E_6 and the bipartite entanglement of three qutrits
arXiv:0704.0507v2  [hep-th]  9 Oct 2007
Imperial/TP/2007/mjd/1
CERN-PH-TH/2007-78
UCLA/07/TEP/7
E6 and the bipartite entanglement of three qutrits
M. J. Duff 1† and S. Ferrara2‡
† The Blackett Laboratory, Imperial College London,
Prince Consort Road, London SW7 2AZ
‡Physics Department,Theory Unit, CERN, CH1211, Geneva23, Switzerland
Department of Physics & Astronomy, Universuty of California,Los Angeles, USA
INFN-Laboratori Nazionale di Frascati, Via E. Fermi 40, 00044 Frascati, Italy
ABSTRACT
Recent investigations have established an analogy between the entropy of four-dimensional
supersymmetric black holes in string theory and entanglement in quantum information the-
ory. Examples include: (1) N = 2 STU black holes and the tripartite entanglement of
three qubits (2-state systems), where the common symmetry is [SL(2)]3 and (2) N = 8
black holes and the tripartite entanglement of seven qubits where the common symmetry
is E7 ⊃ [SL(2)]
7. Here we present another example: N = 8 black holes (or black strings)
in five dimensions and the bipartite entanglement of three qutrits (3-state systems), where
the common symmetry is E6 ⊃ [SL(3)]
3. Both the black hole (or black string) entropy and
the entanglement measure are provided by the Cartan cubic E6 invariant. Similar analogies
exist for “magic” N = 2 supergravity black holes in both four and five dimensions.
1m.duff@imperial.ac.uk
2Sergio.Ferrara@cern.ch
http://arxiv.org/abs/0704.0507v2
Contents
1 D = 4 black holes and qubits 3
1.1 N = 2 black holes and the tripartite entanglement of three qubits . . . . . . 3
1.2 N = 2 black holes and the bipartite entanglement of two qubits . . . . . . . 4
1.3 N = 8 black holes and the tripartite entanglement of seven qubits . . . . . . 4
1.4 Magic supergravities in D = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Five-dimensional supergravity 8
3 D = 5 black holes and qutrits 10
3.1 N = 2 black holes and the bipartite entanglement of two qutrits . . . . . . . 10
3.2 N = 8 black holes and the bipartite entanglement of three qutrits . . . . . . 11
3.3 Magic supergravities in D = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Conclusions 13
5 Acknowledgements 14
1 D = 4 black holes and qubits
It sometimes happens that two very different areas of theoretical physics share the same
mathematics. This may eventually lead to the realisation that they are, in fact, dual de-
scriptions of the same physical phenomena, or it may not. Either way, it frequently leads
to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy
between the entropy of certain four-dimensional supersymmetric black holes in string the-
ory and entanglement measures in quantum information theory. In this paper we extend
the analogy from four dimensions to five which also involves going from two-state systems
(qubits) to three-state systems (qutrits).
We begin by recalling the four-dimensional examples:
1.1 N = 2 black holes and the tripartite entanglement of three
qubits
The three qubit system (Alice, Bob, Charlie) is described by the state
|Ψ〉 = aABC |ABC〉 (1.1)
where A = 0, 1, so the Hilbert space has dimension 23 = 8. The complex numbers aABC
transforms as a (2, 2, 2) under SL(2, C)A×SL(2, C)B ×SL(2, C)C . The tripartite entangle-
ment is measured by the 3-tangle [7, 8]
τ3(ABC) = 4|Det aABC |. (1.2)
where Det aABC is Cayley’s hyperdeterminant [9].
Det a = −
ǫA1A2ǫB1B2ǫA3A4ǫB3B4ǫC1C4ǫC2C3aA1B1C1aA2B2C2aA3B3C3aA4B4C4 (1.3)
The hyperdeterminant is invariant under SL(2)A × SL(2)B × SL(2)C and under a triality
that interchanges A, B and C.
In the context of stringy black holes the 8 aABC are the 4 electric and 4 magnetic charges
of the N = 2 STU black hole [10] and hence take on real (integer) values. The STU model
corresponds to N = 2 supergravity coupled to three vector multiplets, where the symmetry
is [SL(2, Z)]3. The Bekenstein-Hawking entropy of the black hole, S, was first calculated in
[11]. The connection to quantum information theory arises by noting [1] that it can also be
expressed in terms of Cayley’s hyperdeterminant
S = π
|Det aABC |. (1.4)
One can then establish a dictionary between the classification of various entangled states
(separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite
entangled GHZ) and the classfication of various “small” and “large” BPS and non-BPS black
holes [1, 2, 3, 4, 5, 6]. For example, the GHZ state [12]
|Ψ〉 ∼ |111〉+ |000〉 (1.5)
with Det aABC ≥ 0 corresponds to a large non-BPS 2-charge black hole; the W-state
|Ψ〉 ∼ |100〉+ |010〉+ |001〉 (1.6)
with Det aABC = 0 corresponds to a small-BPS 3-charge black hole; the GHZ-state
|Ψ〉 = −|000〉+ |011〉+ |101〉+ |110〉 (1.7)
corresponds to a large BPS 4-charge black hole.
1.2 N = 2 black holes and the bipartite entanglement of two qubits
An even simpler example [2] is provided by the two qubit system (Alice and Bob) described
by the state
|Ψ〉 = aAB|AB〉 (1.8)
where A = 0, 1, and the Hilbert space has dimension 22 = 4. The aAB transforms as a (2, 2)
under SL(2, C)A × SL(2, C)B. The entanglement is measured by the 2-tangle
τ2(AB) = C
2(AB) (1.9)
where
C(AB) = 2 |det aAB| (1.10)
is the concurrence. The determinant is invariant under SL(2, C)A × SL(2, C)B and under
a duality that interchanges A and B. Here it is sufficient to look at N = 2 supergravity
coupled to just one vector multiplet and the 4 aAB are the 2 electric and 2 magnetic charges
of the axion-dilaton black hole with entropy
S = π|det aAB| (1.11)
For example, the Bell state
|Ψ〉 ∼ |11〉+ |00〉 (1.12)
with det aAB ≥ 0 corresponds to a large non-BPS 2-charge black hole.
1.3 N = 8 black holes and the tripartite entanglement of seven
qubits
We recall that in the case of D = 4, N = 8 supergravity, the the 28 electric and 28 magnetic
charges belong to the 56 of E7(7). The black hole entropy is [15, 18]
S = π
|J4| (1.13)
where J4 is Cartan’s quartic E7 invariant [13, 14]. It may be written
J4 = P
ijQjkP
klQli −
P ijQijP
klQkl
ǫijklmnopQijQklQmnQop + ǫijklmnop P
ijP klPmnP op
. (1.14)
where P ij and Qjk are 8× 8 antisymmetric matrices.
The qubit interpretation [4] relies on the decomposition
E7(C) ⊃ [SL(2, C)]
7 (1.15)
under which
(2, 2, 1, 2, 1, 1, 1)
+(1, 2, 2, 1, 2, 1, 1)
+(1, 1, 2, 2, 1, 2, 1)
+(1, 1, 1, 2, 2, 1, 2)
+(2, 1, 1, 1, 2, 2, 1)
+(1, 2, 1, 1, 1, 2, 2)
+(2, 1, 2, 1, 1, 1, 2) (1.16)
suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma,
Fred and George) described by the state.
|Ψ〉 =
aABD|ABD〉
+bBCE |BCE〉
+cCDF |CDF 〉
+dDEG|DEG〉
+eEFA|EFA〉
+fFGB|FGB〉
+gGAC |GAC〉 (1.17)
where A = 0, 1, so the Hilbert space has dimension 7.23 = 56. The a, b, c, d, e, f, g transform
as a 56 of E7(C). The entanglement may be represented by a heptagon where the vertices
A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG,
EFA, FGB, GAC represent the tripartite entanglement. See Figure 1. Alternatively, we can
use the Fano plane. See Figure 2. The Fano plane also corresponds to the multiplication
table of the octonions3
The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle
τ3(ABCDEFG) = 4|J4| (1.18)
J4 ∼ a
4 + b4 + c4 + d4 + e4 + f 4 + g4+
3Not the “split” octonions as was incorrectly stated in the published version of [4].
Figure 1: The E7 entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G rep-
resents a qubit and each of the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC
describes a tripartite entanglement.
2[a2b2 + b2c2 + c2d2 + d2e2 + e2f 2 + f 2g2 + g2a2+
a2c2 + b2d2 + c2e2 + d2f 2 + e2g2 + f 2a2 + g2b2+
a2d2 + b2e2 + c2f 2 + d2g2 + e2a2 + f 2b2 + g2c2]
+8[bcdf + cdeg + defa+ efgb+ fgac+ gabd+ abce] (1.19)
where products like
a4 = (ABD)(ABD)(ABD)(ABD)
= ǫA1A2ǫB1B2ǫD1D4ǫA3A4ǫB3B4ǫD2D3aA1B1D1aA2B2D2aA3B3D3aA4B4D4 (1.20)
exclude four individuals (here Charlie, Emma, Fred and George), products like
a2b2 = (ABD)(ABD)(FGB)(FGB)
= ǫA1A2ǫB1B3ǫD1D2ǫF3F4ǫG3G4ǫB2B4aA1B1D1aA2B2D2bF3G3B3bF4G4B4 (1.21)
exclude two individuals (here Charlie and Emma), and products like
abce = (ABD)(BCE)(CDF )(EFA)
= ǫA1A4ǫB1B2ǫC2C3ǫD1D3ǫE2E4ǫF3F4aA1B1D1bB2C2E2cC3D3F3eE4F4A4 (1.22)
exclude one individual (here George)4.
Once again large non-BPS, small BPS and large BPS black holes correspond to states
with J4 > 0, J4 = 0 and J4 < 0, respectively.
4This corrects the corresponding equation in the published version of [4] which had the wrong index
contraction.
Figure 2: The Fano plane has seven points, representing the seven qubits, and seven lines
(the circle counts as a line) with three points on every line, representing the tripartite
entanglement, and three lines through every point.
1.4 Magic supergravities in D = 4
The black holes described by Cayley’s hyperdeterminant are those of N = 2 supergravity
coupled to three vector multiplets, where the symmetry is [SL(2, Z)]3. In [4] the following
four-dimensional generalizations were considered:
1) N = 2 supergravity coupled to l vector multiplets where the symmetry is SL(2, Z)×
SO(l − 1, 2, Z) and the black holes carry charges belonging to the (2, l + 1) representation
(l + 1 electric plus l + 1 magnetic).
2) N = 4 supergravity coupled to m vector multiplets where the symmetry is SL(2, Z)×
SO(6, m, Z) where the black holes carry charges belonging to the (2, 6 +m) representation
(m+ 6 electric plus m+ 6 magnetic).
3) N = 8 supergravity where the symmetry is the non-compact exceptional group
E7(7)(Z) and the black holes carry charges belonging to the fundamental 56-dimensional
representation (28 electric plus 28 magnetic).
In all three case there exist quartic invariants akin to Cayley’s hyperdeterminant whose
square root yields the corresponding black hole entropy. In [4] we succeeded in giving a
quantum theoretic interpretation in the N = 8 case together with its truncations to N = 4
(with m = 6) and N = 2 (with l = 3, the case we already knew [1]).
However, as suggested by Levay [5], one might also consider the “magic” supergravi-
ties [22, 23, 24]. These correspond to the R, C, H, O (real, complex, quaternionic and
octonionic) N = 2, D = 4 supergravity coupled to 6, 9, 15 and 27 vector multiplets with
symmetries Sp(6, Z), SU(3, 3), SO∗(12) and E7(−25), respectively. Once again, as has been
shown just recently [20], in all cases there are quartic invariants whose square root yields the
corresponding black hole entropy.
Here we demonstrate that the black-hole/qubit correspondence does indeed continue to
hold for magic supergravities. The crucial observation is that, although the black hole
charges aABC are real (integer) numbers and the entropy (1.13) is invariant under E7(7)(Z),
the coefficients aABC that appear in the qutrit state (1.17) are complex. So the three tangle
(1.18) is invariant under E7(C) which contains both E7(7)(Z) and E7(−25)(Z) as subgroups.
To find a supergravity correspondence therefore, we could equally well have chosen the magic
octonionic N = 2 supergravity rather than the conventional N = 8 supergravity. The fact
E7(7)(Z) ⊃ [SL(2)(Z)]
7 (1.23)
E7(−25)(Z) 6⊃ [SL(2)(Z)]
7 (1.24)
is irrelevant. All that matters is that
E7(C) ⊃ [SL(2)(C)]
7 (1.25)
The same argument holds for the magic real, complex and quaternionic N = 2 supergravities
which are, in any case truncations of N = 8 (in contrast to the octonionic) .
Having made this observation, one may then revisit the conventional N = 2 and N = 4
cases (1) and (2) above. When we looked at the seven qubit subsector E7(C) ⊃ SL(2, C)×
SO(12, C), we gave an N = 4 supergravity interpretationwith symmetry SL(2, R)×SO(6, 6)
[4], but we could equally have given an interpretation in terms of N = 2 supergravity coupled
to 11 vector multiplets with symmetry SL(2, R)× SO(10, 2).
Moreover, SO(l−1, 2) is contained in SO(l+1, C) and SO(6, m) is contained in SO(12+
m,C) so we can give a qubit interpretation to more vector multiplets for both N = 2 and
N = 4, at least in the case of SO(4n, C) which contains [SL(2, C)]2n.
2 Five-dimensional supergravity
In five dimensions we might consider:
1)N = 2 supergravity coupled to l+1 vector multiplets where the symmetry is SO(1, 1, Z)×
SO(l, 1, Z) and the black holes carry charges belonging to the (l+1) representation (all elec-
tric) .
2)N = 4 supergravity coupled tom vector multiplets where the symmetry is SO(1, 1, Z)×
SO(m, 5, Z) where the black holes carry charges belonging to the (m+5) representation (all
electric).
3) N = 8 supergravity where the symmetry is the non-compact exceptional group
E6(6)(Z) and the black holes carry charges belonging to the fundamental 27-dimensional
representation (all electric).
The electrically charged objects are point-like and the magnetic duals are one-dimensional,
or string-like, transforming according to the contrgredient representation. In all three cases
above there exist cubic invariants akin to the determinant which yield the corresponding
black hole or black string entropy.
In this section we briefly describe the salient properties of maximal N = 8 case, following
[16]. We have 27 abelian gauge fields which transform in the fundamental representation of
E6(6). The first invariant of E6(6) is the cubic invariant [13, 17, 16, 18, 19]
J3 = qijΩ
jlqlmΩ
mnqnpΩ
pi (2.1)
where qij is the charge vector transforming as a 27 which can be represented as traceless
Sp(8) matrix. The entropy of a black hole with charges qij is then given by
S = π
|J3| (2.2)
We will see that a configuration with J3 6= 0 preserves 1/8 of the supersymmetries. If J3 = 0
and ∂J3
6= 0 then it preserves 1/4 of the supersymmetries, and finally if ∂J3
= 0 (and the
charge vector qi is non-zero), the configuration preserves 1/2 of the supersymmetries. We
will show this by choosing a particular basis for the charges, the general result following by
U-duality.
In five dimensions the compact group H is USp(8). We choose our conventions so that
USp(2) = SU(2). In the commutator of the supersymmetry generators we have a central
charge matrix Zab which can be brought to a normal form by a USp(8) transformation. In
the normal form the central charge matrix can be written as
eab =
s1 + s2 − s3 0 0 0
0 s1 + s3 − s2 0 0
0 0 s2 + s3 − s1 0
0 0 0 −(s1 + s2 + s3)
(2.3)
we can order si so that s1 ≥ s2 ≥ |s3|. The cubic invariant, in this basis, becomes
J3 = s1s2s3 (2.4)
Even though the eigenvalues si might depend on the moduli, the invariant (2.4) only depends
on the quantized values of the charges. We can write a generic charge configuration as UeU t,
where e is the normal frame as above, and the invariant will then be (2.4). There are three
distinct possibilities
J3 6= 0 s1, s2, s3 6= 0
J3 = 0,
6= 0 s1, s2 6= 0, s3 = 0
J3 = 0,
= 0 s1 6= 0, s2, s3 = 0 (2.5)
Taking the case of type II on T 5 we can choose the rotation in such a way that, for example,
s1 corresponds to solitonic five-brane charge, s2 to fundamental string winding charge along
some direction and s3 to Kaluza-Klein momentum along the same direction. We can see that
in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymme-
tries. The respective orbits are
E6(6)
F4(4)
E6(6)
SO(5, 4)×T16
E6(6)
SO(5, 5)×T16
(2.6)
This also shows that one can generically choose a basis for the charges so that all others
are related by U-duality. The basis chosen here is the S-dual of the D-brane basis usually
chosen for describing black holes in type II B on T 5 . All others are related by U-duality to
this particular choice. Note that, in contrast to the four-dimensional case where flipping the
sign of J4 (1.14) interchanges BPS and non-BPS black holes, the sign of the J3 (2.4) is not
important since it changes under a CPT transformation. There is no non-BPS orbit in five
dimensions.
In five dimensions there are also string-like configurations which are the magnetic duals of
the configurations considered here. They transform in the contragredient 27′ representation
and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous
way. We could also have configurations where we have both point-like and string-like ch the
point-like charge is uniformly distributed along the string, it is more natural to consider this
configuration as a point-like object in D = 4 by dimensional reduction.
It is useful to decompose the U-duality group into the T-duality group and the S-duality
group. The decomposition reads E6 → SO(5, 5)× SO(1, 1), leading to
27 → 161 + 10−2 + 14 (2.7)
The last term in (2.7) corresponds to the NS five-brane charge. The 16 correspond to the
D-brane charges and the 10 correspond to the 5 directions of KK momentum and the 5
directions of fundamental string winding, which are the charges that explicitly appear in
string perturbation theory. The cubic invariant has the decomposition
(27)3 → 10−2 10−2 14 + 161 161 10−2 (2.8)
This is saying that in order to have a non-zero area black hole we must have three NS charges
(more precisely some “perturbative” charges and a solitonic five-brane); or we can have two
D-brane charges and one NS charge. In particular, it is not possible to have a black hole
with a non-zero horizon area with purely D-brane charges.
Notice that the non-compact nature of the groups is crucial in this classification.
3 D = 5 black holes and qutrits
So far, all the quantum information analogies involve four-dimensional black holes and qubits.
In order to find an analogy with five-dimensional black holes we invoke three state systems
called qutrits.
3.1 N = 2 black holes and the bipartite entanglement of two
qutrits
The two qutrit system (Alice and Bob) is described by the state
|Ψ〉 = aAB|AB〉
where A = 0, 1, 2, so the Hilbert space has dimension 32 = 9. The aAB transforms as a (3, 3)
under SL(3)A × SL(3)B. The bipartite entanglement is measured by the concurrence [21]
C(AB) = 33/2|det aAB|. (3.1)
The determinant is invariant under SL(3, C)A × SL(3, C)B and under a duality that inter-
changes A and B.
The black hole interpretation is provided by N = 2 supergravity coupled to 8 vector
multiplets with symmetry SL(3, C) where the black hole charges transform as a 9. The
entropy is given by
S = π|det aAB| (3.2)
3.2 N = 8 black holes and the bipartite entanglement of three
qutrits
As we have seen in section (2) in the case of D = 5, N = 8 supergravity, the black hole
charges belong to the 27 of E6(6) and the entropy is given by (2.2).
The qutrit interpretation now relies on the decomposition
E6(C) ⊃ SL(3, C)A × SL(3, C)B × SL(3, C)C (3.3)
under which
27 → (3, 3, 1) + (3′, 1, 3) + (1, 3′, 3′) (3.4)
suggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the
larger symmetry requires that they undergo at most bipartite entanglement of a very specific
kind, where each person has bipartite entanglement with the other two:
|Ψ〉 = aAB|AB〉+ b
C |BC〉+ c
CA|CA〉 (3.5)
where A = 0, 1, 2, so the Hilbert space has dimension 3.32 = 27. The three states trans-
forms as a pair of triplets under two of the SL(3)’s and singlets under the remaining one.
Individually, therefore, the bipartite entanglement of each of the three states is given by the
determinant (3.1). Taken together however, we see from (3.4) that they transform as a com-
plex 27 of E6(C). The entanglement diagram is a triangle with vertices ABC representing
the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The
N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B.
Note that:
1) Any pair of states has an individual in common
2) Each individual is excluded from one out of the three states
The entanglement measure will be given by the concurrence
C(ABC) = 33/2|J3| (3.6)
J3 being the singlet in 27× 27× 27:
J3 ∼ a
3 + b3 + c3 + 6abc (3.7)
Figure 3: The entanglement diagram is a triangle with vertices ABC representing the qutrits
and the lines AB, BC and CA representing the entanglements.
where the products
a3 = ǫA1A2A3ǫB1B2B3aA1B1aA2B2aA3B3 (3.8)
b3 = ǫB1B2B3ǫ
C1C2C3bB1C1b
C3 (3.9)
c3 = ǫC1C2C3ǫA1A2A3c
C1A1cC2A2cC3A3 (3.10)
exclude one individual (Charlie, Alice, and Bob respectively), and the product
abc = aABb
CA (3.11)
excludes none.
3.3 Magic supergravities in D = 5
Just as in four dimensions, one might also consider the “magic” supergravities [22, 23,
24]. These correspond to the R, C, H, O (real, complex, quaternionic and octonionic)
N = 2, D = 5 supergravity coupled to 5, 8, 14 and 26 vector multiplets with symmetries
SL(3, R), SL(3, C), SU∗(6) and E6(−26) respectively. Once again, in all cases there are cubic
invariants whose square root yields the corresponding black hole entropy [20].
Here we demonstrate that the black-hole/qubit correspondence continue to hold for these
D = 5 magic supergravities, as well as D = 4 . Once again, the crucial observation is that,
although the black hole charges aAB are real (integer) numbers and the entropy (2.2) is
invariant under E6(6)(Z), the coefficients aAB that appear in the wave function (3.5) are
complex. So the 2-tangle (3.6) is invariant under E6(C) which contains both E6(6)(Z) and
E6(−26)(Z) as subgroups. To find a supergravity correspondence therefore, we could equally
well have chosen the magic octonionic N = 2 supergravity rather than the conventional
N = 8 supergravity. The fact that
E6(6)(Z) ⊃ [SL(3)(Z)]
3 (3.12)
E6(−26)(Z) 6⊃ [SL(3)(Z)]
3 (3.13)
is irrelevant. All that matters is that
E6(C) ⊃ [SL(3)(C)]
3 (3.14)
The same argument holds for the magic real, complex and quaternionic N = 2 supergravities
which are, in any case truncations of N = 8 (in contrast to the octonionic). In fact, the
example of section 3.1 corresponds to the complex case.
Having made this observation, one may then revisit the conventional N = 2 and N = 4
cases (1) and (2) of section (2). SO(l, 1) is contained in SO(l + 1, C) and SO(m, 5) is
contained in SO(5 +m,C), so we can give a qutrit interpretation to more vector multiplets
for both N = 2 and N = 4, at least in the case of SO(6n, C) which contains [SL(3, C)]n.
4 Conclusions
We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of
the 33-dimensional three qutrit Hilbert space given by (3, 3, 3), but rather a direct sum of
three 32-dimensional Hilbert spaces. It is, however, a subspace of the 73-dimensional three
7-dit Hilbert space given by (7, 7, 7). Consider the decomposition
SL(7)A × SL(7)B × SL(7)C → SL(3)A × SL(3)B × SL(3)C
under which
(7, 7, 7) →
(3′, 3′, 3′) + (3′, 3′, 3) + (3′, 3, 3′) + (3, 3′, 3′) + (3′, 3, 3) + (3, 3′, 3) + (3, 3, 3′) + (3, 3, 3)
+(3′, 3′, 1) + (3′, 1, 3′) + (1, 3′, 3′) + (3′, 1, 3) + (3′, 3, 1) + (1, 3, 3′)
+(3, 3, 1) + (3, 1, 3) + (1, 3, 3) + (3, 1, 3′) + (3, 3′, 1) + (1, 3′, 3)
+(3′, 1, 1) + (1, 3′, 1) + (1, 1, 3′) + (3, 1, 1) + (1, 3, 1) + (1, 1, 3)
+(1, 1, 1)
This contains the subspace that describes the bipartite entanglement of three qutrits, namely
(3′, 3, 1) + (3, 1, 3) + (1, 3′, 3′)
So the triangle entanglement we have described fits within conventional quantum information
theory.
Our analogy between black holes and quantum information remains, for the moment,
just that. We know of no physics connecting them.
Nevertheless, just as the exceptional group E7 describes the tripartite entanglement of
seven qubits [4, 5], we have seen is this paper that the exceptional group E6 describes the
bipartite entanglement of three qutrits. In the E7 case, the quartic Cartan invariant provides
both the measure of entanglement and the entropy of the four-dimensional N = 8 black hole,
whereas in the E6 case, the cubic Cartan invariant provides both the measure of entanglement
and the entropy of the five-dimensional N = 8 black hole.
Moreover, we have seen that similar analogies exist not only for the N = 4 and N = 2
truncations, but also for the magic N = 2 supergravities in both four and five dimensions
(In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat
Gunaydin has suggested (private communication) that the appearance of octonions implies a
connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at
least to us) in the four-dimensional N = 8 case [4], but the appearance in the five dimensional
magic N = 2 case of SL(3, R), SL(3, C), SL(3, H) and SL(3, O) is more suggestive.
5 Acknowledgements
MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull,
Martin Plenio and Tony Sudbery. This work was supported in part by the National Science
Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings
and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation. The work of S.F.
has been supported in part by the European Community Human Potential Program under
contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the
universe, in association with INFN Frascati National Laboratories and by the D.O.E grant
DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under
rolling grant PPA/G/O/2002/00474, PP/D50744X/1.
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