Quantum superpositions and entanglement of thermal states at high
  temperatures and their applications to quantum information processing
Quantum superpositions and entanglement of thermal states at high temperatures
and their applications to quantum information processing
Hyunseok Jeong and Timothy C. Ralph
Centre for Quantum Computer Technology, Department of Physics,
University of Queensland, St Lucia, Qld 4072, Australia
(Dated: October 26, 2018)
We study characteristics of superpositions and entanglement of thermal states at high tempera-
tures and discuss their applications to quantum information processing. We introduce thermal-state
qubits and thermal-Bell states, which are a generalization of pure-state qubits and Bell states to
thermal mixtures. A scheme is then presented to discriminate between the four thermal-Bell states
without photon number resolving detection but with Kerr nonlinear interactions and two single-
photon detectors. This enables one to perform quantum teleportation and gate operations for
quantum computation with thermal-state qubits.
I. INTRODUCTION
In many problems considered within the framework of
quantum physics, physical systems are treated as pure
states that can be represented by state vectors, or equiva-
lently, by wave functions. Even though such an approach
is simple and useful to address certain problems, it could
often be quite different from real conditions of physical
systems. This may be particularly true when one deals
with macroscopic physical systems in terms of quantum
physics. A macroscopic object is a complex open sys-
tem which cannot avoid continuous interactions with the
environment. Such a physical system is generally in a
significantly mixed state and cannot be represented by
a state vector. In general, mixed states are subtle ob-
jects whose properties are significantly more difficult to
characterize than pure states.
Schrödinger’s famous cat paradox is a typical example
where a massive classical object was assumed to be a pure
state. It describes a counter-intuitive feature of quantum
physics which dramatically appears when the principle of
quantum superposition is applied to macroscopic objects.
In the original paradox and its various explanations, the
initial cat isolated in the steel chamber is considered a
pure state that can be represented by a state vector such
as |alive〉 (or a wave function such as ψalive). The cat
isolated from the environment is then assumed to inter-
act with a microscopic superposition state, (|g〉+|e〉)/
where |g〉 and |e〉 are the ground and excited states of a
two-level atom. The cat will be dead if the atom is found
in the excited state, |e〉, while it will remain alive if other-
wise. Thus in Schrödinger’s gedanken experiment the cat
is entangled with the atom as (|g〉|alive〉+ |e〉|dead〉)
where the alive and dead statuses of the cat are described
by the state vectors |alive〉 and |dead〉. If one mea-
sures out the atomic system on the superposed basis,
(|g〉 ± |e〉)/
2, the cat will be in a superposition of alive
and dead states such as (|alive〉± |dead〉)/
2. It is often
argued that such superposed states and entangled states
can theoretically exist but are virtually impossible to ob-
serve because one cannot perfectly isolate a macroscopic
object such as the cat from its environment [4].
However, this explanation is not fully satisfactory be-
cause the cat, a macroscopic object, is a complex open
system which cannot be represented by a state vector.
One may argue that the cat could be assumed to be in an
unknown pure state such that the cat was certainly alive
but the exact state of the cat was unknown. However,
the interactions between the cat and its environment can
cause the cat to become entangled with the environment
[5]. In such a case, even though one can perfectly iso-
late the cat in the steel chamber from the enviroment,
the cat will remain entangled with the environment due
to its pre-interactions with the environment. Therefore,
strictly speaking, even to assume a cat as an unknown
pure state in the steel chamber is not legitimate. Thus
a key point here is that it is unsatisfactory to describe
the cat by a pure state such as |alive〉 and |dead〉. We
may need a more realistic assumption that the “cat” in
Schrödinger’s paradox was in a significantly mixed clas-
sical state. An intriguing question is then whether the
quantum properties of the resulting state would still re-
main or diminish under such an assumption.
Recently, such an analogy of Schrödinger’s cat para-
dox, where the state corresponding to the virtual cat
is a significantly mixed thermal state, was investigated
[6]. A thermal state with a high temperature is consid-
ered a classical state in quantum optics. As the tem-
perature of the thermal state increases, the degree of
mixedness, which can be quantified by linear entropy,
rapidly approaches the maximum value. When the tem-
perature approaches infinity, the thermal state does not
show any quantum properties. As a comparison, coher-
ent states with large amplitudes are known as the most
classical pure states [7], and their superposition is of-
ten regarded as a superposition of classical states [8].
However, coherent states are still pure states which may
not well represent truly classical systems, and they dis-
play some nonclassical features [9]. In Ref. [6], it was
shown that prominent quantum properties can actually
be transferred from a microscopic superposition to a sig-
nificantly mixed thermal state (i.e. a thermal state of
which the degree of mixedness is close to the maximum
value) at a high temperature through an experimentally
http://arxiv.org/abs/0704.0523v2
feasible process. This result clarifies that unavoidable ini-
tial mixedness of the cat does not preclude strong quan-
tum phenomena.
One of the results in Ref. [6] is that quantum entan-
glement can be produced between thermal states with
nearly the maximum Bell-inequality violation when the
temperatures of both modes goes to infinity. In previous
related results, Bose et al. showed that entanglement
can arise when two systems interact if one of the system
are pure even when the other system is extremely mixed
[10]. There is an interesting previous example shown by
Filip et al. for the maximum violation of Bell’s inequal-
ity when one of the modes is an extremely mixed thermal
state [11]. Very recently, Ferreira et al. showed that en-
tanglement can be generated at any finite temperature
between high Q cavity mode field and a movable mirror
thermal state [12]. However, in these example [10, 11, 12]
only one of the modes is considered a large thermal state
[10, 11, 12] and entanglement vanishes in the infinite tem-
perature limit [10, 12], which is obviously in contrast to
the result presented in Ref. [6]. Entanglement for both
of the modes at the thermal limit of the infinitely high
temperature has not been found before. Remarkably, the
violation of Bell’s inequality in our examples reaches up
to Cirel’son’s bound [13] even in this infinite-temperature
limit for both modes. As Vedral [14] and Ferreira et al.
[12] pointed out it is believed that high temperatures re-
duce entanglement and all entanglement vanishes if the
temperature is high enough, which is obviously not the
case in Ref. [6].
The purpose of this paper is twofold. Firstly, we review
and further investigate various properties of superposi-
tions and entanglement of thermal states at high tem-
peratures [6]. In particular, we investigate two classes of
highly mixed symmetric states in the phase space. Both
the classes of these states do not show typical interference
patterns in the phase space while they manifest strong
singular behaviors. Interestingly, the first class of states
has neither squeezing properties nor negative values in
their Wigner functions, however, they are found to be
highly nonclassical states. The second class of states
has the maximum negativity in the Wigner function.
Further, we discuss the possibility of quantum informa-
tion processing with thermal-state qubits. We introduce
thermal-state qubits and thermal-Bell states, which are
a generalization of pure Bell states. We show that four
thermal-Bell states can be well discriminated by nonlin-
ear interactions without photon number resolving mea-
surements. Quantum teleportation and gate operations
for thermal-state qubits can be realized using the Bell
measurement scheme.
This paper is organized as follows. In Sec. II, we review
the generation process of superpositions of thermal states
and study their characteristics. In Sec. III, we study en-
tanglement of thermal states, i.e., Bell inequality viola-
tions. In Sec. IV, we discuss the possibility of quantum
information processing using thermal states. We first de-
fine the thermal-state qubit and the Bell-basis states us-
ing thermal-state entanglement. We then show that the
four Bell states can be well discriminated by homodyne
detection and two Kerr nonlinearities. It follows that
quantum teleportation and quantum gate operations can
be realized with thermal-state qubits. We conclude with
final remarks in Sec. V.
II. SUPERPOSITIONS OF THERMAL STATES
A. Generation of thermal-state superpositions
Let us first consider a two-mode harmonic oscillator
system. A displaced thermal state can be defined as
ρth(V, d) =
d2αP th(V, d)|α〉〈α| (1)
where |α〉 is a coherent state of amplitude α and
P thα (V, d) =
π(V − 1) exp[−
2|α− d|2
V − 1 ] (2)
with variance V and displacement d in the phase space.
The thermal temperature τ increases as V increases as
e~ν/τ = (V + 1)/(V − 1), where ~ is Planck’s constant
and ν is the frequency [15]. Suppose that a microscopic
superposition state
|ψ〉a =
(|0〉a + |1〉a), (3)
where |0〉 and |1〉 are the ground and first excited states
of the harmonic oscillator, interacts with a thermal state
ρthb (V, d) and the interaction Hamiltonian is
HK = λâ†âb̂†b̂ (4)
which corresponds to the cross Kerr nonlinear interac-
tion. The resulting state is then
ρentab =
d2αP th(V, d)
|0〉〈0| ⊗ |α〉〈α|
+ |1〉〈0| ⊗ |αeiϕ〉〈α|+ |0〉〈1| ⊗ |α〉〈αeiϕ|
+ |1〉〈1| ⊗ |αeiϕ〉〈αeiϕ|
and ϕ is determined by the strength of the nonlinearity
λ and the interaction time. The Wigner representation
of ρentab is
W entab (α, β) =
e−2|α|
W th(β; d) + 2αV c(β; d) + 2[αV c(β; d)]∗ + (4|α|2 − 1)W th(β; deiϕ)
where α and β are complex numbers parametrizing the phase spaces of the microscopic and macroscopic systems
respectively and
W th(α; d) =
exp[−2|α− d|
], (7)
V c(α; d) =
exp[− 2
(1 − eiϕ)d2 − 1
(α− 2e
)(α∗ − 2d
)], (8)
K = 2+ (V − 1)(1− eiϕ), J = (sinϕ/2 + iV cosϕ/2)/(2V sinϕ/2 + 2i cosϕ/2), and d has been assumed real without
loss of generality. If one traces ρentab over mode a, the remaining state will be simply in a classical mixture of two
thermal states and its Wigner function will be positive everywhere. However, if one measures out the “microscopic
part” on the superposed basis, i.e., (|0〉a ± |1〉a)/
2, the “macroscopic part” for mode b may not lose its nonclassical
characteristics. Such a measurement on the the superposed basis will reduce the remaining state to
ρsup(±) = N±s
d2αP th(V, d)
|α〉〈α| ± |αeiϕ〉〈α| ± |α〉〈αeiϕ|+ |αeiϕ〉〈αeiϕ|
, (9)
where N±s are the normalization factors, and its Wigner function is
W sup(±)(α) = N±s {W th(α; d)± V c(α; d) ± {V c(α; d)}∗ +W th(α; deiϕ)}. (10)
The ± signs in Eqs. (8) and (9) correspond to the two
possible results from the measurement of the microscopic
system. The state in Eq. (10) is a superposition of two
thermal states.
A feasible experimental setup to generate superposi-
tions of thermal states is atom-field interactions in cavi-
ties, where a π/2 pulse can be used to prepare the atom
in a superposed state. This type of experiment has al-
ready been performed to produce a superposition of co-
herent states [16]. In our cases, simply thermal states
can be used instead of coherent states. Another pos-
sible setup is an all-optical scheme with free-traveling
fields and a cross-Kerr medium, where a standard single-
photon qubit could be used as the microscopic superpo-
sition. Recently, there have been theoretical and experi-
mental efforts to produce and observe giant Kerr nonlin-
earities using electromagnetically induced transparency
[17]. Furthermore, it was shown that a weak Kerr non-
linearity can still be useful if a initially strong field is
employed in this type of experiment [18]. We shall fur-
ther explain this with examples in Sec. III.
B. Negativity of the Wigner function
The negativity of the Wigner function is known as an
indicator of non-classicality of quantum states. In order
to observe negativity of the Wigner function in a real
experiment, its absolute minimum negativity should be
large enough. The minimum negativity of the Wigner
function in Eq. (6) for V = 1 is −0.144 for d = 0 and
−0.246 for d → ∞. Now suppose the initial state can
be considered a classical thermal state by letting V ≫ 1.
One might expect that the negativity would be washed
out as the initial state becomes mixed, but this is not
the case. The minimum negativity actually increases as
V gets larger. If V → ∞, the minimum negativity of the
Wigner function (6) is −0.246 regardless of d: no matter
how mixed the initial thermal state was, the minimum
negativity of Wigner function is found to be a large value.
The point in the phase space which gives the minimum
negativity when V ≫ 1 or d ≫ 0 is (− 1
, 0) and has
negativity
Wneg ≡W entab (−
, 0) =
2(−2 + 1
exp[− 2d
. (11)
It can be shown that Wneg approaches −4/(π2
−0.246 when either d→ ∞ or V → ∞.
This effect is obviously due to the interaction between
the microscopic superposition and the macroscopic ther-
mal state. If the initial microscopic state is not super-
posed, e.g., |ψ〉a = |1〉a, the resulting state will be a
simple direct product, (|1〉〈1|)a ⊗ ρthb (V,−d). Whilst for
V = 1 this state will exhibit negativity, this is washed out
and tends to zero as V → ∞. Needless to say, if it was
|0〉a instead of |1〉a, the resulting Wigner function will be
a direct product of two Gaussian states whose Wigner
fucntion can never be negative. The superpositon state
(3) plays the crucial role in making the minimum negativ-
ity of the resulting Wigner function always saturate to a
certain negative value no matter how mixed and classical
the initial state of the other mode becomes.
− −0.04
100 0.15 0
0.040
0.005
0.015
FIG. 1: The probability distributions of x (left) and p (right)
for a “superposition” of two distant thermal states. A thermal
state with a large mixedness is converted to such a “thermal-
state superposition” by interacting with a microscopic super-
potion (see text). The variance V and displacement d for the
thermal state are chosen as (a) V = 100 and d = 100, and (b)
V = 1000 and d = 300. The fringe visibility is 1 regardless of
V and the fringe spacing (the distance between the fringes)
does not depend on the variance (i.e. mixedness) but only on
the distance d between the two component thermal states.
The Wigner functions of the single-mode states,
W sup(±)(α), in Eq. (10) show large negative values. The
minimum negativity of the Wigner function W sup(−)(α)
is W sup(−)(0) = 2/π regardless of the values of V and
d. On the other hand, the minimum negativity of the
Wigner function W sup(+)(α) approaches 2/π for d→ ∞
and disappears when d = 0.
C. Quantum interference in the phase space
When ϕ = π, the state (9) becomes
ρ± = N(ρth(V, d)±σ(V, d)±σ(V,−d)+ρth(V,−d)), (12)
where σ(V, d) =
d2αP th(V, d)| − α〉〈α| and
N = 2
exp[− 2d
. (13)
If the initial state for mode b is a pure coherent state,
i.e., V = 1, the measurement on the superposed basis for
mode a will produce a superposition of two pure coherent
states as
|Ψ̃±〉 =
1± e−2|α|2
(|α〉 ± | − α〉), (14)
where α = d. The probability P± to obtain the state ρ±
is obtained as [19]
P± = 〈ψ±|Trb[ρentab ]|ψ±〉 =
exp[− 2d2
), (15)
1998 2000 −2002 5 5
FIG. 2: The probability distributions P for a “superposition”
of thermal states where V = 5, d = 2000, ϕ = π/1000. The
x′ (p′) axis in this figure has been rotated by π/2000 from the
x (p) axis for clarity.
where |ψ±〉 = (|0〉±|1〉)/
2. The probability approaches
P± = 1/2 when either d or V becomes large.
As an analogy of Schrödinger’s cat paradox, the vari-
ance V corresponds to the size the initial “cat”, and the
distance d between the two thermal component states
corresponds to distinguishability between the “alive cat”
and the “dead cat”. Suppose that both V and d are very
large for the initial thermal state. The two thermal states
ρth(V,±d) become macroscopically distinguishable when
V , and our example may become a more realis-
tic analogy of the cat paradox in this limit. Both the
states ρ± in this case show probability distributions with
two Gaussian peaks and interference fringes [6]. Figure 1
presents the probability distributions of x (≡ Re[α]) and
p (≡ Im[α]) for ρ− (a) when V = 100 and d = 100 and
(b) when V = 1000 and d = 300. The probability dis-
tribution of x (p) for ρ± can be obtained by integrating
the Wigner function of ρ± over p (x). The two Gaussian
peaks along the x axis and interference fringes along the
p axis shown in Fig. 1 are a typical signature of a quan-
tum superposition between macroscopically distinguish-
able states. The visibility v of the interference fringes is
defined as [15]
Imax − Imin
Imax + Imin
, (16)
where I =
dxW sup(−)(α) and the maximum should be
taken over p. It can be simply shown that the visibil-
ity v is always 1 regardless of the value of V . Note that
d should increase proportionally to
V to maintain the
condition of classical distingushability between the two
component thermal states ρth(V,±d). The interference
fringes with high visibility are incompatible with classical
physics and evidence of quantum coherence. The fringe
spacing (the distance between the fringes) does not de-
pend on V but only on d, i.e., a pure superposition of
coherent states shows the same fringe spacing for a given
d. We emphasize that the states shown in Fig. 1 are
“superpositions” of severely mixed thermal states.
An experimental realization of a nonlinear effect cor-
responding to ϕ = π is very demanding particularly in
the presence of decoherence. Here we point out that the
method using a weak nonlinear effect (ϕ≪ π) combined
with a strong field (d≫ 1) [18] can be useful to generate a
thermal-state superposition with prominent interference
(a) (b) (c)
0.005
0.005
0.015
0.005
0.015
(d) (e) (f)
FIG. 3: (Color online) The time dependent Wigner functions of the thermal state of V = 100 at the origin (d = 0) after an
interaction with a microscopic superposition and a conditional measurement. The measurement result on the microscopic part
was supposed to be (|0〉 + |1〉)/
2. The interaction times are (a) θ = λt = 0, (b) θ = λt = π/32, (c) θ = π/16, (d) θ ≈ 3.102,
(e) θ ≈ 3.122 and (f) θ = π.
patterns. In Fig. 2, we have used experimentally acces-
sible values, V = 5, d = 2000 and ϕ = π/1000, but the
fringe visibility is still 1. In this case, decoherence during
the nonlinear interaction would be significantly reduced
because of the decrease of the interaction time [18]. Note
also that, if required, the state in Fig. 2 can be moved to
the center of the phase space, for example, using a biased
beam splitter (BS) and a strong coherent field [18].
D. Symmetric macroscopic quantum states
Let us assume that d = 0, i.e., the initial state is the
thermal state, ρth(V, 0), at the origin of the phase space.
In this case, the thermal-state superpositions, ρ±, are
produced with probabilities, P± = (1/2){1± (1/V )}, re-
spectively. Figure 3 shows the Wigner functions of ρ+
dependent on the interaction time between the macro-
scopic thermal state and the microscopic superposition
in a cross Kerr medium. The state is always symmetric
in the phase space regardless of the interaction time as
shown in Fig. 3. In this figure, the initial state is a ther-
mal state of V = 100 (Fig. 3(a)). In a relatively short
time (θ = π/32 and θ = π/16), the state shows some in-
terference patterns. When θ = π, the evolved state looks
very localized around the origin as shown in Fig 3. The
generated state at θ = π does not show negativity of the
Wigner function nor squeezing properties. On the other
hand, a well defined P function does not exist for this
state.
In the case of ρ−, with the same assumption d = 0, the
Wigner function at ϕ = π has the minimum negativity
(−2/π) at the origin regardless of V . As a result of the
interaction with the microscopic superposition, a deep
hole to the negative direction below zero has been formed
around the origin for ρ− as shown in Fig. 4 .
III. ENTANGLEMENT BETWEEN THERMAL
STATES
Entanglement between macroscopic objects and its
Bell-type inequality tests are an important issue. In this
section, we shall show that entanglement can be gener-
ated between high-temperature thermal states even when
the temperature of each mode goes to infinity.
−1 −0.5 0 0.5 1
FIG. 4: (Color online) The Wigner function of the thermal
state of V = 100 at the origin (d = 0) after an interac-
tion with a microscopic superposition and a conditional mea-
surement. The measurement result on the microscopic part
was supposed to be (|0〉 − |1〉)/
2 with the interaction time
θ = λt = π.
2001 400 600 800 1000
50 100 150
FIG. 5: (a) The optimized violation, B ≡ |B+|max, of Bell-
CHSH inequality for the “thermal-state entenglement”, ρ+,
of V = 1000 (solid curve) and V = 100 (dashed curve).
The Bell-violation of a pure entangled coherent state, i.e.,
V = 1, has been plotted for comparison (dotted curve). The
Bell-violation B approaches its maximum bound, 2
2, when
V regardless of the level of the mixedness V . (b)
The optimized Bell-violation B against d for the different type
of thermal-state entanglement generated using a 50:50 beam
splitter from ρ+. V = 1000 (solid curve), V = 100 (dashed
curve) and V = 1 (dotted curve).
A. Entanglement using two initial thermal states
If the microscopic superposition interacts with two
thermal states, ρthb (V, d) and ρ
c (V, d), and the micro-
scopic particle is measured out on the superposed basis,
the resulting state will be
ρtm(±) = Nt
ρth(V, d)⊗ ρth(V, d)± σ(V, d) ⊗ σ(V, d)
± σ(V,−d)⊗ σ(V,−d) + ρth(V,−d)⊗ ρth(V,−d)
where
Nt = 2
exp[− 4d
. (18)
Such two-mode thermal-state entanglement can be gener-
ated using two cavities and an atomic state detector [20].
Extending the two cavities to N cavities, entanglement
of N -mode thermal states can also be generated. Such
a state is an analogy of the N -mode pure GHZ state
[21] but each mode is extremely mixed. Here we shall
consider the Bell-CHSH inequality [22, 23] with photon
number parity measurements [20, 24]. The parity mea-
surements can be performed in a high-Q cavity using a
far-off-resonant interaction between a two-level atom and
the field [25]. The Bell-CHSH inequality can be repre-
sented in terms of the Winger function as [24]
|B(±)| = π
|W tm(±)(α, β) +W tm(±)(α, β′)
+W tm(±)(α′, β)−W tm(±)(α′, β′)| ≤ 2,
where W tm(±)(α, β) is the Wigner function of ρtm(±) in
Eq. (17). As shown in Fig. 5, the Bell-violation ap-
proaches the maximum bound for a bipartite measure-
ment, 2
2 [13], when d ≫
V regardless of the level of
the mixedness V , i.e., the temperatures of the thermal
states. Note that it is true for both of ρ+ and ρ− even
though only the case of ρ+ has been plotted in Fig. 5(a).
This implies that entanglement of nearly 1 ebit has been
produced between the two significantly mixed thermal
states for d ≫
V , and such “thermal-state entangle-
ment” cannot be described by a local theory.
B. Entanglement using a beam splitter
A different type of macroscopic entanglement can be
generated by applying the beam splitter operation
exp[θ/2(eiφâ†sâd − e−iφâ
dâs)], (20)
on the “thermal-state superpositions” in Eq. (9). The
state after passing through a 50:50 beam splitter can be
represented as
d2αP thα (V, d)
,− α√
〉 ± | − α√
,− α√
| ± 〈− α√
, (21)
200 400 600 800 1000
FIG. 6: The optimized Bell-violation B against V for the
slightly different type of thermal-state entanglement gener-
ated using a 50:50 beam splitter using ρ+ when d = 0.
3.13 3.14 3.15 3.16
3.13 3.14 3.15 3.16
FIG. 7: (a) The Bell-CHSH function B against θ (= λt) for
V = 1 (solid curve), V = 10 (dashed curve) and V = 20
(dotted curve) for d = 30. (b) The Bell-CHSH function for
d = 10 (solid curve), d = 20 (dashed curve) and d = 30
(dotted curve) for V = 10. The Bell violations are more
sensitive to the interaction time as either V or d increases.
where N is defined in Eq. (13). When d is large, this
state violates the Bell-CHSH inequality to the maximum
bound 2
2 regardless of the level of mixedness V as
shown in Fig. 5(b). Again, it is true for both of ρ+ and
ρ− even though only the case of ρ+ has been plotted
in Fig. 5(b). Furthermore, these states severely violate
Bell’s inequality even when d = 0 as V increases as shown
in Fig. 6. We have found that the optimized Bell violation
of these states approaches 2.32449 for V → ∞. Interest-
ingly, this value is exactly the same as the optimized
Bell-CHSH violation for a pure two-mode squeezed state
in the infinite squeezing limit [26]. Note that multilmode
entangled states can be generated using multiple beam
splitters.
It should be noted that the Bell violations are more
sensitive to the interaction time when either V or d is
larger. Figure 7 clearly shows this tendency. Therefore,
in order to observe the Bell violations using the mixed
state of V (and d) large, the interaction time in the Kerr
medium should be more accurate.
IV. QUANTUM INFORMATION PROCESSING
WITH THERMAL-STATE QUBITS
In this section, we discuss the possibility of quan-
tum information processing with thermal-state qubits
and thermal-state entanglement.
A. Qubits and Bell-state measurements
We introduce a thermal-state qubit
ρψ = |a|2ρth(V, d)± ab∗σ(V, d)± a∗bσ(V,−d) + |b|2ρth(V,−d),
where a and b are arbitrary complex numbers. The ba-
sis states, ρth(V, d) and ρth(V,−d), can be well discrimi-
nated by a homodyne measurement when d is larger than
V . The thermal state qubit (22) can be re-written as
d2αP thα (V, d)
a|α〉+ b| − α〉
a∗〈α| + b∗〈−α|
which can be understood as a generalization of the co-
herent state qubit, a|d〉+ b| − d〉, where |d〉 is a coherent
state of amplitude d. The thermal-state qubit (23) be-
comes identical to the coherent-state qubit when V = 1.
We also define four thermal-Bell states as
ρΦ(±) = Nt
ρth(V, d) ⊗ ρth(V, d)± σ(V, d)⊗ σ(V, d)± σ(V,−d)⊗ σ(V,−d) + ρth(V,−d)⊗ ρth(V,−d)
ρΨ(±) = Nt
ρth(V, d)⊗ ρth(V,−d)± σ(V, d) ⊗ σ(V,−d)± σ(V,−d)⊗ σ(V, d) + ρth(V,−d)⊗ ρth(V, d)
where Nt was defined in Eq. (18). The thermal-Bell states can be written as
ρΦ(±) = Nt
dα2dβ2P thα (V, d)P
β (V, d)
|α, β〉 ± | − α,−β〉
〈α, β| ± 〈−α,−β|
, (26)
ρΨ(±) = Nt
dα2dβ2P thα (V, d)P
β (V, d)
|α,−β〉 ± | − α, β〉
〈α,−β| ± 〈−α, β|
. (27)
Homodyne 
detector C
FIG. 8: A schematic of the thermal-Bell state measurement
(a) using photon number resolving detection and (b) using ho-
modyne measurements with cross-Kerr nonlinear interactions
(NL). See text for details.
For quantum information processing applications, it is
an important task to discriminate between the four Bell
states. Here we discuss two possible ways to discrimi-
nate between the thermal-Bell states (25). We shall only
briefly describe the first scheme using photon number re-
solving measurements and focus on the second scheme
using nonlinear interactions.
The first method is to simply use a 50-50 beam splitter
and two photon number resolving detectors as shown in
Fig. 8(a). This scheme is basically the same as the Bell-
state measurement scheme with pure entangled coherent
states [27, 28]. Let us suppose that the amplitude, d, is
large enough, i.e., d ≫
V . If the incident state was
ρΦ(+) or ρΦ(−), most of the photons are detected on de-
tector A in in Fig. 8(a). Meanwhile, most of the photons
are detected on detector B when the incident state was
ρΨ(+) or ρΨ(−). The average photon numbers between
the “many-photon case” and the “few-photon case” are
compared in Fig. 9. Furthermore, the states ρΨ(+) and
ρΦ(+) contain only even numbers of photons while ρΨ(−)
and ρΦ(−) contain only odd numbers of photons. There-
fore, all the four Bell states can be well discriminated by
analyzing numbers of photons detected at detectors A
and B. For example, if detector A detects many photons
while detector B detects few and the total photon num-
ber detected by the two detectors are even, this means
that state ρΦ(+) was measured by the thermal-Bell mea-
surement. The nonzero failure probability can be made
arbitrarily small by increasing d.
However, the average photon numbers of the thermal-
Bell states are high when V ≫ 1 and d≫ 1. In this case,
it would be unrealistic to use photon number resolving
detectors. It would be an interesting question whether
0.5 1 1.5 2 2.5 3
0.5 1 1.5 2 2.5 3
FIG. 9: The average photon number N for the “many-photon
case” (solid line) and the “few-photon case” (dashed line) for
V = 10 against d (a) when the input state is either ρΦ(+) or
ρΨ(+) and (b) when the input state is either ρΦ(−) or ρΨ(−).
these four thermal-Bell states can be distinguished by
classical measurements, such as homodyne detection, in-
stead of photon number resolving detection. Our alterna-
tive scheme employs cross-Kerr nonlinearities and single
photon detectors as shown in Fig. 8(b). Let us first sup-
pose that the input field was ρΦ(+). The incident two-
mode state passes through a 50-50 beam splitter, BS1.
The state after passing through the 50:50 beam splitter,
BS1, is
ρB = Nt
d2αd2βP thα (V, d)P
β (V, d)
|η,−ξ〉〈η,−ξ|
+ |η,−ξ〉〈−η, ξ|+ | − η, ξ〉〈η,−ξ| + | − η, ξ〉〈−η, ξ|
where η = (α+β)/
2 and ξ = (α−β)/
2. Two dual-rail
single photon qubits, |ψ+〉ee′ and |ψ+〉ff ′ , where
|ψ+〉 =
(|0〉|1〉+ |1〉|0〉), (29)
are prepared using two single photons and 50:50 beam
splitters, BS2 and BS3, as shown in Fig. 8(b). Then,
traveling fields at modes c and d interacts with those
of modes e and f , respectively, in cross-Kerr nonlinear
media. We suppose that the interaction time is t = π/λ,
and the resulting state is then
= UceUdfρ
ff ′U
df (30)
where Uce = exp[iπHKce/λ~] and ρq = |ψq〉〈ψq |. An ex-
plicit form of Eq. (30) can then be simply obtained using
the identity
Uce|α〉c|0〉e = |α〉c|0〉e,
Uce|α〉c|1〉e = | − α〉c|1〉e
where |α〉 is a coherent state. However, we omit such an
explicit expression in this paper for it is too lengthy.
After the nonlinear interactions, the qubit parts,
modes e, e′, f and f ′, should be measured with the mea-
surement basis
{|++〉, |+−〉, | −+〉, | − −〉} (32)
where |+ +〉 = |ψ+〉ee′ |ψ+〉ff ′ , | + −〉 = |ψ+〉ee′ |ψ−〉ff ′ ,
| − +〉 = |ψ−〉ee′ |ψ+〉ff ′ , | − −〉 = |ψ−〉ee′ |ψ−〉ff ′ , and
|ψ−〉 = (|0〉|1〉 − |1〉|0〉)/
2. This measurement can be
performed using two 50:50 beam splitters, BS4 and BS5,
and four detectors, A1, A2, B1 and B2, as shown in
Fig. 8(b). If detector A1 and B1 click, i.e., the mea-
surement result is | + +〉, the resulting state at modes c
and d is
ρ++ =
d2αd2βP thα (V, d)P
β (V, d)
(|η〉 + | − η〉)(〈η| + 〈−η|)
(|ξ〉+ | − ξ〉)(〈ξ| + 〈−ξ|)
Note that state ρ++ is not normalized, which implies that
the probability of obtaining the corresponding measure-
ment result is not unity. The probability of obtaining
this result is
P++ =
(V + 1)(V + e−
2(V 2 + e−
. (34)
When the result is either |+−〉 or | −+〉, the result is
〈ψ2|ρB
|ψ2〉 = 〈ψ3|ρB
|ψ3〉 = 0, (35)
which obviously means that the probability of the ob-
taining this result is zero. When the result is | −−〉, i.e.,
detector A2 and B2 click,
ρ−− =
d2αd2βP thα (V, d)P
β (V, d)
(|η〉 − | − η〉)(〈η| − 〈−η|)
(|ξ〉 − | − ξ〉)(〈ξ| − 〈−ξ|)
which is not normalized. The probability of obtaining
this result is
P−− =
(V − 1)(V − e− 4d
2(V 2 + e−
, (37)
and it can be simply verified that P+++P−− = 1. There-
fore, only the measurement results |++〉 and | −−〉 can
be obtained in the case of the input state ρΦ(+). This is
exactly the same for the case of ρΨ(+). In the same way,
it can be shown that if either the input state was ρΦ(−)
or ρΨ(−), only the measurement results |+−〉 and | −+〉
−20 −10 10 20
Probability
−20 −10 10 20
Probability
FIG. 10: (a) The probability distributions, P++
(solid curve)
and P++
(dashed curve), for homodyne measurements at de-
tector C. (b) The probability distributions, P++
(solid curve)
and P++
(dashed curve), for homodyne measurements at de-
tector C.
can be obtained. In other words, the parity of the to-
tal incoming state is perfectly well discriminated by the
measurements on single-photon qubits.
Subsequently, a homodyne measurement is performed
for mode c by homodyne detector C as shown in Fig. 8(b).
We assume that ideal homodyne measurements are per-
formed, i.e., when a homodyne measurement is per-
formed the state is projected onto eigenstate |x〉 of oper-
ator X with eigenvalue x, where
(a+ a†). (38)
Let us first consider the case when the measurement re-
sult for the single photon qubits is | + +〉. In this case,
the remaining state is ρ++ in Eq. (33). The probabil-
ity distribution P++
for the homodyne measurement at
detector C is
= 〈x|Trd[ρ++]|x〉 =
2 (e−V x
2 (V + 1)
. (39)
Note that the superscript, ++, denotes that the qubit
measurement result was |++〉, and the subscript, Φ(+),
denotes that the input state was ρΦ
. These notations
will be used also for the other cases in this section. The
same analysis can be performed for the other possible
measurement outcome | − −〉:
= 〈x|Trd[ρ−−]|x〉 =
2 (e−V x
2 − e−x
2 (V − 1)
. (40)
In the same way, for another input state, ρΦ(−), it is
straightforward to show:
= P++
, P−+
= P−−
, (41)
and P++
= P−−
= 0. On the other hand, if the input
state was ρΨ(+), the probability distributions P++
at detector C are
= 〈x|Trc[ρ++]|x〉 =
x{4d+(2+V 2)x}
(1+V 2)x2
V + 2e
2x(2d+x)
V + e
x(8d+x+V 2x)
V V ) + 1
, (42)
= 〈x|Trc[ρ−−]|x〉 =
x{4d+(2+V 2)x}
(1+V 2)x2
V − 2e
2x(2d+x)
V + e
x(8d+x+V 2x)
V V )− 1
. (43)
20 4 6 8 10 d
FIG. 11: The distinguishability Ps between states ρ
Ψ(+) and
ρΦ(+) by a homodyne measurement against for V = 10 (solid
curve) and V = 20 (dashed curve) against distance d. See
text for details.
It is straightforward to show for the other input state
ρΨ(−):
= P−−
, P−+
= P++
. (44)
The probability distributions P++
and P++
are plotted
in Fig. 10. Figure 10 shows that when the input state was
ρΦ(+) or ρΦ(−), the homodyne measurement outcome by
detector C, characterized by P++
and P−−
, is located
around the origin. However, when the input state was
ρΨ(+) or ρΨ(−), the homodyne measurement outcome by
detector C, characterized by P++
and P−−
, is located
far from the origin. Therefore, two of the Bell states,
ρΦ(+) or ρΦ(−), can be well distinguished from the other
two by the homodyne detector C for the case of the mea-
surement outcome |++〉. Finally, by combining the ho-
modyne measurement result and the qubit measurement
result, all four Bell states can be effectively distinguished.
For example, let us assume that the measurement out-
come of the single photon detectors was | + +〉 and the
homodyne detection outcome was around the origin, i.e.,
x ≈ 0. Then, one can say that state ρΨ(−) has been mea-
sured for the result of the thermal-Bell measurement.
As implied in Fig. 10, the overlaps between the proba-
bility distributions around the origin, P++
and P−−
and the other distributions, P++
and P−−
, are ex-
tremely small for a sufficiently large d. In other words,
the distinguishability by the homodyne detection rapidly
approaches 1 as d increases. As an example, we can cal-
culate the distinguishability between the states ρΨ(+) and
ρΦ(+) by the homodyne measurement by detector C. The
distinguishability by homodyne detection is
|x|<d
dxP++c (x) +
|x|≥d
dxP++d (x)
which is plotted in Fig. 11. The distinguishability is Ps ≈
0.99 for d = 5.5 (d = 7.8) when V = 10 (V = 20), and
it becomes as high as Ps > 0.99999 for d = 10 (d = 15)
when V = 10 (V = 20). If necessary, another homodyne
measurement can be performed for mode d to enhance
distinguishability of the Bell measurement. When the
probability distribution at detector C is around the origin
that of detector D is far from the origin and vice versa.
Note also that the second scheme using homodyne de-
tection is robust to detection inefficiency compared with
the first scheme using photon number resolving measure-
ments. In the first scheme, even if a detector misses only
one photon, it will result in a completely wrong mea-
surement outcome. In the second scheme, however, the
measurement outcome will not be affected in that way. If
a single photon detector misses a photon, it will be imme-
diately recognized. Such a case can simply be discarded
so that it will only degrade the success probability of the
Bell measurement. The homodyne detection inefficiency
will not significantly affect the result when the distribu-
tions around the origin and the distributions far from the
origin are well separated, i.e., when d ≫
V , as shown
in Fig. 10. On the other hand, loss in the Kerr medium
will have a detrimental affect.
B. Quantum teleportation and computation
Quantum teleportation of a thermal-state qubit can be
performed using one of the Bell states as the quantum
channel. Let us assume that Alice needs to teleport a
thermal-state qubit, ρψ, to Bob using a thermal-state
entanglement, ρΨ(−), shared by the two parties. The total state can be represented as
1 ⊗ ρ
23 = Nt
dα2dβ2dγ2P thα (V, d)P
β (V, d)P
γ (V, d)
(a|α〉+ b| − α〉)1(|β,−γ〉 − | − β, γ〉)23
. (46)
Alice first needs to perform the thermal-Bell measure-
ment described in the previous subsection. To complete
the teleportation process, Bob should perform an appro-
priate unitary transformation on his part of the quantum
channel according to the measurement result sent from
Alice via a classical channel. It is straightforward to show
that the required transformations are exactly the same
to those for the coherent-state qubit [27]. When the mea-
surement outcome is ρΨ(−), Bob obtains a perfect replica
of the original unknown qubit without any operation.
When the measurement outcome is ρΦ(−), Bob should
perform |α〉 ↔ | − α〉 on his qubit in Eq. (23). Such a
phase shift by π can be done using a phase shifter whose
action is described by P (ϕ) = eiϕa
†a, where a and a† are
the annihilation and creation operators. When the out-
come is ρΨ(+), the transformation should be performed
as |α〉 → |α〉 and | − α〉 → −| − α〉. It is known that the
displacement operator is a good approximation of this
transformation for d ≫ 1 [29]. This transformation can
also be achieved by teleporting the state again locally
and repeating until the required phase shift is obtained
[30]. When the outcome is ρΦ(+), σx and σz should be
successively applied.
V. CONCLUSION
In this paper, we have studied characteristics of su-
perpositions and entanglement of thermal states at high
temperatures and discussed their applications to quan-
tum information processing. The superpositions and en-
tanglement of thermal states show various nonclassical
properties such as interference patterns, negativity of the
Wigner functions, and violations of the Bell-CHSH in-
equality. The Bell violations are more sensitive to the
interaction time during the generation process when the
thermal temperature (i.e. mixedness) of the thermal-
state entanglement is larger. Therefore, in order to ob-
serve the Bell violations using the mixed state at a high
temperature, the interaction time in the Kerr medium
should be accurate. We have pointed out that certain
superpositions of high-temperature thermal states, sym-
metric in the phase space, can also be generated. Some
of these states have neither squeezing properties nor neg-
ative values in their Wigner functions but they are found
to be highly nonclassical.
We have introduced the thermal-state qubit and
thermal-Bell states for applications to quantum informa-
tion processing. We have presented two possible methods
for the Bell-state measurement. The Bell-state measure-
ment enables one to perform quantum teleportation and
gate operations for quantum computation with thermal-
state qubits. The first scheme uses two photon number
resolving detectors and a 50-50 beam splitter to discrim-
inate the thermal-Bell states. Using the second scheme,
it is possible to effectively discriminate the thermal-Bell
states without photon number resolving detection. The
required resources for the second scheme are two Kerr
nonlinear interactions, two single photon detectors, two
50:50 beam splitters and one homodyne detector. The
second scheme is more robust to inefficiency of the de-
tectors: the inefficiency of the single photon detectors
only degrades the success probability of the Bell mea-
surement.
Acknowledgments
This work was supported by the DTO-funded U.S.
Army Research Office Contract No. W911NF-05-0397,
the Australian Research Council and Queensland State
Government.
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http://arxiv.org/abs/quant-ph/0504186
http://arxiv.org/abs/quant-ph/0410210
