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Black holes Qubits STU black holes Wrapped D3-branes and 3 qubits

ZIF LECTURESLECTURE I

Black hole/qubit correspondence

M. J. Duff

Physics DepartmentImperial College

October 2009Bielefeld

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Abstract

Quantum entanglement lies at the heart of quantuminformation theory, with applications to quantumcomputing, teleportation, cryptography andcommunication. In the apparently separate world ofquantum gravity, the Bekenstein-Hawking entropy of blackholes has also occupied center stage.Here we describe a correspondence between theentanglement measures of qubits in quantum informationtheory and black hole entropy in string theory.

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Lecture I: Black hole/qubit correspondence

Black holesQubitsSTU black holesBlack hole/qubit correspondenceWrapped D3-branes and 3 qubits

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Black holes

BLACK HOLES

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Reissner-Nordström solution

Static, spherically symmetric four-dimensional line element

ds2 = −e2h(r)dt2 + e2k(r)dr2 + r2(dθ2 + sin2θdφ2). (1.1)

The most general static black hole solution ofEinstein-Maxwell theory is given by Reissner-Nordströmsolution

e2h(r) = e−2k(r) = 1− 2Mr

+Q2

r2 , (1.2)

with M and Q the mass and electric charge of the blackhole.

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Surface gravity and the area

Besides the mass and the charge which are measured atinfinity there are two other quantities, surface gravity andthe area, measured on the event horizon that are given by

κS =

√M2 −Q2

2M(M +√

M2 −Q2)−Q2, A = 4π(M+

√M2 −Q2)2.

(1.3)The R-N solution has two horizons determined by

r± = M ±√

M2 −Q2. (1.4)

Cosmic censorship: M ≥ |Q|. These black holes havesmooth geometries at the horizon and free-fallingobservers would not feel anything as they fall through thehorizon.

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Bekenstein-Hawking

Temperature

TH =~κS

2π(1.5)

Entropy

SBH =A

4~G4, (1.6)

where G4 is the four dimensional Newton constant.

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Extremal black holes

“ Extremal” black holes M = |Q|.Two horizons, r+ and r−, coincide and the area reduces to

A = 4πQ2 (1.7)

Therefore the entropy which is proportional to the area iscompletely determined in terms of the charges of theextremal black hole.

TH = 0 (1.8)

Since the Hawking radiation is proportional to thetemperature, one can conclude that the extremal blackholes are stable.

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Magnetic charge

Like the mass, the charge of the black hole is measured atasymptotic region by

Q =1

4π

∮S2∞

?F , (1.9)

where S2∞ is a spacelike sphere at infinity and ?F is the

field strength dual.The black hole can also carry magnetic charge

P =1

4π

∮S2∞

F , (1.10)

and we just need to replace Q2 with Q2 + P2 in all theformulae.

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More general black holes

The supergravity theories we shall consider have morethan the one photon of Einstein-Maxwell theory. The(D = 4,N = 2) STU model has 4; the (D = 4,N = 8)model has 28, so the black holes will carry 8 or 56 electricand magnetic charges, respectively.Similarly, the (D = 5,N = 8) black hole and black stringhave 27 electric and 27 magnetic charges, respectively.Both also involve scalar fields.

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BPS black holes

The extremal black holes obey generalised mass = chargeconditions.A black hole that preserves some unbrokensupersymmetry (admitting one or more Killing spinors) issaid to be BPS (after Bogomol’nyi-Prasad-Sommerfield)and non-BPS otherwise.All BPS black holes are extremal but extremal black holescan be BPS or non-BPS.

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Qubits

QUBITS

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Two qubits

The two qubit system Alice and Bob (where A,B = 0,1) isdescribed by the state

|Ψ〉 = aAB|AB〉= a00|00〉+ a01|01〉+ a10|10〉+ a11|11〉.

The bipartite entanglement of Alice and Bob is given by

τAB = 4|det ρA| = 4|det aAB|2,

whereρA = TrB|Ψ〉〈Ψ|

τAB is invariant under SL(2)A × SL(2)B, with aABtransforming as a (2,2), and under a discrete duality thatinterchanges A and B.

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Two qubits: examples

Example, separable state:

|Ψ〉 =1√2|00〉+

1√2|01〉

τAB = 0

Example, Bell state:

|Ψ〉 =1√2|00〉+

1√2|11〉

τAB = 1

EPR “paradox”

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Three qubits

The three qubit system Alice, Bob and Charlie (whereA,B,C = 0,1) is described by the state

|Ψ〉 = aABC |ABC〉= a000|000〉+ a001|001〉+ a010|010〉+ a011|011〉+ a100|100〉+ a101|101〉+ a110|110〉+ a111|111〉.

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Cayley’s hyperdeterminant

The tripartite entanglement of Alice, Bob and Charlie isgiven by

τABC = 4|Det aABC |,

Coffman et al: quant-ph/9907047Det aABC is Cayley’s hyperdeterminant

Det aABC = −12εA1A2εB1B2εC1C4εC2C3εA3A4εB3B4

· aA1B1C1aA2B2C2aA3B3C3aA4B4C4

Miyake and Wadati: quant-ph/0212146

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Symmetry

Explicitly

Det aABC =

a2000a2

111 + a2001a2

110 + a2010a2

101 + a2100a2

011

− 2(a000a001a110a111 + a000a010a101a111

+ a000a100a011a111 + a001a010a101a110

+ a001a100a011a110 + a010a100a011a101)

+ 4(a000a011a101a110 + a001a010a100a111).

It is invariant under SL(2)A × SL(2)B × SL(2)C , with aABCtransforming as a (2,2,2), and under a discrete triality thatinterchanges A, B and C.

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Local entropy

Another useful quantity is the local entropy SA, which is ameasure of how entangled A is with the pair BC:

SA = 4det ρA ≡ τA(BC)

where ρA is the reduced density matrix

ρA = TrBC |Ψ〉〈Ψ|,

and with similar formulae for B and C.

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Tangles

Tangles

ΤABC

ΤBC

ΤACΤAB

A

B C

A

B C

2-tangles τAB, τBC , and τCA give bipartite entanglementsbetween pairs in 3-qubit system3-tangle τABC is a measure of the genuine 3-wayentanglement:

τABC = τA(BC) − τAB − τCA

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Entanglement classes

Entanglement classes

Class τA(BC) τB(AC) τ(AB)C τABC

A-B-C 0 0 0 0A-BC 0 > 0 > 0 0B-CA > 0 0 > 0 0C-AB > 0 > 0 0 0

W > 0 > 0 > 0 0GHZ > 0 > 0 > 0 6= 0

Dur, Vidal, Cirac: quant-ph/0005115

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LOCC

Local Operations and Classical Communication=LOCCTwo states are said to be LOCC equivalent if and only ifthey may be transformed into one another with certaintyusing LOCC protocols. Reviews of the LOCC paradigmand entanglement measures may be found inPlenio:2007,Horodecki:2007.

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Orbits

Two states of a composite quantum system are regardedas LOCC equivalent if they are related by a unitarytransformation which factorizes into separatetransformations on the component parts (Bennett:1999),so-called local unitaries. The Hilbert space decomposesinto equivalence classes, or orbits under the action of thegroup of local unitaries.In the case of n qubits the group of local unitaries is given(up to a global phase) by [SU(2)]n.

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SLOCC

Stochastic Local Operations and ClassicalCommunication=SLOCCTwo quantum states are said to be SLOCC equivalent ifand only if they may be transformed into one another withsome non-vanishing probability using LOCC operations(Bennett:1999, Dur:2000). The set of SLOCCtransformations relating equivalent states forms a group(which we will refer to as the SLOCC equivalence group).For n qubits the SLOCC equivalence group is given (up toa global complex factor) by the n-fold tensor product,[SL(2,C)]n, one factor for each qubit (Dur:2000). Note, theLOCC equivalence group forms a compact subgroup of thelarger SLOCC equivalence group.

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Complex qubit parameters

For unnormalized three-qubit states, the number ofparameters [Linden and Popescu: quant-ph/9711016]needed to describe inequivalent states or, what amounts tothe same thing, the number of algebraically independentinvariants [Sudbery: quant-ph/0001116] is given by thedimension of the space of orbits

C2 × C2 × C2

U(1)× SU(2)× SU(2)× SU(2)

namely, 16− 10 = 6.

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Real qubit parameters

However, for subsequent comparison with the STU blackhole [Duff, Liu and Rahmfeld: hep-th/9508094; Behrndt etal: hep-th/9608059], we restrict our attention to states withreal coefficients aABC .In this case, one can show that there are five algebraicallyindependent invariants: Det a, SA, SB, SC and the norm〈Ψ|Ψ〉 , corresponding to the dimension of

R2 × R2 × R2

SO(2)× SO(2)× SO(2)

namely, 8− 3 = 5.

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Representatives

Representatives from each class are:Class A-B-C (product states):

N0|111〉.

Classes A-BC, (bipartite entanglement):

N0|111〉 − N1|100〉,

and similarly B-CA, C-AB.Class W (maximizes bipartite entanglement):

−N1|100〉 − N2|010〉 − N3|001〉.

Class GHZ (genuine tripartite entanglement):

N0|111〉 − N1|100〉 − N2|010〉 − N3|001〉.

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STU black holes

STU BLACK HOLES

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STU model

The STU model consists of N = 2 supergravity coupled tothree vector multiplets interacting through the special Kahlermanifold [SL(2)/SO(2)]3:

SSTU =1

16πG

∫e−η

[(

R +14

(Tr[∂M−1

T ∂MT

]+ Tr

[∂M−1

U ∂MU

] ))? 1

+ ? dη ∧ dη − 12? H[3] ∧ H[3] −

12? F T

S[2] ∧ (MT ⊗MU) FS[2]

]MS =

1=(S)

(1 <(S)<(S) |S|2

)etc.

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STU parameters

A general static spherically symmetric black hole solutiondepends on 8 charges denoted q0,q1,q2,q3,p0,p1,p2,p3,but the generating solution depends on just 8− 3 = 5parameters [Cvetic and Youm: hep-th/9512127; Cvetic andHull: hep-th/9606193], after fixing the action of the isotropysubgroup [SO(2)]3.

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Black hole entropy

Black hole entropy S given by the one quarter the area ofthe event horizon.Hawking: 1975The STU black hole entropy is a complicated function ofthe 8 charges :

(S/π)2 = −(p · q)2

+4[(p1q1)(p2q2) + (p1q1)(p3q3) + (p3q3)(p2q2)

+q0p1p2p2 − p0q1q2q3

]Behrndt et al: hep-th/9608059

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Qubit correspondence

By identifying the 8 charges with the 8 components of thethree-qubit hypermatrix aABC ,

p0

p1

p2

p3

q0q1q2q3

=

a000−a001−a010−a100a111a110a101a011

one finds that the black hole entropy is related to the3-tangle as in

S = π√|Det aABC | =

π

2√τABC

Duff: hep-th/060113432 / 47


BH/qubit correspondence

The measure of tripartite entanglement of three qubits(Alice, Bob and Charlie), known as the 3-tangle τABC , andthe entropy S of the 8-charge STU black hole ofsupergravity are both given by Cayley’s hyperdeterminant.

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Further developments

Further papers have written a more complete dictionary,which translates a variety of phenomena in one languageto those in the other:

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Further developments contd

The attractor mechanism on the black hole side is relatedto optimal local distillation protocols on the QI side.Moreover, supersymmetric and non-supersymmetric blackholes corresponding to the suppression ornon-suppression of bit-flip errors .

Levay:[arXiv:0708.2799 [hep-th]]

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Wrapped D3-branes and 3 qubits

WRAPPED D3-BRANES AND 3 QUBITS

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Microscopic analysis

String interpretation:N = 4 supergravity with symmetry SL(2)×SO(6,22) is thelow-energy limit of the heterotic string compactified on T 6.N = 8 supergravity with symmetry E7(7) is the low-energylimit of the Type IIA or Type IIB strings, compactified on T 6

or M-theory on T 7.Black holes are now 0-branes obtained by wrappingp-branes around p of the compactifying circles.

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Stringy version

The stringy version of the STU black hole is not uniquesince there are many ways of embedding the STU modelin string/M-theory, but a useful one from our point of view isthat of four D3-branes wrapping the(579), (568), (478), (469) cycles of T 6 (intersecting over astring) with wrapping numbers N0,N1,N2,N3.Klebanov and Tseytlin: hep-th/9604166The wrapped circles are denoted by a cross and theunwrapped circles by a nought as shown in the followingtable:

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4 5 6 7 8 9 macro charges micro charges |ABC〉

x o x o x o p0 0 |000〉

o x o x x o q1 0 |110〉

o x x o o x q2 −N3sinθcosθ |101〉

x o o x o x q3 N3sinθcosθ |011〉

o x o x o x q0 N0 + N3sin2θ |111〉

x o x o o x −p1 −N3cos2θ |001〉

x o o x x o −p2 −N2 |010〉

o x x o x o −p3 −N1 |100〉

Table: Three qubit interpretation of the 8-charge D = 4 black holefrom four D3-branes wrapping around the lower four cycles of T 6 withwrapping numbers N0,N1,N2,N3.

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Fifth parameter

The fifth parameter θ is obtained by allowing the N3 braneto intersect at an angle which induces additional effectivecharges on the (579), (569), (479) cycles[Balasubramanian and Larsen: hep-th/ 9704143;Balasubramanian: hep-th/9712215; Bertolini and Trigiante:hep-th/0002191].The microscopic calculation of the entropy consists oftaking the logarithm of the number of microstates andyields the same result as the macroscopic one [Bertoliniand Trigiante: hep-th/0008201].

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Qubit interpretation

To make the black hole/qubit correspondence we associatethe three T 2 with the SL(2)A × SL(2)B × SL(2)C of thethree qubits Alice, Bob, and Charlie. The 8 different cyclesthen yield 8 different basis vectors |ABC〉 as in the lastcolumn of the Table, where |0〉 corresponds to xo and |1〉to ox.We see immediately that we reproduce the five parameterthree-qubit state |Ψ〉:

|Ψ〉 = −N3cos2θ|001〉 − N2|010〉+ N3sinθcosθ|011〉−N1|100〉 − N3sinθcosθ|101〉+ (N0 + N3sin2θ)|111〉.

Note from the Table that the GHZ state describes fourD3-branes intersecting over a string, or groups of 4wrapping cycles with just one cross in common.

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IIA and IIB

Performing a T-duality transformation, one obtains a TypeIIA interpretation with zero D6-branes, N0 D0-branes,N1,N2,N3 D4-branes plus effective D2-brane charges,where |0〉 now corresponds to xx and |1〉 to oo.

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Summary

Our Type IIB microscopic analysis of the D = 4 black holehas provided an explanation for the appearance of thequbit two-valuedness (0 or 1) that was lacking in theprevious treatments: The brane can wrap one circle or theother in each T 2.To wrap or not to wrap? That is the qubit.The number of qubits is three because of the number ofextra dimensions is six.The five parameters of the real three-qubit state are seento correspond to four D3-branes intersecting at an angle.

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L. Borsten, D. Dahanayake, M. J. Duff, H. Ebrahim andW. Rubens, “Black Holes, Qubits and Octonions,” Phys.Rep. (to appear), arXiv:0809.4685 [hep-th].

M. J. Duff, “String triality, black hole entropy and Cayley’shyperdeterminant,” Phys. Rev. D 76, 025017 (2007)[arXiv:hep-th/0601134]

V. Coffman, J. Kundu and W. Wooters, “Distributedentanglement,” Phys. Rev. A61 (2000) 52306,

M. J. Duff, J. T. Liu and J. Rahmfeld, “Four-dimensionalstring-string-string triality,” Nucl. Phys. B 459, 125 (1996)[arXiv:hep-th/9508094].

K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova andW. K. Wong, “STU black holes and string triality,” Phys. Rev.D 54, 6293 (1996) [arXiv:hep-th/9608059].

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http://arxiv.org/abs/hep-th/0601134


R. Kallosh and A. Linde, “Strings, black holes, and quantuminformation,” Phys. Rev. D 73, 104033 (2006)[arXiv:hep-th/0602061].

P. Levay, “Stringy black holes and the geometry ofentanglement,” Phys. Rev. D 74, 024030 (2006)[arXiv:hep-th/0603136].

M. J. Duff and S. Ferrara, “E7 and the tripartiteentanglement of seven qubits,” Phys. Rev. D 76, 025018(2007) [arXiv:quant-ph/0609227].

P. Levay, “Strings, black holes, the tripartite entanglementof seven qubits and the Fano plane,” Phys. Rev. D 75,024024 (2007) [arXiv:hep-th/0610314].

M. J. Duff and S. Ferrara, “E6 and the bipartiteentanglement of three qutrits,” Phys. Rev. D 76, 124023(2007) [arXiv:0704.0507 [hep-th]].

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http://arxiv.org/abs/hep-th//0610314

http://arxiv.org/abs/0704.0507


P. Levay, “A three-qubit interpretation of BPS and non-BPSSTU black holes,” Phys. Rev. D 76, 106011 (2007)[arXiv:0708.2799 [hep-th]].

L. Borsten, D. Dahanayake, M. J. Duff, W. Rubens andH. Ebrahim, “Wrapped branes as qubits,”Phys.Rev.Lett.100:251602,2008 arXiv:0802.0840 [hep-th].

P. Levay, M. Saniga and P. Vrana, “Three-Qubit Operators,the Split Cayley Hexagon of Order Two and Black Holes,”arXiv:0808.3849 [quant-ph].

P. Lévay and P. Vrana, “Special entangled quantumsystems and the Freudenthal construction,”arXiv:0902.2269 [quant-ph].

P. Levay, M. Saniga, P. Vrana, and P. Pracna, “Black HoleEntropy and Finite Geometry,” Phys. Rev. D79 (2009)084036, arXiv:0903.0541 [hep-th].

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M. Saniga, P. Levay, P. Pracna, and P. Vrana, “TheVeldkamp Space of GQ(2,4),” arXiv:0903.0715 [math-ph].

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