: A → B : a 7→ P
B
(Ja) ,
(228)
where the projections P
A
: V → A and P
B
: V →
B with P
A
+ P
B
= 1 provide the unique decom-
position of a vector v = a + b into its part in
a = p
A
(v) ∈ A and b = p
B
(v) ∈ B.
Given such a decomposition, the spectrum of J
2
A
and J
2
B
is the same except for the number of
eigenvalues equal to −1. Put differently, given
an eigenvalue λ of J
2
A
, we have λ = −1 or J
2
B
also has λ as eigenvalue.
Proof. We write J
2
= −1 in blocks to find
J
2
=
J
2
A
+ J
AB
J
BA
J
A
J
AB
+ J
AB
J
B
J
B
J
BA
+ J
BA
J
A
J
2
B
+ J
BA
J
AB
!
=
−1
A
0
0 −1
B
!
. (229)
Let us consider the eigenvector a ∈ A with J
2
A
a =
α a. From the first diagonal block equation, we
find that a must also be eigenvector of J
AB
J
BA
with J
AB
J
BA
a = −(1 + α)a. Let us define b :=
J
BA
a. Now there are two possibilities: Either
b = 0 which implies α = −1 or b 6= 0, in which
case we must have J
BA
J
AB
b = −(1 + α)J
BA
a =
−(1 + α) b. The second diagonal block equation
then implies J
2
B
b = α b.
A.1 Compatible symplectic form (bosons)
For bosonic systems, we are interested in linear
complex structures that are compatible with the
underlying symplectic form or vice versa.
Definition 2. An antisymmetric, non-
degenerate bilinear form ω : V × V → R
is called compatible to a complex structure
J : V → V if and only if it satisfies the following
conditions:
(I) Skew-symmetry: The symplectic form satis-
fies ω(Jv, w) = −ω(v, Jw).
(II) Taming: The bilinear form g(v, w) :=
ω(v, Jw) is positive definite.
We will refer to ω as a J-compatible symplectic
form and to J as a ω-compatible complex struc-
ture.
The compatibility conditions implies the fol-
lowing invariance property.
Proposition 3. A J-compatible symplectic form
ω is invariant under J meaning
ω(Jv, Jw) = ω(v, w) (230)
for all v, w ∈ V , thus J ∈ Sp(V, ω). The bilinear
form
g(v, w) := ω(v, Jw) (231)
is symmetric and positive definite and thus a
proper metric.
Proof. Straightforward computation leads to
ω(Jv, Jw) = −ω(v, −1w) = ω(v, w) . (232)
Every linear map M : V → V that preserves the
symplectic form ω in the sense of ω(Mv, Mw) =
ω(v, w) is part of the symplectic group Sp(V, ω).
The form g(v, w) = ω(v, Jw) is symmetric due to
g(v, w) = ω(v, Jw) = ω(w, Jv) = g(w, v) .
(233)
The taming property (II) of ω ensures that g is
positive definite, and thus a proper positive defi-
nite metric.
The compatibility conditions characterize the
possible spectra of restrictions of the linear com-
plex structure to symplectic subspaces.
Proposition 4. Let the complex structure J and
the symplectic form ω on a vector space V be
compatible, and let V = A ⊕ B a decomposition
of V into a direct sum of even dimensional sym-
plectic complements. Then, the restricted com-
plex structure J
A
is diagonalizable and its spec-
trum consists of complex conjugate pairs ±ic
i
with c
i
∈ [1, ∞). Both J
A
and ω
A
can simul-
taneously be brought into the block diagonal form
J
A
≡
N
A
M
i=1
0 c
i
−c
i
0
!
and ω
A
≡
N
A
M
i=1
0 −1
1 0
!
.
(234)
Accepted in Quantum 2019-07-10, click title to verify 31
Proof. The restricted linear complex structure
J
A
is diagonalizable due to the fact that it
squares to the diagonalizable matrix J
2
A
= −1
A
−
J
AB
J
BA
. Given a non-zero eigenvector a ∈ A
with J
2
A
a = α a, we have α ∈ (−∞, −1] by the
following argument: From
α g(a, a)
| {z }
>0
= g(a, J
2
A
a) = −g(J
A
a, J
A
a)
| {z }
≥0
(235)
follows α ≤ 0. Moreover, we can compute
g(J
A
a, J
A
a) = g(Ja − J
BA
a, Ja − J
BA
a)
= g(Ja, Ja)
| {z }
=g(a,a)≥0
+ g(J
BA
a, J
BA
a)
| {z }
≥0
−2 g(Ja, J
BA
a)
| {z }
=0
,
(236)
where we used in the last step that A and B
are symplectic complements with g(Ja, J
BA
a) =
ω(Ja, JJ
BA
a) = −ω(J
2
a, J
BA
a) = ω(a, J
BA
a) =
0. We thus find g(J
A
a, J
A
a) ≥ g(a, a) which im-
plies α ∈ (−∞, −1].
From this, we find that J
A
must have eigenvalues
appearing in complex conjugate pairs ±ic with
c =
√
−α, meaning that every eigenvalue α of
J
2
A
appears with even multiplicity. Since J
A
is a
linear real map with complex conjugated eigen-
values ±ic, it can can be brought into the block
diagonal form of (234). In particular, in such a
basis, J
2
A
is diagonal. Next, we show that ω
A
si-
multaneously takes the form claimed above.
While the full J satisfies ω(Jv, Jw) = ω(v, w)
for v, w ∈ V , the restricted J
A
does in gen-
eral not satisfy ω
A
(J
A
a, J
A
a
0
) = ω
A
(a, a
0
) for
a, a
0
∈ A. However, we can use the relation
ω(Jv, w) = −ω(v, Jw) for v, w ∈ V to derive
the relation
ω
A
(J
A
a, a
0
) = ω(Ja, a
0
) = −ω(a, Ja
0
)
= ω
A
(a, J
A
a
0
) ,
(237)
which is the condition for J
A
to represent a sym-
plectic algebra element, i.e., J
A
∈ sp(2N
A
, R).
For a diagonalizable symplectic algebra element
with purely imaginary eigenvalues, we can always
choose a basis, e.g., see proposition 3.1.18 in [88],
such that the matrix representations of J
A
and
ω
A
are given by (234).
This result is central for computing the entan-
glement spectrum of bosonic systems. In par-
ticular, the fact that the magnitude of possible
eigenvalues lies in the interval [1, ∞) indicates
that the entanglement entropy in bosonic systems
can be arbitrarily large. This can also be re-
lated to the non-compactness of symplectic group
whose group elements relate different linear com-
plex structures.
Before we construct the standard form for the
matrix representation of J for a general symplec-
tic decomposition V = A ⊕ B, we show that
if J
2
A
= −1
A
, i.e., when the restricted complex
structure defines a complex structure on the sub-
space A, then the complex structure leaves A and
B invariant.
Proposition 5. In the setup of the previous
Proposition 4, if J
2
A
= −1, then J
BA
= 0 van-
ishes, furthermore J
2
B
= −1 and J
AB
= 0.
Proof. Because V = A ⊕ B is a symplectic de-
composition, for all a ∈ A, b ∈ B
0 = ω(a, b) = ω(Ja, Jb)
= ω
A
(Ja, Jb) + ω
B
(Ja, Jb)
= ω
A
(J
A
a, J
AB
b) + ω
B
(J
BA
a, J
B
b) , (238)
hence ω
A
(J
A
a, J
AB
b) = −ω
B
(J
BA
a, J
B
b). For
J
2
A
= −1
A
, we find J
AB
J
BA
= 0 due to −1 = J
2
and −1
A
= J
2
A
+J
AB
J
BA
. To show that J
BA
= 0,
let b := J
BA
a ∈ B leading to
g(b, b) = ω(b, Jb) = ω
B
(J
BA
a, JJ
BA
a)
= ω
B
(J
BA
a, J
B
J
BA
a) = −ω
A
(J
A
a, J
AB
J
BA
a) .
(239)
Hence g(b, b) = 0 which implies b = 0 and since
a ∈ A was arbitrary, this means J
BA
= 0.
Since J
2
A
= −1, by Proposition 2 we have J
2
B
=
−1. Hence, by repeating the argument above, we
obtain J
AB
= 0.
The derived form for J
A
above induces a stan-
dard representation of J which yields the stan-
dard form of the bosonic covariance matrix pre-
sented in (39).
Proposition 6. As before, let the complex struc-
ture J and the symplectic form ω on a vector
space V be compatible, and let V = A ⊕ B be
a decomposition of V into a direct sum of even
dimensional symplectic complements. Without
loss of generality assume that the dimension of
A is lower than, or equal to the dimension of B.
Then there exists a symplectic basis with respect
to which J takes the form
Accepted in Quantum 2019-07-10, click title to verify 32
J ≡
ch(2r
1
) A
2
··· 0 sh(2r
1
) S
2
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· ch(2r
N
A
) A
2
0 ··· sh(2
N
A
) S
2
0 ··· 0
sh(2r
1
) S
2
··· 0 ch(2r
1
) A
2
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· sh(2r
N
A
) S
2
0 ··· ch(2r
N
A
) A
2
0 ··· 0
0 ··· 0 0 ··· 0 A
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· 0 0 ··· 0 0 ··· A
2
, (240)
for some r
i
≥ 0, where A
2
and S
2
are given by
A
2
=
0 1
−1 0
!
, S
2
=
0 1
1 0
!
. (241)
Proof. The form of the upper diagonal block, rep-
resenting J
A
, follows from the previous proposi-
tion, by choosing ch(2r
i
) = c
i
. Now for each i ∈
{1, . . . , N
A
}, we denote the basis vectors bringing
J
A
into this standard form by a
(i)
1
, a
(i)
2
∈ A, i.e.,
J
A
a
(i)
2
= ch(2r
i
)a
1
, J
A
a
(i)
1
= −ch(2r
i
)a
2
.
(242)
Now, if r
i
> 0, define
b
(i)
1
=
1
sh(2r
i
)
J
BA
a
2
, b
(i)
2
=
1
sh(2r
i
)
J
BA
a
1
.
(243)
These vectors lie in the eigenspace of J
2
B
with
eigenvalue −ch(2r
i
)
2
. By definition, they bring
the lower left off-diagonal block, representing
J
BA
into the claimed form. Using J
B
J
BA
=
−J
BA
J
A
which follows from (229), we find that
J
B
b
(i)
1
=
J
B
J
BA
sh(2r
i
)
a
2
= −
J
BA
J
A
a
2
sh(2r
i
)
= −ch(2r
i
)
J
BA
a
1
sh(2r
i
)
= −ch(2r
i
)b
(i)
2
,
(244)
J
B
b
(i)
2
= ch(2r
i
)b
(i)
1
. (245)
This means that the vectors also bring J
B
into
standard form on the eigenspaces of J
2
B
of eigen-
value −ch(2r
i
)
2
. They also bring ω into standard
form, which follows from
ω(b
(i)
1
, b
(i)
2
) =
ω (J
BA
a
2
, J
BA
a
1
)
sh
2
(2r
i
)
=
ω (Ja
2
, J
BA
a
1
)
sh
2
(2r
i
)
=
−ω (a
2
, JJ
BA
a
1
)
sh
2
(2r
i
)
=
−ω (a
2
, J
AB
J
BA
a
1
)
sh
2
(2r
i
)
= −ω(a
2
, a
1
) = −1 ,
(246)
where we used that J
AB
J
BA
a
i
= sh
2
2r
i
a
i
which
follows from −1 = J
2
A
+ J
AB
J
BA
in (229). This
relation we also use to confirm that the defined
vectors bring the lower off-diagonal block, repre-
senting J
BA
into the claimed standard form:
J
AB
b
(i)
1
= sh(2r
i
)a
2
, J
AB
b
2
= sh(2r
i
)a
1
.
(247)
In the case where r
i
= 0, Proposition 5 ensures
that the matrix entries of off-diagonal blocks van-
ish. Hence in the subspaces of B corresponding
to r
i
= 0, and in the remaining dimensions of
B, we can choose any basis which brings J
B
in
its standard form. Thus, we obtain the claimed
matrix representation for J. Finally, this result
implies the standard form (39) of the bosonic co-
variance matrix G by applying G = −JΩ with
Ω ≡ Ω from (2).
To conclude this subsection, let us comment
on the connection to the standard form of the
bosonic covariance matrix and partner formula of
Section 2. The symplectic form ω and the metric
g used in this section act on the phase space V .
In fact, they are the inverses of Ω
ab
and G
ab
used
in the main body of the article which act on V
∗
,
the co-vector space of the phase space.
With respect to a basis in which J is repre-
sented by the matrix form which we just derived,
G is then represented, due to equation (17), by
the matrix G = −JΩ which is exactly as in (39).
Accepted in Quantum 2019-07-10, click title to verify 33
Each pair of vectors a
(i)
1
, a
(i)
1
∈ V , appearing
in the proof above, is dual to a pair of covectors
x, k ∈ V
∗
which defines a mode. From the proof
it follows that the mode’s partner mode is defined
by the covectors (¯x,
¯
k) given by
¯x =
1
sh(2r
i
)
J
|
BA
k =
1
sh(2r
i
)
(−J
|
A
+ J
|
) k
=
1
sh(2r
i
)
(ch(2r
i
)x + J
|
k) , (248)
¯
k =
1
sh(2r
i
)
(−ch(2r
i
)k + J
|
x) , (249)
which is precisely the partner formula (47).
A.2 Compatible metric (fermions)
For fermionic systems the analysis of the com-
plex structure is very similar. Here, we study the
properties of a linear complex structure which is
compatible with a positive definite metric.
Definition 3. A positive definite symmetric bi-
linear form g : V × V → R is called compatible
to a complex structure J : V → V if and only if
it satisfies
g(Jv, w) = −g(v, Jw) for all v, w ∈ V .
(250)
We will refer to g as a J-compatible metric and
to J as a g-compatible complex structure.
The compatibility conditions implies the fol-
lowing invariance property.
Proposition 7. A J-compatible metric g is in-
variant under J meaning g(Jv, Jw) = g(v, w) for
all v, w ∈ V which implies J ∈ O(V, g). More-
over, we can define the antisymmetric bilinear
form ω(v, w) = g(Jv, w) which is a well-defined
symplectic form on V .
Proof. Straightforward computation gives
g(Jv, Jw) = −g(v, J
2
w) = −g(v, −1w)
= g(v, w) . (251)
Every linear map M : V → V that preserves
metric g in the sense of g(Mv, M w) = g(v, w) is
an element of the orthogonal group O(V, g). A
symplectic form on a vector space V is required
to be (a) antisymmetric and (b) non-degenerate:
(a) Antisymmetry: We compute ω(v, w) =
g(Jv, w) = g(w, Jv) = −g(Jw, v) =
−ω(w, v).
(b) Non-Degeneracy: We need to show that for
every non-zero vector v, there is at least one
vector w, such that ω(v, w) 6= 0. This can
easily be done by choosing w = Jv, where
we find ω(v, w) = g(Jv, Jv) > 0.
This proves that every compatible every pair of
a complex structure J and a metric g defines a
symplectic form ω.
Again, the compatibility condition character-
izes the possible spectra of restrictions of the lin-
ear complex structure to even dimensional sub-
spaces.
Proposition 8. Given a compatible pair of a
complex structure J and a metric g, we can de-
compose the vector space V into a direct sum
of orthogonal sum of orthogonal complements
V = A ⊕ B. In this case, the restricted com-
plex structure J
A
is diagonalizable and its spec-
trum consists of complex conjugate pairs ±ic
i
with c
i
∈ [0, 1]. If A is odd-dimensional, J
A
also
has the eigenvalue 0. Provided that we choose a
decomposition into even-dimensional orthogonal
complements with dimensions 2N
A
, we can bring
J
A
and g
A
simultaneously in the following block
diagonal form:
J
A
≡
N
A
M
i=1
0 c
i
−c
i
0
!
and g
A
≡
N
A
M
i=1
1 0
0 1
!
.
(252)
Proof. The restricted complex structure J
A
is it-
self anti-hermitian with respect to the metric g
A
because we have for a
1
, a
2
∈ A
g
A
(J
A
a
1
, a
2
) = −g(a
1
, Ja
2
) = −g
A
(a
1
, J
A
a
2
) .
(253)
Therefore, J
A
is normal with respect to g
A
. This
ensures that J
A
is diagonalizable (with a com-
plete set of eigenvectors) and so is J
2
A
. Given a
non-zero eigenvector a ∈ A with J
2
A
a = α a, we
have α ∈ [−1, 0] due to the following argument.
Let us compute
α g(a, a)
| {z }
>0
= g(a, J
2
A
a) = −g(J
A
a, J
A
a)
| {z }
≥0
. (254)
This already implies that α ≤ 0. Moreover, we
can also compute
g(a, a) = g(J
A
a + J
BA
a, J
A
a + J
BA
a)
= g(J
A
a, J
A
a) + g(J
BA
a, J
BA
a) ,
(255)
Accepted in Quantum 2019-07-10, click title to verify 34
where we used in the last step that A and B
are orthogonal which elimates crossing terms.
This equation implies the inequality g(a, a) ≥
g(J
A
a, J
A
a) = |α|g(a, a) leading to α ∈ [−1, 0].
From here, we find that J
A
must have eigenvalues
appearing in complex conjugate pairs ±ic with
c =
√
−α meaning that also every eigenvalue α of
J
2
A
appears with even multiplicity unless α = 0.
While the full J satisfies g(Jv, Jw) = g(v, w)
for v, w ∈ V , the restricted J
A
does in gen-
eral not satisfy g
A
(J
A
a, J
A
a
0
) = ω
A
(a, a
0
) for
a, a
0
∈ A. However, we can use the relation
g(Jv, w) = −g(v, Jw) for v, w ∈ V to derive the
relation
g
A
(J
A
a, a
0
) = g(Ja, a
0
) = −g(a, Ja
0
)
= g
A
(a, J
A
a
0
) ,
(256)
which is well-known to be the condition for J
A
to represent an orthogonal algebra element, i.e.,
J
A
∈ so(2N
A
). An orthogonal algebra element is
antisymmetric with respect to g
A
and has purely
imaginary eigenvalues. It is well-known that we
can always choose an orthonormal basis, such
that the matrix representations of J
A
and g
A
are
given by (252).
Proposition 9. In the setup of Proposition 8,
if J
2
A
= −1 this implies J
BA
= 0, furthermore
J
2
B
= −1 and J
AB
= 0.
Proof. For any two vectors a, a
0
∈ A, we have
g(J
BA
a, J
BA
a
0
) = −g(a, J
AB
J
BA
a
0
) , (257)
which is shown using g(v, Jw) = −g(Jv, w) and
the fact that A and B are orthogonal subspaces:
g(J
BA
a, J
BA
a
0
) = g(Ja, J
BA
a
0
)
= −g(a, JJ
BA
a
0
) = −g(a, J
AB
J
BA
a
0
) . (258)
If J
2
A
= −1, then from (229) follows J
AB
J
BA
=
−1 − J
2
A
= 0. This however means that for all
a ∈ A we have
g(J
BA
a, J
BA
a) = −g(a, J
AB
J
BA
a) = 0. (259)
Since g is non-degenerate this implies J
BA
= 0.
By Proposition 2, J
2
B
= −1 follows from J
2
A
=
−1, then repeating the argument above shows
J
AB
= 0.
This result is central for computing the entan-
glement spectrum of fermionic systems. In par-
ticular, the fact that the magnitude of possible
eigenvalues lies in the interval [0, 1] implies that
the entanglement entropy of fermionic systems is
bounded by log 2 per fermionic degree of freedom.
Proposition 10. As before, let the complex
structure J and the metric g on a vector space
V be compatible, and let V = A ⊕B be a decom-
position of V into even-dimensional orthogonal
complements. Without loss of generality, assume
that the dimension of A is lower than, or equal
to the dimension of B. Then there exists an or-
thonormal basis with respect to which J takes the
form
J ≡
cos(2r
1
) A
2
··· 0 sin(2r
1
) S
2
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· cos(2r
N
A
) A
2
0 ··· sin(2r
N
A
) S
2
0 ··· 0
−sin(2r
1
) S
2
··· 0 cos(2r
1
) A
2
··· 0 0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· −sin(2r
N
A
) S
2
0 ··· cos(2r
N
A
) A
2
0 ··· 0
0 ··· 0 0 ··· 0 A
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· 0 0 ··· 0 0 ··· A
2
,
(260)
Accepted in Quantum 2019-07-10, click title to verify 35
with r
i
∈ [0,
π
4
] and the two-by-two matrices A
2
and S
2
defined in (241).
Proof. By Proposition 8 the upper left diago-
nal block, representing J
A
, can be brought into
the claimed form by choosing cos 2r
i
= c
i
. Let
a
(i)
1
, a
(i)
2
denote the pair of basis vectors spanning
the space corresponding to r
i
. Then, if r
i
> 0,
define the vectors
b
(i)
1
=
1
sin 2r
i
J
BA
a
(i)
2
, b
(i)
2
=
1
sin 2r
i
J
BA
a
(i)
1
.
(261)
These vectors are orthonormal
g
b
(i)
1
, b
(i)
2
=
−g
a
(i)
2
, J
AB
J
BA
a
(i)
1
sin
2
2r
i
= g
a
(i)
2
, a
(i)
1
= 0 , (262)
g
b
(i)
1
, b
(i)
1
=
−g
a
(i)
2
, J
AB
J
BA
a
(i)
2
sin
2
2r
i
= g
a
(i)
2
, a
(i)
2
= 1 , (263)
g(b
(i)
2
, b
(i)
2
) = 1 , (264)
where we used J
AB
J
BA
a = (−1 + cos
2
2r
i
)a =
−sin
2
(2r
i
)a which follows from (229). Similarly,
one also shows that g(b
(i)
n
, b
(j)
m
) = 0 if i 6= j. By
definition, these vectors bring the lower left di-
agonal block, representing J
BA
into the claimed
standard form. To see that the vectors also bring
the diagonal blocks of J
B
into the claimed form,
we calculate
J
B
b
(i)
1
=
J
B
J
BA
a
(i)
2
sin 2r
i
=
−J
BA
J
A
a
(i)
2
sin 2r
i
=
−cos(2r
i
)J
BA
a
(i)
1
sin 2r
i
= −cos 2r
i
b
(i)
2
,
(265)
J
B
b
(i)
2
= cos 2r
i
b
(i)
1
. (266)
Hence, by extending the set of all these vectors
with an orthonormal set of vectors which brings
the eigenspace of J
2
B
with eigenvalue −1 into the
standard form, we obtain an orthonormal basis
of B which brings the matrix representing J into
the claimed form.
Finally, this result implies the standard
form (144) of the fermionic covariance ma-
trix Ω by applying Ω = JG with G ≡ 1.
The propositions of these section imply both
the normal form of the covariance matrix for
fermions (141), as well as the fermionic partner
mode. In fact, the matrix Ω representing the
fermionic covariance matrix Ω
ab
and the matrix
J representing the complex structure J
a
B
are in
fact identical. This is due to Ω = JG where the
metric G, which is the inverse of the metric g used
in this subsection, is represented by the identity
matrix G = 1. The partner formula for fermions
(150) follows by the same argument as discussed
above for bosons.
B Spectra of restricted Hamiltonian
Here, we show how the spectrum of the restric-
tion of a Hamiltonian to a subspace of partner
modes, is related to the spectrum of the unre-
stricted Hamiltonian operator, as a consequence
of the Courant–Fischer–Weyl min-max principle.
Proposition 11. Let
ˆ
H be a quadratic Hamil-
tonian on a (bosonic or fermionic) system of N
modes, and denote the excitation energies of
ˆ
H
by 0 < ω
1
≤ ω
2
≤ ··· ≤ ω
N
. Assuming that
ˆ
H
A
is the restriction of
ˆ
H onto a subsystem A
spanned by N
A
< N modes such that the ground
state |0i of
ˆ
H is a product state
|0i = |ψi
A
⊗ |φi
B
(267)
between A and its complement B, the excitation
energies
1
≤ ··· ≤
N
A
of
ˆ
H
A
fulfill
ω
i
≤
i
≤ ω
N−N
A
+i
. (268)
Proof. We note that the excitation energies of
a quadratic Hamiltonian
ˆ
H =
1
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
+ f
a
ˆ
ξ
a
are given by the eigenvalues of the linear map
L
a
c
= G
ab
h
bc
obtained by composing the co-
variance matrix G
ab
of the Hamiltonian’s ground
state with the Hamiltonian bilinear form h
ab
.
This is easily seen by observing that with respect
to a basis of energy eigenmodes, where G
ab
≡ 1
is represented by the identity matrix, both L and
h are represented by the same diagonal matrix.
In general, in a basis which is adapted to the
bipartition of the N modes into subsystems A
and B, the map L may not be represented by a
diagonal matrix. However, since we assume the
ground state of
ˆ
H to be a product state between
A and B, the covariance matrix takes block diag-
onal form, and we find that L is represented by
the matrix
L ≡ L = Gh =
G
A
0
0 G
B
!
h
A
h
AB
h
|
AB
h
B
!
Accepted in Quantum 2019-07-10, click title to verify 36
=
G
A
h
A
G
A
h
AB
G
B
h
|
AB
G
B
h
B
!
. (269)
The min-max principle now warrants that the
eigenvalues ω
1
≤ ··· ≤ ω
N
of L (with double mul-
tiplicity) bound the eigenvalues of G
A
h
A
, which
we denote by α
1
≤ ··· ≤ α
2N
A
, as
ω
1
≤ α
1
≤ ω
N−N
A
+1
, ω
N
A
≤ α
2N
A
≤ ω
N
.
(270)
The α
i
with double multiplicity represent the ex-
citation energies of
ˆ
H
A
. We can see this from
the ground state of
ˆ
H, that is a product state
between A and B such that |ψi
A
must be the
ground state of
ˆ
H
A
and its covariance matrix is
represented by the matrix G
A
. Therefore, the
spectrum of L
A
= G
A
h
A
yields the excitation
energies of
ˆ
H
A
.
The equivalent statement for a fermionic
Hamiltonian
ˆ
H =
i
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
follows from the same
line of argument, but the linear map L is now
defined by L
a
c
= −Ω
ab
h
bc
, where Ω
ab
represents
the covariance matrix of the ground state of the
fermionic Hamiltonian
ˆ
H =
i
2
h
ab
ˆ
ξ
a
ˆ
ξ
b
.
References
[1] A. Einstein, B. Podolsky, and N. Rosen.
Can Quantum-Mechanical Description
of Physical Reality Be Considered Com-
plete? Phys. Rev., 47(10):777–780, May
1935. DOI: 10.1103/PhysRev.47.777. URL
https://link.aps.org/doi/10.1103/
PhysRev.47.777.
[2] J. S. Bell. On the Einstein Podolsky
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