Acta Polytechnica


https://doi.org/10.14311/AP.2025.65.0515
Acta Polytechnica 65(5):515–519, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence

Published by the Czech Technical University in Prague

UNITARY GENERATION OF GHZ STATES

Arno Bandillaa, Goce Chadzitaskosb,∗, Igor Jexb

a Humboldt-Universität zu Berlin, Institut für Physik, Arbeitsgruppe “Nichtklassische Strahlung”,
Invalidenstr. 110, 10115 Berlin, Germany

b Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Jugoslávských
partyzánů 3, 160 00 Prague, Czech Republic

∗ corresponding author: chadzgoc@cvut.cz

Abstract. Entangled states represent fascinating elements of quantum mechanics and quantum
technology. Their generation is an interesting problem combining many techniques developed in
quantum optics. Using algebraic properties of special transforms we derive the explicit form of the
effective Hamiltonian that enables the generation of highly nonclassical states of the GHZ type. Such
Hamiltonians belong to the higher order Hamiltonians often discussed in nonlinear quantum optics.
We derive the form of the effective Hamiltonian that enables the generation of highly nonclassical states
of the GHZ type.

Keywords: Quantum optics.

The basic idea of the manuscript was the result of the
discussions with Dr. A. Bandilla. He passed away

some time ago, however, his ideas and dedication to
physics are still with us.

1. Introduction
GHZ states and their generalizations found their ap-
plications far beyond the initial intension proving non-
classicality without statistical tests. GHZ states are
at present used for the design of multipartite commu-
nication across quantum networks with several parties
involved, multipartite secret sharing. The GHZ states
are also considered elementary building blocks for
quantum computing where out of these states more
complicated states are formed on which, or by means
of, computations are carried out. There are multi-
partite (higher than three parties) generalizations of
GHZ states. They are becoming indispensable for
the latest developments in quantum computing and
quantum technologies [1, 2]. Among other studies, the
generation of GHZ states is a useful subject of theories.

The generation and manipulation of quantum states
is one of the standard tasks for physics, particularly in
quantum optics. There are several reasons for being
interested in the problem of quantum state genera-
tion. Firstly, specially prepared quantum states are
tools for performing experiments and measurements.
Certain experiments can be performed only with the
aid of certain classes of states. The sensitivity and
precision of measurements is dependent on the “qual-
ity” of the used states. Quantum states can be by
itself of interest. On the level of quantum states two
of the crucial properties of quantum mechanics can be
demonstrated [3, 4]. First of all the quantum theory
enables the linear superposition of allowed states. This
quite obvious statement gets an exotic flavour when
we begin to be interested on states that are almost

classical. It is not too difficult to prepare a (Glauber)
coherent state with an almost arbitrary amplitude.
However, it is an experimental challenge to prepare
the (coherent) superposition of two coherent states
with two different large amplitudes. Such a state was
named Schrödinger cat in the recent literature. It ex-
emplifies how difficult it is to preserve the quantum
coherence on the “classical” level as such superposi-
tions are extremely fragile towards dissipation and
leads almost inevitably into the real of decoherence
in quantum theory. The other important feature of
quantum states is the entanglement [5]. When we
deal with systems composed from several quantum
objects the quantum state of the whole system can
exhibit correlations that have no classical analogue.
This feature is at the heart of quantum theory and
was pointed out already by the founders of quantum
theory. So since the early days of quantum mechanics
considerable effort was invested into the generation
of entangled pairs of particles (atoms, photons). This
couples of objects were named Bell pairs and with
their aid several experiments have been performed to
test quantum mechanics, especially to discriminate
between quantum mechanics and alternative local hid-
den variable theories. The Bell pairs are states having
strong two-particle quantum correlations. It is an
interesting task to search for generalizations involving
more then two quantum objects [6]. This might be
atoms, photons or their combinations. The higher
order analogues of Bell pairs have been named GHZ
state describing a special superposition of three parti-
cles. It enables a more sophisticated check of quantum
mechanical predictions. Recently, several proposals
have been put forward to solve the problem how to
generate such states and some of them have been re-
alized experimentally. The aim of the present paper
is a proposal of one additional method to this list.

515

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A. Bandilla, G. Chadzitaskos, I. Jex Acta Polytechnica

2. Quantum state generation
The various schemes of quantum state generation com-
bines in principle two different building blocks. The
first one is a unitary transformation between two given
states (the source and the target), the other one is the
projection of the desired state from an ensemble of
states [7]. As a third case we could point out the shap-
ing of a specially chosen potential in which the desired
state (wave function) would be defined [8]. However,
this implementation is technically quite difficult to
realize and implicitly incorporates the two previous
ones.

The first case is a unitary transformation of an
initial state. Let us suppose we wish to generate
the target state |ψout⟩. To generate this particular
state by unitary transformation we have to specify the
initial state |ψi⟩ and the unitary transformation Û .
The two states are then related:

|ψout⟩ = Û |ψi⟩. (1)

The problem to get the final state is transferred to
the choice of the initial state and the transformation
operator. Let us, for the illustration, limit our con-
sideration to optical field or to harmonic oscillators.
As a natural choice we can choose the vacuum state
of the harmonic oscillator |0⟩ for the initial state |ψi⟩.
Then we have to specify the unitary transformation
with respect to this choice. In quantum optics a large
class of final states is easily related to the vacuum.

These are the coherent and squeezed states [9]. The
unitary operators in the first case is:

Û = exp{α↠− α∗â}, (2)

where â, ↠are the annihilation and creation operators
of the harmonic oscillator. For the squeezed states
the transformation is defined as:

Û = exp{ηâ†2 − η∗â2}. (3)

Both transformations can be induced by a Hamilto-
nian that is at most quadratic in the annihilation and
creation operators:

Ĥ = κl(â+ â†) + κn(â2 + â†2) (4)

and describes the two basic (effective) models in quan-
tum optics – the driven oscillator and the parametric
down-conversion. In both cases we deal with para-
metric processes, i.e. a strong external field drives
a medium that can be effectively described by the
Hamiltonian just given. In general, the basic limita-
tion of this method is the rather limited spectrum of
interaction that can be used and so this limits also
the possible states |ψout⟩. The interactions are mainly
bounded by the possibilities of atom field interaction.
In addition, when a certain type of interaction is al-
lowed, it needs not to be very efficient, i.e. it might
be difficult to obtain (from the experimentalists point
of view) a “strong enough output”.

The second possibility is the use of projection tech-
niques. This method is commonly used in cavity
electrodynamics experiments. The first step is the
preparation of an appropriate state. Let us assume we
have a state |ψi⟩ and we wish to prepare a state |ϕ⟩.
The initial state can be in the number state basis
expressed in the form:

|ψi⟩ =
∞∑

n=0
cn|n⟩. (5)

By applying a projection operator |ϕ⟩⟨ϕ| on the
state |ψ⟩ we can project out the desired number state.
In the optical domain it is almost impossible to re-
alize such general projectors. To manipulate state
by projection techniques is an alternative used way.
The field to be manipulated is coupled to another
system (atom) and by projecting (manipulating) on
the atoms we reshape the field state. The basic model
of atom field interaction in a cavity is the Jaynes-
Cumming model [10, 11]. It describes the interaction
of a two-level atom with a single–mode cavity field.
The (interaction) Hamiltonian has the form:

Ĥ = κ(âσ++ â†σ−). (6)

For an initially excited atom (described by the
state |+⟩) and a cavity field with the number state ex-
pansion |ψc⟩ =

∑
cn|n⟩ the state of the whole system

at time t is given as:

|ψ(t)⟩ = exp(−iĤt)|ψ(0)⟩

=
∞∑

n=0
cn{cos(κ

√
n+ 1t)|n⟩|+⟩

− i sin(κ
√
n+ 1t)|n+ 1⟩|−⟩}.

(7)

When me measure the atom in the excited state the
field collapses into the corresponding state:

|ψ(t)⟩ =
∞∑

n=0
cn cos(κ

√
n+ 1t)|n⟩. (8)

For a strong enough initial coherent state (choice of
the expansion parameters cn) and sufficient interac-
tion time the just given state can be expressed as
a superposition of two coherent states with identical
absolute value of the amplitude but with different
phases, i.e. it has the form of a “Schödinger cat”. The
just described procedure gives the heart of the basic
of state engineering in a cavity. By sending a sequence
of atoms into the cavity (not necessarily in the excited
state but in a controlled superposition) we can force
the cavity field into any prescribed superposition. The
basic drawback of this form of engineering is that it
depends crucially on the outcome of the conditional
measurement. When the favourable outcome is not
received we have to either start the whole procedure
anew or we have to prepare the whole strategy with
the inclusion of correction steps which naturally makes

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vol. 65 no. 5/2025 Unitary generation of GHZ states

the procedure longer, because it contains more steps.
This can make the method more fragile towards losses
for instance.

In the next Section we will analyse the possibility
to generate highly entangled states of more then two
“particles” using methods of nonlinear optics. We
will start from entangled pairs of particles and ask
for unitary transformations that enable to generate
chains of this elementary entangled units.

3. GHZ state generation
Progress in nonlinear optics enabled to generate in
a rather efficient way entangled photons forming the
Bell-pairs. The state is generated via spontaneous
nondegenerate parametric down-conversion and can
be used for the creation of higher order entangled
states like for instance GHZ states. In the following
we will identify the Hamiltonian that enables to couple
entangled states to each other in an unitary fashion.
The Bell-state emerging from a nonlinear crystal can
be written as:

|ψB⟩ = 1√
2

{|↑a⟩|↓b⟩ + |↓a⟩|↑b⟩}, (9)

where the states |↑x⟩, |↓x⟩ describe the polarization
state of the entangled modes a and b. The Bell-pair
state can be rewritten in the (scalar) harmonic os-
cillator bases. We use always two scalar modes for
one mode with polarization. The Bell state can be
written as:

|ψB⟩ = 1√
2

{|11⟩|02⟩|13⟩|04⟩ + |01⟩|12⟩|03⟩|14⟩}. (10)

We assume that we are able to mix this field with an
additional photon in the state:

|ψP ⟩ = 1√
2

{|15⟩|06⟩ + |05⟩|16⟩}. (11)

We use the product state:

|ψin⟩ = |ψB⟩|ψP ⟩ (12)

as the input state for an unitary transformation to
generate the desired GHZ state. It can be a passive
transformation as the initial state |ψin⟩ has the suffi-
cient number of excitations. The GHZ state is in our
notation as:

|ψGHZ⟩ = 1√
2

{|11⟩|02⟩|13⟩|04⟩|15⟩|06⟩

+ |01⟩|12⟩|03⟩|14⟩|15⟩|06⟩}.
(13)

The input state |ψin|⟩ can be written as the coherent
superposition of two GHZ states, namely:

|ψin⟩ = |ψB⟩|ψP ⟩ = 1√
2

(|ψGHZ⟩ + |ψGHZ1⟩), (14)

where the second GHZ state takes the form:

|ψGHZ1⟩ = 1√
2

{|11⟩|02⟩|13⟩|04⟩|05⟩|16⟩

+ |01⟩|12⟩|13⟩|04⟩|15⟩|06⟩}.
(15)

It is easily checked that the two states are mutually
orthogonal, i.e. ⟨ψGHZ |ψGHZ1⟩ = 0. Due to this fact
we have to search for an orthogonal transformation Û
which realizes the transformation:

|ψGHZ⟩ = Û
1√
2

(|ψGHZ⟩ + |ψGHZ1⟩) (16)

in the basis of the two GHZ states this transformation
is an rotation, i.e. an SU(2) transformation. In matrix
notation this transformation can be written in the
form:

U = 1√
2

(
1 1

−1 1

)
. (17)

Because this transformation is cyclic, i.e. U2 = 1
we can formally associate a Hamiltonian with the
transformation in the form:

H = π

2 (1 − U). (18)

This is still written in the basis of the GHZ states.
We need to rewrite the transformation operator in
the basis of creation and annihilation operators of the
participating modes âi, â

†
i . A brief inspection reveals

that the transformation can be expressed in terms of
one photon projection operators in the Bell state |ψB⟩,
and the transformation which interchange the two
photon state of the additional entangle state |ψP ⟩:

U = 1√
2

[1 + (â†
5â6 − â5â

†
6)(â†

i âi − â†
j âj)]. (19)

The index i can take values 1 or 3 and the index j
the values 2 or 4. It guarantees the projection of Bell
state |ψB⟩ on the first or second GHZ states, whose
superposition the Bell state is. These relations express
the Hamiltonian in terms of mode of creation and an-
nihilation operators. The Hamiltonian belongs to the
class of third order nonlinear optical Hamiltonians.
It could be realized either within the frame of inter-
acting optical modes in an nonlinear medium or in
systems with trapped ions. The interaction naturally
expresses also effective all the processes that generate
entangled states from nonentangled states. Note that
this strategy is possible to use for quite a general class
of states. We use the observation that only four modes
out of six are involved in the transformation. We can
consider generally, the initial state is the state that
has the form:

|ψin⟩ = |ψ1⟩|ψ2⟩, (20)

where:

|ψ1⟩ = 1√
2

(|Φ10⟩|13⟩|04⟩ + (|Φ01⟩|03⟩|14⟩), (21)

and:
|ψ2⟩ = 1√

2
(|15⟩|06⟩ + (|05⟩|16⟩). (22)

Because all involved states are mutually orthogonal
as those in the original GHZ state generation the
same procedure can be applied and the result for the

517



A. Bandilla, G. Chadzitaskos, I. Jex Acta Polytechnica

unknown Hamiltonian will be the same. The results
after the action of the unitary transformation:

U = 1√
2

[1 + (â†
5â6 − â5â

†
6)(â†

3â3 − â†
4â4)], (23)

on the initial state will be:

|ψout⟩ = 1√
2

(|Ψ10⟩|13⟩|04⟩|15⟩|06⟩

+ |Ψ01⟩|03⟩|14⟩|05⟩|16⟩).
(24)

This means also that the original Bell states can be
generated in this manner. However, the order of the
nonlinearity is in such a case higher. The reason is
that in this case we start initially with a vacuum and
we need a parametric process to pump energy into the
manipulated system.

In connection with the more general additional
state:

|ψ3⟩ = 1√
2

(|Ψ20⟩|15⟩|06⟩ + |Ψ02⟩|05⟩|16⟩), (25)

we might ask for the transformation that will decom-
pose the initial state:

|ψin3⟩ = |ψ1⟩|ψ3⟩ (26)

into the form:

|ψout1⟩ = 1
2(|Ψ10⟩|Ψ20⟩|13⟩|04⟩|15⟩|06⟩

+ |Ψ01⟩|Ψ02⟩|03⟩|14⟩|05⟩|16⟩).
(27)

We can proceed along the same type of argumen-
tation as we done before. However, to get an explicit
form we would have to specify in more detail the form
of the states |Ψij⟩ to find the transformation operator
between |Ψ02⟩ and |Ψ20⟩ and to express it in the form
the annihilation and creation operators. If we have
such transformation:

T (|Ψ02⟩) = |Ψ20⟩, (28)

then the asking operator is:

U = 1√
2

[1 + (T â†
5â6 −T−1 â5â

†
6)(â†

3â3 − â†
4â4)], (29)

so:
U (|ψin3⟩) = |ψout1⟩. (30)

Applying the transformation operator:

U = 1√
2

(1 − T−1 â†
3â3â

†
6â5 − T â†

4â4â
†
5â6

+ T−1 â†
5â5â

†
3â4 + T â†

6â6â
†
4â3)

(31)

on the initial state |ψin3⟩, we get the decomposition:

|ψout2⟩ = 1
2(|Ψ01⟩|Ψ02⟩)(|13⟩|04⟩|15⟩|06⟩

+ |Ψ10⟩|Ψ20⟩ + |03⟩|14⟩|05⟩|16⟩).
(32)

For Bell-type states the already derived Hamilto-
nian would work well. However, for different states
it will lead to Hamiltonians of higher order and will
involve nonlinearities of higher order.

Entanglement presents one of the most fascinating
parts of quantum mechanics. It is currently and very
close inspection as it is not only of theoretical but
also of practical interest. Entanglement is the essence
of quantum computing and secret information trans-
mission (quantum cryptography). It can be expected,
that at least some of the effects in the realm of entan-
glement will become used in non quantum life in the
future.

4. Conclusion
Entanglement presents one of the most fascinating
parts of quantum mechanics. It is currently and very
close inspection as it is not only of theoretical but also
of practical interest. Entanglement is the essence of
quantum computing and secret information transmis-
sion (quantum cryptography). Entanglement is rather
commonly associated with the microscopic world. It
can be expected, that at least some of the effects in
the realm of entanglement will become used in non-
quantum life in the future. In the present manuscript
we reported on a way how the enigmatic entangled
GHZ state can be generated. We derived an explicit
form for generating optical GHZ states and discussed
some details. The corresponding Hamiltonians (or
generating operators) have the form of sums of higher
order products of annihilation and creation (harmonic
oscillator) operations indicating situation when such
transform arise.

Acknowledgements
The work was supported by the Czech Science Founda-
tion (GAČR) project number 23-07169S, RVO14000, and
by the project “Centre for Advanced Applied Sciences”,
Registry No. CZ.02.1.01/0.0/0.0/16_019/0000778, sup-
ported by the Operational Programme Research, Develop-
ment and Education, co-financed by the European Struc-
tural and Investment Funds and the state budget of the
Czech Republic. The support by the European Union’s
research and innovation program under EPIQUE Project
GA No. 101135288 is acknowledged.

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	Acta Polytechnica 65(5):515–519, 2025
	1 Introduction
	2 Quantum state generation
	3 GHZ state generation
	4 Conclusion
	Acknowledgements
	References