Singularity of type D4 arising from four qubit systems

Singularity of type D4 arising from four qubit systems Frédéric Holweck

a)

b)

, Jean-Gabriel Luque

Michel Planat

c)

An intriguing correspondence between four-qubit systems and simple singularity of type

D4

is established. We rst consider the algebraic variety

within the projective Hilbert space hyperplane

X,

H,

we prove that the

P(H) = P15 .

X -hypersurface,

has an isolated singularity of type

D4 ;

X

Then, cutting

of separable states

X

with a specic

dened from the section

X ∩H ⊂

it is also shown that this is the worst-

possible isolated singularity one can obtain by this construction.

Moreover, it is

demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of of type

2 × 2 × 2 × 2,

X , which is nothing but the Cayley hyperdeterminant

can be expressed in terms of the SLOCC invariant polynomials

as the discriminant of the miniversal deformation of the

D4 -singularity.

Keywords: Quantum Information Theory, Entangled states, Simple singularities of hypersurfaces, Dynkin diagrams, Hyperdeterminant.

PACS :02.10.Xm, 02.40.-k, 03.65.Aa , 03.65.Fd, 03.65.Ud Msc :

a)

32S25,32S30,15A69,14M17,15A72

[email protected], Laboratoire IRTES-M3M, Université de Technologie de Belfort-Montbéliard,

90010 Belfort Cedex, FR

b)

[email protected], Université de Rouen, Laboratoire d'Informatique, du Traitement de

l'Information et des Systèmes (LITIS), Avenue de l'Université - BP 8 6801 Saint-étienne-du-Rouvray

Cedex, FR

c)

[email protected], Institut FEMTO-ST, CNRS, 32 Avenue de l'Observatoire, 25044 Besançon,

FR

1

I.

INTRODUCTION Several branches of geometry and algebra tend to play an increasing role in quantum in-

formation theory. We have in mind algebraic geometry for describing entanglement classes

6,11,12,22

of multiple qubits

, representation theory and Jordan algebras for entanglement and

35

the black-hole/qubit correspondence

, and geometries over nite elds/rings for deriv-

18,23,24

ing point-line congurations of observables relevant to quantum contextuality

.

The

topology of hypersurface singularitites, and the related Coxeter-Dynkin diagrams, represent another eld worthwhile to be investigated in quantum information, as shown in this paper. Dynkin diagrams are well known for classifying simple Lie algebras, Weyl groups, subgroups of

SU (2) and simple singularities, i.e.

isolated singularities of complex hypersurfaces

that are stable under small perturbations. More precisely, if we consider simple-laced Dynkin

i.e.

diagrams,

diagrams of type

A − D − E,

we nd objects of dierent nature classied by

the same diagrams: Type Lie algebra

Subgroup of

SU (2)

Hypersurface with simple singularity

An

sln+1 (C)

cyclic group

xn+1 + x22 + · · · + x2k = 0 1

Dn

so2n (C)

binary dihedral group

x1n−1 + x1 x22 + x23 + · · · + x2k = 0

E6

e6

binary tetrahedral

x41 + x32 + x23 + · · · + x2k = 0

E7

e7

binary octahedral

x31 x2 + x32 + x23 + · · · + x2k = 0

E8

e8

binary icosahedral

x51 + x32 + x23 + · · · + x2k = 0

A challenging question in mathematics is to understand these

ADE -correspondences

by

establishing a direct construction from one class of objects to the other. For instance, the construction of surfaces with simple singularities from the corresponding subgroup of

SU (2)

is called the McKay correspondence. A construction due to Grothendieck allows us to recover the simple singularities of a given type from the nullcone (the set of nilpotent elements) of the corresponding simple Lie algebra. For an overview of such Ref

25,26

ADE

correspondences, see

and references therein.

Another construction connecting simple Lie algebras and simple singularities is due to

14

Knop

.

In his construction, Knop considers a unique smooth orbit,

action of Lie group

G

on the projectivization of its Lie algebra

by a specic hyperplane. The resulting same type as

X -hypersurface

g. 2

P(g)

X,

for the adjoint

and cuts this variety

has a unique singular point of the

Looking at

ADE -correspondences

in the context of QIT is a way to understand the

role played by those diagrams in this eld. In dierent classication schemes of four-qubit systems, the Dynkin diagram Lie algebra

so(8)

D4

7

For instance, Verstraete

et al's

classication

SO(4) × SO(4) ⊂ SO(8) orbits on M4 (C).

27

is

Chterental and

7

use the same group action and refer to (Remark 5.3 of Ref ) the Hilbert space of

four qubits as a subspace of

SO(8)

D4 ).

(that is the type

based on the classication of the Djokovic

has already appeared thanks to the role played by the

so(8)

whose SLOCC orbits arise from the trace of the adjoint

orbits. In their study of the four-qubit classication from the string theory point of

view, Borsten form of

so(8)

et al2

employ a correspondence between nilpotent orbits of

with signature

least, the relation between

(4, 4))

so(8)

so(4, 4)

(the real

and nilpotent orbits of four-qubit systems. Last but not

and four-qubit systems has been pointed out by Lévay

16,17

in his papers on the black-hole/qubit correspondence. In these papers Lévay describes the Hilbert space of four qubits as the tangent space of

SO(4, 4)/(SO(4) × SO(4)).

In the present paper, we will establish a correspondence between four-qubit systems and

D4 -singularities establish an

by using a construction inspired by Knop's paper. In other words, we will

ADE -type

correspondence between

SO(4, 4)

and singularities of type

D4

using

the Hilbert space of four qubits. Let

H = C2 ⊗ C2 ⊗ C2 ⊗ C2

|Ψi ∈ H

multiplication, a four-qubit

P15 = P(H).

can be considered as a point of the projective space

The set of separable states in

which can be factorized as

{|0i, |1i}

be the Hilbert space of four-qubit systems. Up to scalar

H corresponds to tensors of rank one, i.e.

|Ψi = v1 ⊗ v2 ⊗ v3 ⊗ v4

with

for the single-qubit computational basis and

vi ∈ C2 .

tensors

Adopting the notation

|ijkli = |ii ⊗ |ji ⊗ |ki ⊗ |li

for the

four-qubit basis, a general four-qubit state can be expressed as

|Ψi =

X

aijkl |ijkli

with

aijkl ∈ C.

0≤i,j,k,l≤1 Let

G

be the group of Stochastic Local Operation and Classical Communication (SLOCC)

of four qubits [acting on known that

G

P(H)], i.e. G = SL2 (C) × SL2 (C) × SL2 (C) × SL2 (C).

It is well

acts transitively on the set of separable states. The projectivization of the

corresponding orbit  also called the highest weight orbit  is the unique smooth orbit for the action of

G

on

P(H),

that is

X = P(G.|0000i) = {The

set of separable states}

3

⊂ P15 .

X

A parametrization of

φ:

 

X

is given by the Segre embedding of four projective lines

P1 × P1 × P1 × P1

11,12

P15

→

 ([w0 : w1 ], [x0 : x1 ], [y0 : y1 ], [z0 : z1 ]) 7→ [w0 x0 y0 z0 : · · · : WJ : · · · : w1 x1 y1 z1 ] where

WJ = wi xj yk zl

WJ1 ≺ WJ2

if

X

J = {i, j, k, l} ∈ {0, 1}4

H ⊂ P(H)

H ⊂ P(H),

Given

H ⊂ P(H) there

|Φi ∈ P(H)

is the set of states

the hyperplane section

dened by the restriction of exists a state

LH

to

X.

H = hΨ| =

equivalently,

P

0≤i,j,k,l≤1

|Ψi ∈ P(H) such

X ∩ hΨ|,

hijkl hijkl|

on which a linear form

X∩H ⊂ X

LH ∈ H∗

is the hypersurface of

Due to the duality of Hilbert spaces, for any that

what follows, we will often identify the hyperplane write

and the monomial order is such that

8i1 + 4j1 + 2k1 + l1 ≤ 8i2 + 4j2 + 2k2 + l2 .

A hyperplane vanishes.

for

with

is dened by the linear form

H

and the linear form dening it, and

hijkl ∈ C.

will be the hypersurface of

hΨ|φ(P1 × P1 × P1 × P1 )i =

hΨ|.

H

X

The hyperplane section

X ∩ H,

In

or,

given by

X

hijkl wi xj yk zl = 0.

(1)

0≤i,j,k≤1 To state our main Theorem, let us recall that the ring of polynomials invariant under is generated by

4 invariants19 .

Let us denote by

I˜1 , I˜2 , I˜3 , I˜4

a choice of four generators of the

ring of invariants (that choice will be explained in Section III B), The quotient map

Φ : H → C4

is dened by

G

i.e. C[H]G = C[I˜1 , I˜2 , I˜3 , I˜4 ].

Φ(x) = (I˜1 (x), I˜2 (x), I˜3 (x), I˜4 (x)).

The main

result of this article is the following theorem:

Theorem 1. Let H = hΨ| be a hyperplane of P(H) tangent to X and such that X ∩ H has only isolated singular points. Then the singularities are either of types A1 , A2 , A3 , A4 , or of type D4 , and there exist hyperplanes realizing each type of singularity. Moreover, if b ∗ ⊂ H the cone over the dual variety of X , we denote by X

i.e.

the zero locus of the Cayley

hyperdeterminant of format 2 × 2 × 2 × 2, then the quotient map Φ : H → C4 is such b ∗ ) = ΣD4 , where ΣD4 is the discriminant of the miniversal deformation of the that Φ(X D4 -singularity. The paper is organized as follows. In Section II, we will give the denition of a simple

1

singularity and the invariants that follow from the Arnol'd classication

(Section II A).

Then we will compute the singularity type of any hyperplane section of the set of separable states featuring only isolated singularities (see Section II B Proposition II.1).

4

In Section

III, we will establish a dual version of Proposition II.1. We will rst dene the notion of discriminant of a singularity (see Section III A) and then show how it allows us to give a

∆4

new expression for the Cayley hyperdeterminant III.1 about the relation between

∆4

and

ΣD4 .

(Section III B) and prove Proposition

Propositions II.1 and III.1 lead to the proof

of Theorem 1.

II.

SIMPLE SINGULARITIES AND HYPERPLANE SECTIONS OF

SEPARABLE STATES A.

Simple singularities following Arnol'd classication Simple singularities have been studied from an algebraic geometrical viewpoint as ratio-

nal double points of algebraic surfaces, Du Val singularities, and from a complex analytic perspective as critical points of holomorphic functions in several variables. These approaches

9

lead to many equivalent characterizations of what a simple singularity is . Here, we select the complex analytic approach introduced by Vladimir Arnol'd. We rst recall the ingredients

1

of Arnol'd classication of simple singularities . Let us denote by and by

Ok

(f, 0)

the germ of a holomorphic function,

the set of all those germs.

We consider the group

f : (Ck , 0) → (C, 0)

Dk

at

0,

of biholomorphic maps

Ok such that g.f = f ◦ g −1 . A singularity is an equivalence ∂f (0) = 0 for i = 1, . . . , k . In other words, a singularity is class of a germ (f, 0) such that ∂xi an orbit in Ok and we will write [(f, 0)] for the orbit of the representative (f, 0). We denote g : (Ck , 0) → (Ck , 0)

by

Sk ⊂ Ok

the set of all singular germs.  2 

let us denote by germ

(f, 0)

acting of

A=

∂ f (0) ∂xi ∂xj

Let

f

be a representative of a singularity and

the corresponding Hessian matrix. The corank of the

i,j is the dimension of the kernel of

A.

From the denition of the action of

Dk

it

follows that equivalent germs will have the same corank, which means that the corank is an invariant of a singularity.

Denition II.1. A singularity is said to be non-degenerate, or quadratic, or of the Morse type, if, and only if, its corank is zero. The Morse Lemma

x21 + · · · + x2k .

20

ensures that if

(f, 0)

is a non-degenerate singular germ, then

The non-degenerate singularity is a dense orbit in

singularity of corank l , a generalization of Morse's Lemma

5

1

Sk .

Assume that

tells us that

f ∼

[(f, 0)] is a

f ∼ h(x1 , . . . , xl ) +

x2l+1 + · · · + x2k

and leads to an equivalence relation between germs of distinct number of

variables.

Denition II.2. Two function germs f : (Ck , 0) → (C, 0) and g : (Cm , 0) → (C, 0), with k < m, are said to be stably equivalent if, and only if, f (x1 , . . . , xk ) + x2k+1 + · · · + x2m ∼ g(x1 , . . . , xm ).

Remark II.1.

In terms of the last denition we can compare singularities of functions

which do not have the same number of variables. Adding quadratic terms of full rank in new variables do not aect the classication of the singular type.

20

Another important invariant of singular germs is the famous Milnor number be a singular germ and consider

I∇f

∂f ∂f = Ok < (0), . . . , (0) > ∂x1 ∂xk

. Let

(f, 0)

the gradient ideal.

Denition II.3. The Milnor number µ of a singular germ (f, 0) is equal to the dimension of the local algebra of (f, 0),

i.e.

the quotient of the algebra Ok by I∇f , µ = dimC (Ok I∇f ) .

The critical point

0

of the function

f

will be isolated if, and only if, its Milnor number is

nite. Let us now state what, in the sense of Vladimir Arnol'd, a simple singularity is .

Denition II.4. The orbit [(f, 0)] is a simple singularity if, and only if, a suciently small neighborhood of (f, 0) intersects Sk with a nite number of non-equivalent orbits.

Remark II.2. · · · + x2k ,

If we consider a representative of a non-degenerate singularity

a small perturbation of

of full rank for



small. Thus

f

in

Sk , i.e. f + εh

f ∼ f + εh,

with

h ∈ Sk ,

f ∼ x21 +

will still have a Hessian

which means that non-degenerate singularity is

the most stable type of singularity. We can rephrase Denition II.4 by saying that a simple singularity if, and only if, a small perturbation of a representative

f

[(f, 0)] is

will only lead

to a nite number of non-equivalent singularities.

1

In his classication of simple singularities , Arnol'd proved that being simple is equivalent to the following conditions:

ˆ µ < +∞, 6

 ∂ 2f (0) ≤ 2, ˆ corank ∂xi ∂xj  2  ∂ f ˆ if corank (0) = 2 ∂xi ∂xj 

the cubic term in the degenerate direction of the Hessian

is non-zero,

ˆ

if corank

=2

and the cubic term is a cube then

µ < 9.

With these conditions Arnol'd obtained the classication of simple singularities into ve dierent types (Table I).

An

Type

Dn

E6

E7

E8

xn+1 xn−1 + xy 2 x3 + y 4 x3 + xy 3 x3 + y 5

Normal forms Milnor number

n

n

6

7

8

Table I. Simple singularities.

Remark II.3.

The functions given in Table I are stably equivalent to the hypersurfaces

given in the introduction. They are also clearly equivalent to the rational double points of algebraic surfaces.

The classication given by Arnol'd furnishes an algorithm to test if a singularity is simple or not.

Algorithm II.1. Let (f, 0) be a singularity. ˆ Compute µ; if µ = ∞ the singularity is not isolated (and not simple), ˆ If not, compute r = corank(Hess(f, 0)).

 if r ≥ 3, the singularity is not simple,  if r = 1, the singularity is of type Aµ ,  if r = 2, then * if the cubic term in the degenerate directions is non-zero and is not a cube,

then the singularity is of type Dµ , * if the cubic term in the degenerate directions is a cube and µ < 9, then the

singularity is of type Eµ , 7

* if not, the singularity is not simple. In the next section we will follow this algorithm to compute the singular type of a given hyperplane section.

B.

Computing singularities of hyperplane sections Before we prove the rst proposition, let us consider two examples in order to explain

how we calculate the singular type of a hyperplane section.

Example II.1. h0011| + h1100|.

Let

be a hyperplane, or a linear form, given by

The corresponding hyperplane section

a tangent vector to

δ|1110i

H ∈ P(H∗ )

X

at

|1111i

and it is clear that

will be of the form

hΨ1 |vi = 0.

X ∩H

is tangent to

H = hΨ1 | =

|1111i.

|vi = α|0111i + β|1011i + γ|1101i +

The homogeneous form of the linear section

corresponds to its restriction to (the cone over)

X,

Indeed,

X ∩H

that is to

f (w0 , w1 , x0 , x1 , y0 , y1 , z0 , z1 ) = w0 x0 y1 z1 + w1 x1 y0 z0 . In a non-homogeneous form as

f

can be written in the chart corresponding to

f (w0 , x0 , y1 , z1 ) = w0 x0 + y0 z0 .

In this chart the point

|1111i

w1 , x1 , y1 , z1 = 1

has coordinates

(0, 0, 0, 0)

and (we can forget about the subscripts) the hyperplane section is a hypersurface of

X

dened (locally) by the equation

f (w, x, y, z) = wx + yz = 0.  This hypersurface has a unique singularity

a = (0, 0, 0, 0),

which corresponds to

|1111i,  0  1   0  0

is of the full rank. One concludes that and we denote it by

 ∂f ∂f ∂f ∂f (a), (a), (a), (a) = (0, 0, 0, 0) ⇔ ∂w ∂x ∂y ∂z

and the Hessian matrix

 1 0 0  0 0 0   0 0 1  0 1 0

(X ∩ H, |1111i)

(X ∩ H, |1111i) ∼ A1 ,

is an isolated singularity of type

or, equivalently, by

8

(X ∩ hΨ1 |, |1111i) ∼ A1 .

A1

Example II.2.

Let us consider the hyperplane section dened by

h1011| + h1101| + h1110| ∈ H∗ . that a tangent vector to

γ|0101i + |0110i is

and

X

This section

|0111i

at

H|vi = 0.

X ∩H

H = hΨ2 | = h0000| +

is tangent to

will be of the form

|0111i.

|vi = α|1111i + β|0011i +

The homogeneous linear form corresponding to

f (w0 , w1 , x0 , x1 , y0 , y1 , z0 , z1 ) = w0 x0 y0 z0 + w1 x0 y1 z1 + w1 x1 y0 z1 + w1 x1 y1 z0 .

w0 = x1 = y1 = z1 = 1

It is clear

X∩H

In the chart

the form becomes a hypersurface dened by

xyz + wx + wy + wz = 0 and

8

(0, 0, 0, 0)

is the only singularity of this hypersurface. Using the software SINGULAR

one can check that

µx=(0,0,0,0) (f ) = 4

and the rank of the Hessian



 0 1 1 1   1 0 0 0     1 0 0 0   1 0 0 0 is

2.

i.e.

(

Thus, we conclude that

(X ∩ H, |0111i) ∼ D4 , or, equivalently, (X ∩ hΨ2 |, |0111i) ∼ D4

the unique isolated singularity where the corank equals

2

and

µ = 4).

We can now prove our rst proposition.

Proposition II.1. Let X ∩ H be a singular hyperplane section of the variety of separable states for four-qubit systems,

i.e.

X = P1 × P1 × P1 × P1 , with an isolated singularity

x ∈ X ∩ H . Then the singularity (X ∩ H, x) will be of type A1 , A2 , A3 or D4 and each type

can be obtained by such a linear section of X . Proof.

To prove Proposition II.1, we compute the singular type of all possible hyperplane

sections of

X.

As the variety

X

is

G-homogeneous,

identical for any representative of the hyperplane

H

by Verstraete

corresponds to a point

et al.27

orbit of

h ∈ P(H).

will be

By the duality of the Hilbert space, a

But the

G orbits of P(H) have been classied 7

et al.'s

classication, the

G-orbits

of the four-qubit Hilbert space

9 families (3 families are parameter free and 6 of them depend on parameters) and 7,27

normal forms for each family are known

|Ψi

H.

X∩H

(with a corrected version provided by Chterental and Djokovic ).

According to Verstraete consist of

G

the singular type of

. From each of Verstraete

we compute the corresponding hyperplane section

9

X ∩ hΨ|.

et al.'s

normal forms

Then we look at isolated

singular points of each hyperplane section and we calculate the corresponding singular type with a formal algebra system following the procedure described in examples II.1, II.2 and Algorithm II.1.

For the normal forms depending on parameters, the singular type of the

hyperplane sections will depend on values of the parameters. The results of our calculations are given in Tables II and III and provide a proof of the proposition.2

Verstraete et al.'s notation

Hyperplane

L07⊕1

h0000| + h1011| + h1101| + h1110|

L05⊕3

h0000| + h0101| + h1000| + h1110|

non-isolated

L03⊕1 03⊕1

h0000| + h0111|

non-isolated

Singular type of the hyperplane section

D4

(a unique singularity )

Table II. Hyperplanes and the corresponding sections which do not depend on parameters.

Remark II.4.

Tables III, IV, V show that the classication of entangled states into 9

families can be rened according to the singular type of the corresponding section. singular type of the linear section

X ∩ hΨ|

is an invariant of the

G-orbit

be used to distinguish two non-equivalent classes of entanglement.

of

|Ψi

The

and may

Thus, the values of

the parameters which distinguish the sections indicate how we can decompose further the classication. However, to fully distinguish non-equivalent sections from their singular type, it would be necessary to investigate more precisely the non-isolated singular sections.

Remark II.5.

It is worthwile to point out that the dierent isolated singular types we

obtain by this construction (A1 , A2 , A3 and the

D4 -singularity.

meet, in

Sk ,

D4 )

are exactly the possible degenerations of

In particular, any small neighborhood of the singularity of type

the orbits corresponding to the singular types

A1 , A2

and

1

A3

D4

will

as shown in the

adjacency diagrams of Arnold's classication (Corollary 8.7 in Ref ). The fact that

D4

is

the worst-possible isolated singularity we get from the hyperplane sections of the set of separable states will be lighted with Proposition III.1.

10

Verstraete's

Hyperplane

parameters

Singular type

notation

La2 03⊕1

a(h0000| + h1111|) + h0011| + h0101| + h0110|

a

A1

generic

a=0 La4

a(h0000| + h0101| + h1010| + h1111|)

a

+ih0001| + h0110| − ih1011| Lab3

a(h0000| + h1111|) +

a+b 2 (h0101|

generic

non-isolated

A3

a=0

+ h1010|)

+ a−b 2 (h0110| + h1001|)

a, b

generic

(a unique singularity) non-isolated

A2

a=b=0

(a unique singularity) non-isolated

+ √i2 (h0001| + h0010| − h0111| − h1011|) La2 b2

a(|0000i + |1111i) + b(|0101i + |1010i)

a, b

+|0110i + |0011i

a=0

generic or

smooth section

b=0

non-isolated

a=b=0 Labc2

a+b 2 (h0000|

+ h1111|) +

a−b 2 (h0011|

+ h1100|)

c(h1010| + h0101|) + h0110|

a, b, c

generic

+ |1111i) +

a−d 2 (|0011i

+ b+c 2 (|0101i + |1010i) +

+ |1100i)

b−c 2 (|0110i

+ |1001i)

(a unique singularity)

A1

c=0

A1

a = ±b = ±c

non-isolated

or

b=c=0

a=b=c=0 a+d 2 (|0000i

A1

a = ±b

a=c=0

Gabcd

non-isolated

a, b, c, d

generic

non-isolated non-isolated smooth section

see Table IV

A1

see Table V

non-isolated

Table III. Hyperplanes and the corresponding sections which do depend on parameters.

III. THE CAYLEY 2 × 2 × 2 × 2 HYPERDETERMINANT AND THE D4 -DISCRIMINANT

Another fundamental concept associated with a simple singularity is its discriminant,

i.e.

the locus that parametrizes the deformation of the singular germs. In this section, we will show that the discriminant of the

D4 -singularity

is linked to the dual variety, in the sense

of the projective duality, of the set of separable four-qubit states.

11

A.

Discriminant of the miniversal deformation of the singularity Consider a holomorphic germ

Milnor number

µ(f, 0) = n.

A

f : (Ck , 0) → (C, 0)

miniversal deformation1 f+

where

(g1 , . . . , gn )

with a simple isolated singularity of

is a basis of

X

of the germ

f

is given by

λi gi ,

Ok I∇f .

Denition III.1. The discriminant Σ ⊂ Cn is the subset of values (λ1 , . . . , λn ) ∈ Cn such that the miniversal deformation f +

P

λi gi is singular, n

Σ = {(λ1 , . . . , λn ) ∈ C , ∆(f +

i.e.

n X

λi gi ) = 0},

i=1

where ∆ is the usual notion of discriminant.

Remark III.1. 29

known

The discriminant parametrizes all singular deformations of

(f, 0).

It is

that for hypersurfaces endowed with a simple singularity, the discriminant of the

singularity characterizes its type.

Example III.1. 1, x, . . . , xn−1 >.

Let

(f, 0) be a singularity of type An , i.e. f ∼ xn+1 .

Thus, a miniversal deformation of

f

Then

O1 I∇xn+1 =