Existence of Universal Entangler
Jianxin Chen,1, ∗ Runyao Duan,1, † Zhengfeng Ji,1, ‡ Mingsheng Ying,1, § and Jun Yu2, ¶
1 State Key Laboratory of Intelligent Technology and Systems,
Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
2 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
(Dated: November 18, 2018)
A gate is called an entangler if it transforms some (pure) product states to entangled states. A
universal entangler is a gate which transforms all product states to entangled states. In practice,
a universal entangler is a very powerful device for generating entanglements, and thus provides
important physical resources for accomplishing many tasks in quantum computing and quantum
information. This Letter demonstrates that a universal entangler always exists except for a degen-
erated case. Nevertheless, the problem how to find a universal entangler remains open.
PACS numbers: 02.20.-a, 02.20.Uw, 03.65.Ud, 03.67.Lx
Introduction. It is a common sense in the quantum
computation and quantum information community that
entanglement is an extremely important kind of phys-
ical resources. How to generate this kind of resources
therefore becomes an important problem. One possibil-
ity one may naturally conceive is to generate entangle-
ment from product states which can be prepared spatially
separately. If a gate can transform some product states
to entangled states, then we call it an entangler. For
some given product states, it is not difficult to find an
entangler that maps them to entangled states. Here we
consider a more challenging problem: Does there exist
some entangler that can transform all product states to
entangled states. We call such an entangler a universal
entangler [9, 17]. One may even wonder at the existence
of universal entanglers.
Formally, the problem of existence of universal entan-
glers in bipartite systems can be stated as follows:
Problem 1. Suppose Alice and Bob have two quan-
tum systems with state spaces HA and HB respectively.
Whether there exists a unitary operator U acting on
HA ⊗HB such that U(|φ〉 ⊗ |ψ〉) is always entangled for
any |φ〉 ∈ HA and |ψ〉 ∈ HB?
The purpose of this letter is to present a complete so-
lution to the above problem. We have our main theorem
as follows:
Theorem 1. Given a bipartite quantum system
Hm ⊗H\, where Hm and H\ are Hilbert space with di-
mension m and n respectively, then there exists a uni-
tary operator U which maps every product state of this
system to an entangled bipartite state if and only if
min(m,n) ≥ 3 and (m,n) 6= (3, 3).
There are many literatures and different approaches
which are potentially relevant to the topic of entanglers.
Besides universal entangler, Zhang et al. discussed per-
fect entanglers which are defined as the unitary opera-
tions that can generate maximal entangled states from
some initially separable states [16, 19, 20]. Bužek et
al. [9] considered the entangler that entangles a qubit
in unknown state with a qubit in a reference, also they
proved the nonexistence of universal entangler for qubits,
which answers a special case of our problem. Further-
more, nonexistence of universal entangler in 2 ⊗ n bi-
partite system can be derived straightforwardly from
Parthasarathy’s recent work on the maximal dimension
of completely entangled subspace [14], which is tightly
connected to the unextendible product bases introduced
by Bennett et al. [6, 7, 10].
But the proof of the above theorem requires some
mathematical results from basic algebraic geometry. We
will first review some basic notions in algebraic geometry
in the next section. To make a clear presentation, Theo-
rem 1 is divided into two parts, namely Theorems 2 and 3
below, and detailed proofs of them are given. Finally, a
brief conclusion is drawn and some open problems are
proposed.
Preliminaries. For the convenience of the reader, we
recall some basic definitions in algebraic geometry [3, 11].
The n× n general linear group and the n× n unitary
group are denoted by GL(n,C) and U(n), respectively.
An affine n-space, denoted by An, is the set of all n-
tuples of complex numbers. An element of An is called
a point, and if point P = (a1, a2, · · · , an) with ai ∈ C,
then the ai’s are called the coordinates of P .
The polynomial ring in n variables, denoted by
C[x1, x2, · · · , xn], is the set of polynomials in n variables
with coefficients in a ring.
A subset Y of An is an algebraic set if it is the common
zeros of a finite set of polynomials f1, f2, · · · , fr with fi ∈
C[x1, x2, · · · , xn] for 1 ≤ i ≤ r, which is also denoted by
Z(f1, f2, · · · , fr).
It is not hard to check that the union of a finite number
of algebraic sets is an algebraic set, and the intersection
of any family of algebraic sets is again an algebraic set.
Thus by taking the open subsets to be the complements
of algebraic sets, we can define a topology, called the
Zariski topology on An.
http://arxiv.org/abs/0704.1473v2
A nonempty subset Y of a topological spaceX is called
irreducible if it cannot be expressed as the union Y =
Y1 ∪ Y2 of two proper closed subsets Y1, Y2. The empty
set is not considered to be irreducible.
Let X , Y be two topological spaces, then we have two
useful facts:
Fact 1. If X is irreducible and F : X → Y be a contin-
uous function, then F (X) with induced topology is also
irreducible.
Fact 2. Let Y ⊂ X be a subset, if X is irreducible and
Y is open, then Y is irreducible; if Y is irreducible and
dense in X, then X is irreducible.
An affine algebraic variety is an irreducible closed sub-
set of some An, with respect to the induced topology.
We define projective n-space, denoted by Pn, to be the
set of equivalence classes of (n + 1)−tuples (a0, · · · , an)
of complex numbers, not all zero, under the equivalence
relation given by (a0, · · · , an) ∼ (λa0, · · · , λan) for all
λ ∈ C, λ 6= 0.
A notion of algebraic variety may also be introduced
in projective spaces, called projective algebraic variety:
a subset Y of Pn is an algebraic set if it is the com-
mon zeros of a finite set of homogeneous polynomials
f1, f2, · · · , fr with fi ∈ C[x0, x1, · · · , xn] for 1 ≤ i ≤ r.
We call open subsets of irreducible projective varieties as
quasi-projective varieties.
We will mainly use the following two varieties:
Example 1. A1 is irreducible, because its only proper
closed subsets are finite, yet it is itself infinite. This fact
is somewhat trivial. If T is an algebraic subset of A1,
then we should say, there is a finite set F of polynomi-
als over C, such that T = Z(F ). For all f ∈ C[x], if
degree(f) is 0 or f is a constant function, then T should
be the degenerate subset of C, i.e.,empty set ∅ or C. Oth-
erwise, we can choose a f ∈ C[x], such that f is not a
constant function, and degree(f) is no less than 1. From
the fundamental theorem of algebra, we know the num-
ber of its roots is at most its degree, so its solution set
should be a finite set. Thus, a subset of the solution set
must be finite too, and we proved that T is a finite subset
of C.
Example 2. The Segre embedding is defined as the map:
σ : Pm−1 × Pn−1 → Pmn−1
taking a pair of points ([x], [y]) ∈ Pm−1 × Pn−1 to their
product
σ : ([x0 : x1 : · · · : xm−1], [y0 : y1 : · · · : yn−1])
7−→ [x0y0 : x0y1 : · · · : xm−1yn−1]
Here, Pm−1 and Pn−1 are projective vector spaces over
some arbitrary field,
[x0 : x1 : · · · : xn−1]
is the homogeneous coordinates of x, and similarly for y.
The image of the map is a variety, called Segre variety,
written as Σm−1,n−1.
What concerns us is that Segre variety represents the
set of product states [8, 12, 13].
If X is a topological space, we define the dimension
of X, denoted by dim(X), to be the supremum of all
integers n such that there exists a chain Z0 ⊂ Z1 ⊂ · · · ⊂
Zn of n+1 distinct irreducible closed subsets of X. The
dimension of a quasi-projective variety is then defined
according to the Zariski topology.
Main results. With the notations introduced above,
Problem 1 can be restated as follows:
Problem 2. Let Z denote the Segre variety Σm−1,n−1.
Whether there exists a gate Φ ∈ U(mn) s.t. Φ(Z)∩Z =
Note that Z is the set of product states, and thus Φ(Z)
is the set of states generated by gate Φ from product
states. So, Φ(Z)∩Z = ∅means that all states in Φ(Z) are
entangled, and the above problem coincides with Prob-
lem 1.
We claim that the answer to the above question is af-
firmative except for some degenerated cases. First, we
consider the degenerated cases that min(m,n) ≤ 2 or
(m,n) = (3, 3). The following theorem gives a negative
answer for this case:
Theorem 2. Given a bipartite quantum system
Hm ⊗H\, where Hm and H\ are Hilbert spaces with di-
mension m and n respectively, if min (m,n) ≤ 2 or
(m,n) = (3, 3), then ∀Φ ∈ U(mn), we have
Φ(Z) ∩ Z 6= ∅
In other words, universal entangler does not exist.
To prove the above theorem, we need the following:
Lemma 1. [Projective Dimension Theorem] Let Y , Z
be varieties of dimensions r, s in Pn. Then every irre-
ducible component of Y ∩ Z has dimension ≥ r + s− n.
Furthermore, if r + s− n ≥ 0, then Y ∩ Z is nonempty.
For the proof of Lemma 1, see [11], chapter I, theorem
7.2. Then we can easily prove Theorem 2:
Proof. For any Φ ∈ U(mn), dim(Z) = m − 1 + n − 1 =
m+n−2. Then we have dim(Φ(Z))+dim(Z)−dim(Y ) =
2 dim(Z)−dim(Y ) = 2(m+n−2)− (mn−1) = 1− (m−
2)(n − 2) ≥ 0. So we have Φ(Z) ∩ Z 6= ∅ according to
Lemma 1.
We now turn to consider the general case where
min(m,n) ≥ 3 and (m,n) 6= (3, 3).
The following lemmas are needed in the proof of our
main theorem.
Lemma 2. U(k) is Zariski dense in GL(k,C)
Proof. For the case k = 1, the only Zariski closed subsets
of G = GL(1,C) = C\{0} ∼= {(z1, z2) ∈ C
2 : z1z2 = 1}
are ∅, G and non-empty finite subsets of G. This follows
immediately from Example 1. Since GL(1,C) is an open
subset of affine line C, its closed subsets are intersections
of closed subsets of C and C\{0}. Now we see that U(1)
is infinite, thus U(1) = G.
In general, H(k) = {diag(z1, z2, · · · , zk) :
z1, z2, · · · , zk ∈ C\{0}} ∼= (C\{0})
k is the Fi-
bre product of k copies of C\{0} [11]. Then
H(k) ∩ U(k) = {diag(z1, z2, · · · , zk) : |z1| = |z2| =
· · · = |zk| = 1} ∼= U(1)
By the results of the case k = 1, we also have
H(k) ∩ U(k) = H(k) ⊇ A(k) = {diag(z1, z2, · · · , zk) :
z1 > 0, z2 > 0, · · · , zk > 0}.
Now for any B ∈ U(k), LB : X → B · X and RB :
X → X ·B are two isomorphisms of G, and LB(U(k)) =
U(k) = RB(U(k)). Thus LB(U(k)) = U(k) = RB(U(k)).
Since A(k) ⊆ H(k) ∩ U(k) ⊆ U(k), we have
U(k)A(k)U(k) ⊆ U(k).
By singular value decomposition for GL(k,C), we get
U(k)A(k)U(k) = GL(k,C) ⊆ U(k). Thus U(k) =
GL(k,C).
The following lemma [18] establishes a connection be-
tween the dimensions of domain and codomain of a va-
riety morphism. A morphism Φ : Z1 → Z2 is called a
dominant morphism if Φ(Z1) is dense in Z2.
Lemma 3.
1 Z1 and Z2 are both irreducible varieties over C,
and φ : Z1 → Z2 is a dominant morphism, then
dim(Z2) ≤ dim(Z1).
2 Z1 and Z2 are both varieties over C, and φ : Z1 →
Z2 is a morphism, r = max
dim(φ−1(z)), then
dim(Z1) ≤ r + dim(Z2).
3 If V = ∪si=1Vi is a finite open covering, and ∀i,
Vi is irreducible, ∀i, j, Vi ∩ Vj 6= ∅, then V is irre-
ducible.
Let X = {Φ|Φ ∈ GL(mn,C),Φ(Z) ∩ Z 6= ∅}. We
are able to give an upper bound on the dimension of the
closure of X with respect to the Zariski topology.
Lemma 4. dim(X) ≤ m2n2 − (m− 2)(n− 2)+ 1, where
X is the Zariski closure of X.
Proof. We have a morphism F : G×Y → Y which is just
the left action of G on Y , defined by
F (g, [w]) = [g · w]
where G = GL(k,C), Y = Pmn−1, k = mn.
Let y0 = (1, 0, · · · , 0)
T be a column vector with k
entries. For any given y1, y2 ∈ Y , we choose g1,
g2 ∈ GL(k,C), such that [g1 ·y0] = [y1] and [g2 ·y0] = [y2].
Then we have
[g · y1] = [y2]
⇐⇒ [gg1 · y0] = [g2 · y0]
⇐⇒ [g−1
gg1 · y0] = [y0]
From above observations, F has the following prop-
erty: for any y1, y2 ∈ Y , F
−1(y2) ∩ {G × {y1}} ∼=
: z1 ∈ C\{0}, g
′ ∈ GL(k − 1,C), α ∈
k−1 is a row vector.}. Hence dim(F−1(y2) ∩
G× {y1}) = m
2n2 − (mn− 1).
Let P1, P2 be projections of G×Y to G, Y respectively.
Now we only look atG×Z ⊆ G×Y , to get F : G×Z → Y .
Then we have a characterization of X: X = P1F
−1(Z).
In fact,
g ∈ X
⇐⇒ g(Z) ∩ Z 6= ∅
⇐⇒ ∃z1, z2 ∈ Z, s.t.g(z1) = z2
⇐⇒ ∃z1, z2 ∈ Z, s.t.(g, z1) ∈ F
−1(z2)
⇐⇒ ∃z2 ∈ Z, s.t.g ∈ P1F
−1(z)
⇐⇒ g ∈ P1F
−1(Z)
Let X ⊆ G be the Zariski closure of X in G, then P1 :
F−1(Z) → X is a dominant morphism.
Furthermore, consider Ψ : F−1(Z) → Z × Z given by
Ψ(g, [z]) = ([z], [g · z])
For all z1, z2 ∈ Z, we have Ψ
−1(z1, z2) = (g2Tg
1 , z1),
where T = {
: z0 ∈ C\{0}, g
′ ∈ GL(k −
1,C), α ∈ Ck−1 is a row vector}, and g1, g2 ∈ GL(k,C),
s.t. g1(y0) = z1, g2(y0) = z2. Ψ is a dominant morphism
since G acts transitively on Pk−1. Then we obtain
dim(F−1(Z)) ≤dim(T ) + dim(Z × Z)
=m2n2 −mn+ 1 + 2 · dim(Z)
It is required in Lemma 3.1 that varieties Z1 and Z2 are
irreducible, but we haven’t proved F−1(Z) is irreducible.
Actually, this condition can be weakened: Lemma 3.1
is still true for the more general case that Z1 and Z2
are closed subsets of irreducible varieties [11]. Thus, we
can fill out the gap and apply this lemma. Indeed, the
irreducibility of F−1(Z) andX really holds, but the proof
is not easy (see [1] for a brief proof ). From Lemma 3.1,
we have
dim(X) ≤ dim(F−1(Z))
≤(m2n2 − (mn− 1)) + 2 · dim(Z)
=m2n2 − (mn− 1) + 2(m+ n− 2)
=m2n2 − (m− 2)(n− 2) + 1
With the above lemmas, we can now derive the main
result:
Theorem 3. Given a bipartite quantum system
Hm ⊗H\, where Hm and H\ are Hilbert spaces with di-
mensions m and n respectively, if min (m,n) > 2 and
(m,n) 6= (3, 3), then there exists an unitary operator
Φ ∈ U(mn), s.t.
Φ(Z) ∩ Z = ∅
That is, Φ is a universal entangler.
Proof. If U(mn) ⊂ X , then it follows that GL(mn,C) =
U(mn) ⊂ X from Lemma 2. And in this assumption, we
also have dim(X) ≤ m2n2− (m−2)(n−2)+1< m2n2 =
dim(GL(mn,C)) from Lemma 4 [2]. It’s a contradiction.
So U(mn) 6⊂ X , i.e. a unitary operator Φ ∈ U(mn) with
Φ(Z) ∩ Z = ∅ exists.
Conclusion. In summary, it is shown that a universal
entangler of bipartite system exists except for the cases
of 1⊗n, 2⊗n, m⊗ 1, m⊗ 2 and 3⊗ 3. So we have com-
pletely determined when a universal entangler exists. It
seems that the method employed in this Letter can be
extended to the multipartite case. This extension will
lead us to explore the geometric structure of entangled
states and other related objects. Furthermore, we can
consider some more practical questions: (1) How to con-
struct such a universal entangler for 3⊗4 bipartite system
explicitly? (2) Given a universal entangler, what is the
minimum entanglement it guarantees to output with re-
spect to the definition of entanglement measure for pure
states [4, 5, 15]? (3) Furthermore, what is the optimal
universal entangler which maximized the minimally pos-
sible entanglement the entangler outputs. Intuitively, for
a bipartite system of sufficiently large dimensions,a ran-
domly chosen unitary operator seems to be a universal
entangler with high probability. Nevertheless, we failed
to give a proof of this conjecture.
Acknowledgements. We are thankful to the colleagues
in the Quantum Computation and Information Research
Group of Tsinghua University for helpful discussions.
And one of us (J.-X. Chen) enjoyed delightful discus-
sions with W. Huang, L. Xiao, Z. Q. Zhang and Q. Lin.
Jun Yu thank Professor X.-J. Tan and Q.-C. Tian who
ask him to study Hartshorne ’s book. This work was
partly supported by the Natural Science Foundation of
China(Grant Nos.60621062 and 60503001) and the Hi-
Tech Research and Development Program of China(863
project)(Grant No.2006AA01Z102).
∗ Electronic address: chenkenshin@gmail.com
† Electronic address: dry@tsinghua.edu.cn
‡ Electronic address: jizhengfeng98@mails.thu.edu.cn
§ Electronic address: yingmsh@tsinghua.edu.cn
¶ Electronic address: majunyu@ust.hk
[1] Here, we will present a short proof of the irreduciblity
of F−1(Z). ∀i, j: 0 ≤ i ≤ m− 1, 0 ≤ j ≤ n− 1,
let Pm,i = {[x0, x1, · · · , xi, · · · , xm−1]|xi 6= 0}, and
Pn,j = {[y0, y1, · · · , yj , · · · , yn−1]|yi 6= 0}. Let Zi,j =
σ(Pm,i × Pn,j), we will have Z = ∪i,jZi,j , and for
∀i, j, Zi,j ∼= Pm,i × Pn,j ∼= A
m−1 × An−1 ∼= A
m+n−2,
and each Zi,j is open in Z. ∀i, j, we can construct a
morphism Ti,j : Zi,j → GL(mn,C) s.t. ∀y ∈ Zi,j ,
(Ti,j(y))(y0) = y. Then ∀i, j, î, ĵ, (g, y) ∈ Ψ
î,ĵ
Zi,j) ⇐⇒ g·y ∈ Zi,j and y ∈ Zî,ĵ ⇐⇒ (g·Tî,ĵ(y1))y0 =
Ti,j(y2)(y0) for some y2 ∈ Zi,j , y1 ∈ Zî,ĵ ⇐⇒ g ∈
Ti,j(Zi,j)·H ·(Tî,ĵ(Zî,ĵ))
−1. Thus we have F−1(Z)
i,j,̂i,ĵ
{(g, y) ∈ Ψ−1(Z
î,ĵ
× Zi,j)} is irreducible since it’s the
image of a morphism Zi,j × H × Zî,ĵ → F
−1(Z). Each
−1(Z)
i,j,̂i,ĵ
is also open in F−1(Z), then the irreducib-
lity follows from lemma 3.3.
[2] In fact, we proved Ψ : F−1(Z) → Z × Z is an alge-
braic fibre bundle thus dim(F−1(Z)) = m2n2 − mn +
1 + 2 · dim(Z). We also conjecture that dim(X) =
dim(F−1(Z)) and X = X.
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  A gate is called entangler if it transforms some (pure) product states to
entangled states. A universal entangler is a gate which transforms all product
states to entangled states. In practice, a universal entangler is a very
powerful device for generating entanglements, and thus provides important
physical resources for accomplishing many tasks in quantum computing and
quantum information. This Letter demonstrates that a universal entangler always
exists except for a degenerated case. Nevertheless, the problem how to find a
universal entangler remains open.

Introduction. It is a common sense in the quantum
computation and quantum information community that
entanglement is an extremely important kind of phys-
ical resources. How to generate this kind of resources
therefore becomes an important problem. One possibil-
ity one may naturally conceive is to generate entangle-
ment from product states which can be prepared spatially
separately. If a gate can transform some product states
to entangled states, then we call it an entangler. For
some given product states, it is not difficult to find an
entangler that maps them to entangled states. Here we
consider a more challenging problem: Does there exist
some entangler that can transform all product states to
entangled states. We call such an entangler a universal
entangler [9, 17]. One may even wonder at the existence
of universal entanglers.
Formally, the problem of existence of universal entan-
glers in bipartite systems can be stated as follows:
Problem 1. Suppose Alice and Bob have two quan-
tum systems with state spaces HA and HB respectively.
Whether there exists a unitary operator U acting on
HA ⊗HB such that U(|φ〉 ⊗ |ψ〉) is always entangled for
any |φ〉 ∈ HA and |ψ〉 ∈ HB?
The purpose of this letter is to present a complete so-
lution to the above problem. We have our main theorem
as follows:
Theorem 1. Given a bipartite quantum system
Hm ⊗H\, where Hm and H\ are Hilbert space with di-
mension m and n respectively, then there exists a uni-
tary operator U which maps every product state of this
system to an entangled bipartite state if and only if
min(m,n) ≥ 3 and (m,n) 6= (3, 3).
There are many literatures and different approaches
which are potentially relevant to the topic of entanglers.
Besides universal entangler, Zhang et al. discussed per-
fect entanglers which are defined as the unitary opera-
tions that can generate maximal entangled states from
some initially separable states [16, 19, 20]. Bužek et
al. [9] considered the entangler that entangles a qubit
in unknown state with a qubit in a reference, also they
proved the nonexistence of universal entangler for qubits,
which answers a special case of our problem. Further-
more, nonexistence of universal entangler in 2 ⊗ n bi-
partite system can be derived straightforwardly from
Parthasarathy’s recent work on the maximal dimension
of completely entangled subspace [14], which is tightly
connected to the unextendible product bases introduced
by Bennett et al. [6, 7, 10].
But the proof of the above theorem requires some
mathematical results from basic algebraic geometry. We
will first review some basic notions in algebraic geometry
in the next section. To make a clear presentation, Theo-
rem 1 is divided into two parts, namely Theorems 2 and 3
below, and detailed proofs of them are given. Finally, a
brief conclusion is drawn and some open problems are
proposed.
Preliminaries. For the convenience of the reader, we
recall some basic definitions in algebraic geometry [3, 11].
The n× n general linear group and the n× n unitary
group are denoted by GL(n,C) and U(n), respectively.
An affine n-space, denoted by An, is the set of all n-
tuples of complex numbers. An element of An is called
a point, and if point P = (a1, a2, · · · , an) with ai ∈ C,
then the ai’s are called the coordinates of P .
The polynomial ring in n variables, denoted by
C[x1, x2, · · · , xn], is the set of polynomials in n variables
with coefficients in a ring.
A subset Y of An is an algebraic set if it is the common
zeros of a finite set of polynomials f1, f2, · · · , fr with fi ∈
C[x1, x2, · · · , xn] for 1 ≤ i ≤ r, which is also denoted by
Z(f1, f2, · · · , fr).
It is not hard to check that the union of a finite number
of algebraic sets is an algebraic set, and the intersection
of any family of algebraic sets is again an algebraic set.
Thus by taking the open subsets to be the complements
of algebraic sets, we can define a topology, called the
Zariski topology on An.
http://arxiv.org/abs/0704.1473v2
A nonempty subset Y of a topological spaceX is called
irreducible if it cannot be expressed as the union Y =
Y1 ∪ Y2 of two proper closed subsets Y1, Y2. The empty
set is not considered to be irreducible.
Let X , Y be two topological spaces, then we have two
useful facts:
Fact 1. If X is irreducible and F : X → Y be a contin-
uous function, then F (X) with induced topology is also
irreducible.
Fact 2. Let Y ⊂ X be a subset, if X is irreducible and
Y is open, then Y is irreducible; if Y is irreducible and
dense in X, then X is irreducible.
An affine algebraic variety is an irreducible closed sub-
set of some An, with respect to the induced topology.
We define projective n-space, denoted by Pn, to be the
set of equivalence classes of (n + 1)−tuples (a0, · · · , an)
of complex numbers, not all zero, under the equivalence
relation given by (a0, · · · , an) ∼ (λa0, · · · , λan) for all
λ ∈ C, λ 6= 0.
A notion of algebraic variety may also be introduced
in projective spaces, called projective algebraic variety:
a subset Y of Pn is an algebraic set if it is the com-
mon zeros of a finite set of homogeneous polynomials
f1, f2, · · · , fr with fi ∈ C[x0, x1, · · · , xn] for 1 ≤ i ≤ r.
We call open subsets of irreducible projective varieties as
quasi-projective varieties.
We will mainly use the following two varieties:
Example 1. A1 is irreducible, because its only proper
closed subsets are finite, yet it is itself infinite. This fact
is somewhat trivial. If T is an algebraic subset of A1,
then we should say, there is a finite set F of polynomi-
als over C, such that T = Z(F ). For all f ∈ C[x], if
degree(f) is 0 or f is a constant function, then T should
be the degenerate subset of C, i.e.,empty set ∅ or C. Oth-
erwise, we can choose a f ∈ C[x], such that f is not a
constant function, and degree(f) is no less than 1. From
the fundamental theorem of algebra, we know the num-
ber of its roots is at most its degree, so its solution set
should be a finite set. Thus, a subset of the solution set
must be finite too, and we proved that T is a finite subset
of C.
Example 2. The Segre embedding is defined as the map:
σ : Pm−1 × Pn−1 → Pmn−1
taking a pair of points ([x], [y]) ∈ Pm−1 × Pn−1 to their
product
σ : ([x0 : x1 : · · · : xm−1], [y0 : y1 : · · · : yn−1])
7−→ [x0y0 : x0y1 : · · · : xm−1yn−1]
Here, Pm−1 and Pn−1 are projective vector spaces over
some arbitrary field,
[x0 : x1 : · · · : xn−1]
is the homogeneous coordinates of x, and similarly for y.
The image of the map is a variety, called Segre variety,
written as Σm−1,n−1.
What concerns us is that Segre variety represents the
set of product states [8, 12, 13].
If X is a topological space, we define the dimension
of X, denoted by dim(X), to be the supremum of all
integers n such that there exists a chain Z0 ⊂ Z1 ⊂ · · · ⊂
Zn of n+1 distinct irreducible closed subsets of X. The
dimension of a quasi-projective variety is then defined
according to the Zariski topology.
Main results. With the notations introduced above,
Problem 1 can be restated as follows:
Problem 2. Let Z denote the Segre variety Σm−1,n−1.
Whether there exists a gate Φ ∈ U(mn) s.t. Φ(Z)∩Z =
Note that Z is the set of product states, and thus Φ(Z)
is the set of states generated by gate Φ from product
states. So, Φ(Z)∩Z = ∅means that all states in Φ(Z) are
entangled, and the above problem coincides with Prob-
lem 1.
We claim that the answer to the above question is af-
firmative except for some degenerated cases. First, we
consider the degenerated cases that min(m,n) ≤ 2 or
(m,n) = (3, 3). The following theorem gives a negative
answer for this case:
Theorem 2. Given a bipartite quantum system
Hm ⊗H\, where Hm and H\ are Hilbert spaces with di-
mension m and n respectively, if min (m,n) ≤ 2 or
(m,n) = (3, 3), then ∀Φ ∈ U(mn), we have
Φ(Z) ∩ Z 6= ∅
In other words, universal entangler does not exist.
To prove the above theorem, we need the following:
Lemma 1. [Projective Dimension Theorem] Let Y , Z
be varieties of dimensions r, s in Pn. Then every irre-
ducible component of Y ∩ Z has dimension ≥ r + s− n.
Furthermore, if r + s− n ≥ 0, then Y ∩ Z is nonempty.
For the proof of Lemma 1, see [11], chapter I, theorem
7.2. Then we can easily prove Theorem 2:
Proof. For any Φ ∈ U(mn), dim(Z) = m − 1 + n − 1 =
m+n−2. Then we have dim(Φ(Z))+dim(Z)−dim(Y ) =
2 dim(Z)−dim(Y ) = 2(m+n−2)− (mn−1) = 1− (m−
2)(n − 2) ≥ 0. So we have Φ(Z) ∩ Z 6= ∅ according to
Lemma 1.
We now turn to consider the general case where
min(m,n) ≥ 3 and (m,n) 6= (3, 3).
The following lemmas are needed in the proof of our
main theorem.
Lemma 2. U(k) is Zariski dense in GL(k,C)
Proof. For the case k = 1, the only Zariski closed subsets
of G = GL(1,C) = C\{0} ∼= {(z1, z2) ∈ C
2 : z1z2 = 1}
are ∅, G and non-empty finite subsets of G. This follows
immediately from Example 1. Since GL(1,C) is an open
subset of affine line C, its closed subsets are intersections
of closed subsets of C and C\{0}. Now we see that U(1)
is infinite, thus U(1) = G.
In general, H(k) = {diag(z1, z2, · · · , zk) :
z1, z2, · · · , zk ∈ C\{0}} ∼= (C\{0})
k is the Fi-
bre product of k copies of C\{0} [11]. Then
H(k) ∩ U(k) = {diag(z1, z2, · · · , zk) : |z1| = |z2| =
· · · = |zk| = 1} ∼= U(1)
By the results of the case k = 1, we also have
H(k) ∩ U(k) = H(k) ⊇ A(k) = {diag(z1, z2, · · · , zk) :
z1 > 0, z2 > 0, · · · , zk > 0}.
Now for any B ∈ U(k), LB : X → B · X and RB :
X → X ·B are two isomorphisms of G, and LB(U(k)) =
U(k) = RB(U(k)). Thus LB(U(k)) = U(k) = RB(U(k)).
Since A(k) ⊆ H(k) ∩ U(k) ⊆ U(k), we have
U(k)A(k)U(k) ⊆ U(k).
By singular value decomposition for GL(k,C), we get
U(k)A(k)U(k) = GL(k,C) ⊆ U(k). Thus U(k) =
GL(k,C).
The following lemma [18] establishes a connection be-
tween the dimensions of domain and codomain of a va-
riety morphism. A morphism Φ : Z1 → Z2 is called a
dominant morphism if Φ(Z1) is dense in Z2.
Lemma 3.
1 Z1 and Z2 are both irreducible varieties over C,
and φ : Z1 → Z2 is a dominant morphism, then
dim(Z2) ≤ dim(Z1).
2 Z1 and Z2 are both varieties over C, and φ : Z1 →
Z2 is a morphism, r = max
dim(φ−1(z)), then
dim(Z1) ≤ r + dim(Z2).
3 If V = ∪si=1Vi is a finite open covering, and ∀i,
Vi is irreducible, ∀i, j, Vi ∩ Vj 6= ∅, then V is irre-
ducible.
Let X = {Φ|Φ ∈ GL(mn,C),Φ(Z) ∩ Z 6= ∅}. We
are able to give an upper bound on the dimension of the
closure of X with respect to the Zariski topology.
Lemma 4. dim(X) ≤ m2n2 − (m− 2)(n− 2)+ 1, where
X is the Zariski closure of X.
Proof. We have a morphism F : G×Y → Y which is just
the left action of G on Y , defined by
F (g, [w]) = [g · w]
where G = GL(k,C), Y = Pmn−1, k = mn.
Let y0 = (1, 0, · · · , 0)
T be a column vector with k
entries. For any given y1, y2 ∈ Y , we choose g1,
g2 ∈ GL(k,C), such that [g1 ·y0] = [y1] and [g2 ·y0] = [y2].
Then we have
[g · y1] = [y2]
⇐⇒ [gg1 · y0] = [g2 · y0]
⇐⇒ [g−1
gg1 · y0] = [y0]
From above observations, F has the following prop-
erty: for any y1, y2 ∈ Y , F
−1(y2) ∩ {G × {y1}} ∼=
: z1 ∈ C\{0}, g
′ ∈ GL(k − 1,C), α ∈
k−1 is a row vector.}. Hence dim(F−1(y2) ∩
G× {y1}) = m
2n2 − (mn− 1).
Let P1, P2 be projections of G×Y to G, Y respectively.
Now we only look atG×Z ⊆ G×Y , to get F : G×Z → Y .
Then we have a characterization of X: X = P1F
−1(Z).
In fact,
g ∈ X
⇐⇒ g(Z) ∩ Z 6= ∅
⇐⇒ ∃z1, z2 ∈ Z, s.t.g(z1) = z2
⇐⇒ ∃z1, z2 ∈ Z, s.t.(g, z1) ∈ F
−1(z2)
⇐⇒ ∃z2 ∈ Z, s.t.g ∈ P1F
−1(z)
⇐⇒ g ∈ P1F
−1(Z)
Let X ⊆ G be the Zariski closure of X in G, then P1 :
F−1(Z) → X is a dominant morphism.
Furthermore, consider Ψ : F−1(Z) → Z × Z given by
Ψ(g, [z]) = ([z], [g · z])
For all z1, z2 ∈ Z, we have Ψ
−1(z1, z2) = (g2Tg
1 , z1),
where T = {
: z0 ∈ C\{0}, g
′ ∈ GL(k −
1,C), α ∈ Ck−1 is a row vector}, and g1, g2 ∈ GL(k,C),
s.t. g1(y0) = z1, g2(y0) = z2. Ψ is a dominant morphism
since G acts transitively on Pk−1. Then we obtain
dim(F−1(Z)) ≤dim(T ) + dim(Z × Z)
=m2n2 −mn+ 1 + 2 · dim(Z)
It is required in Lemma 3.1 that varieties Z1 and Z2 are
irreducible, but we haven’t proved F−1(Z) is irreducible.
Actually, this condition can be weakened: Lemma 3.1
is still true for the more general case that Z1 and Z2
are closed subsets of irreducible varieties [11]. Thus, we
can fill out the gap and apply this lemma. Indeed, the
irreducibility of F−1(Z) andX really holds, but the proof
is not easy (see [1] for a brief proof ). From Lemma 3.1,
we have
dim(X) ≤ dim(F−1(Z))
≤(m2n2 − (mn− 1)) + 2 · dim(Z)
=m2n2 − (mn− 1) + 2(m+ n− 2)
=m2n2 − (m− 2)(n− 2) + 1
With the above lemmas, we can now derive the main
result:
Theorem 3. Given a bipartite quantum system
Hm ⊗H\, where Hm and H\ are Hilbert spaces with di-
mensions m and n respectively, if min (m,n) > 2 and
(m,n) 6= (3, 3), then there exists an unitary operator
Φ ∈ U(mn), s.t.
Φ(Z) ∩ Z = ∅
That is, Φ is a universal entangler.
Proof. If U(mn) ⊂ X , then it follows that GL(mn,C) =
U(mn) ⊂ X from Lemma 2. And in this assumption, we
also have dim(X) ≤ m2n2− (m−2)(n−2)+1< m2n2 =
dim(GL(mn,C)) from Lemma 4 [2]. It’s a contradiction.
So U(mn) 6⊂ X , i.e. a unitary operator Φ ∈ U(mn) with
Φ(Z) ∩ Z = ∅ exists.
Conclusion. In summary, it is shown that a universal
entangler of bipartite system exists except for the cases
of 1⊗n, 2⊗n, m⊗ 1, m⊗ 2 and 3⊗ 3. So we have com-
pletely determined when a universal entangler exists. It
seems that the method employed in this Letter can be
extended to the multipartite case. This extension will
lead us to explore the geometric structure of entangled
states and other related objects. Furthermore, we can
consider some more practical questions: (1) How to con-
struct such a universal entangler for 3⊗4 bipartite system
explicitly? (2) Given a universal entangler, what is the
minimum entanglement it guarantees to output with re-
spect to the definition of entanglement measure for pure
states [4, 5, 15]? (3) Furthermore, what is the optimal
universal entangler which maximized the minimally pos-
sible entanglement the entangler outputs. Intuitively, for
a bipartite system of sufficiently large dimensions,a ran-
domly chosen unitary operator seems to be a universal
entangler with high probability. Nevertheless, we failed
to give a proof of this conjecture.
Acknowledgements. We are thankful to the colleagues
in the Quantum Computation and Information Research
Group of Tsinghua University for helpful discussions.
And one of us (J.-X. Chen) enjoyed delightful discus-
sions with W. Huang, L. Xiao, Z. Q. Zhang and Q. Lin.
Jun Yu thank Professor X.-J. Tan and Q.-C. Tian who
ask him to study Hartshorne ’s book. This work was
partly supported by the Natural Science Foundation of
China(Grant Nos.60621062 and 60503001) and the Hi-
Tech Research and Development Program of China(863
project)(Grant No.2006AA01Z102).
∗ Electronic address: chenkenshin@gmail.com
† Electronic address: dry@tsinghua.edu.cn
‡ Electronic address: jizhengfeng98@mails.thu.edu.cn
§ Electronic address: yingmsh@tsinghua.edu.cn
¶ Electronic address: majunyu@ust.hk
[1] Here, we will present a short proof of the irreduciblity
of F−1(Z). ∀i, j: 0 ≤ i ≤ m− 1, 0 ≤ j ≤ n− 1,
let Pm,i = {[x0, x1, · · · , xi, · · · , xm−1]|xi 6= 0}, and
Pn,j = {[y0, y1, · · · , yj , · · · , yn−1]|yi 6= 0}. Let Zi,j =
σ(Pm,i × Pn,j), we will have Z = ∪i,jZi,j , and for
∀i, j, Zi,j ∼= Pm,i × Pn,j ∼= A
m−1 × An−1 ∼= A
m+n−2,
and each Zi,j is open in Z. ∀i, j, we can construct a
morphism Ti,j : Zi,j → GL(mn,C) s.t. ∀y ∈ Zi,j ,
(Ti,j(y))(y0) = y. Then ∀i, j, î, ĵ, (g, y) ∈ Ψ
î,ĵ
Zi,j) ⇐⇒ g·y ∈ Zi,j and y ∈ Zî,ĵ ⇐⇒ (g·Tî,ĵ(y1))y0 =
Ti,j(y2)(y0) for some y2 ∈ Zi,j , y1 ∈ Zî,ĵ ⇐⇒ g ∈
Ti,j(Zi,j)·H ·(Tî,ĵ(Zî,ĵ))
−1. Thus we have F−1(Z)
i,j,̂i,ĵ
{(g, y) ∈ Ψ−1(Z
î,ĵ
× Zi,j)} is irreducible since it’s the
image of a morphism Zi,j × H × Zî,ĵ → F
−1(Z). Each
−1(Z)
i,j,̂i,ĵ
is also open in F−1(Z), then the irreducib-
lity follows from lemma 3.3.
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braic fibre bundle thus dim(F−1(Z)) = m2n2 − mn +
1 + 2 · dim(Z). We also conjecture that dim(X) =
dim(F−1(Z)) and X = X.
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mailto:chenkenshin@gmail.com
mailto:dry@tsinghua.edu.cn
mailto:jizhengfeng98@mails.thu.edu.cn
mailto:yingmsh@tsinghua.edu.cn
mailto:majunyu@ust.hk
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