Hamiltonian Quantum Dynamics With
Separability Constraints
Nikola Burić ∗
Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia.
November 9, 2018
Abstract
Schroedinger equation on a Hilbert space H, represents a linear
Hamiltonian dynamical system on the space of quantum pure states,
the projective Hilbert space PH. Separable states of a bipartite quan-
tum system form a special submanifold of PH. We analyze the Hamil-
tonian dynamics that corresponds to the quantum system constrained
on the manifold of separable states, using as an important example
the system of two interacting qubits. The constraints introduce non-
linearities which render the dynamics nontrivial. We show that the
qualitative properties of the constrained dynamics clearly manifest the
symmetry of the qubits system. In particular, if the quantum Hamil-
ton’s operator has not enough symmetry, the constrained dynamics is
nonintegrable, and displays the typical features of a Hamiltonian dy-
namical system with mixed phase space. Possible physical realizations
of the separability constraints are discussed.
PACS: 03.65.-w
∗e-mail: buric@phy.bg.ac.yu
http://arxiv.org/abs/0704.1359v1
1 Introduction
Classical and quantum descriptions of a physical system that is considered
as composed of interacting subsystems have radically different features. The
typical feature of quantum dynamics is the creation of specifically quantum
correlations, the entanglement, among the subsystems. On the other hand,
the typical property of classical description is the occurrence of chaotic or-
bits and fractality of the phase space portrait, which can be considered as
typically classical type of correlations between the subsystems. The type
of correlations introduced by the dynamical entanglement does not occur in
the classical description, and likewise, the type of correlations introduced
by the chaotic orbits with fractal structures does not occur in the quantum
description. This intriguing complementarity of the two descriptions repre-
sents a problem that is expected to be solved by a detailed formulation of
the correspondence principle.
Comparison of typical features of classical and quantum mechanics is fa-
cilitated if the same mathematical framework is used in both theories. It is
well known, since the work of Kibble [1],[2],[3], that the quantum evolution,
determined by the linear Schroedinger equation, can be represented using the
typical language of classical mechanics, that is as a Hamiltonian dynamical
system on an appropriate phase space, given by the Hilbert space geometry
of the quantum system. This line of research was later developed into the
full geometric Hamiltonian representation of quantum mechanics. [4]-[12].
Such geometric formulation of quantum mechanics has recently inspired nat-
ural definitions of measures of the entanglement [13], and has been used to
model the spontaneous collapse of the state vector [14],[15], and dynamics of
decoherence [16].
It is our goal to use the geometric Hamiltonian formulation of quantum
mechanics to study the relation between the dynamical entanglement and
typical qualitative properties of Hamiltonian dynamics. Motivated by the
fact that the Schroedinger equation can always be considered as a Hamil-
tonian dynamical system, and that for Hamiltonian systems the definitions
and properties of the dynamical chaos are well understood, we shall seek
for a formal condition that when imposed on the Hamiltonian system repre-
senting the Schroedinger equation of the compound quantum system renders
the Hamiltonian dynamics nonintegrable and chaotic. It is well known that
the linear Schroedinger equation of quantum mechanics represents always
an integrable Hamiltonian dynamical system, irrespective of the dynamical
symmetries of the system. This is in sharp contrast with the Hamiltonian
formulation of classical systems, where enough symmetry implies integrabil-
ity and the lack of it implies the chaotic dynamics. Linearity of the quantum
Hamiltonian dynamics, and the consequent integrability, is introduced in the
Hamiltonian formulation by a very large dimensionality of the phase space of
the quantum system. This high dimensionality can be considered as a conse-
quence of two reasons. For a single quantum system, say a one dimensional
particle in a potential, linear evolution and with it the principle of state
superposition require infinite dimensional phase space of the Hamiltonian
formulation. If the classical mechanical model is linear, say the harmonic os-
cillator, the quantum Hamiltonian dynamics can be exactly describe on the
reduced finite-dimensional phase space, the real plane in the case of the har-
monic oscillator. The other related reason that increases the dimensionality
of the quantum phase space compared to the classical model is in the way
the state space of the compound systems are formed out of the components
state spaces in the two theories. In order to represents the entangled states as
points of the quantum phase space the dimensionality of the quantum phase
space is much larger than just the sum of the dimensions of the components
phase spaces. The points in the Cartesian product of the components phase
spaces represent the separable quantum states and form a subset of the full
quantum phase space. Needles to say, although the separable states are the
most classical-like states of the compound system, they still are quantum
states with nonclassical properties like nonzero dispersion of some subsys-
tem’s variables. Our main result will be that when the quantum dynamics,
represented as a Hamiltonian system, is constrained on the manifold of sepa-
rable quantum states the relation between the symmetry and the qualitative
properties of the dynamics such as integrability or chaotic motion is reestab-
lished. Thus, suppression of dynamical entanglement is enough to enable
manifestations of the qualitative differences in dynamics of quantum systems
and the relation between integrability and symmetry, traditionally related
with classical mechanical models.
In order to study the relation between the dynamical entanglement, sep-
arability and the properties of Hamiltonian formulation of the quantum dy-
namics we shall use, in this paper, the simplest quantum system that displays
the dynamical entanglement, that is a system of two interacting qubits:
H = ωσ1 + ωσ2 + µxσx
2 + µyσy
2 + µzσz
2, (1)
where σx,y,z
i are the three Pauli matrices of the i-th qubit, and satisfy the
usual SU(2) commutation relations. In particular we shall compare the dy-
namics of the system (1) in the case µz 6= 0, µx = µy = 0 with the case
when µx 6= 0, µz = µy = 0. The former case is symmetric with respect to
SO(2) rotations around z-axis and the later lacks this symmetry. Besides
its simplicity, the systems of the form (1) are of considerable current interest
because the hamiltonian of the universal quantum processor is of this form
[17],[18].
Various lines of research, during the last decade, improved the under-
standing of the relation between dynamical entanglement and properties of
the dynamics. Strong impetus to the study of all aspects of quantum entan-
glement came from the theory of quantum computation [18]. Quantization
of classical non-integrable systems, and various characteristic properties of
resulting quantum systems, have been studied for a long time [19]. The de-
pendence of the dynamical entanglement, between a quantum system and its
environment, on the qualitative properties of the dynamics of the system was
studied indirectly, within the theory of environmental decoherence [20]. The
relation between the rates of dynamical entanglement and the qualitative
properties of the dynamics in the semi-classical regime was initiated in the
reference [21] and various aspects of this relations have been studied since
[22]-[31]. The relation between the symmetry of the genuinely quantum sys-
tem (1) and the degree of dynamical entanglement was studied in reference
[32]. As we shall see, our present analyzes is related to the quoted works,
but the relation between the dynamical entanglement and symmetry is here
approached from a very different angle
The structure of the paper is as follows. We shall first recapitulate the
necessary background such as: the complex symplectic and Riemannian ge-
ometry of CP n; Hamiltonian formulation on CP n of the quantum dynamics;
geometric formulation of the set of separable pure states and Hamiltonian for-
mulation of the constrained dynamics. In parallel with the general reminder,
the explicit formulas for the system of two interacting qubits will be given.
These are then applied, in section 3, to the study the qualitative properties
of the separability constrained dynamics for the qubits systems. The main
results are summarized and discussed in section 4. There we also discuss a
model of an open quantum system with dynamics that clearly differentiates
between the symmetric and the nonsymmetric systems.
2 Geometry of the state space CP n
Hamiltonian formulation of quantum mechanics is based on the fact that the
scalar product of vectors |ψ > in the Hilbert space of a quantum system can
be used to represent the linear Schroedinger equation of quantum mechanics
in the form of Hamilton’s equations. The canonical phase space structure of
this equations is determined by the imaginary part of the scalar product, and
the Hamilton’ s function is given by the quantum expectation < ψ|H|ψ > of
the quantum hamiltonian.
However, due to phase invariance and arbitrary normalization the proper
space of pure quantum states is not the Hilbert space used to formulate the
Schroedinger equation, but the projective Hilbert space which is the manifold
to be used in the Hamiltonian formulation of quantum mechanics. In general,
the resulting Hamiltonian dynamical system is infinite-dimensional, but we
shall need the general definitions only for the case of quantum system with
finite-dimensional Hilbert space, like the finite collection of qubits, in which
case the quantum phase space is also finite-dimensional. We shall first review
the definition of the complex projective space CP n, and then briefly state
the basic definitions and recapitulate the formulas which are needed for the
Hamiltonian formulation of the quantum dynamics on the state space and
its restriction on the separable state subset. The general reference for the
mathematical aspects of complex differential geometry is [33]. All concepts
and formulas will be illustrated using the system of two interacting qubits.
Differential geometry of the state space CP n is discussed by viewing it
as a real 2n dimensional manifold endowed with complex, Riemannian and
symplectic structure. In the case ofCP n this three structures are compatible.
2.1 Definition and intrinsic coordinates of CP n
States of a collection of N = n + 1 qubits are represented using normalized
vectors of the complex Hilbert space CN . Since all quantum mechanical
predictions are given in terms of the Hermitian scalar product on CN , and
this is invariant under multiplication by a constant (vector independent)
phase factor, the states of the quantum system are actually represented by
equivalence classes of vectors in CN . Two vectors ψ1 and ψ2 are equivalent:
ψ2 ∼ ψ1 if there is a complex scalar a 6= 0 such that ψ2 = aψ1. This set of
equivalence classes defines the complex projective space: CP n :≡ (Cn+1 −
0)/ ∼. It is the state space of the system of N qubits. Global coordinates
(c1, . . . cN) of a vector in CN that represent an equivalence class [ψ], that
is an element of CP n, are called homogeneous coordinates on CP n. The
complex projective space is topologically equivalent to S2n+1/S1, where the
2n+ 1-dimensional sphere comes from normalization and the circle S1 takes
care of the unimportant overall phase factor.
The projective space CP n is locally homeomorphic with Cn. Intrinsic
coordinates on CP n are introduced as follows. A chart Uµ consists of equiv-
alence classes of all vectors in (Cn+1 − 0) such that cµ 6= 0. In the chart Uµ
the local ( so called inhomogeneous) coordinates ζν, ν = 1, 2 . . . n are given
ζν = ξν (ν ≤ µ− 1), ζν = ξν+1 (ν > µ), (2)
where
ξν = cν/cµ ν = 1, 2, . . . µ− 1, µ+ 1, . . . n+ 1. (3)
The coordinates ζνµ(c) and ζ
µ′(c) of a point c which belongs to the domain
where two charts Uµ and Uµ′ overlap are related by the following holomorphic
transformation
ζνµ′(c) = (c
)ζνµ(c) (4)
As an illustration consider the system of two qubits. The Hilbert space
is H = H1
⊗H2 = C2 ⊗ C2 = C4. As a basis we can choose the set
of separable vectors | ↑↑>, | ↑↓>, | ↓↑>, | ↓↓> or any other four orthogonal
vectors. The coordinates of a vector in C4 with respect to a basis are denoted
(c1, c2, c3, c4). The corresponding projective space is CP 3 ≡ S7/S1. At
least two charts are needed to define the intrinsic coordinates over all CP 3.
Consider first all vectors with a nonzero component along |1 >= | ↑↑> that
is c1 6= 0, i.e. all vectors except the vector | ↓↓>. Then the numbers ξν1
are defined as ξ11 = c
1/c1 = 1, ξ21 = c
2/c1, ξ31 = c
3/c1, ξ41 = c
4/c1 and
finally the three intrinsic coordinates (ζ11 , ζ
1 , ζ
1 ) are given by relabelling of
ξν1 : ζ
1 = ξ
1 , ζ
1 = ξ
1 , ζ
1 = ξ
1. To coordinatize the vector |4 >= | ↓↓> we
need another chart.
Quantum mean values of linear operators on C4 are indeed reduced to
functions on CP 3. For example, consider the following Hamiltonian operator
H = ωσz ⊗ 1+ ω1⊗ σz + µσx ⊗ σx (5)
In the separable bases the normalized quantum expectation < ψ|H|ψ > / <
ψ|ψ > is given by the following function of (c1, c2 . . . , c̄4)
2ω(c1c̄1 − c4c̄4) + µ(c̄2c3 + c̄3c2 + c̄1c4 + c̄4c1)
c1c̄1 + c2c̄2 + c3c̄3 + c4c̄4
. (6)
In the intrinsic coordinates ζ1, ζ2, ζ3 and their conjugates this expression is
given by
2ω(1− ζ3ζ̄3) + µ(ζ̄1ζ2 + ζ̄2ζ1 + ζ3 + ζ̄3)
1 + ζ1ζ̄1 + ζ2ζ̄2 + ζ3ζ̄3
. (7)
We shall also analyze the following Hamiltonian
H = ωσz ⊗ 1+ ω1⊗ σz + µσz ⊗ σz , (8)
whose normalized mean value is given by
2ω(c1c̄1 − c4c̄4) + µ(c1c̄1 + c4c̄4 − c2c̄2 − c3c̄3)
c1c̄1 + c2c̄2 + c3c̄3 + c4c̄4
. (9)
The corresponding function on CP 3 is, in the intrinsic coordinates, given by
ω(1− ζ3ζ̄3) + µ(1 + ζ3ζ̄3 − ζ1ζ̄1 − ζ2ζ̄2)
1 + ζ1ζ̄1 + ζ2ζ̄2 + ζ3ζ̄3
. (10)
2.1.1 Submanifold of separable states
Consider two quantum systems A and B with the corresponding Hilbert
spaces HA and HB. Taken together, the systems A and B form another
quantum system. The statistics of measurements that could be performed
on this compound system requires that the Hilbert space of the compound
system is given by the direct product HAB = HA ⊗ HB. The space of pure
states of the compound system is the projective Hilbert space PHAB. In the
case of finite dimensional state spaces PHn+1A = CP n and PHm+1A = CPm the
state space of the compound system is CP (m+1)(n+1)−1. Vectors inHAB of the
form ψA ⊗ ψB where ψA/B ∈ HA/B are called separable. The corresponding
separable states form the (m + n)-dimensional submanifold CPm × CP n
embedded in CP (m+1)(n+1)−1.
In the case of two qubits the submanifold of the separable states CP 1 ×
CP 1 forms a quadric in the full state space CP 3, given in terms of the
homogeneous coordinates (c1, c2, c3, c4) of CP 3 by the following formula
c1c4 = c2c3. (11)
In terms of the intrinsic coordinates ζ1, ζ2, ζ3, in the chart with c1 6= 0, i.e.
ξ1 = 1, the equation (11) is
ζ1ζ2 = ζ3. (12)
2.2 Complex structure on CP n
Consider a complex manifold M with complex dimension dimC M = n (in
particular CP n ). We can look at M as a real manifold with dimR M = 2n.
The real coordinates (x1, . . . x2n) are related to the holomorphic (ζ1, . . . ζn)
and anti-holomorphic (ζ̄1, . . . ζ̄n) coordinates via the following formulas:
(xν + ıxν+n)/
2 = ζν, ν = 1, 2, . . . n,
(xν − ıxν+n)/
2 = ζ̄ν, ν = 1, 2, . . . n, (13)
qν ≡ xν = (ζν + ζ̄ν)/
2, ν = 1, 2, . . . n,
pν ≡ xν+n = (ζ̄ν − ζ̄ν)/
2, ν = 1, 2, . . . n. (14)
The tangent space TxM is spanned by 2n vectors:
, . . .
, . . .
} (15)
or by the basis
, . . .
, . . .
}. (16)
An almost complex structure on a real 2n-dimensional manifold is given
by a (1, 1) tensor J satisfying J2 = 1, i.e. Jac J
b = −δab . Locally, the almost
complex structure J is given in the real coordinates by the following matrix
, (17)
where 1 is n-dimensional unit matrix. If the real 2n manifold is actually a
complex manifold, like in our case, the almost complex structure is defined
globally and is called the complex structure.
2.3 Riemannian structure on CP n
Hermitian scalar product induces a complex Euclidean metric on CN . The
metric induced on CP n is the Fubini-Study metric, and is given, in (ζ, ζ̄)
coordinates, using an n× n matrix with following entries
gµ,ν̄(ζ, ζ̄) =
δµ,ν(1 + ζζ̄)− ζµζ̄ν
(1 + ζζ̄)2
, µ, ν = 1, 2 . . . n, (18)
where ζζ̄ ≡ ∑nµ ζµζ̄µ.
The Fubini-Study metric in (ζ, ζ̄) coordinates is then given by 2n × 2n
matrix
G(ζ, ζ̄) = 1
0 gµ,ν̄
gµ̄,ν 0
. (19)
In the real coordinates the Fubini-Study metric is given by the standard
transformation formulas
Gi,j(q, p̄) = Gk,l(ζ(q, p), ζ̄(q, p))
, (20)
where we used Z = (ζ1, . . . ζ̄n and X = (q
1 . . . pn).
In the example of two qubits the Fubini-Study metric on CP 3 is
0 0 0
(1+ζζ̄)−ζ1ζ̄1
(1+ζζ̄)2
−ζ1ζ̄2
(1+ζζ̄)2
−ζ1ζ̄3
(1+ζζ̄)2
0 0 0 −ζ
(1+ζζ̄)2
(1+ζζ̄)−ζ2ζ̄2
(1+ζζ̄)2
−ζ2ζ̄3
(1+ζζ̄)2
0 0 0 −ζ
(1+ζζ̄)2
−ζ3ζ̄2
(1+ζζ̄)2
(1+ζζ̄)−ζ3ζ̄3
(1+ζζ̄)2
(1+ζζ̄)−ζ1ζ̄1
(1+ζζ̄)2
−ζ2ζ̄1
(1+ζζ̄)2
−ζ3ζ̄1
(1+ζζ̄)2
0 0 0
−ζ1ζ̄2
(1+ζζ̄)2
(1+ζζ̄)−ζ2ζ̄2
(1+ζζ̄)2
−ζ3ζ̄2
(1+ζζ̄)2
0 0 0
−ζ1ζ̄3
(1+ζζ̄)2
−ζ2ζ̄3
(1+ζζ̄)2
(1+ζζ̄)−ζ3ζ̄3
(1+ζζ̄)2
0 0 0
Transformation to the real coordinates, by application of the formula (20),
gives
−p1p2+q1q2
−p1p3+q1q3
1q2−p2q1
p1q3−p3q1
−p1p2+q1q2
p2p3+q2q3
p2q1−p1q2
2q3−p3q2
−p1p2+q1q2
p2p3+q2q3
p3q1−p1q3
p3q2−p2q3
2q1−p1q2
p3q1−p1q3
−p1p2+q1q2
−p1p3q1q3
p1q2−p2q1
3q2−p2q3
−p1p2+q1q2
−p2p3+q2q3
p1q3−p3q1
p2q3−q2p3
0 −p1p3−q1q3
−p2p3+q2q3
where
a = (p1)2+(p2)2+(p3)2+(q1)2+(q2)2+(q3)2+2, b = (p1)2+(p3)2+(q1)2+(q3)2+2.
Obviously, G is positive definite and symmetric.
2.4 Symplectic structure on CP n
The Hermitian scalar product on CN is also used to define the symplectic
structure on CN and this induces the symplectic structure on CP n. The
symplectic structure is the closed nondegenerate two form Ω on CP n, which
is, in (ζ, ζ̄) coordinates given by
ω = ıg(ζ, ζ̄)µ,ν̄dζ
µ ∧ ζ̄ν (23)
where gµ,ν̄ is the Fubini-Study metric (18). In real coordinates, the symplectic
structure is given by Ω(q, p) = JG(q, p) where G(q, p) is given by (20) and J
by (17).
The symplectic form on the two qubits state space is in the real bases
given by the product of matrices (17) and (22). The results is
2q1+p1q2
−p3q1+p1q3
p1p2+q1q2
p1p3+q1q3
p2q1−p1q2
2q3−p3q2
p2p1+q1q2
p2p3+q2q3
p3q1−p1q3
p3q2−p2q3
1p3+q1q3
p2p3+q2q3
−p2p1+q1q2
−p1p3+q1q3
1q2−p2q1
p1q3−p3q1
−p1p2+q1q2
−p2p3+q2q3
p2q1−p1q2
2q3−p3q2
−p1p3+q1q3
−p2p3+q2q3
p3q1−p1q3
p3q2−p2q3
3 Quantum Hamiltonian dynamical system
on CP n
The Schroedinger equation on CN is in some basis {|ψi >, i = 1, 2 . . .N}
given by:
=< ψj |H|ψi > cj. (25)
In the real coordinates this equation assumes the form of a Hamiltonian
dynamical system on R2N with a global gauge symmetry corresponding to
the invariance |ψ >→ exp(ix)|ψ >. Reduction with respect to this symmetry
results in the Hamiltonian system onCP n, considered as a real manifold with
the symplectic structure given by (23). The Hamilton equation on CP n, that
are equivalent to the Schroedinger equation (25), are
= 2Ωl,k∇kH(x), (26)
where Ωl,k is the inverse of the symplectic form, and H(x) is given by the
normalized quantum expectation of the Hamilton’s operator < ψ|H|ψ >
/ < ψ|ψ > expressed in terms of the real coordinates (14). For example,
the hamiltonian (7) is given in terms of the real coordinates qi ≡ xi, pi ≡
xi+n, i = 1, . . . n by
[2− (p3)2 − (q3)2] + µ
(p1p2 + q1q2 +
2q3). (27)
and the symmetric hamiltonian (9) is given by
[2−(p3)2−(q3)2]−µ
[(p1)2+(p2)2+(q1)2+(q2)2−(p3)2−(q3)2−2] (28)
The Hamilton’s equations (26) with the hamiltonian (27) and the sym-
plectic form (24) assume the following form
q̇1 = −2ωp1 + µp2 − µ(p3q1 + p1q3)/
q̇2 = −2ωp2 + µp2 − µ(p3q2 + p2q3)/
q̇3 = −4ωp3 −
2µp3q3
ṗ1 = 2ωq1 − µq2 + µ(q3q1 − p1p3)/
ṗ2 = 2ωq2 − µq1 + µ(q3q2 − p2p3)/
ṗ3 = 4ωq3 + µ((q3)2 − (p3)2 − 2)/
2. (29)
The equations of motion with the symmetric hamiltonian (28) on CP 3
are quite simple
q̇1 = −2(ω + µ)p1
q̇2 = −2(ω + µ)p2
q̇2 = −4ωp3
ṗ1 = 2(ω + µ)q1
ṗ2 = 2(ω + µ)q1
ṗ3 = 4ωq3. (30)
3.1 Quantum Hamiltonian system with imposed sep-
arability constraints
Dynamics of a constrained Hamiltonian system is usually described by the
method of Lagrange multipliers [34],[35]. Consider a Hamiltonian system
given by a symplectic manifold M with the symplectic form Ω and the
Hamilton’s function H on M. Suppose that besides the forces described
by H the dynamics of the system is affected also by forces whose sole effect
is to constrain the motion on a submanifold N ∈ M determined by a set
functional relations
f1(q, p) = . . . fk(q, p) = 0 (31)
The method of Lagrange multiplies assumes that the dynamics on N is de-
termined by the following set of differential equations
Ẋ = Ω(∇X,∇H ′), H ′ = H +
, λjfj (32)
which should be solved together with the equations of the constraints (31).
The Lagrange multipliers λj are functions of (p, q) that are to be determined
from the following, so called compatibility, conditions.
ḟl = Ω(∇fl,∇H ′) (33)
onN . The equations (33) uniquely determine the functions λ1(p, q), . . . λk(p, q)
if and only if the matrix of Poison brackets {fi, fj} = Ω(∇fi,∇fj) is nonsin-
gular. If this is the case then all constraints (31) are called primary, and N
is symplectic manifold with the symplectic structure determined by the so
called Dirac-Poison brackets
{F1, F2}′ = {F1, F2}+
{fi, F1}{fi, fj}−1{fj, F2} (34)
As we shall see, this is the case in the examples of pairs of interacting qubits
constrained on the manifold of separable states that we shall analyze. On the
other hand, if some of the compatibility equations do not contain multipliers,
than for that constrain ḟj = {fj, H} = 0, which represents an additional
constraint. These are called secondary constraints, and they must be added
to the system of original constraints (31). If this enlarged set of constraints
is functionally independent one can repeat the procedure. At the end one
either obtains a contradiction, in which case the original problem has no
solution, or one obtains appropriate multipliers λk such that the system (33)
is compatible. In the later case the solution for λk might not be unique in
which case the orbits of (32) and (31) are not uniquely determined by the
initial conditions.
Let us apply the formalism of Lagrange multipliers on the system of
two interacting qubits additionally constrained to remain on the manifold
of separable pure state. The real and imaginary parts of (12) give the two
constraints in terms of real coordinates (q1, q2, q3, p1, p2, p3)
f1 = p
1p2 − q1q2 +
2q3, f2 =
2q3 − p2q1 − p1q2 (35)
The compatibility conditions (33) assume the following form
ḟ1 = Ω(∇f1,∇H) + λ2Ω(∇f1,∇f2) = 0,
ḟ2 = Ω(∇f2,∇H) + λ1Ω(∇f2,∇f1) = 0. (36)
where Ω is the symplectic form (24) and Ω(∇f1,∇H) = Ωa,b∇af1∇bH .
The matrix of Poisson brackets {fi, fj} on N is
0 [2 + (p1)2 + (q1)2][2 + (p2)2 + (q2)2]/8
−[2 + (p1)2 + (q1)2][2 + (p2)2 + (q2)2]/8 0
and is nonsingular. Thus the compatibility conditions can be solved for the
Lagrange multipliers λ1(q, p), λ2(q, p),
λ1 = 4µ
4p1p2q1q2 + [(q1)2 − 2][2 + (p2)2 − (q2)2] + (p1)2[(q2)2 − (p2)2 − 2]
[2 + (p1)2 + (q1)2)2(2 + (p2)2 + (q2)2]2
λ2 = 8µ
(p1)2p2q2 − p2q2[(q1)2 − 2] + p1q1[2 + (p2)2 − (q2)2]
[2 + (p1)2 + (q1)2)2(2 + (p2)2 + (q2)2]2
Finally, the dynamics of the constrained system is described by the equa-
tions (32) and (31) with λ1(q, p), λ2(q, p) and f1(q, p), f2(q, p) given by (38)
and (35). For the Hamiltonian (27) the resulting equations of motion for
q1, q2, p1, p2 are
q̇1 = −4µp
1q1q2 + 2ωp1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
q̇2 = −4µp
2q1q2 − 2ωp2[2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
ṗ1 =
2µq2[(q1)2 − (p1)2 − 2] + 2ωq1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
ṗ2 =
2µq1[(q2)2 − (p2)2 − 2] + 2ωq2[2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
. (39)
The same procedure for the symmetric hamiltonian (28) results with the
following equations of motion
q̇1 =
2µp1[(p2)2 + (q2)2 − 2)]− 2ωp1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
q̇2 =
2µp2[(p1)2 + (q1)2 − 2]− 2ωp2[(2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
ṗ1 =
−2µq1[(q2)2 + (p2)2 − 2] + 2ωq1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
ṗ2 =
−2µq2[(q1)2 + (p1)2 − 2] + 2ωq2[2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
. (40)
There are also the equations expressing q̇3 and ṗ3 in terms of q1, q2, p1, p2,
but the solutions of these are already given by the constraints.
3.2 Qualitative properties of the constrained dynam-
ics of two interacting qubits
In this section we present the results of numerical analyzes of the qualita-
tive properties of the dynamics generated by the constrained equations (40)
and (39), corresponding to the quantum Hamiltonians (28) with the SO(2)
symmetry and (27) without such symmetry.
It is well known that any quantum system is integrable when considered
as the Hamiltonian dynamical system on the symplectic space H, and that
the reduction on the symplectic manifold PH preserves this property. This is
simply a consequence of the form of the quantum Hamiloton’s function, which
is always defined as the mean value of the Hamiltonian operator. Contrary
to the case of classical Hamiltonian systems, the symmetry of the physical
system has no relevance for the property of integrability in the Hamiltonian
formulation of the Schroedinger equation. We illustrate this fact, in figures
-1.50 -0.75 0.00 0.75 1.50
-0.4 -0.2 0.0 0.2 0.4
-0.30 -0.15 0.00 0.15 0.30
-0.30
-0.15
-0.30 -0.15 0.00 0.15 0.30
-0.30
-0.15
Figure 1: Projections on (q1, p1) plane of a typical orbit for the hamiltonian
systems (28) (a) and (27) (b) on CP 3 and on the submanifold of separable
states (c) for (40) and d for (39). The values of the parameters are ω = 1
and µ = 1.7
1a,b, by projections on (q1, p1) plane of a typical orbit for the symmetric and
nonsymmetric hamiltonians of the pair of qubits. The motion on CP 3 in
the symmetric case has further degeneracy compared with the nonsymmetric
case, but both cases generate integrable, regular Hamiltonian dynamics.
On the other hand, the qualitative properties of the dynamics constrained
by the separability conditions, are quite different. Typical orbits in the sym-
metric and nonsymmetric cases are illustrated in figure 1c,d. Symmetric
dynamics constrained by separability is still regular, while the nonsymmetric
Hamiltonian generates the constrained dynamics with typical chaotic orbits.
This is further illustrated in figures 2, where we show Poincaré surfaces of
section, defined by q2 = 0, p2 > 0 and H(p1, q1, p2, q2) = h for different values
of the coupling µ. Obviously, the constrained system displays the transition
from predominantly regular to predominantly chaotic dynamics, with all the
intricate structure of the phase portrait, characteristic for typical Hamilto-
nian dynamical systems. Thus, we can conclude that the quantum system
constrained on the manifold of separable state behaves as typical classical
Hamiltonian systems. If there is enough symmetry, i.e. enough integrals of
motion, the constrained dynamics is integrable, otherwise the constrained
quantum dynamics is that of typical chaotic Hamiltonian system.
-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9
-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9
Figure 2: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
4 Summary and discussion
We have studied Hamiltonian formulation of quantum dynamics of two in-
teracting qubits. Hamiltonian dynamical system on the state space CP 3
as the phase space, is integrable irrespective of the different symmetries of
the quantum system. We have then studied the dynamics of the quantum
Hamiltonian system constrained on the manifold of separable states. The
main result of this analyzes, and of the paper, is that the quantum Hamilto-
nian system without symmetry generates nonintegrable chaotic dynamics on
the set of separable states, while the constrained symmetric dynamics gives
an integrable system. It is important to bare on mind that neither the system
nor the separable states that lie on an orbit of the constrained system have
an underlining classical mechanical model. Thus, forcing a non-degenerate
quantum system to remain on the manifold of separable states is enough to
generate a dynamical system with typical properties of Hamiltonian chaos.
Our analyzes of the separability constrained quantum dynamics has been
rather formal. In order to inquire into possible interpretation of our results we
need a model of a physical realization of the separability constraints. To this
end we consider an open quantum system of two interacting qubits, whose
dynamics satisfies the Markov assumption [36], and we choose a Hermitian
Lindblad operator of the following form
L = l11σ
⊗ σ2+σ2− + l12σ1+σ1− ⊗ σ2−σ2+ + l21σ1−σ1+ ⊗ σ2+σ2− + l22σ1−σ1+ ⊗ σ2−σ2+
i,j=1
li,j|i >< j|1 ⊗ |i >< j|2 (41)
where |1 >≡ | ↑> and |2 >≡ | ↓>.
The dynamics of a pure state of the open system under the action of a
Hamiltonian H and the Linblad γL is described by the following stochastic
nonlinear Schroedinger equation [36],[37]
|dψ > = −iH|ψ > dt+ γ
(L− < ψ|L|ψ >)2|ψ > dt
+ γ(L− < ψ|L|ψ >)|ψ > dW, (42)
where dW is the increment of complex Wiener c-number process W (t).
The equation (42) represent a diffusion process on a complex Hilbert
space, and is central in the ”Quantum State Diffusion” (QSD) theory of
open quantum systems [37]. It has been used to study the systems of in-
teracting qubits in various environments for example in [32],[38], and the
effect of the Linblad operator (41) on the entanglement between two qubits
was considered in [16]. The influence of the non-Hamiltonian terms of drift
(proportional to γ2) and the diffusion (proportional to γ)), with the Linblad
operator of the form (41), is to drive an entangled state towards one of the
separable states with the corresponding probability. This process occurs on
the time scale proportional to γ−1. So, for large γ there occurs an almost
instantaneous collapse of an entangled state into a separable one. We believe
that with a proper choice of the parameters li,j the long term dynamics of
a pure state described by (42) can have the same qualitative properties as
the separability constrained quantum dynamics. In particular, the difference
between the qualitative properties of symmetric and nonsymmetric systems,
reflected in the constrained Hamiltonian system, should also manifest in the
dynamics of (42) for a proper choice of li,j. This expectations are supported
by figures 3, which illustrate the dynamics of (< σ1x >,< σ
y >) for the Hamil-
tonian operators (5) and (8) as calculated using the constrained Hamiltonian
equations (39) and (40) (figures 3b and 3a ), or the QSD equation (42) (fig-
ures 3d and 3c) for a particular choice of li,j and large γ = 5. Of course, the
choice of optimal values for li,j should be according to some criterion, which
is the problem we are currently investigating.
-1.0 -0.5 0.0 0.5 1.0
1.0 d)
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
Figure 3: Figures illustrate the dynamics of (< σx >,< σy >) for the
constrained Hamiltonian systems (40) (a) and (39) (b) and for the stochastic
Schroedinger equation (42) with the Linblad (41) and the hamiltonians (8) (c)
and (5) (d). the parameters are ω = 1, µ = 1.7, γ = 5 and l1,1 = 0.21, l1, 2 =
0.21, l2,1 = 0.215, l2,2 = 0.205.
The pair of coupled qubits, analyzed in this paper, is the simplest quan-
tum system exhibiting dynamical entanglement. We intend to investigate
the effects of suppression of the dynamical entanglement in systems with
spacial degrees of freedom, obtained by quantization of classically chaotic
systems, for example a pair of coupled nonlinear oscillators. In this case,
the Hamiltonian formulation of the quantum dynamics requires an infinite-
dimensional phase space, and the analyzes of the separability constrained
dynamics is more complicated. However, it wold be interesting to compare
the dynamics obtained by separability constraints with that of some more
standard semi-classical approximation.
Acknowledgements This work is partly supported by the Serbian Min-
istry of Science contract No. 141003. I should also like to acknowledge the
support and hospitality of the Abdus Salam ICTP.
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FIGURE CAPTIONS
Figure 1 Projections on (q1, p1) plane of a typical orbit for the hamil-
tonian systems (28) (a) and (27) (b) on CP 3 and on the submanifold of
separable states (c) for (40) and d for (39). The values of the parameters are
ω = 1 and µ = 1.7
Figure 2 Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a)µ = 1.1,
(b) µ = 1.3, (c) µ = 1.5 and (d) µ = 1.7
Figure 3 Figures illustrate the dynamics of (< σx >,< σy >) for the
constrained Hamiltonian systems (40) (a) and (39) (b) and for the stochastic
Schroedinger equation (42) with the Linblad (41) and the hamiltonians (8) (c)
and (5) (d). the parameters are ω = 1, µ = 1.7, γ = 5 and l1,1 = 0.21, l1, 2 =
0.21, l2,1 = 0.215, l2,2 = 0.205.
	Introduction
	Geometry of the state space CPn
	Definition and intrinsic coordinates of CPn 
	 Submanifold of separable states
	Complex structure on CPn
	Riemannian structure on CPn
	Symplectic structure on CPn
	Quantum Hamiltonian dynamical system on  CPn 
	 Quantum Hamiltonian system with imposed separability constraints
	 Qualitative properties of the constrained dynamics of two interacting qubits 
	 Summary and discussion

  Schroedinger equation on a Hilbert space ${\cal H}$, represents a linear
Hamiltonian dynamical system on the space of quantum pure states, the
projective Hilbert space $P {\cal H}$. Separable states of a bipartite quantum
system form a special submanifold of $P {\cal H}$.
  We analyze the Hamiltonian dynamics that corresponds to the quantum system
constrained on the manifold of separable states, using as an important example
the system of two interacting qubits. The constraints introduce nonlinearities
which render the dynamics nontrivial. We show that the qualitative properties
of the constrained dynamics clearly manifest the symmetry of the qubits system.
In particular, if the quantum Hamilton's operator has not enough symmetry, the
constrained dynamics is nonintegrable, and displays the typical features of a
Hamiltonian dynamical system with mixed phase space. Possible physical
realizations of the separability constraints are discussed.

Introduction
Classical and quantum descriptions of a physical system that is considered
as composed of interacting subsystems have radically different features. The
typical feature of quantum dynamics is the creation of specifically quantum
correlations, the entanglement, among the subsystems. On the other hand,
the typical property of classical description is the occurrence of chaotic or-
bits and fractality of the phase space portrait, which can be considered as
typically classical type of correlations between the subsystems. The type
of correlations introduced by the dynamical entanglement does not occur in
the classical description, and likewise, the type of correlations introduced
by the chaotic orbits with fractal structures does not occur in the quantum
description. This intriguing complementarity of the two descriptions repre-
sents a problem that is expected to be solved by a detailed formulation of
the correspondence principle.
Comparison of typical features of classical and quantum mechanics is fa-
cilitated if the same mathematical framework is used in both theories. It is
well known, since the work of Kibble [1],[2],[3], that the quantum evolution,
determined by the linear Schroedinger equation, can be represented using the
typical language of classical mechanics, that is as a Hamiltonian dynamical
system on an appropriate phase space, given by the Hilbert space geometry
of the quantum system. This line of research was later developed into the
full geometric Hamiltonian representation of quantum mechanics. [4]-[12].
Such geometric formulation of quantum mechanics has recently inspired nat-
ural definitions of measures of the entanglement [13], and has been used to
model the spontaneous collapse of the state vector [14],[15], and dynamics of
decoherence [16].
It is our goal to use the geometric Hamiltonian formulation of quantum
mechanics to study the relation between the dynamical entanglement and
typical qualitative properties of Hamiltonian dynamics. Motivated by the
fact that the Schroedinger equation can always be considered as a Hamil-
tonian dynamical system, and that for Hamiltonian systems the definitions
and properties of the dynamical chaos are well understood, we shall seek
for a formal condition that when imposed on the Hamiltonian system repre-
senting the Schroedinger equation of the compound quantum system renders
the Hamiltonian dynamics nonintegrable and chaotic. It is well known that
the linear Schroedinger equation of quantum mechanics represents always
an integrable Hamiltonian dynamical system, irrespective of the dynamical
symmetries of the system. This is in sharp contrast with the Hamiltonian
formulation of classical systems, where enough symmetry implies integrabil-
ity and the lack of it implies the chaotic dynamics. Linearity of the quantum
Hamiltonian dynamics, and the consequent integrability, is introduced in the
Hamiltonian formulation by a very large dimensionality of the phase space of
the quantum system. This high dimensionality can be considered as a conse-
quence of two reasons. For a single quantum system, say a one dimensional
particle in a potential, linear evolution and with it the principle of state
superposition require infinite dimensional phase space of the Hamiltonian
formulation. If the classical mechanical model is linear, say the harmonic os-
cillator, the quantum Hamiltonian dynamics can be exactly describe on the
reduced finite-dimensional phase space, the real plane in the case of the har-
monic oscillator. The other related reason that increases the dimensionality
of the quantum phase space compared to the classical model is in the way
the state space of the compound systems are formed out of the components
state spaces in the two theories. In order to represents the entangled states as
points of the quantum phase space the dimensionality of the quantum phase
space is much larger than just the sum of the dimensions of the components
phase spaces. The points in the Cartesian product of the components phase
spaces represent the separable quantum states and form a subset of the full
quantum phase space. Needles to say, although the separable states are the
most classical-like states of the compound system, they still are quantum
states with nonclassical properties like nonzero dispersion of some subsys-
tem’s variables. Our main result will be that when the quantum dynamics,
represented as a Hamiltonian system, is constrained on the manifold of sepa-
rable quantum states the relation between the symmetry and the qualitative
properties of the dynamics such as integrability or chaotic motion is reestab-
lished. Thus, suppression of dynamical entanglement is enough to enable
manifestations of the qualitative differences in dynamics of quantum systems
and the relation between integrability and symmetry, traditionally related
with classical mechanical models.
In order to study the relation between the dynamical entanglement, sep-
arability and the properties of Hamiltonian formulation of the quantum dy-
namics we shall use, in this paper, the simplest quantum system that displays
the dynamical entanglement, that is a system of two interacting qubits:
H = ωσ1 + ωσ2 + µxσx
2 + µyσy
2 + µzσz
2, (1)
where σx,y,z
i are the three Pauli matrices of the i-th qubit, and satisfy the
usual SU(2) commutation relations. In particular we shall compare the dy-
namics of the system (1) in the case µz 6= 0, µx = µy = 0 with the case
when µx 6= 0, µz = µy = 0. The former case is symmetric with respect to
SO(2) rotations around z-axis and the later lacks this symmetry. Besides
its simplicity, the systems of the form (1) are of considerable current interest
because the hamiltonian of the universal quantum processor is of this form
[17],[18].
Various lines of research, during the last decade, improved the under-
standing of the relation between dynamical entanglement and properties of
the dynamics. Strong impetus to the study of all aspects of quantum entan-
glement came from the theory of quantum computation [18]. Quantization
of classical non-integrable systems, and various characteristic properties of
resulting quantum systems, have been studied for a long time [19]. The de-
pendence of the dynamical entanglement, between a quantum system and its
environment, on the qualitative properties of the dynamics of the system was
studied indirectly, within the theory of environmental decoherence [20]. The
relation between the rates of dynamical entanglement and the qualitative
properties of the dynamics in the semi-classical regime was initiated in the
reference [21] and various aspects of this relations have been studied since
[22]-[31]. The relation between the symmetry of the genuinely quantum sys-
tem (1) and the degree of dynamical entanglement was studied in reference
[32]. As we shall see, our present analyzes is related to the quoted works,
but the relation between the dynamical entanglement and symmetry is here
approached from a very different angle
The structure of the paper is as follows. We shall first recapitulate the
necessary background such as: the complex symplectic and Riemannian ge-
ometry of CP n; Hamiltonian formulation on CP n of the quantum dynamics;
geometric formulation of the set of separable pure states and Hamiltonian for-
mulation of the constrained dynamics. In parallel with the general reminder,
the explicit formulas for the system of two interacting qubits will be given.
These are then applied, in section 3, to the study the qualitative properties
of the separability constrained dynamics for the qubits systems. The main
results are summarized and discussed in section 4. There we also discuss a
model of an open quantum system with dynamics that clearly differentiates
between the symmetric and the nonsymmetric systems.
2 Geometry of the state space CP n
Hamiltonian formulation of quantum mechanics is based on the fact that the
scalar product of vectors |ψ > in the Hilbert space of a quantum system can
be used to represent the linear Schroedinger equation of quantum mechanics
in the form of Hamilton’s equations. The canonical phase space structure of
this equations is determined by the imaginary part of the scalar product, and
the Hamilton’ s function is given by the quantum expectation < ψ|H|ψ > of
the quantum hamiltonian.
However, due to phase invariance and arbitrary normalization the proper
space of pure quantum states is not the Hilbert space used to formulate the
Schroedinger equation, but the projective Hilbert space which is the manifold
to be used in the Hamiltonian formulation of quantum mechanics. In general,
the resulting Hamiltonian dynamical system is infinite-dimensional, but we
shall need the general definitions only for the case of quantum system with
finite-dimensional Hilbert space, like the finite collection of qubits, in which
case the quantum phase space is also finite-dimensional. We shall first review
the definition of the complex projective space CP n, and then briefly state
the basic definitions and recapitulate the formulas which are needed for the
Hamiltonian formulation of the quantum dynamics on the state space and
its restriction on the separable state subset. The general reference for the
mathematical aspects of complex differential geometry is [33]. All concepts
and formulas will be illustrated using the system of two interacting qubits.
Differential geometry of the state space CP n is discussed by viewing it
as a real 2n dimensional manifold endowed with complex, Riemannian and
symplectic structure. In the case ofCP n this three structures are compatible.
2.1 Definition and intrinsic coordinates of CP n
States of a collection of N = n + 1 qubits are represented using normalized
vectors of the complex Hilbert space CN . Since all quantum mechanical
predictions are given in terms of the Hermitian scalar product on CN , and
this is invariant under multiplication by a constant (vector independent)
phase factor, the states of the quantum system are actually represented by
equivalence classes of vectors in CN . Two vectors ψ1 and ψ2 are equivalent:
ψ2 ∼ ψ1 if there is a complex scalar a 6= 0 such that ψ2 = aψ1. This set of
equivalence classes defines the complex projective space: CP n :≡ (Cn+1 −
0)/ ∼. It is the state space of the system of N qubits. Global coordinates
(c1, . . . cN) of a vector in CN that represent an equivalence class [ψ], that
is an element of CP n, are called homogeneous coordinates on CP n. The
complex projective space is topologically equivalent to S2n+1/S1, where the
2n+ 1-dimensional sphere comes from normalization and the circle S1 takes
care of the unimportant overall phase factor.
The projective space CP n is locally homeomorphic with Cn. Intrinsic
coordinates on CP n are introduced as follows. A chart Uµ consists of equiv-
alence classes of all vectors in (Cn+1 − 0) such that cµ 6= 0. In the chart Uµ
the local ( so called inhomogeneous) coordinates ζν, ν = 1, 2 . . . n are given
ζν = ξν (ν ≤ µ− 1), ζν = ξν+1 (ν > µ), (2)
where
ξν = cν/cµ ν = 1, 2, . . . µ− 1, µ+ 1, . . . n+ 1. (3)
The coordinates ζνµ(c) and ζ
µ′(c) of a point c which belongs to the domain
where two charts Uµ and Uµ′ overlap are related by the following holomorphic
transformation
ζνµ′(c) = (c
)ζνµ(c) (4)
As an illustration consider the system of two qubits. The Hilbert space
is H = H1
⊗H2 = C2 ⊗ C2 = C4. As a basis we can choose the set
of separable vectors | ↑↑>, | ↑↓>, | ↓↑>, | ↓↓> or any other four orthogonal
vectors. The coordinates of a vector in C4 with respect to a basis are denoted
(c1, c2, c3, c4). The corresponding projective space is CP 3 ≡ S7/S1. At
least two charts are needed to define the intrinsic coordinates over all CP 3.
Consider first all vectors with a nonzero component along |1 >= | ↑↑> that
is c1 6= 0, i.e. all vectors except the vector | ↓↓>. Then the numbers ξν1
are defined as ξ11 = c
1/c1 = 1, ξ21 = c
2/c1, ξ31 = c
3/c1, ξ41 = c
4/c1 and
finally the three intrinsic coordinates (ζ11 , ζ
1 , ζ
1 ) are given by relabelling of
ξν1 : ζ
1 = ξ
1 , ζ
1 = ξ
1 , ζ
1 = ξ
1. To coordinatize the vector |4 >= | ↓↓> we
need another chart.
Quantum mean values of linear operators on C4 are indeed reduced to
functions on CP 3. For example, consider the following Hamiltonian operator
H = ωσz ⊗ 1+ ω1⊗ σz + µσx ⊗ σx (5)
In the separable bases the normalized quantum expectation < ψ|H|ψ > / <
ψ|ψ > is given by the following function of (c1, c2 . . . , c̄4)
2ω(c1c̄1 − c4c̄4) + µ(c̄2c3 + c̄3c2 + c̄1c4 + c̄4c1)
c1c̄1 + c2c̄2 + c3c̄3 + c4c̄4
. (6)
In the intrinsic coordinates ζ1, ζ2, ζ3 and their conjugates this expression is
given by
2ω(1− ζ3ζ̄3) + µ(ζ̄1ζ2 + ζ̄2ζ1 + ζ3 + ζ̄3)
1 + ζ1ζ̄1 + ζ2ζ̄2 + ζ3ζ̄3
. (7)
We shall also analyze the following Hamiltonian
H = ωσz ⊗ 1+ ω1⊗ σz + µσz ⊗ σz , (8)
whose normalized mean value is given by
2ω(c1c̄1 − c4c̄4) + µ(c1c̄1 + c4c̄4 − c2c̄2 − c3c̄3)
c1c̄1 + c2c̄2 + c3c̄3 + c4c̄4
. (9)
The corresponding function on CP 3 is, in the intrinsic coordinates, given by
ω(1− ζ3ζ̄3) + µ(1 + ζ3ζ̄3 − ζ1ζ̄1 − ζ2ζ̄2)
1 + ζ1ζ̄1 + ζ2ζ̄2 + ζ3ζ̄3
. (10)
2.1.1 Submanifold of separable states
Consider two quantum systems A and B with the corresponding Hilbert
spaces HA and HB. Taken together, the systems A and B form another
quantum system. The statistics of measurements that could be performed
on this compound system requires that the Hilbert space of the compound
system is given by the direct product HAB = HA ⊗ HB. The space of pure
states of the compound system is the projective Hilbert space PHAB. In the
case of finite dimensional state spaces PHn+1A = CP n and PHm+1A = CPm the
state space of the compound system is CP (m+1)(n+1)−1. Vectors inHAB of the
form ψA ⊗ ψB where ψA/B ∈ HA/B are called separable. The corresponding
separable states form the (m + n)-dimensional submanifold CPm × CP n
embedded in CP (m+1)(n+1)−1.
In the case of two qubits the submanifold of the separable states CP 1 ×
CP 1 forms a quadric in the full state space CP 3, given in terms of the
homogeneous coordinates (c1, c2, c3, c4) of CP 3 by the following formula
c1c4 = c2c3. (11)
In terms of the intrinsic coordinates ζ1, ζ2, ζ3, in the chart with c1 6= 0, i.e.
ξ1 = 1, the equation (11) is
ζ1ζ2 = ζ3. (12)
2.2 Complex structure on CP n
Consider a complex manifold M with complex dimension dimC M = n (in
particular CP n ). We can look at M as a real manifold with dimR M = 2n.
The real coordinates (x1, . . . x2n) are related to the holomorphic (ζ1, . . . ζn)
and anti-holomorphic (ζ̄1, . . . ζ̄n) coordinates via the following formulas:
(xν + ıxν+n)/
2 = ζν, ν = 1, 2, . . . n,
(xν − ıxν+n)/
2 = ζ̄ν, ν = 1, 2, . . . n, (13)
qν ≡ xν = (ζν + ζ̄ν)/
2, ν = 1, 2, . . . n,
pν ≡ xν+n = (ζ̄ν − ζ̄ν)/
2, ν = 1, 2, . . . n. (14)
The tangent space TxM is spanned by 2n vectors:
, . . .
, . . .
} (15)
or by the basis
, . . .
, . . .
}. (16)
An almost complex structure on a real 2n-dimensional manifold is given
by a (1, 1) tensor J satisfying J2 = 1, i.e. Jac J
b = −δab . Locally, the almost
complex structure J is given in the real coordinates by the following matrix
, (17)
where 1 is n-dimensional unit matrix. If the real 2n manifold is actually a
complex manifold, like in our case, the almost complex structure is defined
globally and is called the complex structure.
2.3 Riemannian structure on CP n
Hermitian scalar product induces a complex Euclidean metric on CN . The
metric induced on CP n is the Fubini-Study metric, and is given, in (ζ, ζ̄)
coordinates, using an n× n matrix with following entries
gµ,ν̄(ζ, ζ̄) =
δµ,ν(1 + ζζ̄)− ζµζ̄ν
(1 + ζζ̄)2
, µ, ν = 1, 2 . . . n, (18)
where ζζ̄ ≡ ∑nµ ζµζ̄µ.
The Fubini-Study metric in (ζ, ζ̄) coordinates is then given by 2n × 2n
matrix
G(ζ, ζ̄) = 1
0 gµ,ν̄
gµ̄,ν 0
. (19)
In the real coordinates the Fubini-Study metric is given by the standard
transformation formulas
Gi,j(q, p̄) = Gk,l(ζ(q, p), ζ̄(q, p))
, (20)
where we used Z = (ζ1, . . . ζ̄n and X = (q
1 . . . pn).
In the example of two qubits the Fubini-Study metric on CP 3 is
0 0 0
(1+ζζ̄)−ζ1ζ̄1
(1+ζζ̄)2
−ζ1ζ̄2
(1+ζζ̄)2
−ζ1ζ̄3
(1+ζζ̄)2
0 0 0 −ζ
(1+ζζ̄)2
(1+ζζ̄)−ζ2ζ̄2
(1+ζζ̄)2
−ζ2ζ̄3
(1+ζζ̄)2
0 0 0 −ζ
(1+ζζ̄)2
−ζ3ζ̄2
(1+ζζ̄)2
(1+ζζ̄)−ζ3ζ̄3
(1+ζζ̄)2
(1+ζζ̄)−ζ1ζ̄1
(1+ζζ̄)2
−ζ2ζ̄1
(1+ζζ̄)2
−ζ3ζ̄1
(1+ζζ̄)2
0 0 0
−ζ1ζ̄2
(1+ζζ̄)2
(1+ζζ̄)−ζ2ζ̄2
(1+ζζ̄)2
−ζ3ζ̄2
(1+ζζ̄)2
0 0 0
−ζ1ζ̄3
(1+ζζ̄)2
−ζ2ζ̄3
(1+ζζ̄)2
(1+ζζ̄)−ζ3ζ̄3
(1+ζζ̄)2
0 0 0
Transformation to the real coordinates, by application of the formula (20),
gives
−p1p2+q1q2
−p1p3+q1q3
1q2−p2q1
p1q3−p3q1
−p1p2+q1q2
p2p3+q2q3
p2q1−p1q2
2q3−p3q2
−p1p2+q1q2
p2p3+q2q3
p3q1−p1q3
p3q2−p2q3
2q1−p1q2
p3q1−p1q3
−p1p2+q1q2
−p1p3q1q3
p1q2−p2q1
3q2−p2q3
−p1p2+q1q2
−p2p3+q2q3
p1q3−p3q1
p2q3−q2p3
0 −p1p3−q1q3
−p2p3+q2q3
where
a = (p1)2+(p2)2+(p3)2+(q1)2+(q2)2+(q3)2+2, b = (p1)2+(p3)2+(q1)2+(q3)2+2.
Obviously, G is positive definite and symmetric.
2.4 Symplectic structure on CP n
The Hermitian scalar product on CN is also used to define the symplectic
structure on CN and this induces the symplectic structure on CP n. The
symplectic structure is the closed nondegenerate two form Ω on CP n, which
is, in (ζ, ζ̄) coordinates given by
ω = ıg(ζ, ζ̄)µ,ν̄dζ
µ ∧ ζ̄ν (23)
where gµ,ν̄ is the Fubini-Study metric (18). In real coordinates, the symplectic
structure is given by Ω(q, p) = JG(q, p) where G(q, p) is given by (20) and J
by (17).
The symplectic form on the two qubits state space is in the real bases
given by the product of matrices (17) and (22). The results is
2q1+p1q2
−p3q1+p1q3
p1p2+q1q2
p1p3+q1q3
p2q1−p1q2
2q3−p3q2
p2p1+q1q2
p2p3+q2q3
p3q1−p1q3
p3q2−p2q3
1p3+q1q3
p2p3+q2q3
−p2p1+q1q2
−p1p3+q1q3
1q2−p2q1
p1q3−p3q1
−p1p2+q1q2
−p2p3+q2q3
p2q1−p1q2
2q3−p3q2
−p1p3+q1q3
−p2p3+q2q3
p3q1−p1q3
p3q2−p2q3
3 Quantum Hamiltonian dynamical system
on CP n
The Schroedinger equation on CN is in some basis {|ψi >, i = 1, 2 . . .N}
given by:
=< ψj |H|ψi > cj. (25)
In the real coordinates this equation assumes the form of a Hamiltonian
dynamical system on R2N with a global gauge symmetry corresponding to
the invariance |ψ >→ exp(ix)|ψ >. Reduction with respect to this symmetry
results in the Hamiltonian system onCP n, considered as a real manifold with
the symplectic structure given by (23). The Hamilton equation on CP n, that
are equivalent to the Schroedinger equation (25), are
= 2Ωl,k∇kH(x), (26)
where Ωl,k is the inverse of the symplectic form, and H(x) is given by the
normalized quantum expectation of the Hamilton’s operator < ψ|H|ψ >
/ < ψ|ψ > expressed in terms of the real coordinates (14). For example,
the hamiltonian (7) is given in terms of the real coordinates qi ≡ xi, pi ≡
xi+n, i = 1, . . . n by
[2− (p3)2 − (q3)2] + µ
(p1p2 + q1q2 +
2q3). (27)
and the symmetric hamiltonian (9) is given by
[2−(p3)2−(q3)2]−µ
[(p1)2+(p2)2+(q1)2+(q2)2−(p3)2−(q3)2−2] (28)
The Hamilton’s equations (26) with the hamiltonian (27) and the sym-
plectic form (24) assume the following form
q̇1 = −2ωp1 + µp2 − µ(p3q1 + p1q3)/
q̇2 = −2ωp2 + µp2 − µ(p3q2 + p2q3)/
q̇3 = −4ωp3 −
2µp3q3
ṗ1 = 2ωq1 − µq2 + µ(q3q1 − p1p3)/
ṗ2 = 2ωq2 − µq1 + µ(q3q2 − p2p3)/
ṗ3 = 4ωq3 + µ((q3)2 − (p3)2 − 2)/
2. (29)
The equations of motion with the symmetric hamiltonian (28) on CP 3
are quite simple
q̇1 = −2(ω + µ)p1
q̇2 = −2(ω + µ)p2
q̇2 = −4ωp3
ṗ1 = 2(ω + µ)q1
ṗ2 = 2(ω + µ)q1
ṗ3 = 4ωq3. (30)
3.1 Quantum Hamiltonian system with imposed sep-
arability constraints
Dynamics of a constrained Hamiltonian system is usually described by the
method of Lagrange multipliers [34],[35]. Consider a Hamiltonian system
given by a symplectic manifold M with the symplectic form Ω and the
Hamilton’s function H on M. Suppose that besides the forces described
by H the dynamics of the system is affected also by forces whose sole effect
is to constrain the motion on a submanifold N ∈ M determined by a set
functional relations
f1(q, p) = . . . fk(q, p) = 0 (31)
The method of Lagrange multiplies assumes that the dynamics on N is de-
termined by the following set of differential equations
Ẋ = Ω(∇X,∇H ′), H ′ = H +
, λjfj (32)
which should be solved together with the equations of the constraints (31).
The Lagrange multipliers λj are functions of (p, q) that are to be determined
from the following, so called compatibility, conditions.
ḟl = Ω(∇fl,∇H ′) (33)
onN . The equations (33) uniquely determine the functions λ1(p, q), . . . λk(p, q)
if and only if the matrix of Poison brackets {fi, fj} = Ω(∇fi,∇fj) is nonsin-
gular. If this is the case then all constraints (31) are called primary, and N
is symplectic manifold with the symplectic structure determined by the so
called Dirac-Poison brackets
{F1, F2}′ = {F1, F2}+
{fi, F1}{fi, fj}−1{fj, F2} (34)
As we shall see, this is the case in the examples of pairs of interacting qubits
constrained on the manifold of separable states that we shall analyze. On the
other hand, if some of the compatibility equations do not contain multipliers,
than for that constrain ḟj = {fj, H} = 0, which represents an additional
constraint. These are called secondary constraints, and they must be added
to the system of original constraints (31). If this enlarged set of constraints
is functionally independent one can repeat the procedure. At the end one
either obtains a contradiction, in which case the original problem has no
solution, or one obtains appropriate multipliers λk such that the system (33)
is compatible. In the later case the solution for λk might not be unique in
which case the orbits of (32) and (31) are not uniquely determined by the
initial conditions.
Let us apply the formalism of Lagrange multipliers on the system of
two interacting qubits additionally constrained to remain on the manifold
of separable pure state. The real and imaginary parts of (12) give the two
constraints in terms of real coordinates (q1, q2, q3, p1, p2, p3)
f1 = p
1p2 − q1q2 +
2q3, f2 =
2q3 − p2q1 − p1q2 (35)
The compatibility conditions (33) assume the following form
ḟ1 = Ω(∇f1,∇H) + λ2Ω(∇f1,∇f2) = 0,
ḟ2 = Ω(∇f2,∇H) + λ1Ω(∇f2,∇f1) = 0. (36)
where Ω is the symplectic form (24) and Ω(∇f1,∇H) = Ωa,b∇af1∇bH .
The matrix of Poisson brackets {fi, fj} on N is
0 [2 + (p1)2 + (q1)2][2 + (p2)2 + (q2)2]/8
−[2 + (p1)2 + (q1)2][2 + (p2)2 + (q2)2]/8 0
and is nonsingular. Thus the compatibility conditions can be solved for the
Lagrange multipliers λ1(q, p), λ2(q, p),
λ1 = 4µ
4p1p2q1q2 + [(q1)2 − 2][2 + (p2)2 − (q2)2] + (p1)2[(q2)2 − (p2)2 − 2]
[2 + (p1)2 + (q1)2)2(2 + (p2)2 + (q2)2]2
λ2 = 8µ
(p1)2p2q2 − p2q2[(q1)2 − 2] + p1q1[2 + (p2)2 − (q2)2]
[2 + (p1)2 + (q1)2)2(2 + (p2)2 + (q2)2]2
Finally, the dynamics of the constrained system is described by the equa-
tions (32) and (31) with λ1(q, p), λ2(q, p) and f1(q, p), f2(q, p) given by (38)
and (35). For the Hamiltonian (27) the resulting equations of motion for
q1, q2, p1, p2 are
q̇1 = −4µp
1q1q2 + 2ωp1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
q̇2 = −4µp
2q1q2 − 2ωp2[2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
ṗ1 =
2µq2[(q1)2 − (p1)2 − 2] + 2ωq1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
ṗ2 =
2µq1[(q2)2 − (p2)2 − 2] + 2ωq2[2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
. (39)
The same procedure for the symmetric hamiltonian (28) results with the
following equations of motion
q̇1 =
2µp1[(p2)2 + (q2)2 − 2)]− 2ωp1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
q̇2 =
2µp2[(p1)2 + (q1)2 − 2]− 2ωp2[(2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
ṗ1 =
−2µq1[(q2)2 + (p2)2 − 2] + 2ωq1[2 + (p2)2 + (q2)2]
2 + (p2)2 + (q2)2
ṗ2 =
−2µq2[(q1)2 + (p1)2 − 2] + 2ωq2[2 + (p1)2 + (q1)2]
2 + (p1)2 + (q1)2
. (40)
There are also the equations expressing q̇3 and ṗ3 in terms of q1, q2, p1, p2,
but the solutions of these are already given by the constraints.
3.2 Qualitative properties of the constrained dynam-
ics of two interacting qubits
In this section we present the results of numerical analyzes of the qualita-
tive properties of the dynamics generated by the constrained equations (40)
and (39), corresponding to the quantum Hamiltonians (28) with the SO(2)
symmetry and (27) without such symmetry.
It is well known that any quantum system is integrable when considered
as the Hamiltonian dynamical system on the symplectic space H, and that
the reduction on the symplectic manifold PH preserves this property. This is
simply a consequence of the form of the quantum Hamiloton’s function, which
is always defined as the mean value of the Hamiltonian operator. Contrary
to the case of classical Hamiltonian systems, the symmetry of the physical
system has no relevance for the property of integrability in the Hamiltonian
formulation of the Schroedinger equation. We illustrate this fact, in figures
-1.50 -0.75 0.00 0.75 1.50
-0.4 -0.2 0.0 0.2 0.4
-0.30 -0.15 0.00 0.15 0.30
-0.30
-0.15
-0.30 -0.15 0.00 0.15 0.30
-0.30
-0.15
Figure 1: Projections on (q1, p1) plane of a typical orbit for the hamiltonian
systems (28) (a) and (27) (b) on CP 3 and on the submanifold of separable
states (c) for (40) and d for (39). The values of the parameters are ω = 1
and µ = 1.7
1a,b, by projections on (q1, p1) plane of a typical orbit for the symmetric and
nonsymmetric hamiltonians of the pair of qubits. The motion on CP 3 in
the symmetric case has further degeneracy compared with the nonsymmetric
case, but both cases generate integrable, regular Hamiltonian dynamics.
On the other hand, the qualitative properties of the dynamics constrained
by the separability conditions, are quite different. Typical orbits in the sym-
metric and nonsymmetric cases are illustrated in figure 1c,d. Symmetric
dynamics constrained by separability is still regular, while the nonsymmetric
Hamiltonian generates the constrained dynamics with typical chaotic orbits.
This is further illustrated in figures 2, where we show Poincaré surfaces of
section, defined by q2 = 0, p2 > 0 and H(p1, q1, p2, q2) = h for different values
of the coupling µ. Obviously, the constrained system displays the transition
from predominantly regular to predominantly chaotic dynamics, with all the
intricate structure of the phase portrait, characteristic for typical Hamilto-
nian dynamical systems. Thus, we can conclude that the quantum system
constrained on the manifold of separable state behaves as typical classical
Hamiltonian systems. If there is enough symmetry, i.e. enough integrals of
motion, the constrained dynamics is integrable, otherwise the constrained
quantum dynamics is that of typical chaotic Hamiltonian system.
-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9
-0.9 -0.6 -0.3 0.0 0.3 0.6 0.9
Figure 2: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
4 Summary and discussion
We have studied Hamiltonian formulation of quantum dynamics of two in-
teracting qubits. Hamiltonian dynamical system on the state space CP 3
as the phase space, is integrable irrespective of the different symmetries of
the quantum system. We have then studied the dynamics of the quantum
Hamiltonian system constrained on the manifold of separable states. The
main result of this analyzes, and of the paper, is that the quantum Hamilto-
nian system without symmetry generates nonintegrable chaotic dynamics on
the set of separable states, while the constrained symmetric dynamics gives
an integrable system. It is important to bare on mind that neither the system
nor the separable states that lie on an orbit of the constrained system have
an underlining classical mechanical model. Thus, forcing a non-degenerate
quantum system to remain on the manifold of separable states is enough to
generate a dynamical system with typical properties of Hamiltonian chaos.
Our analyzes of the separability constrained quantum dynamics has been
rather formal. In order to inquire into possible interpretation of our results we
need a model of a physical realization of the separability constraints. To this
end we consider an open quantum system of two interacting qubits, whose
dynamics satisfies the Markov assumption [36], and we choose a Hermitian
Lindblad operator of the following form
L = l11σ
⊗ σ2+σ2− + l12σ1+σ1− ⊗ σ2−σ2+ + l21σ1−σ1+ ⊗ σ2+σ2− + l22σ1−σ1+ ⊗ σ2−σ2+
i,j=1
li,j|i >< j|1 ⊗ |i >< j|2 (41)
where |1 >≡ | ↑> and |2 >≡ | ↓>.
The dynamics of a pure state of the open system under the action of a
Hamiltonian H and the Linblad γL is described by the following stochastic
nonlinear Schroedinger equation [36],[37]
|dψ > = −iH|ψ > dt+ γ
(L− < ψ|L|ψ >)2|ψ > dt
+ γ(L− < ψ|L|ψ >)|ψ > dW, (42)
where dW is the increment of complex Wiener c-number process W (t).
The equation (42) represent a diffusion process on a complex Hilbert
space, and is central in the ”Quantum State Diffusion” (QSD) theory of
open quantum systems [37]. It has been used to study the systems of in-
teracting qubits in various environments for example in [32],[38], and the
effect of the Linblad operator (41) on the entanglement between two qubits
was considered in [16]. The influence of the non-Hamiltonian terms of drift
(proportional to γ2) and the diffusion (proportional to γ)), with the Linblad
operator of the form (41), is to drive an entangled state towards one of the
separable states with the corresponding probability. This process occurs on
the time scale proportional to γ−1. So, for large γ there occurs an almost
instantaneous collapse of an entangled state into a separable one. We believe
that with a proper choice of the parameters li,j the long term dynamics of
a pure state described by (42) can have the same qualitative properties as
the separability constrained quantum dynamics. In particular, the difference
between the qualitative properties of symmetric and nonsymmetric systems,
reflected in the constrained Hamiltonian system, should also manifest in the
dynamics of (42) for a proper choice of li,j. This expectations are supported
by figures 3, which illustrate the dynamics of (< σ1x >,< σ
y >) for the Hamil-
tonian operators (5) and (8) as calculated using the constrained Hamiltonian
equations (39) and (40) (figures 3b and 3a ), or the QSD equation (42) (fig-
ures 3d and 3c) for a particular choice of li,j and large γ = 5. Of course, the
choice of optimal values for li,j should be according to some criterion, which
is the problem we are currently investigating.
-1.0 -0.5 0.0 0.5 1.0
1.0 d)
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
-1.0 -0.5 0.0 0.5 1.0
Figure 3: Figures illustrate the dynamics of (< σx >,< σy >) for the
constrained Hamiltonian systems (40) (a) and (39) (b) and for the stochastic
Schroedinger equation (42) with the Linblad (41) and the hamiltonians (8) (c)
and (5) (d). the parameters are ω = 1, µ = 1.7, γ = 5 and l1,1 = 0.21, l1, 2 =
0.21, l2,1 = 0.215, l2,2 = 0.205.
The pair of coupled qubits, analyzed in this paper, is the simplest quan-
tum system exhibiting dynamical entanglement. We intend to investigate
the effects of suppression of the dynamical entanglement in systems with
spacial degrees of freedom, obtained by quantization of classically chaotic
systems, for example a pair of coupled nonlinear oscillators. In this case,
the Hamiltonian formulation of the quantum dynamics requires an infinite-
dimensional phase space, and the analyzes of the separability constrained
dynamics is more complicated. However, it wold be interesting to compare
the dynamics obtained by separability constraints with that of some more
standard semi-classical approximation.
Acknowledgements This work is partly supported by the Serbian Min-
istry of Science contract No. 141003. I should also like to acknowledge the
support and hospitality of the Abdus Salam ICTP.
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FIGURE CAPTIONS
Figure 1 Projections on (q1, p1) plane of a typical orbit for the hamil-
tonian systems (28) (a) and (27) (b) on CP 3 and on the submanifold of
separable states (c) for (40) and d for (39). The values of the parameters are
ω = 1 and µ = 1.7
Figure 2 Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a)µ = 1.1,
(b) µ = 1.3, (c) µ = 1.5 and (d) µ = 1.7
Figure 3 Figures illustrate the dynamics of (< σx >,< σy >) for the
constrained Hamiltonian systems (40) (a) and (39) (b) and for the stochastic
Schroedinger equation (42) with the Linblad (41) and the hamiltonians (8) (c)
and (5) (d). the parameters are ω = 1, µ = 1.7, γ = 5 and l1,1 = 0.21, l1, 2 =
0.21, l2,1 = 0.215, l2,2 = 0.205.
	Introduction
	Geometry of the state space CPn
	Definition and intrinsic coordinates of CPn 
	 Submanifold of separable states
	Complex structure on CPn
	Riemannian structure on CPn
	Symplectic structure on CPn
	Quantum Hamiltonian dynamical system on  CPn 
	 Quantum Hamiltonian system with imposed separability constraints
	 Qualitative properties of the constrained dynamics of two interacting qubits 
	 Summary and discussion
