Entanglement Efficiencies in PT -Symmetric Quantum Mechanics
Christian Zielinski
1, ∗
and Qing-hai Wang
2, †
1
Institut für Theoretische Physik, Universität Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany
2
Department of Physics, National University of Singapore,
117542, Singapore
(Dated: November 12, 2018)
The degree of entanglement is determined for an arbitrary state of a broad class of PT -symmetric
bipartite composite systems. Subsequently we quantify the rate with which entangled states are
generated and show that this rate can be characterized by a small set of parameters. These relations
allow one in principle to improve the ability of these systems to entangle states. It is also noticed
that many relations resemble corresponding ones in conventional quantum mechanics.
PACS numbers: 03.67.Bg, 03.65.Ud, 03.65.Ta
Keywords: entanglement generation, entanglement efficiency, entanglement entropy, PT -symmetric quantum
mechanics
I. INTRODUCTION
In conventional quantum mechanics, one demands that
the Hamiltonian H generating the time evolution has a
real spectrum and that the corresponding time evolution
operator U is unitary. These conditions are fulfilled if the
Hamiltonian is Hermitian, i.e. H = H
†
, which is usually
considered an axiom of quantum mechanics. However,
the condition of Hermiticity can be weakened. In the
class of so-called PT -symmetric Hamiltonians, one can
ensure real eigenvalues and a unitary time evolution even
for explicitly non-Hermitian Hamiltonians [2, 3]. A thor-
ough review of the foundations of PT -symmetric quan-
tum mechanics can be found in [1].
In the following we will investigate entanglement phe-
nomena in bipartite systems within this framework. An
entangled state is a quantum state where two or more
degrees of freedom are intertwined, so that they are not
independent anymore. In this context a couple of histori-
cal discussions took place and gave deep insights into the
nature of quantum mechanics, like the Einstein-Podolsky-
Rosen paradox [8]. These states have a wide range of
applications, for example in quantum information theory
and quantum computing. The question of entanglement
generation and entanglement efficiencies for conventional
quantum mechanics was addressed earlier in [7].
Using PT -symmetric quantum mechanics to describe
entanglement phenomena was also done in [13]. Espe-
cially for the case of bipartite systems, relations for the
degree of entanglement of given states and the entangle-
ment capability of certain systems could be found. Al-
though we confirmed many of these results, we have some
different and new findings. For a particular initial state,
as well as for general states, we give relations between
the efficiency of the system to generate entangled states
∗
Electronic address: zielinski@thphys.uni-heidelberg.de
†
Electronic address: phywq@nus.edu.sg
and the parameters of the Hamiltonian describing the
dynamics of the system.
We will first introduce a measure for entanglement and
a certain class of PT -symmetric Hamiltonians. We then
quantify the degree of entanglement of an arbitrary and
of a generalized Einstein-Podolsky-Rosen state. Subse-
quently, we are dealing with the question of entangle-
ment generation of a particular PT -symmetric state and
generalize for arbitrary states. The question we try to
answer is how to characterize the rate of entanglement
generation for this class of systems.
II. BIPARTITE SYSTEMS
A. A measure for entanglement
Let H
i=1,...,N
denote a set of Hilbert spaces. We call a
state of a composite system H = H
1
⊗···⊗H
N
entangled,
if there is no decomposition of the form |Ψi = |χ
1
i⊗···⊗
|χ
N
i with suitable |χ
i
i ∈ H
i
. In the following we restrict
ourselves to bipartite systems, i.e. N = 2.
A measure of entanglement are the entropies
E(Ψ) = −tr
1
(ρ
1
log
2
ρ
1
) = −tr
2
(ρ
2
log
2
ρ
2
) , (1)
where ρ
1
= tr
2
ρ and ρ
2
= tr
1
ρ are the reduced den-
sity matrices, ρ = |ΨihΨ| is the density matrix itself and
tr
i
denotes the partial trace over the i
th
subsystem [12].
Here E(Ψ) ∈ [0, 1] and E(Ψ) = 0 if and only if |Ψi ∈ H
is not entangled.
B. General entanglement content
Consider the Hamiltonian
H =
re
iΘ
s
s re
−iΘ
, r, s, Θ ∈ R, (2)
with s
2
> r
2
sin
2
Θ. Observe that H is in general not
represented by a Hermitian matrix, H
†
6= H. But it
arXiv:1106.3856v3 [hep-th] 15 May 2012
2
obeys PT -symmetry with
P =
0 1
1 0
(3)
and T complex conjugation [4] (see also erratum [5]),
i.e. [H, PT ] = 0.
Define sin ϕ ≡ r/s · sin Θ with ϕ ∈ [−π/2, π/2]. The
simultaneous eigenstates of H and PT are given by
|φ
+
i =
1
√
2 cos ϕ
e
iϕ/2
e
−iϕ/2
,
|φ
−
i =
i
√
2 cos ϕ
e
−iϕ/2
−e
iϕ/2
, (4)
with eigenvalues E
±
= r cos Θ ± s cos ϕ. These eigen-
states are orthonormal with respect to the positive CPT -
inner product hΨ|Φi
CPT
≡ hΨ|
CPT
·|Φi, where hΨ|
CPT
≡
(CPT |Ψi)
|
, see [4]. For this Hamiltonian, the C operator
has the form
C =
i tan ϕ sec ϕ
sec ϕ −i tan ϕ
(5)
with [H, C] = 0.
In the following we will consider the composite system
H = H
1
× H
2
of Hilbert spaces H
i=1,2
with dynamics
governed by the Hamiltonian H = H
1
⊗ H
2
, where H
i
is of the form of (2) with r
i
, s
i
, Θ
i
∈ R, s
2
i
> r
2
i
sin
2
Θ
i
,
sin ϕ
i
≡ r
i
/s
i
·sin Θ
i
and |φ
±
i
i
∈ H
i
being the eigenstates
of H
i
for i = 1, 2. Our respective C operator reads C
1
⊗
C
2
, where C
i
is of the form of (5) with corresponding
parameters. As
[H
1
⊗ H
2
, PT ⊗PT ] = [H
1
⊗ H
2
, C
1
⊗ C
2
]
= [C
1
⊗ C
2
, PT ⊗PT ]
= 0 (6)
holds, H obeys PT symmetry and we can define a CPT -
inner product as above. The decomposition of the op-
erators means the same direct-product form of the met-
ric operator η in [14]. According to proposition 2 and
proposition 3 in the same paper, this permits a proper
quantum mechanical description of the bipartite system
with a unitary time evolution.
Now consider the density matrix ρ = |Ψ(t)ihΨ(t)|
CPT
of a system in the state
|Ψ(t)i = α(t)|φ
+
i
1
⊗ |φ
+
i
2
+ β(t)|φ
+
i
1
⊗ |φ
−
i
2
+ γ(t)|φ
−
i
1
⊗ |φ
+
i
2
+ δ(t)|φ
−
i
1
⊗ |φ
−
i
2
(7)
with |Ψ(t)i ∈ H. The eigenvalues of ρ
1
(t) = tr
2
ρ(t) are
given by λ
±
(t) =
1
2
±
1
2
p
1 − Ξ(t) with Ξ(t) = 4|α(t)δ(t)−
β(t)γ(t)|
2
, which is a simplification of the result in [13].
Hence the entanglement content is
E(t) ≡ E(Ψ(t))
= −λ
+
(t) log
2
λ
+
(t) − λ
−
(t) log
2
λ
−
(t). (8)
Note that |Ψ(t)i only separates if α(t
0
)δ(t
0
) = β(t
0
)γ(t
0
)
for t = t
0
.
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
1
1
1
1
1
-Π
-
Π
2
0
Π
2
Π
-Π
-
Π
2
0
Π
2
Π
-Π
-
Π
2
0
Π
2
Π
-Π
-
Π
2
0
Π
2
Π
j
2
Figure 1: Entanglement content E(Ψ
−
) as a function of ϕ
1
and ϕ
2
.
C. The Einstein-Podolsky-Rosen state
Consider the Einstein-Podolsky-Rosen state from con-
ventional quantum mechanics and normalize them with
respect to the CPT -inner product:
|↑i
i
=
√
cos ϕ
i
1
0
, |↓i
i
=
√
cos ϕ
i
0
1
. (9)
Overall normalization yields
|Ψ
−
i = κ (|↑i
1
⊗ |↓i
2
− |↓i
1
⊗ |↑i
2
) (10)
with κ = [2 (1 −sin ϕ
1
sin ϕ
2
)]
−1/2
. The eigenvalues of
the reduced density matrix of the first subsystem are
given by
λ
±
=
1
2
±
sin ϕ
1
− sin ϕ
2
2 (1 − sin ϕ
1
sin ϕ
2
)
. (11)
Our results are in disagreement with [13] mainly due to
the author’s use of non-CPT -normalized states.
We can consider the entanglement content
E(Ψ
−
) = −λ
+
log
2
λ
+
− λ
−
log
2
λ
−
(12)
as a function of ϕ
1
and ϕ
2
, i.e. of the Hamiltonians H
1
and H
2
. The result can be seen in figure 1. Note the
cases ϕ
1
= ϕ
2
or ϕ
1
+ ϕ
2
= ±π, where (10) has the form
of a PT -symmetric Bell state.
3
Π
2 Π
tΤ
-1
-0.5
0.5
1
EHtL, GHtLΤ
-1
Figure 2: Entanglement content E(t) (solid) and entangle-
ment rate Γ(t)/τ
−1
(dashed) of |Ψ(t)i with τ = 2~/ (ω
1
ω
2
).
III. ENTANGLEMENT GENERATION
A. Entanglement capability
We investigate the question how to increase the capa-
bility of a system to entangle states. More precisely we
want to understand the dependencies of the entanglement
capability Γ(t) ≡ dE(t)/dt of the parameters of the sys-
tem. In order to determine the time evolution operator
of the Hamiltonian H
1
⊗ H
2
(see section II B) define
−→
n
i
≡
2
ω
i
(s
i
, 0, ir
i
sin Θ
i
)
|
, ω
i
≡ 2s
i
cos ϕ
i
. (13)
Rewrite H as
H =
r
1
cos Θ
1
1 +
ω
1
2
−→
n
1
·
−→
σ
⊗
r
2
cos Θ
2
1 +
ω
2
2
−→
n
2
·
−→
σ
, (14)
with
−→
σ = (σ
1
, σ
2
, σ
3
)
|
, σ
i
denoting the Pauli matrices
and 1 the 2×2 identity matrix. Expanding the expression
for H yields four terms of which only one can generate
entanglement. We restrict ourselves to this term and
define
˜
H = ω
1
ω
2
/4 ·(
−→
n
1
·
−→
σ ⊗
−→
n
2
·
−→
σ ) resulting in a time
evolution operator U (t) = exp(−i
˜
Ht/~) given by
U(t) = cos
ω
1
ω
2
t
4~
(1 ⊗ 1)
− i sin
ω
1
ω
2
t
4~
(
−→
n
1
·
−→
σ ⊗
−→
n
2
·
−→
σ ) . (15)
Consider now a simple initial state, namely |Ψ(t =
0)i = |↑i⊗ |↑i, and apply U(t) to find
|Ψ(t)i = α(t) |↑i⊗ |↑i + β(t) |↑i⊗ |↓i
+ γ(t) |↓i⊗ |↑i + δ(t) |↓i⊗ |↓i (16)
with
α(t) = cos
ω
1
ω
2
t
4~
+
4ir
1
r
2
sin Θ
1
sin Θ
2
ω
1
ω
2
sin
ω
1
ω
2
t
4~
,
β(t) =
4s
2
r
1
sin Θ
1
ω
1
ω
2
sin
ω
1
ω
2
t
4~
,
γ(t) =
4s
1
r
2
sin Θ
2
ω
1
ω
2
sin
ω
1
ω
2
t
4~
,
δ(t) = −
4is
1
s
2
ω
1
ω
2
sin
ω
1
ω
2
t
4~
. (17)
Up to here our results are in agreement with [13]. We
now find the entanglement content to be given by
E(t) = −λ
+
(t) log
2
λ
+
(t) − λ
−
(t) log
2
λ
−
(t) (18)
with
λ
±
(t) =
1
2
±
1
2
cos
ω
1
ω
2
t
2~
. (19)
We find the entanglement rate to be
Γ(t) =
dE(t)
dt
=
ω
1
ω
2
~
sin
ω
1
ω
2
t
2~
log
16
cot
2
ω
1
ω
2
t
4~
. (20)
The maximal entanglement capability is Γ
max
=
max
t
Γ(t) = 0.4781 ω
1
ω
2
/~. Therefore by changing ω
i
due to an adjustment of the parameters of the Hamil-
tonians H
i
, we can control Γ
max
. The typical time de-
pendency of the entanglement content and entanglement
rate can be seen in figure 2.
B. Efficiency of general systems
If we have a given Hamiltonian one may ask how to
maximize the entanglement rate of the system. We want
to generalize some results of conventional quantum me-
chanics addressed in [7] to the PT -symmetric case.
Consider an arbitrary PT -symmetric two qubit sys-
tem. Using the Schmidt-decomposition theorem we
rewrite an arbitrary state |Ψ(t)i ∈ H
1
× H
2
as
|Ψ(t)i =
p
p(t)|ϕ
t
i ⊗ |χ
t
i
+ e
iα
p
1 − p(t)|ϕ
⊥
t
i ⊗ |χ
⊥
t
i (21)
with p ∈ [0, 1] as Schmidt-coefficient and hϕ
t
|ϕ
⊥
t
i
CPT
=
hχ
t
|χ
⊥
t
i
CPT
= 0 as Schmidt-vectors. The entanglement
content is given by
E(t) = −p(t) log
2
p(t) − (1 − p(t)) log
2
(1 − p(t)) (22)
and the entanglement rate factorizes in two terms,
i.e. Γ(t) = dE(t)/dp(t) × dp(t)/dt, where
dE(t)
dp(t)
=
2
log 2
arctanh (1 − 2p(t)) . (23)
After choosing the phase α appropriately the evolution of
the Schmidt-coefficient is determined by the differential
equation
4
dp(t)
dt
=
2
~
p
p(t) (1 − p(t))
× |(hϕ
t
|
CPT
⊗ hχ
t
|
CPT
) H
|ϕ
⊥
t
i ⊗ |χ
⊥
t
i
|. (24)
This relation is known from conventional quantum me-
chanic, but also holds for PT -symmetric systems. Define
~Ω ≡ max
kϕk=1
kχk=1
|(hϕ|
CPT
⊗ hχ|
CPT
) H
|ϕ
⊥
i ⊗ |χ
⊥
i
|. (25)
Then the time evolution of p(t) for an optimally prepared
setup, i.e. a setup forcing the qubit states to be optimal
at every instant of time
|(hϕ
t
|
CPT
⊗ hχ
t
|
CPT
) H
|ϕ
⊥
t
i ⊗ |χ
⊥
t
i
| = ~Ω (26)
for all t, follows from the differential equation
dp
opt
(t)
dt
= 2Ω
q
p
opt
(t) (1 − p
opt
(t)). (27)
We find p
opt
(t) = sin
2
(Ωt + δ
0
) with an integration con-
stant δ
0
, which is in agreement with results from conven-
tional quantum mechanics in [7] (see also [13]). Hence we
can characterize the entanglement rate of an optimally
prepared setup completely in terms of Ω via
Γ
opt
(t) =
dp
opt
(t)
dt
·
dE(t)
dp(t)
p
opt
(t)
(28)
=
2Ω
log 2
arctanh (cos (2Ωt + 2δ
0
)) sin (2Ωt + 2δ
0
) .
and find for the maximal rate Γ
max
= max
t
Γ
opt
(t) =
1.9123 Ω. The parameter Ω is the critical value one needs
to maximize to efficiently entangle states. We remark
that this result also holds in conventional quantum me-
chanics, where Ω is defined with conventional conjugates
instead of CPT -conjugates.
IV. CONCLUSIONS
For the case of a state from a bipartite system we de-
termined the degree of entanglement and saw the emer-
gence of symmetrical patterns (see figure 1) in the case
of the Einstein-Podolsky-Rosen state. We quantified the
capability of a given PT -symmetric system to generate
entangled states in terms of the parameters of the Hamil-
tonian. Their ability to entangle states can be described
by the parameters ω
i=1,2
in (13) or in general by Ω in
(25). Many relations are similar to the corresponding
ones in conventional quantum mechanics after replac-
ing the usual inner product with the CPT -inner prod-
uct. However, these results are not obvious and need
to be checked. For example, the recent discussion of
the quantum brachistochrone problem showed that PT -
symmetric quantum mechanics can give some surprising
results [6, 9–11].
Acknowledgments
This work was carried out at the Department of
Physics of the National University of Singapore, whose
hospitality is gratefully acknowledged. The results were
obtained within the UROPS program and we thank the
National University of Singapore, the University of Hei-
delberg and the German Academic Foundation for finan-
cial support.
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