Pure three-qubit states revisited: symplectic
quotient under Local Unitary transformations
Saeid Molladavoudi1 , Hishamuddin Zainuddin1,2
1
Laboratory of Computational Sciences and Mathematical Physics, Institute for
Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang,
Selangor, Malaysia
2
Department of Physics, Faculty of Science, Universiti Putra Malaysia, 43400
UPM Serdang, Selangor, Malaysia
E-mail: [email protected], [email protected]
Abstract. In this paper, we explicitly construct the space of entanglement classes
of three-qubit pure states in terms of the local unitary invariant polynomials
provided that a specific set of minimal eigenvalues of (shifted) one-qubit reduced
density matrices is fixed. In particular, given the components of the invariant
moment map in the associated moment polytope, by using the symplectic orbit
reduction method and the generalized Schmidt decomposition for three-qubit
pure states we obtain the image under the induced Hilbert map of the resulting
symplectic quotient, as a two dimensional semi-algebraic subset of the Euclidean
space.
PACS numbers: 45.20.Jj, 02.20.Tw, 03.65.Ud
Pure three-qubit states: symplectic quotient under LU transformations
2
1. Introduction
Physically, the question that we address in this paper can be stated as follows:
what information about the types of pure tripartite entanglement can be inferred
by solely referring to the local information encoded in the spectra of the particles’
reduced density matrices? In particular, classification and characterization of the
types of entanglement that a composite quantum system possesses amounts to
distinguishing multipartite states under the action of the Lie group of Local Unitary
(LU) transformations [1]. More precisely, the local unitary group K = SU (nj )×N ,
which is a compact Lie group, acts on the space of pure multipartite states M , as
the complex projective Hilbert space P(H) for H = ⊗N
j=1 Hj , by partitioning it into
orbits. Each orbit contains those states that have the same type of entanglement.
The space of orbits (orbit space) can be studied by using the algebra of K-invariant
polynomials, since for each orbit there exists an invariant polynomial which is
constant on it. In fact, compactness of the Lie group K guarantees that the K-action
on the manifold M is a proper action and so the orbit space M/K is topologically a
Hausdorff space. Then according to the Hilbert’s theorem the algebra of K-invariant
polynomials is finitely generated [2] and the orbit space M/K is mapped into a
semi-algebraic subset of the Euclidean space under the corresponding Hilbert map.
However, the space of pure multipartite states M is a Kähler manifold, which is
acted upon symplectically and in a Hamiltonian fashion by the local unitary group K
[3, 4]. In general, if M is a compact symplectic manifold equipped with a closed nondegenerate 2-form ω, under the proper and Hamiltonian action of the symmetry Lie
group K, there exists an equivariant moment(um) map J : M → k∗ , where k∗ is the
dual of the Lie algebra k of K, whose fibers are conserved with respect to the integral
curves of Hamiltonian vector fields associated to K-invariant Hamiltonian functions,
i.e. the Noether’s theorem. Then (M, ω, K, J) is then called a Hamiltonian Kmanifold. Symmetry reduction of Hamiltonian systems, i.e. studying the geometry
and topology of the quotient space Mξ = J−1 (ξ)/Kξ , for ξ ∈ k∗ , was initiated in
[5] as the Marsden-Weinstein regular reduction and extended in [6, 7, 8] for singular
reductions. In general case, the resulting symplectic quotient Mξ = J−1 (ξ)/Kξ is a
stratified space and the strata are symplectic manifolds. Here Kξ is the stabilizer of
ξ ∈ k∗ with resepect to the coadjoint action of K on k∗ .
Moreover, for every Hamiltonian K-manifold (M, ω, K, J), which is equipped
with an equivariant moment map J : M → k∗ , there exists a convex polytope
Pure three-qubit states: symplectic quotient under LU transformations
3
∆ := J0 (M ), so called the moment (Kirwan) polytope [9, 10]. The corresponding
invariant moment map J0 is defined by J0 : M → t∗+ , p 7→ J0 (p) = J(K . p) ∩ t∗+ ,
in which t∗+ = k∗ /K denotes the positive Weyl chamber. Hence, one can consider
the invariant moment values J0 (p) ≡ ξ0 ∈ ∆ = J0 (M ) ⊂ t∗+ in order to apply a
variant of the symplectic singular reduction method, namely the symplectic singular
orbit reduction. The reduced symplectic quotient MOξ = J−1 (Oξ )/K and its strata
(H)
MOξ can be obtained with respect to the coadjoint orbit Oξ through a given
ξ0 = Oξ ∩ t∗+ ∈ ∆, where ξ = J(p) ∈ k∗ and H < K.
In this paper, we consider symmetry reduction of the Hamiltonian K-manifold
(P(H), ω, K, J) for pure tripartite qubit states, as a Kähler manifold, which is acted
upon by the local unitary group K = SU (2)×3 . Since, for some states (points in
M ) their stabilizers are non-trivial the values ξ = J(p) ∈ k∗ are singular values of
the equivariant moment map J. Therefore, we use the symplectic singular orbit
reduction to obtain the reduced symplectic quotient MOξ = J−1 (Oξ )/K.
The dimension of the quotient space MOξ , for a fixed ξ0 = Oξ ∩ t∗+ ∈ ∆, is
studied in [11] for the case of pure tripartite states for three qubits, in which Sawicki
et al. showed that all the generic states, with trivial isotropy subalgebras kp = {0},
are mapped into the relative interior (rel. int.) of the Kirwan polytope ∆ by the
invariant moment map J0 . By considering the more general action of the Stochastic
Local Operations with Classical Communications (SLOCC) group G = K C , they
proved that closure of the G-orbit of the W-states is a spherical variety, whose
image under the invariant moment map J0 is also contained in the relative interior
of the Kirwan polytope ∆, i.e. the convex hull between the separable state and the
maximally entangled bi-separable states. Therefore, for a given invariant moment
value ξ0 inside that convex hull, the symplectic quotients MOξ = J−1 (Oξ )/K are
single points, namely every SLOCC W-state is uniquely determined by the spectra
of its single-qubit reduced density matrices. Moreover, for an invariant moment value
ξ0 in the boundary of the Kirwan polytope ∆, they illustrated that the closure of
their SLOCC classes are spherical varieties too and so their symplectic quotients
MOξ are isolated points and they, up to local unitary operations, are determined by
the spectra of their single-particle reduced density matrices.
In [12], it is shown that the principal stratum of the reduced symplectic
quotient J−1 (ξ)/Kξ by singular symplectic point reduction, is locally isomorphic
to the symplectic normal space in the Guillemin-Sternberg local normal form for
Pure three-qubit states: symplectic quotient under LU transformations
4
the Hamiltonian K-manifolds [13]. However, in this paper the key ideas are to use
the generalized Schmidt decomposition for pure three-qubit states [14] to find the
reduced symplectic quotient MOξ = J−1 (Oξ )/K by considering symplectic singular
orbit reduction under the local unitary group K-action. In fact, the canonical state
vector in the generalized Schmidt decomposition, whose coefficients are local unitary
invariants, is used to construct explicitly the image under the induced Hilbert map
σ̃ : M/K → Rd of the resulting quotient space MOξ , as a two dimensional stratified
symplectic space, for a specific ξ0 = Oξ ∩ t∗+ ∈ rel. int.(∆) and with respect to
K-invariant polynomials. For the case of three qubits, the image under the σ̃ map
of the orbit space M/K, as a stratified Poisson space σ̃(M/K), is a semi-algebraic
subset of the Euclidean space R5 , where d = 5 is the number of linearly independent
K-invariant polynomials [15]. The legitimacy of use of the generalized Schmidt
decomposition for three qubits under local unitary action is further discussed in
section 4 and remark 4.3.
The outline of the paper is as follows. In section 2, singular symplectic orbit
reduction is briefly reviewed. In section 3, local unitary group action on the projective
Hilbert space of a composite quantum system is introduced. In section 4, singular
symplectic orbit reduction method is applied to the projective Hilbert space under
the local unitary group action, by using the canonical state vector and the local
unitary invariant polynomials. Finally in section 5 we summarize the results.
2. Symplectic Singular Reduction
Let (M, ω) be a symplectic manifold, equipped with a closed nondegenerate 2-form
ω and is acted upon properly by the compact Lie group K. The action of the Lie
group K is defined as Φ : K × M → M, (g, p) 7→ g.p ≡ Φg (p), for every g ∈ K and
p ∈ M . The action of the Lie algebra k, i.e. φ : M × k → X(M ) is then given by
d φX (p) ≡ XM (p) := Φexp(tX) (p), p ∈ M,
dt t=0
where X(M ) denotes the Lie algebra of the smooth vector fields defined on M . Also,
φX ≡ XM ∈ X(M ) denotes the fundamental vector field (or infinitesimal generator)
of X ∈ k. For every point p ∈ M , the subgroup Kp := {g ∈ K| Φg (p) = p} is
called the isotropy subgroup (or stabilizer) of the point p ∈ M . The subgroup
Kp is a compact Lie subgroup of the group K. Therefore, the Lie subalgebra
Pure three-qubit states: symplectic quotient under LU transformations
5
kp := {X ∈ k| φX (p) ≡ XM (p) = 0} is called the stabilizer, or isotropy subalgebra
for p ∈ M . In fact, kp is the Lie algebra of the Lie subgroup Kp .
The K-orbit through the point p ∈ M , i.e. K . p, under the group action Φ
is the set K . p = {Φg (p) : g ∈ K}. Moreover, stabilizers of the points in the same
orbit are all conjugate in K, which establishes an equivalence relation in the space
of stabilizers of the Lie group K, i.e. Kp ∼ Kq if and only if there exists g ∈ K
such that Kp = gKq g −1 . Each equivalence class (Kp ) is called the (Kp )-orbit type
at p ∈ M . The orbit type submanifold M(H) = {p ∈ M : Kp ∈ (H)} is the union of
points having the same orbit type, i.e. there exists g ∈ K such that Kp = gHg −1 ,
and its image under the projection π : M → M/K, which is denoted by M(H) /K, is
called the orbit type stratum of M/K.
Regarding the Noether’s theorem for a Hamiltonian K-manifold (M, ω, K, J),
the components of the associated equivariant moment map J : M → k∗ are preserved
during the Hamiltonian dynamics, i.e. J ◦ ϕt = J, where ϕt is the corresponding
Hamiltonian flow. More precisely, the infinitesimal generator XM (p) of an element
X ∈ k is the Hamiltonian vector field generated by the Hamiltonian function
JX (p) = hJ(p) , Xi, i.e. dJX (.)(p) = ωp (XM , . ). The Hamiltonian K-manifold
(M, ω, K, J), may be reduced to the symplectic quotient Mξ containing equivalence
classes of the fibers of the moment map J−1 (ξ) under the action of the stabilizer Kξ ,
for a fixed ξ ∈ k∗ .
For free action of the Lie group K on manifold M , i.e. when Kp is trivial for
all p ∈ M , the resulting symplectic reduced space Mξ := J−1 (ξ)/Kξ , for ξ ∈ k∗ ,
would be a symplectic manifold. This reduction method is known as the MarsdenWeinstein [16, 5], or the regular symplectic point reduction [8], since the point ξ ∈ k∗
is fixed. However, for a general Hamiltonian K-manifold (M, ω, K, J), the fiber
J−1 (ξ) is a topological space and the orbit space Mξ := J−1 (ξ)/Kξ is endowed with
the quotient topology. In [6], it is shown that the symplectic quotient Mξ=0 is a
stratified symplectic space, i.e. each stratum is a symplectic manifold and the pieces
satisfy the Whitney’s condition. This result is extended to the case of the proper
action of a Lie group K with locally closed coadjoint orbits in [7]. This formulation
can be generalized to the singular symplectic orbit reduction, as follows
Theorem 2.1. [8] Let (M, ω, K, J) be a Hamiltonian K-space, where (M, ω) is a
symplectic manifold acted upon properly and symplectically by the compact Lie group
K and J : M → k∗ is the associated equivariant moment map, such that J(p) = ξ,
Pure three-qubit states: symplectic quotient under LU transformations
6
with p ∈ M and ξ ∈ k∗ as a value of J. Let Kp ≡ H and Oξ = Ad∗g ξ : g ∈ K
denotes the coadjoint orbit through ξ ∈ k∗ of the coadjoint action of K on k∗ .
Then, the quotient space MOξ = J−1 (Oξ )/K is a stratified symplectic space, with
J−1 (Oξ ) = K . J−1 (ξ) = {g . p : g ∈ K, J(p) = ξ}, such that
• J−1 (Oξ ) ∩ M(H) is a submanifold of M .
(H)
• The set MOξ
:= (J−1 (Oξ ) ∩ M(H) )/K has a unique quotient differentiable
(H)
(H)
structure such that the projection map πOξ : J−1 (Oξ ) ∩ M(H) → MOξ is a
surjective submersion.
(H)
(H)
(H)
• (MOξ , ωOξ ) is a symplectic manifold, where the symplectic structure ωOξ is
given by
(H)
(H)
(H)
(H)
(1)
(iOξ )∗ ω = (πOξ )∗ ωOξ + (JOξ )∗ ωOξ ,
(H)
where iOξ : J−1 (Oξ ) ∩ M(H) ,→ M is the inclusion map and ωOξ is the KostantKirillov-Souriau symplectic form [17] of the coadjoint orbit Oξ defined by
ωOξ (ad∗η ξ, ad∗ν ξ) := hξ, [η, ν]i,
where ξ ∈ Oξ and η, ν ∈ k∗ and ad∗η ξ =
d
dt t=0
(2)
(H)
Ad∗exp(tη) ξ. Also, the map JOξ
(H)
(H)
is defined by J|J−1 (Oξ ) ∩ M(H) for every H < K. The manifolds (MOξ , ωOξ ) are
called the singular symplectic orbit stratum.
(prin)
• Since K is a compact Lie group, there exists a unique principal stratum MOξ ,
which is open, connected and dense in the reduced space MOξ ; also known as the
principal stratum.
This is called the singular symplectic orbit reduction.
For more details and proofs one can refer to [6, 7, 8]. For a symplectic manifold
(M, ω) under the Hamiltonian action of a semisimple compact Lie group K, the
orbit space is a Poisson stratified space whose orbit type strata M(H) /K, for H < K,
and their images under the induced Hilbert map σ̃(M(H) /K) are Poisson manifolds,
which are glued together in such a way that the boundary of each stratum is the
union of other lower dimensional strata [18, 19].
Theorem 2.2. [6] Let M be a Hamiltonian K-manifold, where K is a compact Lie
group. The induced Hilbert map σ̃ : M/K → Rd is a proper injective map, which
embeds the orbit space M/K into the Euclidean space Rd , where d is the number of
Pure three-qubit states: symplectic quotient under LU transformations
7
linearly independent K-invariant polynomials. The embedding of symplectic reduced
space MOξ = J−1 (Oξ )/K, for ξ ∈ k∗ , into a subspace of the Euclidean space Rd can
be constructed by restriction of the map σ̃ to MOξ . If σ̃ : MOξ → Rd be the embedding
constructed by restriction, then the images by map σ̃ of the symplectic pieces form a
stratification of the subset σ̃(MOξ ) of Rd satisfying the Whitney’s conditions.
In section 4 we will use the theorem 2.2 to determine the image under the induced
Hilbert map σ̃ of the symplectic quotient MOξ , for a specific given Oξ ∩ t∗+ = ξ0 ≡
J0 (p0 ) ∈ rel. int.(∆), from the corresponding image under the map σ̃ of the orbit
space M/K.
Recall that the adjoint action of the compact Lie group K on its Lie algebra k,
is defined by Ad : K × k → K, (g, X) 7→ Adg X = gXg −1 . The associated coadjoint
action is then given by hAd∗g ξ, Xi = hξ, Adg−1 Xi = hξ, g −1 Xgi, for ξ ∈ k∗ , where
h . , . i represents the natural pairing between k and k∗ . The coadjoint orbit is defined
as the corresponding orbit Oξ ≡ K . ξ = Ad∗g ξ : g ∈ K , which intersects the dual
of the maximal commutative subalgebra t∗ of k∗ in accordance to the action of the
Weyl group N (T )/T , where N (T ) is the normalizer of the maximal torus T of K
[20, 17]. In fact, the maximal commutative subalgebra (Cartan subalgebra) t is the
Lie algebra of the maximal torus T and t ∼
= t∗ . Hence, each coadjoint orbit Oξ ≡ K . ξ
intersects the corresponding positive Weyl chamber t∗+ ∼
= k∗ /K only once up to the
action of the Weyl group N (T )/T on the Cartan subalgebra t∗ .
As mentioned in section 1, for each Hamiltonian K-manifold (M, ω, K, J), with
the equivariant moment map J : M → k∗ , there exists an associated convex polytope
∆ := J(M ) ∩ t∗+ , called the moment (or Kirwan) polytope [9, 10]. The invariant
moment map J0 : M → k∗ → t∗+ , p 7→ J0 (p) = J(K . p) ∩ t∗+ is an open map
onto its image [21], such that all its fibers are connected [10]. Hence, instead of
considering the moment value J(p) ≡ ξ ∈ k∗ we can consider its invariant moment
value J0 (p) ≡ ξ0 ∈ ∆ = J0 (M ) ⊂ t∗+ in order to use the symplectic singular orbit
reduction method 2.1. The reduced symplectic quotients MOξ = J−1 (Oξ )/K and
(H)
their strata MOξ can be obtained with respect to the coadjoint orbit Oξ through a
given ξ0 = Oξ ∩ t∗+ ∈ ∆, where ξ = J(p) ∈ k∗ and H < K.
−1
Remark 2.1. For every ξ0 ∈ t∗+ , we have J0 (ξ0 ) = J−1 (Oξ0 ) = K . J−1 (ξ0 ), since the
K-action is a proper action on (M, ω) [22].
Remark 2.2. Due to the equivariance property of the moment map J, i.e. J(Φg (p)) =
Ad∗g J(p), we have J(K . p) = OJ(p) and J−1 (Oξ ) = J−1 (Oξ0 ), where ξ = J(p) and
Pure three-qubit states: symplectic quotient under LU transformations
8
ξ0 = Oξ ∩ t∗+ = J(K . p) ∩ t∗+ = J0 (p). Here, Oξ = Oξ0 = Ad∗g J(p) : g ∈ K =
J(K . p) denotes the coadjoint orbit through J(p) = ξ ∈ k∗ , whose intersection with
t∗+ is determined by ξ0 = J0 (p). In other words, the equivariance property of the
moment map J guarantees that J(K . p) = Oξ and so ξ0 = J0 (p) ∈ ∆, for every
p ∈ M.
In the sebsequent sections, the notation introduced in section 2 will be adopted.
3. Local Unitary Group Action
The pure states space P(H) of a composite quantum system, consisting of N
Q
distinguishable ni -level quantum subsystems, for i = 1, · · · , N , is a ( N
i=1 ni − 1)dimensional Kähler manifold. The Symplectic and Riemannian structures of the
corresponding complex projective Hilbert space P(H) are induced from the imaginary
N
and real parts of the Hermitian inner product in H = N Hi . Moreover, the special
unitary Lie group SU (H) acts on the Kähler manifold M = P(H) transitively.
Hence, infinitesimal generators XM (p) span the tangent space Tp M , for all X ∈
su(H). In particular, the symplectic structure ω at p ∈ P(H) is given by [23]
ωp (XM , YM ) :=
i
i hψ|[A, B]ψi
= Tr(ρψ [A, B]),
2 hψ|ψi
2
(3)
for two Hermitian operators A, B ∈ su∗ (H) acting linearly on H and where
XM , YM ∈ Tp P(H) are given by
d XM (p) = π(exp(−iAt) ψ) = −i [A, ρψ ], A ∈ su∗ (H),
dt t=0
for all p ∈ P(H), where π : H → P(H), ψ 7→ p ≡ ρψ is the canonical projection.
The local unitary group K = SU (ni )×N acts properly on the Kähler manifold
N
P(H), where H = N Hi and Hi = Cni , for i = 1, · · · , N . This action is induced
from the natural action of the group K, as a compact Lie group, on the Hilbert space
H such that for g = (g1 , · · · , gN ) ∈ K and ψ = ψ1 ⊗ · · · ⊗ ψN ∈ H, with ψi ∈ Hi ,
we have g .ψ = g1 ψ1 ⊗ · · · ⊗ gN ψN ∈ H. Therefore, the action of K on the Kähler
manifold M is determined by
Φ : K × M → M, (g, p) 7→ Φg (p) = gρψ g −1 ,
(4)
Pure three-qubit states: symplectic quotient under LU transformations
9
where p ≡ ρψ = |ψihψ|/hψ|ψi ∈ M . Therefore, the infinitesimal generator for
the Hermitian operator η ∈ k∗ is given by XM (p) = ρ̇ψ = −i [η, ρψ ]. Recall that
the natural pairing between the Lie algebra k and its dual space k∗ induces the
isomorphism k ∼
= k∗ , i.e. for X ∈ k we have iX ≡ η ∈ k∗ such that the Killing-Cartan
metric defined on K, i.e. hX, Y i := −Tr(XY )/2, is satisfied for every Y ∈ k. The
same reasoning leads to the isomorphism su(H) ∼
= su∗ (H).
The K-action Φ is also a symplectic action on the Kähler manifold M = P(H),
since Φ∗g ω = ω, for every element g ∈ K. Hence, (P(H), ω, K, J) is a Hamiltonian
K-manifold, for which the equivariant moment map J : M → k∗ is defined as follows
[23]
i
i hψ|X ψi
= Tr(Xρψ ) ≡ JX (p), X ∈ k,
(5)
hJ(p), Xi =
2 hψ|ψi
2
where JX : M → R is the corresponding Hamiltonian function. In other words,
the quadruple (M, ω, K, J) is a Hamiltonian K-manifold, with the moment map
J : M → k∗ given by [3, 4]
J(p) =
N
M
j=1
(ρ(j) −
1
1nj ) ∈ k∗ ,
nj
(6)
where ρ(j) , for j = 1, · · · , N , denotes the jth-single-particle reduced density matrix,
since each ρ(j) can be decomposed as ρ(j) := n1j 1nj +ξ (j) , such that ξ (j) ∈ su∗ (nj ) with
dim(Hj ) = nj [24]. This concludes our brief introduction to the local unitary group
action on the complex projective Hilbert space of pure quantum states for a composite
quantum system containing N distinguishable ni -level quantum subsystems, for
i = 1, 2, · · · , N .
4. Singular Reduction of Local Unitary Action
As mentioned above, the local unitary group K = SU (2)×3 acts properly and in a
Hamiltonian fashion on the complex projective Hilbert space P(H) = π(C2 ⊗ C2 ⊗
C2 ) ∼
= CP (7) for a quantum system containing three qubits it its pure state. The
Lie group K is a compact Lie subgroup of the natural unitary group SU (H), with
its Lie algebra k = ⊕3i=1 su(2), where Hi ∼
= C2 , for i = 1, 2, 3, and which is spanned
by the matrices X1 ⊗ 12 ⊗ 12 + 12 ⊗ X2 ⊗ 12 + 12 ⊗ 12 ⊗ X3 , where Xj ∈ su(2) are
traceless, skew-Hermitian matrices, for j = 1, 2, 3.
Pure three-qubit states: symplectic quotient under LU transformations
10
Any state ψ ∈ H can be written as
|ψi =
1
X
i1 ,i2 ,i3 =0
Ci1 i2 i3 ei1 ⊗ ei2 ⊗ ei3 ≡
1
X
i1 ,i2 ,i3 =0
Ci1 i2 i3 |i1 i2 i3 i,
(7)
where {eik }s are orthonormal bases for the Hilbert spaces Hk , for k = 1, 2, 3.
Regarding the Eq. (6), the equivariant moment map J : P(H) = M → k∗ , p 7→ J(p)
is given by
3
M
1
(8)
J(p) :=
(ρ(i) − 12 ) ∈ k∗ ,
2
i=1
where
k∗ =
3
M
i=1
su∗ (2) =
( 3
M
k=1
−xk −i ϑ∗k
i ϑk
xk
!
)
: xk ∈ R, ϑk ∈ C, k = 1, 2, 3 ,
since ξ (k) = ρ(k) − 12 12 ∈ su∗ (2). The elements of the kth particle reduced density
matrix are calculated by
(k)
(ρ )mn =
1
X
C̄i1 ,m̂,i2 Ci1 ,n̂,i2 ,
(9)
i1 ,i2 =0
in which the sum is over all pair indices except m̂ and n̂ at the kth place. As an
example, for k = 2 it is shown in Eq. (9). Hence, one can write the associated
Hamiltonian function JX (p) for X ∈ k as
iX X
i
(ρ(k) )ik ,jk heik |Xk ejk i,
JX (p) = Tr(Xρψ ) =
2
2 k=1 i ,j =0
1
3
k
(10)
k
where Xk ∈ su(2). The Hamiltonian functions (10) are in fact the summation of the
expectation values of the skew-Hermitian operators Xk ∈ su(2).
Recall that the following polygonal inequalities describe the moment (Kirwan)
polytope in terms of the minimal eigenvalues γk of the shifted-one-particle reduced
density matrices ξ (k) = ρ(k) − 21 12 , for a system containing N -qubits [25, 11], namely
(
)
N
X
N −2
1
∆ := γk ≤
γj + (
) : − ≤ γl ≤ 0, l = 1, 2, · · · , N ,
(11)
2
2
j=1, j6=k
Pure three-qubit states: symplectic quotient under LU transformations
11
γ3
0
γ1
0
− 21
0
γ2
Figure 1. Kirwan Polytope ∆ = J0 (M ) for three qubits. The γk s denote minimal
eigenvalues of each qubit’s shifted reduced density matrices, for k = 1, 2, 3.
where 0 ≤ pk = γk + 21 ≤ 12 denotes the minimal eigenvalue of the k-th qubit reduced
density matrix ρ(k) . For three qubits it is shown in the Figure 1.
Recalling the singular symplectic orbit reduction theorem 2.1, corresponding to
each coadjoint orbit Oξ = {Ad∗g ξ : g ∈ K} through the singular moment value
ξ = J(p) ∈ k∗ there exists a symplectic reduced space MOξ = J−1 (Oξ )/K, which is a
stratified symplectic space, namely,
[
(H)
MOξ := J−1 (Oξ )/K =
MOξ ,
H<K
(H)
where each stratum MOξ denotes the equivalence class of all the points p ∈
J−1 (Oξ ) = K . J−1 (ξ) whose stabilizers Kp are conjugate to H. Moreover, there
(prin)
exists a unique principal stratum MOξ , which is open, dense and connected in the
reduced space MOξ .
Remark 4.1. Regarding the remarks 2.1 and 2.2 and the theorem 2.2, our goal is to
find the restriction of σ̃(M/K) to σ̃(MOξ0 ), for every orbit type submanifold M(H)
and ξ0 ∈ ∆. This corresponds to the restriction of the equivalence class of K-orbits
with the same orbit type, i.e. every Poisson stratum M(H) /K, to the equivalence class
−1
of K-orbits with the same orbit type in the level set J0 (ξ0 ), namely the symplectic
−1
stratum (J−1 (Oξ0 ) ∩ M(H) )/K, where J0 (ξ0 ) = J−1 (Oξ0 ), for ξ0 ∈ ∆ ⊂ t∗+ .
Remark 4.2. Also, according to the remarks 2.1 and 2.2, finding the intersection of the
coadjoint orbit through J(p) = ξ ∈ k∗ , namely Oξ = Ad∗g J(p) : g ∈ K = J(K . p),
Pure three-qubit states: symplectic quotient under LU transformations
12
with the positive Weyl chamber t∗+ amounts to diagonalizing matrices in the moment
value ξ, given by the equation (8), and then permuting the diagonal eigenvalues to
be ordered non-increasingly.
In case of three-qubit pure states, there exists a generalized Schmidt
decomposition up to the local unitary operations [26]. Consider a generic state‡
p = |ψihψ|/hψ|ψi ∈ P(H) = M , as in Eq. (7), then it can be brought by local
unitary transformations to the canonical state p0 ∈ M given by p0 = |ψ0 ihψ0 |, where
|ψ0 i = λ0 |000i + λ1 eiφ |100i + λ2 |101i + λ3 |110i + λ4 |111i,
(12)
P
with λk ≥ 0, for k = 0, · · · , 4, 4k=0 λ2k = 1 and φ ∈ [0, π]. The canonical state p0 is
non-unique up to permutation of particles for non-generic states [14]. However, for
the canonical states p0 and p1 = k . p0 which are related by the permutation operator
k, we have Kp0 ∼
= Kp1 , since k ∈ N (Kp0 ) ⊂ SU (H), where N (Kp0 ) is the normalizer
of the isotropy subgroup Kp0 . Other non-generic tripartite pure states can also be
decomposed minimally in terms of the product of local bases of the canonical state
vector (12). Their complete classification appeared in [26, 14, 27].
Recalling Eq. (8), the moment value J(p0 ) for the above choice of canonical
state (12) is given by
!
1
iφ
2
1
e
λ
λ
λ
−
0
1
0
2
,
ξ (1) = ρ(1) − 12 =
e−iφ λ0 λ1 λ21 + λ22 + λ23 + λ24 − 21
2
ξ (2)
1
= ρ(2) − 12 =
2
λ20 + λ21 + λ22 − 12 e−iφ λ1 λ3 + λ2 λ4
eiφ λ1 λ3 + λ2 λ4
λ23 + λ24 − 12
ξ (3)
1
= ρ(3) − 12 =
2
λ20 + λ21 + λ23 − 12 e−iφ λ1 λ2 + λ3 λ4
eiφ λ1 λ2 + λ3 λ4
λ22 + λ24 − 12
!
!
,
.
We also need to determine the Kirwan polytope ∆ for the above choice of
canonical state (12), by evaluating the minimal eigenvalues of the shifted local
reduced density matrices ξ (k) s. The diagonalization and permutation of the resulting
diagonal elements of shifted reduced density matrices amounts to finding ξ0 =
(k)
⊕3k=1 ξ0 ∈ t∗+ , by the coadjoint action Ad∗g ξ = g −1 ξg of the local unitary group
‡ A state p ∈ M is called generic if it has a trivial isotropy subalgebra kp = {0}; otherwise it is
called a non-generic state.
Pure three-qubit states: symplectic quotient under LU transformations
13
K on the moment value J(p0 ) = ξ ∈ k∗ ; in other words, to finding the intersection
of coadjoint orbit Oξ with the positive Weyl chamber t∗+ , or J0 (p0 ). More precisely,
(k)
there exist gk ∈ SU (2) such that gk−1 ξ (k) gk = ξ0 ∈ t∗+ (2). Therefore,
0
ξ0 = J (p0 ) = OJ(p0 ) ∩
=
( 3
M
k=1
−γk 0
0 γk
t∗+
!
=
3
M
(k)
ξ0
k=1
)
1
: − ≤ γk ≤ 0, k = 1, 2, 3 ,
2
in which the minimal eigenvalues of ξ (k) in J(p0 ) = ξ ∈ k∗ are given by
1√
γk = −
1 − 4ck ,
2
where
(13)
(14)
c1 = λ20 (λ22 + λ23 + λ24 ),
c2 = λ20 (λ23 + λ24 ) + λ21 λ24 + λ22 λ23 − 2 cos(φ)λ1 λ2 λ3 λ4 ,
c3 = λ20 (λ22 + λ24 ) + λ21 λ24 + λ22 λ23 − 2 cos(φ)λ1 λ2 λ3 λ4 .
(15)
Remark 4.3. Recall that every set of K-invariant coefficients (λj , φ) in the canonical
state p0 of the generalized Schmidt decomposition (12) determines, up to permutation
of particles, a single K-orbit. The values of the K-invariant coefficients and so
the canonical state p0 is changed, to say p00 , by diagonalization and permutation
of eigenvalues of the moment value J(p0 ) = ξ ∈ k∗ to the one with moment value
J(p00 ) = ξ0 ∈ t∗+ . However, according to the remarks 2.1 and 2.2, this is equivalent to
J0 (p0 ) = ξ0 ∈ t∗+ , which is the intersection of the coadjoint orbit Oξ through ξ and the
positive Weyl chamber t∗+ . Hence, the use of the generalized Schmidt decomposition
along with the singular symplectic orbit reduction 2.1 is justified. This is due to
the proper action of the local unitary group K on M = P(H) and the equivariance
property of the moment map J.
4.1. Symplectic Quotient as Semi-algebraic Subset
In [14, 26], by using the generalized Schmidt decomposition (12), the local unitary
invariants corresponding to the K-invariant polynomials are introduced as the
following
β1 = |λ1 λ4 eiφ − λ2 λ3 |2 ,
β4 = λ20 λ24 ,
β2 = λ20 λ22 ,
β3 = λ20 λ23 ,
β5 = λ20 (β1 + λ22 λ23 − λ21 λ24 ).
(16)
Pure three-qubit states: symplectic quotient under LU transformations
such that
1
0 ≤ β4 ≤ ,
4
1
1
β1 β2 + β1 β3 + β2 β3 + β4 (β1 + β2 + β3 ) + β42 ≤ β4 + β5 ,
4
2
1
0 ≤ β1 ≤ ,
4
1
0 ≤ β2 ≤ ,
4
14
1
0 ≤ β3 ≤ ,
4
and
β4 + β5 ≥ 0,
β52 − 4β1 β2 β3 ≤ 0,
(β4 + β5 )2 − 4(β1 + β4 )(β2 + β4 )(β3 + β4 ) ≥ 0,
which determine the image of the Hilbert map σ̃(M/K), as semi-algebraic subsets of
the R5 . The description of σ̃(M/K), as a cell complex, appears in [15] by a hierarchy
of the local unitary invariants βi s, in which β4 plays an important role. In fact, β4 is
the K-invariant that controls GHZ entanglement and ranges from β4 = 0 to β4 = 41 .
The sixth K-invariant polynomial β6 , which is introduced in [14] and subsequently in
[15] for distinguishing the K-orbits associated with |ψi and hψ| in H is redundant in
our case, since both |ψi and hψ| determine the same point p = |ψihψ| in the complex
projective Hilbert space M = P(H).
In terms of the K-invariant polynomials (16), the minimal eigenvalues pk (14)
of the single-qubit reduced density matrices ρ(k) can be re-written as
pk =
1 1√
−
1 − 4ck ,
2 2
ck =
4
X
βl .
(17)
l=1, l6=k
The minimal eigenvalues γk of the moment value matrices ξ (k) = ρ(k) − 21 12 are given
by γk = pk − 12 , such that − 12 ≤ γk ≤ 0. The Kirwan polytope ∆ is then given by
P
γk ≤ j6=k γj + 12 , for j, k = 1, 2, 3, which was shown in Figure 1.
Hence, recalling the remark 4.1, our aim is to find the image under the
induced Hilbert map of the symplectic quotient σ̃(MOξ0 ), for a given ξ0 = J0 (p0 ) ∈
rel. int.(∆). This can be done by restriction of the corresponding orbit space
σ̃(M/K) for every orbit type submanifold M(H) , which is mapped to a semialgebraic subset of the Euclidean space Rd , where d is the number of linearly
independent K-invariant polynomials. As it is mentioned previously, the key ideas
−1
are that J0 (ξ0 ) = J−1 (Oξ0 ) = K . J−1 (ξ0 ), for every ξ0 ∈ J0 (M ) = ∆, and
J−1 (Oξ0 ) = J−1 (Oξ ), since ξ0 = Oξ ∩ t∗+ .
Pure three-qubit states: symplectic quotient under LU transformations
P
Now, consider ξ0 = J0 (p0 ) ∈ rel. int.(∆) ∩ ( 3k=1 γk > − 21 ), where
(
)
X
1
1
rel. int.(∆) = γk <
γj + , − < γk < 0, j, k = 1, 2, 3 ,
2
2
j6=k
15
(18)
P
since the convex hull 3k=1 γk ≤ − 21 , with − 12 < γk < 0 for k = 1, 2, 3, corresponds
to the image under the invariant moment map J0 of the closure of the SLOCC orbit
of W states, which is shown to be a spherical variety in [11]. From the equation (17),
we have
4
X
1
− γk2 =
βl , k = 1, 2, 3,
(19)
4
l=1, l6=k
from which, with the assumption γ1 = γ2 = γ3 without loss of generality, it is implied
that
(
β1 = β2 = β3 , 0 ≤ βk < 18 − 12 γk2 ,
(20)
β4 + 2βk = 41 − γk2 , 0 < β4 < 14 − γk2 .
Recall that the image under the σ̃ map of the 5-dimensional principal stratum
M(prin) , i.e. σ̃(M(prin) /K), is given by the following semi-algebraic subset of R5 as [15]



β1 β2 β3 > 0,
σ̃(M(prin) /K) =
F ≥ 0,


β ≤ 2√β β β ,
5
1 2 3
(21)
√
where F ≡ β4 ( 41 − β4 ) + β1 β2 β3 − β1 β2 − β1 β3 − β2 β3 − β4 (β1 + β2 + β3 ). Our
−1
(prin)
goal is to find its restriction to σ̃(MOξ ) ≡ σ̃((J0 (ξ0 ) ∩ M(prin) )/K), for a given
0
P
ξ0 = J0 (p0 ) ∈ rel. int.(∆) ∩ ( 3k=1 γk > − 21 ). With the assumption γ1 = γ2 = γ3 ≡ γ,
we can proceed as follows:
• From the condition β1 β2 β3 > 0 in Eq. (21), it is implied that 0 < β < 81 − 12 γ 2 ,
i.e. β 6= 0 such that β1 = β2 = β3 ≡ β in Eq. (20). Also, from the Eq. (20), we
have β4 + 2β = 41 − γ 2 , 0 < β4 < 14 − γ 2 .
• The condition F ≥ 0 in Eq. (21) is automatically satisfied for β 6= 0 and
P3
1
− 16 < γ < 0, which is implied from
k=1 γk > − 2 with the assumption
γ1 = γ2 = γ3 ≡ γ.
Pure three-qubit states: symplectic quotient under LU transformations
16
√
• From the conditions β5 ≤ 2 β1 β2 β3 in Eq. (21) and β4 + 2β = 14 − γ 2 in Eq.
(20), it is implied that
β52 − 4(−
β4 1 1 2 3
+ − γ ) ≤ 0,
2
8 2
(22)
for a given − 61 < γ < 0.
In other words, the image under the induced Hilbert map σ̃ of the principal
(prin)
stratum MOξ of the symplectic quotient MOξ0 is given by
0
(prin)
σ̃(MOξ
0
−1
) ≡ σ̃((J0 (ξ0 ) ∩ M(prin) )/K) = β52 − 4(−
β4 1 1 2 3
+ − γ ) < 0,
2
8 2
(23)
P
for a given ξ0 = J0 (p0 ) ∈ rel. int.(∆) ∩ ( 3k=1 γk > − 12 ), with the assumption
γ1 = γ2 = γ3 ≡ γ. More precisely, the linear dependency of the K-invariant
polynomials β4 and βk ≡ β, for k = 1, 2, 3 in Eq. (20), provides us with two equivalent
descriptions in terms of (β4 , β5 ) or (β, β5 ), given ourpparticular invariant moment map
value ξ0 = J0 (p0 ) with − 16 < γ < 0, namely β5 < 2 β 3 for 0 < β < 18 − 12 γ 2 , and the
Eq. (23).
(prin)
In Figure 2, the closure of the principal stratum MOξ
of the symplectic
0
quotient MOξ0 , under the induced Hilbert map σ̃, is shown in terms of (β4 , β5 ). The
(prin)
principal stratum MOξ is the relative interior manifold of the 2-dimensional semi0
algebraic subset of R2 and since it is a 2-dimensional symplectic manifold, other lower
dimensional strata of the symplectic quotient MOξ0 would be zero-dimensional, or
isolated points, whose images under the induced Hilbert map σ̃ are shown in Figure
2 as the following:
• The origin (β4 = 0, β5 = 0) ∈ σ̃(MOξ0 ) represents the separable tripartite pure
state, since from β5 = 0 it is implied that βk ≡ β = 0, for k = 1, 2, 3 and γk ≡ γ.
• The point (β4 = 41 −γ 2 , β5 = 0) ∈ σ̃(MOξ0 ) represents the generalized GHZ state
[27], since again from β5 = 0 it is implied that βk ≡ β = 0, for k = 1, 2, 3 and
γk ≡ γ. In terms of the canonical state (12), this is the state for which only
λ0 , λ4 6= 0, such that |λ0 | =
6 |λ4 |.
q
• The point (β4 = 0, β5 = 2 ( 18 − 21 γ 2 )3 ) ∈ σ̃(MOξ0 ) represents the non-generic
W -state, since from β4 = 0 it is implied that βk ≡ β = 81 − 12 γ 2 6= 0, for
k = 1, 2, 3 and γk ≡ γ. Regarding the canonical state (12), this state can be
denoted by pw = |ψw ihψw |, for which λ1 = λ4 = 0 with λw = √13 .
Pure three-qubit states: symplectic quotient under LU transformations
q
8
2 ( 18 − 21 γ 2 )3
17
·10−2
β5
6
4
2
0
0
5 · 10−2
0.1
β4
0.15
0.2
0.25
1
4
− γ2
Figure 2. Image under σ̃ map of the symplectic quotient MOξ0 , i.e. σ̃(MOξ0 ),
in terms of the local unitary invariant polynomials (β4 , β5 ), for ξ0 = J0 (p0 ) with
− 61 < γ < 0.
For more information on the definitions of tripartite entangled states for three
qubits one can also refer to [26]. The boundaries of σ̃(MOξ0 ) represent the following
states:
• The curve β52 = 4(− β24 + 18 − 12 γ 2 )3 in Figure 2 represents those generic SLOCC
GHZ states with β 6= 0, which are mapped to ξ0 = J0 (p0 ) with − 61 < γ < 0, by
the invariant moment map J0 , or the generic SLOCC GHZ states with β 6= 0
whose (shifted) single-particle reduced density matrices possess equal minimal
eigenvalues − 61 < γ < 0.
• The open line 0 < β4 <
β = 0 and β5 = 0.
1
4
− γ 2 represents the generic SLOCC GHZ states with
3
• The open line 0 < β5 < 2( 81 − 12 γ 2 ) 2 represent the generic W-states. In terms of
the canonical state (12), these are the states for which λ4 = 0 and 0 6= β < 19 .
In other words, the interplay of two real parameters (β4 , β5 ), as local unitary invariant
polynomials, determine the entanglement class for pure tripartite state, provided that
a specific set of eigenvalues of qubits’ (shifted) reduced density matrices is fixed.
Pure three-qubit states: symplectic quotient under LU transformations
18
5. Conclusions and Outlooks
In this paper, given the minimal eigenvalues of (shifted) single-particle reduced
density matrices, as components of the associated invariant moment map J0 , the
restriction to symplectic quotient of the image under induced Hilbert map of the
orbit space is obtained, as a semi-algebraic subset of the Euclidean space. The key
ideas are using the canonical state, or the generalized Schmidt decomposition, and the
corresponding local unitary invariant polynomials. In fact, construction of σ̃(MOξ0 ),
for a specific ξ0 ∈ rel. int.(∆), implies that given equal minimal eigenvalues of shifted
local reduced density matrices such that (− 61 < γ < 0), the non-local properties of
pure three-qubit states are completely determined by two real parameters, such as
(β4 , β5 ).
The singular symplectic orbit reduction of pure tripartite states for a composite
quantum system containing three qubits can obviously be extended by regarding the
reduced Hamiltonian dynamics on the resulting symplectic quotient as a stratified
symplectic space, namely the reduced smooth Hamiltonian functions and their
associated Hamiltonian vector fields. For the case of three qubits, the resulting
reduced Hamiltonian dynamics can be induced from the original manifold M on
(prin)
the principal stratum MOξ , as a two dimensional symplectic manifold. Moreover,
0
stability and persistence of Hamiltonian dynamics around isolated points, i.e. relative
equilibria, are other problems for further investigation.
Furthermore, this can be considered as a manifestation of the Kempf-Ness
theorem in the context of Geometric Invariant Theory (GIT) [28], according to
which the quotient of semi-stable points in the manifold M under the action of the
complexified group G = K C is isomorphic to the corresponding symplectic reduced
space under the action of its maximal compact subgroup K. For instance for the
case of three qubits, it was shown in [29] that there exists two inequivalent classes
of genuine tripartite entanglement under SLOCC group action. In other words, we
propose that the symplectic singular reduction method used in this paper can also
be employed for classification of the entanglement types of semi-stable states under
SLOCC group action in a general pure multipartite case.
Pure three-qubit states: symplectic quotient under LU transformations
19
Acknowledgements
This work is partially supported by the Universiti Putra Malaysia, Research
University Grant Scheme (RUGS) number 05-02-12-1865RU.
References
[1] Linden N and Popescu S 1998 Fortschritte der Physik 46 567–578 ISSN 1521-3978
[2] Weyl H 1997 The Classical Groups: Their Invariants and Representations (Princeton, NJ:
Princeton University Press)
[3] Sawicki A, Huckleberry A and Kuś M 2011 Communications in Mathematical Physics 305(2)
441–468 ISSN 0010-3616
[4] Sawicki A and Kuś M 2011 Journal of Physics A: Mathematical and Theoretical 44 495301
[5] Marsden J and Weinstein A 1974 Reports on Mathematical Physics 5 121 – 130 ISSN 0034-4877
[6] Sjamaar R and Lerman E 1991 The Annals of Mathematics 134 375–422
[7] Bates L and Lerman E 1997 Pacific Journal of Mathematics 181 201–229
[8] Ortega J P and Ratiu T S 2004 Momentum Maps and Hamiltonian Reduction (Progress in
Mathematics vol 222) (Boston: Birkhaüser)
[9] Guillemin V and Sternberg S 1982 Inventiones Mathematicae 67(3) 491–513 ISSN 0020-9910
[10] Kirwan F 1984 Inventiones Mathematicae 77(3) 547–552 ISSN 0020-9910
[11] Sawicki A, Walter M and Kuś M 2013 Journal of Physics A: Mathematical and Theoretical 46
055304
[12] Molladavoudi S On the symplectic reduced space of three-qubit pure states arXiv:1211.1168,
2012
[13] Guillemin V and Sternberg S 1984 A normal form for the moment map Differential geometric
methods in mathematical physics (Jerusalem 1982) Math. Phys. Stud., 6 ed Sternberg S
(Reidel, Dordrecht: D. Reidel Publishing Company) pp 161–175
[14] Acı́n A, Andrianov A, Costa L, Jané E, Latorre J I and Tarrach R 2000 Phys. Rev. Lett. 85(7)
1560–1563
[15] Walck S N, Glasbrenner J K, Lochman M H and Hilbert S A 2005 Phys. Rev. A 72(5) 052324
[16] Meyer K R 1973 Symmetries and integrals of mechanics Dynamical Systems ed Piexoto M
(Academic Press) pp 259–273
[17] Kirillov A A 2004 Lectures on the Orbit Methods (Graduate Studies in Mathematics vol 64)
(USA: American Mathematical Society)
[18] Cushman R and Sniatycki J 2001 Canadian Journal of Mathematics 53 715–755
[19] Arms J M, Cushman R H and Gotay M J 1991 A universal reduction procedure for hamiltonian
group actions The geometry of Hamiltonian systems (Berkeley, CA, 1989) (Math. Sci. Res.
Inst. Publ. vol 22) (New York: Springer) pp 33–51
[20] Bott R 1979 The geometry and representation theory of compact lie groups Representation
Theory of Lie Groups (London Mathematical Society Lecture Note Series no 34) (Cambridge
University Press) pp 65–90
[21] Knop F 2002 J. of Lie Theory 12 571–582
Pure three-qubit states: symplectic quotient under LU transformations
20
[22] Ortega J P 1998 Symmetry, Reduction, and Stability in Hamiltonian Systems Ph.D. thesis
University of California, Santa Cruz
[23] Benvegnú A, Sansonetto N and Spera M 2004 Journal of Geometry and Physics 51 229 – 243
ISSN 0393-0440
[24] Chruściński D and Jamiolkowski A 2004 Geometric Phases in Classical and Quantum
Mechanics (Progress in Mathematical Physics vol 36) (Boston: Birkhüser)
[25] Higuchi A, Sudbery A and Szulc J 2003 Phys. Rev. Lett. 90(10) 107902
[26] Acı́n A, Andrianov A, Jané E and Tarrach R 2001 Journal of Physics A: Mathematical and
General 34 6725
[27] Carteret H A and Sudbery A 2000 Journal of Physics A: Mathematical and General 33 4981
[28] Kempf G and Ness L 1979 The length of vectors in representation spaces Algebraic Geometry
(Lecture Notes in Mathematics vol 732) ed Lonsted K (Springer Berlin Heidelberg) pp 233–
243 ISBN 978-3-540-09527-9
[29] Dür W, Vidal G and Cirac J I 2000 Phys. Rev. A 62(6) 062314